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Hi friends, today we are going to learn one of the most important instructions in the PLC ladder which is MOVE instruction by which we can move data between different memory storage including input, output, marker, and variables. Also, data of different data types and sizes can be transferred from source to destination and source memory locations. For example data types including char, string, integer, floating, time and date can be transferred between source and destination. Memory location like input, output, and marker memory area can be acting as source or destination. Furthermore, a mask can be utilized to customize and control the part of data to be transferred between source and destination. In that move with a mask, the instruction uses a source address, a destination address, and the mask as well. Guys, please do not worry if you are not understanding what is mentioned hundreds percent because you will get it in more detail and practice with the simulator as well. So let’s continue to learn and practice that MOVE and masked MOVE instruction. 

MOVE Instruction in Ladder Logic

Let’s start with the basic MOVE instruction. Figure 1 shows the typical block of a MOVE instruction in Siemens. It shows two sides for the input and output parameters of the block. Simply, the input EN input shows the contact that enables the block and triggers the movement of data between the source specified by the memory address given at input IN and the destination pointed by the memory address given at output OUT1. Also, output ENO denotes the status of the movement process to indicate the completion of the process.

Source and destination of the MOVE instruction 

As shown in the function block of the MOVE instruction shown in figure 1, there are source and destination which are an address of a memory area and can be expected to be byte, word, or double word DWORD or of any datatype like integer, character string, float, time, or date datatype. Some examples of memory size and datatype are listed in table 1 below:

MOVE instruction in ladder logic

Figure 2 shows one very basic ladder logic rung that demonstrates the usage of a MOVE instruction. It consists of a contact “trigger” to control activation of the move instruction, the MOVE instruction block which has source and destination memory address MW10 and MW12 respectively. The source and destination in this example represent integer data types.

Guys! Let's simulate this rung to learn together how the MOVE instruction works. Figure 3 shows the case when the MOVE instruction is not activated, in that case as shown in figure 3 the data is not transferred from source to destination.

But after activating the MOVE instruction by enabling the EN of the MOVE block using the contact M0.1, the data is moved from the sources to the destination.

It is important to let you guys know, this MOVE function block can be used to move different types of data types as listed above by providing the address of both source and destination.

Massive move instruction using ladder logic

Friends! Sometimes we need to move arrays of data. So do your think we are going to move one by one? The answer is yes it is possible but that is too boring to code and the program is getting very lengthy. So there is another version of MOVE instruction that enable us to move a massive amount of variables which is MOVE_BLK instruction. Figure 5 shows the function block of the massive move instruction in a very basic example ladder logic rung. As you see friends, similar to basic MOVE instruction, it has EN and ENO for enabling the instruction and returning its status showing the completion status. Also, it has an input source denoted by the address of the memory storage of the source and the address of the memory location of the destination as well. But you notice that time there is an array of variables or datatypes for source and destination. And there are other parameters called count which denotes the number of items to be moved from source to destination.  Let’s simulate this MOVE_BLK instruction to see how it works and the virtues of moving many data items in one command.

Figure 6 shows the execution having the block not activated. On the right part the memory of the source and the destination show array of 5 integers for each. You can notice guys there is no data moved.

Figure 7 shows the data is moved from source to destination after activation of the MOVE_BLK instruction. But wait for a second! Why not all the data been moved? Yes, you are correct friends, because we use count parameters to control how many variables or data items are to be transferred. And it has 3. So the first three data items have been moved only.

MOVE instruction in Allen Bradley

 Figure 8 shows a simple program that demonstrates the ladder logic rung that uses MOVE instruction. It shows the input contact I:3/0 triggers the move instruction block to activate it and enable data transfer from the 16bits input I:1 to the 16bits output O:2. On the left side, you can see guys the inputs are mirrored at the output by using the MOVE instruction. Now let’s see how we can control the data transfer using a mask to allow only selected bits to be transferred and prevent the others.

Masked move instruction

The idea behind the masked move instruction is that we utilize an array of bits to be used as a mask to filter the data that are being transferred. For example Figure 9 shows a very simple ladder logic rung that uses a masked MOVE instruction to customize the data transfer from source to destination. The MVM instruction has three parameters as shown in figure 9 which are: the source, destination, and mask. All of them are addresses of the memory area. For the very given example here, the source is the 16 bits input at I:1/0 and the output is the 16 bits output at O:2/0 while the mask is the 16bits input at I:3/0. Before we are going to simulate this example, you should expect that the data will be transferred from source to destination based on the status of every bit in the mask data.

Simulating masked move instruction

In the first test case shown in figure 10, the data to be transferred is ready at the source but nothing is transferred to the destination because the activating contact I:3/0 is not enabled. So let’s switch it on and see what is happening!

Now we enabled the activation switch at I:3/0 but still, data are not transferred from source to destination! Is there a problem in coding or logic? No, that is because the mask bits are set to off that is why bits of the source are not enabled to be transferred to the destination. Now, let us enable the masked bits and see what’s going on?

Yes! Now guys you can see all bits of the source have been transferred to the destination as long as their mates at the masked bits are enabled. But those who have their mates of bits at the mask are disabled and are still prohibited from being transferred. Like this, you can see friends how we can move data without control as when we use the MOVE instruction. In addition, you see how to control that data transfer by using mask bits as a filter. Do not forget, the move instruction can be utilized to move different data types, variables, memory sizes, and data blocks. So, it is a great tool for you as an experienced ladder logic programmer to flexibly move data between your data variables and memory location through long the program you are writing.

What’s next???

Guys! I would like to thank you for being here with me up till this point of our tutorial hoping you are really happy with it and seeing it is useful. Now guess what? I have a gift for you which is our next tutorial. It is not like every time we come up with another aspect in ladder logic to learn and enjoy practicing simulation. Now we come to the point that we should be counted as real ladder logic programmers because we have learned a lot and practiced more topics. So, it is time to practice a complete project string from reading the logic chart, understanding the logic narrative, and interpreting it to a complete successful ladder logic program. So next time will come up with a complete process with a logic diagram and requirements and practice together step by step. You can consider it as a great workshop in which you will apply all your knowledge base and what you have learned so far to take the responsibility as a PLC ladder logic programmer to convert the logic narrative and diagrams into a real ladder logic program. The work will include deciding the required logic components including switches, relays, inputs, outputs, digital and analog. Then will do a flow chart showing the logic flow based on which the program will be written. And lastly, the coded program will be applied to the plc simulator to verify the logic and make sure it is functioning correctly according to what has been requested at the beginning from the customer. So please be ready, make some revisions, and be there when we are going to learn by practicing a real project as in the real life and industry problems.

Matrices are an essential topic in different fields of study, especially in mathematics, where you have a bulk of data and want to organize, relate, transfer, and perform different operations on data in a better manner. We have studied a lot of types and operations on the matrices and have worked on different types with the help of MATLAB. Today, we are here to present the applications of the matrices in different fields of study to clarify the importance of this topic. So, have a look at the list of topics we are going to learn. Yet, first of all, I am going to describe what a matrix is.

What is a Matrix?

In the fields of physics and mathematics, there is the use of different types of numbers in groups of various types. In order to organize the data into a manageable format, matrices are used. A matrix is a rectangular array of numbers that are enclosed by a square matrix (or, in some cases, parentheses). 

The information about the numbers of rows and columns is called the order of matrices, and on the basis of this information, we can recognize different types of matrices. The types, in turn, are used in different applications because of their unique behavior. By changing the order of the matrix, the properties and working of the matrices changes according to the changes. Mostly, square matrices are used in different applications.

It is interesting to know that in the early days of matrices, these were considered arrays. With the passage of time, when they were involved more and more in different research and methodologies, matrices gained popularity. Because of the ease of usage, this popularity is not going to end. So, have a look at the different types of fields where matrices are used in different ways but before going into details, you must know, these fields are not only dependent on the matrices but the normal functioning of these fields include the usage of matrices in different ways. 

Applications of Matrices

As we have said earlier, matrices are used to deal with massive amounts of data better. The matrices that we have learned and seen till now are of very minimum order, and these are kept simple for the better concepts in easy ways, but in practice, we have seen that there are gigantic matrices with complex data in them, and at that point, it is difficult to solve and save them manually. Have a look at the different fields of practical life where matrices are used in a routine.

Matrices in Cryptography 

Data cryptography is an important department in data communication in which the encryption of the data is an important process. It is done by using different keys and codes to secure the message and communication in a better way. A large amount of data is sent and received by different parties, and the encryption techniques also require some other space as well. Matrixes are used to make sure that all the data and its codes are stored in a better way. These matrices 

save the keys of the encrypted data to decrypt them on the receiving end and in this way, matrices play a key role in cryptography.

Use of Matrices in Wireless Communication

We all know that in wireless communication, usually air is used as the medium to send and receive messages from one point to the other. In this process, matrices are used to detect, extract, and process the information about the message that is to be delivered. Here are some other uses of matrices in this department:

Signal estimation and the detection of the problem during communication are done with the help of matrices. 

• Sensor array signal processing involves the matrices.

• In the processing and representation of a digital image, matrices have a great role. 

• Radar signals

• Underwater surveillance

With the help of matrices, wireless communication is done efficiently, and understanding the code becomes easy. Think about the case if the data of different queries are not used in the form of a matrix, then finding the data of a simple command would never be organized. 

Matrices in Mathematics

The use of matrices in the field of mathematics is not new to us. We all know that it is a basic concept in mathematics and a great variety of concepts of matrices are used in different ways while solving mathematical problems. One of the major use in mathematics is a solution of linear equations with the help of matrices. The complex and time taking equations can be easily solved with the help of rules of matrices in different ways. 

In engineering and other related fields of mathematics, matrices are the basic concepts and it is used in different ways to make the working of the system better manner. We have seen different cases in which the matrix is used as the alternative method to find the unknown value because it is a more organized way and the great number of research resulted in different theorems and laws therefore, the long calculations are minimized to their result by simply using the theorems.

Matrices in Computer Graphics

One of the amazing applications of matrices is in the form of computer graphics where the pictures and the graphics comprise pixels and the array of these pixels and points are arranged in the form of matrices for easy transformation and working. Overall, you can say that in computer graphics, each digital image is treated as a matrix. Therefore, different types of operations used in the matrices are applied to the graphics with great ease. Not only in the dimensions and the sizes but also for the colors of the images, matrices are used to store and reuse the values for the images and graphics. For example, in the CNN technique, different types of matrices are used. For the greyscale image, only a 2D image is used and if one wants to get the RBG system image, there is a need for a 3D matrix. 

Three Dimensional Games and Matrices

Gaming is one of the most important filed in graphics and when we talk about three-dimensional games, matrices are important there in order to alter the 3D space in different ways. For this purpose, if we use simple words, the conversions between the 2D and 3D matrices are used by different techniques, and therefore, we get the final output. Moreover, the quality of the result depends upon the way you use the data in different ways. 

Matrices and Geology

During the seismic survey in geology, matrices are used in a great way. For real-time surveys of different areas of common real problems such as mortality rate, population, the number of people in different areas of the world, and other specific counting related to real-life problems involve the use of matrices because it becomes easy to deal with great data using different operations on the data.

The Matrices in the Field of IT

In different Information technology organizations, matrices are used to execute and search the different queries. The IT security system needs to have a secure way to deal with all the information and once saved, data is to be retrieved in an efficient way with the help of minimum commands. If the data is not present in the form of tables, or we should say, in the form of matrices, organizing, storing, retrieving, and dealing with the data will be like a nightmare. 

Using Matrices to Find the Collinear Points

It is one of the procedures in mathematics in which the values of collinear points are found with the help of matrices. If it seems simple at this time then you should think of the case where gigantic collinear points are found with the help of matrics with the help of different software and these points are in return used in different ways. 

Pre-allocation of Matrices in MATLAB

As we have said, matrices make the working of daily life data and complex calculations easy. We read different types of commands about the matrices when we were learning about their functioning and therefore, we can now use them in a simple program to prepare the code in which the matrix is used for the pre-allocation of the data in a simple way by using limited lines of codes. Trus, have a look at the code given below:

Code:

p=zeros(5)

p(1,:)= [ 3 6 2 8 7]

for i=3:4

p(i,:)=p(i-1,:) +1

end

Output:

Understanding the MATLAB Code

In this code, we have used a simple function of zeros and used a loop to execute the whole instruction. Let us discuss both of them in detail:

Zeros Function

It is a pre-defined function of MATLAB that is used to make a null matrix of the required dimensions by using just a simple signal command. For the null matrices, there is the condition that the square matrix should be used.

For loop in Matrix

There are certain loops used in the MATLAB and with the help of this for loop, we just have to simply follow the syntax given below:

for index = value

statement

end

Here, the index value is the starting point where we want to start the matrix formation and the statement is the condition that we want to be executed while the formation of our matrix. If it is not clear right now in your mind, just have a look at the flow of the program.

• First of all, we have used the zeros function to allocate the space in the memory according to the requirement. We have used the square matrix which has the order of 5 by 5. 

• I wanted to pre-allocate the values in the matrix row after the row therefore, we simply changed the values of the matrix p formed before from 0 to other values defined by us. If you are not familiar with this function, have a look at the notation given below:

p(a,b)

Where,

a=number of row

b=number of column

As we did not want to alter any value in the columns so we have used a colon in its place. 

• By using the for loops, we have specified that the index value starts from 3 and ends and 4. 

• In the next step, we are using these index values and specifying to MATLAB that we want to change the values of the 3rd and 4th rows only and the program ends. 

• MATLAB does this task step by step and changes the null or zero values of the matrix p into the required pre-allocated values in the matrix.

• For the best concepts, we have changed the values of just two rows and the other matrix remains the same.

Larger versions of these kinds of procedures are used in diverse ways to recognize, store, and use the data in a better way, and with the help of this short program, we have seen a glimpse of a real-time application in which a matrix can be used to pre-allocate the different values and people can have benefit from it.

Trus, today, we have learned a lot about matrices and their applications. We have read great information about matrices in the past lectures and it was quite interesting to know how can you use these basic pieces of information in a better way and how people are working on the matrices to make their daily tasks easy during their professional life. We have seen different departments where matrices are making the work easy and more efficient. Most of them can not work without using matrices. Moreover, one must have the idea that many times we use matrices in our daily life unintentionally. As we have said earlier, 3D games require the involvement of a matrix. So, when your child is playing the game, he or she is enjoying the application of the matrices without knowing it. In the end, the small program was helpful to understand how little programs and the working of matrices are helpful to perform different tasks automatically.

Hello learners. Welcome to The Engineering Projects. We all know that matrices have been used in the engineering field for a long time, and they have a vital role in the calculation of different data. Therefore, we are learning about some very special kinds of matrices that are usually introduced to engineers at a higher level. Till now, we have seen some general operations on matrices and also examined the special kinds of matrices. Yet, today, we are moving a step forward and learning about some complex types of matrices that require strong basic concepts. So, have a glimpse at the topics that you are going to learn, and then, we’ll start practicing.

• What is a matrix?

• What are the different types of matrices that are less common?

• How can we implement some interesting commands in MATLAB?

• What is the procedure for dealing with complex equations in MATLAB?

What is a Matrix?

First of all, it is recommended to learn about the introduction of the matrix. A matrix has different elements in it with different sequences, and we define it as:

“A matrix is a rectangular array in which different types of elements in the different sequences are present, and all the elements are enclosed in a long bracket that is usually a square bracket.”

The sequence and the values of the matrices are important, and the collection of horizontal lines is called a row. On the other hand, the vertical entries are collectively called columns. To find the difference in the types of matrices, it is important to know that the basic kinds of matrices are square matrix and rectangular matrix. Most of the matrices that we are going to discuss here will be square matrices. 

Types of Matrices

Depending upon different parameters in matrices, these are divided into different types and groups. In a square matrix, the numbers of rows and columns are equal to each other, and in a rectangular matrix, the numbers of rows and columns are never equal. You will observe different parameters on the basis of which these matrices are recognized, and then the behavior of each of them is examined. So have a look at these types. 

A Boolean Matrix  

We all know that boolean numbers mean two conditions only, that are, zero and one. So, the boolean matrix is the type of matrix in which only zero and one value are used, and other values are never involved. The order of the matrix may be anything, but the entries must be either zero or one.

One thing that must be kept in mind is that the matrix will surely have these values in a non-specific pattern and it never has only zero or only one. There must be one and zero of each type. We are clearing it because some other types, such as singular matrix, identity matrix, null matrix, and scalar matrix, also involve zero and one, but in a specific manner. 

Boolean numbers and conditions are used in almost every field of mathematics, and therefore, this matrix is also used in different ways in such fields. With the help of a matrix, it becomes easy to deal with a large number of data points at once. The example we have shown here has a small size. Yet, when the boolean matrix is used in practical life, we use a bulk of matrices with greater sizes. 

Orthogonal Matrices

In the previous section, we learned that a transpose of the matrix is obtained when the rows and columns, along with the entries, are interchanged with each other. Keeping this concept in mind, we can define this type of matrix as

A matrix A is called an orthogonal matrix if and only if the multiplication of the matrix A with the transpose of itself provides us with the identity matrix. 

If you are new to the identity matrix, then you must know that an identity matrix is one that contains only value 1 in its diagonal and all the entries other than the diagonal are zero. 

The inverse of a Matrix

As we have read about the orthogonal matrices, they remind us of the inverse of the matrix. It is interesting to know that there exists the inverse of a matrix that, when multiplied with the original matrix, delivers us the identity matrix. We hope you remember that an identity matrix is a type of matrix that always has the value 1 in the diagonal elements and zero in all the remaining elements. So, if we are taking matrix A, then for the inverse matrix, we can say:

A.AT= I

In this way, if we have the value of A and want to know the inverse of it, we use the following equation:

AT= I/A

So, in this way, we have observed that no matter if the inverse of the matrix is multiplied with the original matrix or vice versa, we get the identity matrix. This condition is somehow special because the multiplication of the matrix is a more complex operation than addition and subtraction, yet here we are using the rules of the dot product. There is another way to find the inverse of a matrix with the help of a formula that states that the inverse of a matrix A is equal to the determinant of matrix A with an adjoint of A. 

There are certain conditions for the existence of a square matrix:

• A matrix has the inverse only if it is a square matrix. 

• If the determinant of the matrix is equal to zero, then there is no need to find the adjoint of that matrix because the inverse does not exist for that matrix. 

It is not always necessary that in the inverse of a matrix, the values are reciprocal of the original matrix. The values may also be changed, and by dot multiplication, we can get the identity matrix.

Stochastic Matrix

Here is another interesting type of matrix in which all the values show the probability values. We know that the probability values are so small that they are less than one and have fractional values or in the form of points less than one. So, the stochastic matrix has all the probability values as its elements.

You can see all the values are less than one because the probability is usually in the form of such small numbers. If any of the values is greater, then it is not a stochastic matrix. 

Nilpotent Matrix

This is a less common type of matrix in which Ak = O, that is, for every power of the matrix, we get a null matrix (the one with all the values equal to zero). There are two conditions that must be fulfilled in this type:

1. The matrix A must be square.

2. We get a null matrix every time we apply any power to matrix A, which is a natural number.

For example, when we take the square of each value of A, we get the null matrix. It is a less common type of matrix, but at a higher level of calculations, it has its applications. 

Idemportenet Matrix

It is a type of uncommon matrix that provides us with a matrix equal to itself when any power is applied to it greater than 2. That is, An = A where the value of n is always equal to or greater than 2. Usually, it has been noticed that if the square of the matrix A is equal to itself, then there is no need to check further because the result remains the same for any number greater than 2. You can take it as homework and find examples of such matrices by yourself.

Special Commands in MATLAB

It seems like you have learned a lot about the special kinds of matrices. When we talk about MATLAB, we have observed that there are different MATLAB commands that are not related directly to the matrices yet are used to make the functioning of a group of numbers and matrices easier and more effective in some cases. So, have a look at some of the commands.

The procedure to perform the Complex Number in MATLAB

We all know about complex numbers and have also used them in the matrices when we were talking about hermitian metrics. A special kind of function in MATLAB is the “complex” function, which is used to make the complex equation. So have a look at the step-by-step procedure to perform the complex equation in MATLAB.

• Hit the start button of MATLAB. 

• Go to the command window.

• Start by writing the following command. 

a=3

w=3

• It will provide us with two numbers that we will use in the complex function. We are naming this equation ComplexEqaution, and the command to perform this is

ComplexEqaution=complex(a,w)

• Press the enter button. 

• Now, using this equation, we can also find the real and imaginary parts of the equation using different commands. For real numbers, we are writing:

real(ComplexEqaution)

• It will give us the real part of the equation. Similarly, write the following command for the imaginary number:

imag(ComplexEqaution)

• Similarly, to obtain the absolute value of the equation we have just made, we are using the required command.

abs(ComplexEqaution)

• Finally, to find the angle of the equation, we are using its command.

angle(ComplexEqaution) 

• Now, to perform the inverse of a matrix, we are going to introduce the matrix first. So, write the following command:

B=[1 55 2 7; 4 5 2 8; 4 99 1 6; 3 8 4 1]

• The command for this purpose is:

InversofMatrix= inv(B)

If you are confused about the absolute value of the complex equation, then you must realize that complex numbers are present in the plane or the coordinates of the system, and the absolute value of the complex numbers is defined as the total distance between the origin of the complex number (where the plane is (0,0)) and the complex plane (a,b). The procedure to do so is a little bit complex if we solve it theoretically, but with the help of MATLAB, this can be done with the help of a simple command and all the control in your hand. You should practice more with all these commands by using your own values in the matrix with a different sequence. Once you have followed all the instructions given above, the output at your MATLAB command window should be like the images given next:

So, in this way, we have gotten rid of the long and error-prone calculations, and by using simple commands, we have simply applied the formula of inverse, and in this way, we have not had to find the adjoint and determinant of the matrix A. 

Consequently, we had a little bit of a complex yet interesting lecture today on matrix, in which we defined it first for a solid foundation of the concept. After that, there were different types of matrices that are less common but have interesting patterns of the elements in which the values and positions are important to define the type of matrix. Moreover, we have seen some interesting commands in MATLAB to perform the operations of complex equations and also implement them all practically. In the next lecture, you are going to learn the amazing features and applications of MATLAB in different aspects of the study.

Hey students welcome to another tutorial in The Engineering Projects where we are going to learn a lot about matrices. If you are a beginner to the metrics, then you should go to learn the fundamentals of matrices. Yet, if you know the basic introduction, you are at the right lecture because we are learning about the special kinds of matrices and you are also going to see the matrices in action using MATLAB. So, here is a simple list of today’s topics. 

• What is a matrix?

• How can we identify the matrix with the help of its general form?

• What are the different types of matrices?

• What is the concept of transpose while dealing with matrices?

• How can we implement these types of matrices in MATLAB by different commands?

What is a Matrix?

A matrix is a type of array that stores data of the same kind. It has great importance in the world of technology and it is defined as:

"A matrix takes the form of an ordered rectangular array surrounded by a square bracket and has entries of the same kind in the form of real or complex data."

To perform different operations on the matrices, you just have to apply them to the whole matrix at once; there is no need to apply them to all the entries one after the other. This will make sense when you see the matrices in action. 

Till now, we have learned about the basic introduction of the following types of matrices:

• Row matrix

• Column matric

• Square matrix

• Rectangular matrix

These are the basic types irrespective of the elements or entries present in them, and now, we are going to discuss some types that have the features of these matrices but the entries in them are in a specific pattern, so they are considered the special kinds of matrices. Before going into the details of each of them, there must be the foundation of a concept of a general form of the matrix. 

The General Form of the Matrix

To examine the location of a specific element, we use the terms "rows” and “columns” where

1. Rows are the horizontal entries of the matrix.

2. Columns are the vertical entries of the matrix.

A specific element is the part of a row and column at a time and to mention the location of an element, it is important to know the information about both of them. So, in the general form, a three-by-three matrix is shown as:

Where

• A= name of the matrix
• a= elements of the matrix A

Notice that a has two numbers with them in which the first number denotes the rows and the second one is the number of columns. To make it easier, we denote rows with i and columns with j. So we can say,

• aij=a23=element of the second row and first column

Keeping this concept in mind, now it is easy to understand the types of matrices in which the value of the elements matters. Have a look at some of these types.

Identity Matrix

The identity matrix is a special kind of matrix having the arrangement of entries in such a way that all the diagonal entries are one and the remaining ones are all zero. In this way, we get the matrix of the form:

You can observe that for an identity matrix, the elements where i=j are all ones, and all the elements other than this are zero. There are certain applications of the identity matrix and you will know them in the upcoming lectures. 

Zero or Null Matrix

Here is an interesting type of matrix. This matrix has all the entries zero and no other value can be found in this element. So we assume the zero or null matrix as:

Where the sequence of the matrix can be any. No matter whether it square, row, or rectangular matrix. 

Singular Matrix

A singular matrix is the type of square matrix that has the determinant equal to zero. It is possible in the condition when all the entries of the matrix have the value one.

We all know the procedure to find the determinant. So, taking the determinant of this matrix C, we provide ourselves with the following calculations:

|C|=1x[(1x1) - (1x1)] - 1x [(1x1)-(1x1)] + 1x[(1x1)-(1x1)]

    =1x(1-1) - 1x(1-1) + 1x(1-1)

    =1x(0) - 1x(0) + 1x(0)

    = 0 - 0 + 0

    =0

So in general, we say, in a singular matrix:

|C| =0

Non-singular Matrix

A non-singular matrix, on the other hand, has entries in such a way that the determinant is never zero. So, we can say, most of the matrices are included in a non-singular type of matrix. Here is an example of a non-singular matrix, which we have also mentioned in other categories.

|C|=3x[(-6x7) - (1x-1)] - 0x [(2x7)-(1x4)] + (-5)x[(2x-1)-(-6x4)]

    =3x(-42+1) - 0x(14-4) - 5x(-2+24)

    =1x(-41) - 0x(-10) + 5x(22)

    = -41 - 0 + 110

    =69

Hence, we can conclude for the non-singular matrix that if A is a non-singular matrix, then:

|A|≠ 0

Diagonal Matrix

The concept of diagonal was used in the identity matrix, but it is a slightly different case. A diagonal matrix is one that has all the entries, other than the diagonal, equal to zero. The values in the diagonal may be anything. 

Here, one concept must be clear. In some cases, all the diagonal values except one may be zero, but it will still be called the diagonal matrix. In other words, by changing even one of the values in the null matrix, we can convert it into the diagonal matrix. 

Scalar Matrix

A scalar matrix has all the diagonal values the same, no matter what the value is, and all other values are zero. So, in the scaler matrix, three conditions must be satisfied:

1. The values other than the diagonal are all zero. 

2. The values in the diagonal must be non-zero.

3. All the values of the diagonal are the same, no matter what the value is. 

4. The matrix is a square matrix.

If one of the conditions is not satisfied, it is not a scalar matrix. 

Upper Triangular Matrix

This is an interesting type of matrix. The upper triangular matrix can be recognized when all the entries below the diagonals are zero and the entries above the diagonal are non-zero. So we get the matrix as:

The value in the diagonal must be non-zero for at least one value. And the 

The Lower Triangular Matrix

As you can guess, the lower triangular matrix has non-zero values in the lower elements than the diagonal. The values presented in the upper portion of the diagonal are all zero. So the lower triangular matrix looks like this:

Transpose of a Matrix

This is another concept in matrices that is important to discuss here because some types of matrices depend upon it. We know that a matrix has i rows and j columns. Then, the transpose of the matrix is defined as:

The transpose of matrix A, denoted by AT , is obtained when the rows and columns of matrix A are interchanged no matter what the total number of rows and columns. 

This concept is used in different ways, and the results obtained are important. To understand well, you must know, that a rectangular matrix of order 2 by 3 will be converted into a matrix of order 2 by 3 when the transpose is applied to it. 

Symmetrix Matrix

It is a special kind of matrix that involves the procedure of transpose. A skew-symmetric matrix is one that has the arrangement of elements in such a way that applying the transposed result in the matrix that has the same entries and same order. In other words:

AT=A

So, the skew-symmetric matrix is always a square.

The Skew Symmetrix Matrix

Here is another kind of square matrix that involves the transpose, and it is slightly different from the one discussed just before. This type of matrix, after taking the transpose, results in a matrix with all the negative values. So we define this in a simple way as

AT=-A

Harmatian Matrix

A hermitian matrix involves complex numbers in it. That means some, or all the elements of a matrix A are complex numbers, and the transpose of that matrix A results in the transpose of a conjugate matrix. So, the following conditions are applied to the hermitian matrix:

• It is a square matrix.

• It involves complex numbers.

• A conjugate of the matrix is obtained. 

• The resultant matrix has the conjugate values.

The Skew Harmatian Matrix

A skew hermitian matrix, with the complex conjugate, is the one that involves the negative values of the original matrix ( the one before the transpose procedure). It will be crystal clear to you when you examine the case given below:

Special Matrices in MATLAB

Are you enjoying the different types of matrices? Matrices are fun in MATLAB, it is good practice if we attempt all these types in MATLAB. Notice that, these types follow a specific sequence. Hence, MATLAB has a special function designed for the user with the help of which, one can have all these types by simply writing some commands. But it would be helpful if you understood the commands first.

Follow the procedure given below to perform the tasks.

• Start your MATLAB software. 

• Go to the command window.

• Start typing the codes given below and get your desired matrix.

A=[ 2 7 4; 5 11 5; 3 8 23]

• Notice that we have a square matrix now. It will help us in different operations. So let’s start using different operations. 

• Write your first command.

IdentityMatrix=eye(3,3)

• It will create an identity matrix with the name IdentityMatrix with three rows and columns.

• Similarly, write the command given below:

 UpperTriangularMatrix=triu(A)

• It's time to create the lower triangular matrix. So write the command.

 LowerTriangularMatrix=tril(A)

• For the null matrix, we are going to use:

NullMatrix=null(2,5)

• For the singular matrix, we are using the following command:

SingularMatrix=ones(3,4)

• If you wish to have a diagonal matrix of A, write the command:

DiagonalOfMatrix=diag(A)

• To sort your matrix A, we are using the command given below:

SortMatrix=sort(A)

• If you want to find the hermitian of the matrix, first you have to introduce the complex matrix in MATLAB. So we are writing the following matrix:

B = [1+j; 1-j; 2-j; 1+2j]    

• Now, by simply applying the command, we can get the hermitian of this matrix.

HermitianMatrix=B’

Thus, today’s lecture was full of different types of matrices, and many of them were new to us. We began with the definition of the matrix, then we saw different types of it where the data or the elements of the matrix were important. Many of them have conditions where the order and location of the elements are important. The concept of transport in the matrix was new to us, and by using it, we explored different types. In the end, with the help of different commands in MATLAB and with the help of practice, we sharpened our concepts. There are different and interesting types of matrices, and we are going to explore all of them in our upcoming lectures.

Before starting this important topic there are some of the important terminologies related to the theorem that are frequently used throughout. So let us have a look at them.

System

Some important key points related to the system are as follows:

• In thermodynamics, a system is a quantity of matter of fixed quantity. 

• The system is mostly closed. system. There is a change in the size and shape of the system but due to the closed system, no mass can cross the boundary.

• One of the important points is that the mass of the system remains constant.

Control Volume

Some important key points related to the control volume are as follows:

• The control volume is also known as the open system and it is defined as the region that is specified for a case study. 

• In the case of a control, volume mass can cross and move in and out of the boundary and which is called the control surface.

• One of the important key points related to the control volume is that the mass of the control volume decreases during the process but, the volume remains constant.

Reynold’s Transport Theorem

The important key points related to Reynold’s Transport Theorem are as follows:

• In fluid mechanics, control volumes are easy to handle and many of the laws and theorem are based on it. 

• So Reynold’s Transport theorem is actually a relationship between the rate of change of extensive properties of the system and control volumes.

• A system is considered arbitrary when Reynold’s Theorem is being considered. But here the derivation is also being carried away. 

• Reynold’s Transport Theorem provides a relationship between the system and control volume.

Derivation of Reynold’s Transport Theorem

Let us start the theorem with a background of it:

• First of all, we will consider the flow that is moving from left to right and that flow is passing through an expanding field.

• As the fluid flow the upper and the lower bounds of the fluid flow are considered to be streamlines of flow. Between any two streamlines, it is assumed automatically that the flow will be uniform and so in this situation, it is also considered.

• So there is a diagram that is mentioned below that shows the moving system and control volume (shaded region) of the flow field is considered at the times t and ∆t. the streamlines are also shown.
  

• Here one interesting fact is that the system and the control volume are the same as the system seems to be coinciding with the control volume.

• With reference to the diagram, in part (1) the system moves with uniform speed V1 and so in part (2) it moves with speed V2 uniformly.

• If you look at the diagram, then there is a central region called the hatched region.

• The region that is uncovered by the system during the motion is mentioned as section I and this is part of the control volume. but the new region that is covered by the system is said to be section II and this is not part of the control volume.

• Now let us start the derivation. For derivation let us suppose B as extensive property and that may be mass, energy or momentum.

• Moreover, let us suppose

b= B/m

• Now this b=B/m is the intensive property (we are supposing it). 

• So the extensive property is additive, they can be added easily.

• so the extensive property at times t and ∆t are presented as follows:

Bsys,t=BCV,t

Bsys,t+∆t=BCV,t+∆t - BI,t+∆t+BII,t+∆t

• Now we will get to the final result by taking two steps. The first step will be subtracting equation 1 with equation 2 and then taking the limit ∆t→ 0. So the final result will be:

dBsysdt=dBCVdt-Bin+Bout

• The value Bin and Bout is as follows:

Bin= b11V1A1

Bout= b22V2A2

• Here A1 and A2 are the cross-sectional areas.

• And the values of BI, t+∆t and BII, t+∆t is as follows:

BI, t+∆t = b11V1∆tA1

BII, t+∆t=b22V2∆tA2

• Now we are on the second last step and it is as follows:

Bin=BI=b11V1A1( after taking limit)

Bout=BII=b22V2A2 (after taking the limit)

• The important point here is that property is the extensive property and the time rate change of the system is equal to the time rate of change of B of control volume along with the net flux B of control volume by mass.

• Now the final result will be:

Bnet=Bout-Bin=∫CS ρbV.n dA  (inflow is negative)

• So below mentioned is the diagram that shows the whole working of the derivation:

• In the case of the control volume, the equation will be as follows

BCV=∫CV ρb dV

• Here the following equation presents the time rate of change of property B content of the control volume:

dBCVdt= d/dt∫CV ρb dV

• so the system to control volume in the case of the fixed control system will be as follows:

dBCVdt= ddt∫CV ρb dV +∫CS ρbV.n dA  

So that is the explanation related to Reynold’s Transport Theorem. 

Conclusion:

I hope you have learned a lot through this article. Thank you for reading.

Flow visualization widely uses Computational Fluid Dynamics (CFD) and physical experiments. So following are the types of patterns that can be visualized computationally and experimentally. So without wasting any time, let us start.

Streamlines and Streamtubes

There are two patterns. I will first explain the streamlines. So the definition of streamlines is as follows:

A streamline is a curve that is tangent everywhere to the instantaneous local velocity vector.

You might not understand the definition by reading it, so just for your ease, let me explain to you in few key points:

• A streamline defines fluid’s motion throughout the flow field.

• A streamline acts as an indicator of the instantaneous direction of motion of the fluid. To explain this situation, let me exemplify it with an example.

• For instance, when we throw water on a solid surface, we observe the fluid flow pattern on the wall. That pattern is a streamlined pan in which the water is separated, moved in recirculating motion, or coming off the wall.

• There is an equation of streamline; let me explain the background of the equation to you. Before that, have a look at the diagram.

• As you can see, there is an infinitesimal arc along the streamline. Here the infinitesimal arc length is as follows:

dr=dxi+dyj +dzk

• Here is a condition that dr should be parallel to the local velocity, whose equation is as follows:

V=ui+vj +wk

• By utilising the geometric rules, infinitesimal arc dr is proportional to the local velocity, and the equation of streamline will be like this:

drV=dxu=dyv=dzw

• Here, dr is the magnitude of infinitesimal arc length, and V is the magnitude of velocity.

• To obtain the equation in an (x, y) plane, we will integrate equation 1 and get the equation of streamline in an (x, y) plane. The equation is as follows:

(dydx)along astreamline=vu

Now I will give a brief explanation about the Streamtubes. So the definition of Streamtubes is as follows:

A Streamtubes is a bundle of streamlines similar to the communication cable with optic fibre cables.

You might not understand the definition by reading it, so just for your ease, let me explain to you in few key points:

• As I have discussed earlier, streamlines are parallel to the local velocity, so according to the theoretical information, the fluid cannot cross the streamlines.

• A simple diagram elaborates on tee difference between the streamlines and streamtubes.

Pathlines

The definition of Pathlines is as follows:

A Pathline is an actual path travelled by an individual particle (obviously a fluid particle) at some time period.

The following are some essential key points related to the pathlines:

•  You might have noticed one thing while reading the definition of pathline and Lagrangian are much similar. Both follow a path of an individual particle when the fluid flows.

• There is a technique named Particle Image Velocimetry (PIV) that is used to measure the velocity field in one flow of a specific plane.

• Let me explain to you the PIV technique briefly. So what happens in PIV is that those small particles are released in fluid, as shown in the diagram.

• Then, the flow is observed by a two-flash light to make two spots on a film of every small particle moving. The magnitude and direction of the velocity of each particle location are fixed because the particle size is small. In today’s modern era of science and technology, many modern computers and digital photography have enabled this feature.

• In order to trace the location of particles by the following equation:

x=xstart+tstarttV dt

• So, the diagram mentioned earlier shows the pathline following the actual path of fluid particles.

• There is a condition that if the velocity field is steady, the fluid particles are bound to follow the streamlines.

Streaklines

The definition of a streakline is as follows:

A streakline is the locus of fluid particles passed sequentially through the flow’s specific (prescribed) point.

The following are some important key points related to the streaklines:

• The streaklines are one of the most common flow patterns produced through physical experiments. To explain this point more clearly, let me explain to you through an example. If we insert a tube (specifically in small size) in the flow and then add a continuous stream of tracer fluid, the pattern produced due to the addition will be streaklines. The flowing diagram shows the streaklines produced when the constant stream of tracer fluid (colored fluid) is added to the flow. The fact about the diagram is that the streaklines are similar to the pathline and streamlines.

• Here is an interesting point to be noted: if the flow is steady, then the pathlines, streamline, and streaklines are all identical.

• So you might be confused by the statement that pathlines, streamlines, and streaklines are similar in steady flow. But in the case of unsteady flow, the scenario is entirely different. 

• There is a difference, i.e., streamlines give an instantons flow pattern (by definition) but in the case of streaklines is an instantons snapshot of time-integrated flow patterns. And the pathlines are the time-exposed flow path of an individual particle at some time.

• The equation to find the integrated tracer particle is as follows:

x=xinjection+tinjecttpresentV dt

Timelines

The definition of timelines is as follows:

A timeline is a set of adjacent fluid particles that were marked at the same time.

The following are some key points related to the timeline:

• The fluid flows in which there is uniformity and a steady flow, then timelines are observed. 

• Practically timelines can be generated in any water channel with the help of hydrogen bubble wire.

Plots of Fluid Flow Data

As we all know, flow properties vary from time to time, and in space, it is necessary to plot flow data in various ways. In this article, I am going to explain three kinds of plots that are profile plots, vector plots, and contour plots, respectively.

So without wasting any time, let’s start explaining the plots.

Profile Plots

The definition of a profile plot is as follows:

A profile plot indicates how the scalar property varies along some desired direction in the flow field.

The following are some essential key points related to profile plots:

• It is one of the simplest plots much similar to the XY-plots. 

• As the definition describe that the profile plots are created for the scalar quantities, but in fluid mechanics, velocity profile plots are used. As velocity is a vector quantity. So, to create a profile plot, we either use magnitude or velocity.

Vector Plots

The definition of the vector plots is as follows:

A vector plot is an array of arrows reflecting the magnitude along with the direction of vector quantity at an instant of time.

The following are some essential points related to the vector plots:

• Streamlines are used to present the direction of the instantaneous velocity; here, they do not show the velocity magnitude.

• A flow pattern is a vector plot for experimental and computational fluid flow. They have an array of arrows that indicate both magnitude and direction of a vector quantity.

Contour Plots

The following definition of a contour plot is as follows:

A contour plot shows curves of constant values of scalar property (or magnitude of a vector property) at an instant in time.

The following are some essential key points related to the contour plots:

• The contour plot may have curves indicating various properties called Contour Line Plot.

• Some of the contours are filled with color of grey are called Filled Contour Plot.

Kinematic Descriptions

In fluid mechanics, the elements have four fundamental types of motion or deformation in two dimensional as follows. It is interesting to know the fact that all four of these motions can act at the same time. Yes, you heard it right. Isn’t it amazing? In fluid dynamics, the motion and deformation of liquid elements at different times are described. So the deformation rates are expressed in terms of velocity and derivatives of velocity.

• Translation

• Rotation

• Linear Strain (Extensional Strain)

• Shear Strain

So let us start explaining one by one. 

Translation 

The translator and rotatory motion are one of the most common motions that are observed in our daily life. For the three dimensions, a vector is used to define the translator rate. In Cartesian coordinates the rate of translation is as follows:

V=ui+vj+wk

Rotation

The rate of rotation or the angular velocity is defined to be the average rotation rate of two perpendicular lines that intersects at some point. 

Example:

In order to exemplify this whole situation, let us explain through some examples.

• Let us take an initially square fluid element and consider its bottttttttt.

• Technically, the left and the bottom edge of the element intersect at some point. Thus we can say that they are perpendicular to each other. 

• Now, these two lines tend to rotate in a counterclockwise direction (which is said to be the positive direction). 

• Toto, show a clearer picture to you let me show you a diagram. So the below-mentioned diagram explains the rotational effect.

• One of the main points to observe is that the angle between the two lines remains the same, which is 90 degrees. 

• So the line rotates at the same rate of rotation.

• Here there is also one of the most essential points about the rate of rotation is that when the case is two dimensional then the fluid elements move in translator motion and then rotate. But while rotating, they deform easily.

• In order to calculate the rate of rotation, the whole thing is calculated by the scenarios mentioned above, i.e. two lines are taken named a and b respectively. These two lines intersect at a specific point called P ( as mentioned above that this procedure is in two dimensions. Which Means that they are in the XY plane).

• These lines are being followed and rotated in an infinitesimal increment of time, and that is:

dt=t2-t1

• Line a  rotates at some angle aand line b also rotates at a specific angle b. The average rotation angle will be:

a+b2

• So the final equation will be as follows:

ω=ddt(a+b2)

ω=12(∂v∂x-∂u∂y)

 Linear Strain Rate

The definition of linear strain rate is as follows:

Linear strain rate is defined as the rate of increase in length per unit length.

Some of the important key points related to the linear strain rate are as follows:

• The linear strain rate depends upon the direction of the line segment and this line segment measures the linear strain.

• But here important point should be kept in mind i.e. that the linear strain rate cannot be defined as vector or scalar quantity.

• The linear strain rate can be defined in an arbitrary direction. 

• The linear strain rate can be defined in Cartesian coordinates by the formula as follows:

xx=∂u∂x   yy=∂v∂y   zz=∂w∂z

Volumetric Strain Rate

The definition of the volumetric strain rate is as follows:

The rate of increase in the volume of fluid element per unit volume is called the volumetric strain rate.

Some of the important critical points related to the volumetric strain rate are as follows:

• In an incompressible flow, the volumetric strain rate is zero.

• With the increase in volume, the kinematic property is always positive.

• The other definition word for the volumetric strain rate is called Rate of Volumetric Dilatation. 

• The rate of volumetric dilatation is remembered by an example. Let us take an example of the iris of expands enlarging when there is less light.

• The formula of volumetric strain rate in Cartesian coordinates is as follows:

1VDVDt=1VdVdt=xx+yy+zz=∂u∂x+∂v∂y+∂w∂z

Shear Strain Rate 

The definition of the shear strain rate is as follows:

A shear strain rate is defined at the point as half of the rate of decrease of the angle between two initially perpendicular lines that intersects at some points.

Some of the important key points related to the shear strain rate are as follows:

• In order to explain the definition, let us explain through some examples. Let us see a diagram first that is mentioned below:

• In this diagram, the angle is at first 90 in the lower left corner and upper right corner of the element of fluid the angle decreases so that is a positive shear strain. But the angles at the upper-left and lower-right square fluid element increase so that is negative shear strain.

• In the Cartesian coordinates system, the shear strain rate will be:

xy=12∂u∂x+∂v∂x  zx=12∂w∂x+∂u∂z  yz=12∂v∂z+∂w∂y

Conclusion:

I hope you have learned a lot through this article. Thank you for reading.

Hello friend. In this article, I will cover essential points related to fluid kinematics, i.e., what fluid kinematics is and how the motion of the fluids can be explained without describing the forces acting on them. Further, I will explain Lagrangian and Euler about the motion of fluid and a lot more. So buckle up, and let’s start.

What is Fluid kinematics?

The definition of fluid kinematics is as follows:

Fluid kinematics is related to fluid motion without considering the forces responsible for the motion.

The following are some essential key points related to fluid kinematics:

• The nature of fluid motion is categorized into two types:

  1. Lagrangian
  2. Euler.

• I will extensively explain both types in upcoming topics.

• The acceleration, velocity, flow rate, and nature of fluid flow are considered while working on fluid kinematics.

Lagrangian and Euler Description 

First of all, I will explain the Lagrangian. So the definition of Lagrangian is as follows:

Lagrangian Description of fluid flow deals with individual particles and their nature, and the working trajectory of each particle is calculated separately.

And the definition of Euler is as follows:

Euler’s Description of fluid flow deals with the concentration of the particles, and the number of particles and diffusion are all calculated.

The following are some essential key points related to the Lagrangian Description:

• Lagrangian is all about tracking the path of individual particles. 

• In this regard, Newton’s Laws are beneficial in detecting the pathways. With the help of their kinetic energy, speed, distance velocity, and acceleration.

• As we all know, fluid is Continuum in nature. If you are following my fluid mechanic’s articles series, you have forwarded to this word, where I have extensively described the continuum. So it becomes pretty challenging to follow the path of fluids as they deform whenever a direction changes.

• In the Lagrangian description, the position and velocity of individual particles must be tracked throughout. 

• To describe through a simple example the balls on the snooker table. In this case, each ball is responsible for its path.

The following are some essential key points related to the Euler Description:

• In the case of Euler’s description, the control volume is defined through which the fluid can move in.

• In this description method, there is no need to track the velocity and position of fluid particles.

• In this description, space and time are defined within the domain of control volumes.

• As I have defined in previous articles, pressure is a scalar quantity. So pressure field in case of three-dimensional fluid flow in Cartesian co-ordinates can be defined as follows:

P = P (x, y, z, t)

• The velocity field is mentioned as follows:

V=V(x, y, z, t)

• The acceleration field is as follows:

a= a (x, y, z, t)

• The interesting fact to know is these three field(along with some others but for now only these three fields) variables define the flow field:

V=(u, v, w)

V=ux,y,z,ti  +vx,y, z, tj +w(x, y,z,t)k

Difference between Lagrangian and Euler Description

I will explain it through a simple example to depict the Lagrangian and Euler description.

• The example is as simple as one person standing beside the river. So when he goes through the probe, which will move downstream with water, this situation will be classified as Lagrangian. This will be classified as Euler’s description when he anchors the probe in some fixed location in the water.

Application of Fluid Kinematics

The applications of fluid Kinematics are as follows:

Usage in Vehicle

• The purpose of dealing with Fluid kinematics is that fluid has excellent properties, and they are used almost in every vehicle. So for lubrication purposes and as driving fuel.

Electricity Generation

• Anything that moves has kinetic and potential energy and fluids. They are used in hydroelectric power plants for the generation of electricity.

Air Conditioner and Refrigeration

• In refrigerators and air conditioners, refrigerants used are fluids. The primary purpose of refrigerants is to absorb the heat (from the room or, in the case of refrigerators, anything that is kept inside them) and keep them cool, then release the heat into the atmosphere. In the case of air conditioners, heat is absorbed from the room to keep them cool and release that heat into the atmosphere. 

Renewable Energy

• Surprisingly fluids can be used as a renewable energy resource. One of the simplest and best examples is the water used in tidal power plants for electricity generation. Moreover, a vegetable oil known as biodiesel is used in many vehicles. Wind and air are also used as renewable energy sources. 

Thermal Power Plants

• One of the essential resources of electricity in thermal power plants. In thermal power plants, water (used as fluids) is heated to form steam, which then turns on the turbine. The turbine then turns on the generator, which generates electricity. So the purpose of explaining this whole procedure is to explain the importance of fluid in electricity generation.

So that are the applications of fluid kinematics in our daily life. Dear friends, I will explain two significant ways to describe motion.

As I have discussed the pressure now I will explain fluid mechanics. The definition of fluid mechanics is as follows:

Fluid mechanics deals with the properties and characteristics of fluid at rest.

Some of the important key points related to the fluid statics are as follows:

• Fluid statics has many properties just as hydrostatics when the fluid is in a liquid state and acts as aerostatics when it is in a gaseous state.

• The only stress that is in fluid statics is the normal ones other than that there is no shear stress.

• The main application of fluid statics is to determine the forces that are acting on the floating or the submerged bodies.

That is a brief introduction to fluid statics further I will explain the forces that are acting on submerged and floating bodies. So without wasting any time let us start.

Hydrostatics Forces on submerged plane surfaces

The structure of the surface and the forces that are acting on the surface is explained as follows:

• As the name suggests, there is a plane surface on which liquid flows and the surface is resting. 

• The fluid pressure is equally distributed onto the surface.

• So the set of parallel forces is formed on the surface as a result of hydrostatics forces. 

• As one side of the surface is facing the fluid pressure whereas the other side is sometimes exposed to the atmosphere and as the result is zero resultant.

Working Principle

The working principle along with all the equations that are acting on the surface are explained as follows:

• First of all, as I have discussed earlier that one side of the plate is completely exposed to the atmosphere and has atmospheric pressure whereas the other side of the plate is completely submerged in the fluid as shown in the diagram.

• The horizontal surface intersects the plane surface at some angle so that line is considered to be x-axis.

• Po is the absolute pressure i.e., the pressure that is above the fluid (liquid) also known to be local atmospheric pressure if the fluid or liquid is exposed to the atmosphere.

• The equation of absolute pressure is:

P=P0+ρgh

P=P0+ρg ysinθ (equation 1)

• Here h is the vertical distance of a specific point to a surface and y is the distance of a point from the x-axis from the point O as shown in the diagram.

• Now the next step is to find the resultant hydrostatics force and for that, the equation is as follows:

FR=∫APdA

• Here we will substitute the value of P from equation 1 and equation 2 will look like this:

FR=∫A(P=P0+ρg ysinθ) dA

FR=P=P0A+ρg sinθ∫AydA

• ∫AydA is the first moment of the area as it is in the y-coordinate and the value is as follows:

yc=1A∫AydA

By substituting the value, we get the final result:

FR=P0A+ρgyc sinθA

FR=P0+ρghcA

FR=PcA

FR=PavgA

The final result will be stated theoretically:

The final resultant force is equal to the pressure at the centroid surface and the area.

Hydrostatic Forces on Submerged curved surfaces:

The structure of the surface and the forces that are acting on the surface is explained as follows:

• As the name shows that the curved surface is submerged in liquid and the hydrostatic forces are determined.

• In this part we have applied the integration of pressure forces that fluctuates when the direction of the surfaces changes.

• The final force (resultant) is equal and opposite to the force that is acting on the curved surface.

Working Principle

The working principle along with all the equations that are acting on the surface are explained as follows:

• Here the weight of the enclosed liquid is as follows:

W=ρgυ

• Here V is the volume of fluid and that is in a downward direction as shown in the figure.

• the vertical and horizontal forces will be:

• the horizontal force component on a curved surface:

FH=FX

• the vertical force component on the curved surface

FV=Fy+W

The final result will be:

• the horizontal and the vertical component of the hydrostatic force of a curved surface are both equal in case of magnitude and line of action.

• The horizontal and vertical component is an equal hydrostatic forces acting on horizontal projection along with the weight of the fluid.

Conclusion:

I hope you enjoyed reading the article and have an extensive overview of the pressure and fluid statics. Thanks for reading.

The article will cover essential aspects of fluid statics and pressure forces applied to the fluid. The article will start with the absolute pressure gauge and the variation of pressure with depth in different gravitational field manometers and barometers, respectively. Then I will explain the forces that deal with fluids, such as hydrostatic force, buoyant force, etc. So, dear friends, without wasting any time, let’s start.

Pressure

The definition of pressure according to fluid mechanics is as follows:

Pressure is the normal force applied by a fluid per unit area.

Unit of Pressure

• The units of pressure Pascal is too small to deal with in practical cases. For instance,

• Kilopascal: 1kPa=103Pa

• Mega Pascal: 1MPa=106Pa

• The three common pressure units used are standard atmosphere bar and kilogram-force per square centimeter. 

• 1bar=105Pa=0.1MPa=100kPa

• 1atm=101,325Pa=101.325kPa=101325 bars

• 1 kgf/cm2=9.807 N/ cm2=9.807×m2=9.807×104Pa

=0.9807bar

=0.9679 atm

• The unit of pressure in the English system is pound-force per square inch (psi or lbf/in2)

Absolute and Gage Pressure

• The actual pressure at a given position is called absolute pressure. The pressure is measured relative to an absolute vacuum or absolute zero pressure.

• The difference between the absolute and local atmospheric pressure is called gage pressure.

• The pressure below the atmospheric pressure is called vacuum pressure.

Formula of Gage, Absolute, and Vacuum Pressure

All three of these pressures are interrelated with each. Other, and this is visible with the formulae as mentioned below:

Pvac=Patm-Pabs

Pgage=Patm-Pabs

Diagram Related to Pressure

The following diagram shows the relation of the pressures from the below-mentioned diagram:

Pressure Variation according to Depth

We all know there is no fluid pressure change at rest and in the horizontal position. This phenomenon can further be elaborate on a few points:

• Assuming a thin layer horizontally of fluid followed by force balance at any point. The pressure increases as the depth increase as there are more layers of fluids, and these layers are being balanced by the pressure increasing.

• An example can explain the whole phenomenon. 

• Toto show how pressure changes with the change in depth. We assumed a rectangular fluid element, as shown in the diagram.

• The length, width, and heights are mentioned on the diagram.

• The pressure of the fluid is constant, whereas force balance in z direction vertically will be:

∑Fz=maz=0

P2△x-P1△x-ρg△x△z=0

• Now, we will divide the above equation with delta x △xand will get the below-mentioned equation:

△P=P2-P1=ρg△z=sz

• Here, one of the most important things is that the s is the specific weight of fluid.

• The final statement to be noted is that there is an increase in pressure linearly with an increase in depth.

• It is essential to note that the pressure force applied by the fluid is always normal at some points to the surface.

Pressure at Point

As pressure is a scalar quantity, sometimes it seems to be vector one. The pressure will be the same in any fluid at any point.  To prove this point, let us discuss a scenario.

• Let us consider fluid element (a wedge shape/ right-angled triangle) at equilibrium.

• The mean pressures at three surfaces are mentioned as P1, P2, and P3, respectively. 

• The diagram shows the points clearly.

• Here, the value along the pressure is the surface area.

• According to Newton’s second law, the force in the direction of x and z is as follows as △y=1:

∑Fx=max=0             P1△z-P3Isinθ = 0     (1)

∑Fz=maz=0             P2△x-P3Icosθ-12ρg△x△z = 0         (2)

• Here, the value of w is the weight of the fluid, and the value is as follows:

W=mg

W=ρg△x△z/2

• The value of △x and △z are as follows:

x = I cos θ

△z=I sinθ

• So, by substituting all the values in equations 1 and 2 and then diving equation 1 with △z and equation 2 with △x then, the final result will be:

P1-P3= 0     (3)

P2-P3-12ρg△z=0       (4)

• Here, in equation 3 △, z drops to 0, and the final result will be:

P1=P2=P3=P

Pressure measuring devices

Numerous devices are used to measure fluid pressure and work on different principles. These devices are explained below briefly.

Manometer

The definition of a manometer is explained as follows:

A manometer is a device in which fluid columns measure pressure differences.

The important key points about the manometer are explained below:

• the manometer is used to measure small or moderate pressure differences.

• If pressure is at a specific position inside any gas tank, then the pressure anywhere in the tank will be the same. But here important point to be noted is that it is for gases only as their gravitational effect is negligible.

• The basic diagram of the manometer is shown as follows:

• As we can see, two points in a basic barometer and a height h. So the pressure at point 1 will be the same throughout the tank so  that P1=P2 and the pressure will be by the equation:

P2=Patm+ρgh

• Here, h is the height of column fluid, which is in static equilibrium, and as we can see, the column is exposed to the atmosphere.

• is the density of the fluid, and Patm is the atmospheric pressure. 

• Here, one important point to be noted is that the height of the tube is independent of the area of the tube.

Barometer

The definition of the barometer is as follows:

A barometer is a device that is used for the measurement of atmospheric pressure.

The following are some important key points related to the barometer:

• The structure of the barometer is the inversion of a tube filled with mercury in a container, which is exposed to the atmosphere.

• The diagram shows the structure of the barometer;

• Toto calculates the formula of the atmospheric pressure; we will follow the above diagram. As you can see, there is a point B, so this B equals atmospheric pressure. Point C inside the tube is assumed to have zero pressure as there are only mercury vapors above point C, and pressure above that point is low compared to the atmospheric pressure, so for ease, the pressure is taken to be zero. The formula of pressure is as follows:

Patm=ρgh

Other Pressure Measuring Devices

There are many other pressure-measuring devices that are in use in our daily life, and they are briefly explained as follows:

• Bourdon Tube is a pressure measuring device consisting of a hook like a hollow metal tube that is connected to a dial indicator. There is a fluid inside the tube that is pressurized. When this fluid is being pressurized up to a point then the needle that is on the dial is deflected. The needle is read to zero when the tube is exposed to the atmosphere.

• There are special types of pressure detectors that can convert the pressure effect into electrical energy and they are called pressure transducers.

When the mechanical pressure is created when the electric potential is applied to a crystalline form. Such kinds of devices are called Piezoelectric transducers.

In the introduction, I will extensively explain the property r and how to impact them are so. Let us start. Then I explain properties that are related to fluid flow.

First of all, I will start with the word Property. The definition of property is as follows:

Property

The word property is defined as any characteristic of the system. There are two main types of the property named intensive and extensive.

Example

The examples of property are as follows

• Temperature -T

• Pressure - P

• Mass – m

Types of Properties 

Following is a brief explanation of the types of properties.

Intensive Property

The intensive properties are the ones that are independent of mass, temperature, density, and pressure. Therefore, they are presented in lowercase letters. But here stands an exceptional case, i.e., the temperature and pressure are only denoted with uppercase letters.

Example

The examples of intensive property are as follows:

• Velocity - v

• Specific Internal Energy - u

• Temperature - T

• Charge Density - ne or ρ

• Molality - m or b

Extensive Property

The extensive property is the ones that depend on the size or extent of the system. They are presented by the uppercase letter, but here is an exceptional case for the mass presented by a lowercase letter. 

Example

Extensive properties are usually derived, so the following are some examples.

• Total Energy

e= E/m

• Specific Volume 

v=V/m

Specific Property 

The specific property is the ones that depend on that are derived from either extensive or intensive ones. 

Example

One of the simple examples of a specific property is the density of water; as we know, it is an intensive property. So the extensive property (derived ones) is the mass of water volume divided by the volume. Here both the mass of water and volume are extensive properties.

Continuum

The term continuum is related to matter in this way: the matter is made of atoms, and their distance and compactness vary from state to state. As in solids, atoms are closely packed, whereas, in liquids, there is some distance. However, in gases, the atoms are spaced widely. So in the continuum, the matter is defined as continuous and homogenous with no holes. 

The primary purpose of using a continuum is to treat properties as a point function, and there are no jump discontinuities as the properties vary in space continually. This only applies when the system size is large compared to the space in molecules. It is practically applicable in almost every case except exceptional cases.

Properties related to fluid flow

At the start of the discussion, I discussed the properties such as intensive and extensive observed during the fluid flow. Now some more properties play a distinct role in the fluid flow and their importance. So without wasting any time, let’s start.

1. Density

The density is defined as;

Density is mass per unit volume.

Formula:

ρ=m

Specific Density

It is defined as 

The specific density is the reciprocal of density, i.e., volume per unit mass. 

Formula:

υ=Vm

Factors affecting Density

There are numerous factors on which the density depends. The most important of all is the temperature and the pressure, which change the nature of the substance. 

Gases

In gases, an increase in pressure will decrease the temperature and vice versa.

Solids and Liquids

Compared to gases, solids and liquids are incompressible, so an increase or decrease in pressure is negligible. 

Example 

In solids and liquids, the temperature is an essential factor that changes the density compared to the pressure. For instance, at the pressure of 1atm, the density of water changes from 998kg/m3 to 975kg/m3. So the difference is about 2.3 percent which is negligible in many cases.

Relative Density or Specific Gravity

The definition is as follows:

The density of the substance is relative to the density of the well-known substance.

Formula:

SG=H2O

SI units of Specific Gravity

As relative gravity is a dimensionless quantity, so the SI unit of it is as same as the density one, i.e., kg/L or g/cm3 (0.001 times density in kg/m3)

Specific Gravity of Substances

The following is the list of some substances with specific gravity at 0˚C.

Specific Weight

The definition of specific weight is as follows:

The weight of a unit volume of a substance is known as specific weight.

Formula

s=ρg

Vapor Pressure and Cavitation

Vapor Pressure

The definition of vapor pressure is as follows:

Vapor pressure is the pressure applied by the vapor of a pure substance at the state of equilibrium with its liquid at a given temperature.

Some important key points related to the vapor pressure are as follows:

• It is presented by Pv.

• Vapor pressure is as same as saturation pressure Pv.=Psat.

• Sometimes, vapor and partial pressure are considered one thing, but they have two different definitions and applications. 

• Increase in temperature increases the vapor pressure.

The following table shows water saturation or vapor pressure at different temperatures.

Some important terms are related to vapor pressure, and they are named saturation temperature and saturation pressure, respectively. 

• Saturation pressure is the pressure at which changes in the state of a pure substance occur at a given temperature. It is represented by Psat.

• Saturation temperature is the temperature at which changes in the state of a pure substance occur at a given pressure. It is represented by Tsat.

Partial Pressure

The definition of partial pressure is as follows:

Partial pressure is the pressure of gas or vapors with a combination mixture of gases.

Some important key points related to partial pressure:

• Partial pressure of water vapors has some atmospheric pressure, and the value is 0.3. 

• The absence of liquid equalized or less than the partial pressure of vapors to the vapor pressure. 

• At the state of equilibrium, the presence of partial and vapor pressure in the system then the system is saturated.

• In open lakes, the evaporation rate is managed by the vapor and partial pressure differences.

Cavitation

As we know, vapor bubbles in the liquid collapse when they are moved or swept from low regions. The result is the production of high-pressure waves that creates destruction. The reason for explaining this is that the phenomenon is related to cavitation, and the vapor bubbles formed are called cavitation bubbles. The phenomenon mentioned above is called cavitation as it is the leading cause of destruction and causes performance to drop off.

Some important key points related to cavitation:

• In designing hydraulics pumps and turbines, this phenomenon (cavitation) is prioritized

• In flow systems, cavitation is not considered as they result in poor performance, turbulence and noise.

• For checking whether cavitation is present in the flow system or not, cavitation is observed by a unique tumbling sound.

Energy and Specific Heat

We are well known for the term that energy cannot be created nor destroyed but can be transformed from one form to another. There is various form of energy that we observe in our daily life. 

So I will recall some of the vital energy that is used during the fluid flow:

Kinetic Energy

The basic concept of kinetic energy is that anybody or system (comprised of energy) is in motion relative to the reference frame and is said to have kinetic energy. 

Formula:

K.E = 12v2

• Here, v stands for the velocity of the system or body relative to the reference frame.

Potential Energy

The potential energy is the energy possessed by the body or system due to elevation in the gravitational field.

Formula:

P.E =gz

• Here, g represents the gravitational acceleration, whereas z represents the elevation of the system or body concerning the reference position. 

Thermal Energy

Thermal energy is the energy that is related to the temperature. So it is defined as the energy in the system responsible for temperature.

Formula

Q=c ∆T

• Here ∆T represents the temperature difference, whereas c represents the specific heat.

After discussing all the energy, one important term related to fluid flow is the Enthalpy. So let us discuss enthalpy.

Enthalpy

The enthalpy is the combination of internal energy and the product of volume and pressure. The enthalpy value is calculated with the help of pressure, temperature, and volume, respectively. The energy conservation law says that internal energy changes equal heat transfer. The enthalpy change is assumed to equal the heat transfer if the work changes the volume at constant pressure.

Formula:

H=E +PV

• Here, H is the enthalpy, E stands for internal energy, P stands for pressure, and V stands for the system volume.

Total Energy

The sum of all the energy of flowing fluid is called total energy. 

Formula

eflowing=K.E+P.E+Enthalpy

eflowing=h+V22+gz

Coefficient of Compressibility

There is an expansion in fluids when they are heated, and they contract when they are cooled. This means that temperature and pressure are two essential parameters responsible for contraction and expansion. The temperature and pressure also change the volume or density of the fluids. But volume or density varies differently from fluid to fluid. So, two properties are related to the volume or density changes to the change in pressure. And they are named Bulk Modulus of Elasticity and Coefficient of volume expansion.

So let’s explain these two properties without wasting any time.

Bulk Modulus of Elasticity

There is a similarity in solids’ and liquids' expansion and contraction principles. So, the definition of the bulk modulus is as follows:

Bulk Modulus of Elasticity is a property that is the ratio of change in pressure relative to the volume.

Formula:

κ = -v(Pv)T= ρ(P∂ρ)T     (1)

The formula can also be presented concerning the finite changes, as shown below.

κ≅ -∆P∆vv≅-∆P∆pp

• Here ∆vv and ∆pp are dimensionless. 

• The unit of κ is psi or Pa.

• Here, the temperature is kept constant, so the bulk modulus of elasticity presents the change in pressure to the change in volume and density of the fluid. 

• The above equation proves that the incompressible substance coefficient of compressibility (Bulk modulus of Elasticity) is infinite.

• Increase in pressure decreases the volume.

• The negative sign indicates Pv is a negative quantity, whereas the bulk modulus value is positive.

An important key point related to Bulk Modulus of Elasticity:

• Bulk modulus of elasticity is also known as the coefficient of compressibility or bulk modulus of compressibility.

• If the value of bulk modulus is significant, then firstly, the fluid is incompressible, and then a large amount of pressure is required for a small amount of fractional change in the volume.

• When the equation is being differentiated, then

• From the equation, it is evident an increase in one quantity decreases the other and vice versa. From equation 1, we will differentiate the ρ=1/v, and we will get dρ=-dv/v2. So after cancellation, the final result will be:

dρ=-dvv

• The inverse of the coefficient of compressibility is called isothermal compressibility. 

• The formula is as follows:

α=1=-1v(∂v∂P)T=1(∂ρ∂P)T

Coefficient of Volume Expansion

It is observed that there is a drastic change in liquid density when it is increased or decreased. A property named coefficient of volume expansion is responsible for the fluid density difference at constant pressure and variable temperature.

Formula:

𝛃=-1v(∂v∂T)P=-1(∂ρ∂T)P

• The coefficient of volume expansion is represented by beta.

• At the infinite changes, the formula is shown below:

-∆vv∆T=-∆pp∆T

• Here, the pressure is kept constant.

Important key points related to the coefficient of volume expansion:

• Increase in temperature increases the value of as they have a direct relationship.

• ∆T represents the temperature difference in the equation. 

Viscosity

If we define viscosity, the most reasonable and appropriate term will be resistance. When the fluid flows, there exist layers of viscosity at that point. There are intermolecular forces in fluid when the fluid is more viscous. So the definition of viscosity will be:

Viscosity is a property that represents the internal resistance of a fluid to motion.

Example:

One of the best examples is the pouring of honey. As you observe, honey is dense and very has strong internal resistance.

Formula:

η=2ga2(∆ρ)9v

• We assume a sphere that is dropped onto the fluid, and then the fluid’s viscosity is measured.

• Here ∆ρ is the density difference between the fluid and surface

•  a is the radius of the sphere 

•  G is the acceleration due to gravity

Important key points related to viscosity:

• When one layer moves adjacently to the other, some friction exists, which we named viscosity. The layers are moving at some distance and are named dy. The velocities of the fluids are u and u+du, respectively. 

• The graphical presentation of the layer velocity versus the distance is shown below.

• The graph will explain the trends of velocity and distance. As mentioned, two layers are moving adjacently to each other, so the layer that is on top imposes shear stress on the lower layer, and the lower layer, in response, causes shear stress on the upper one. 

Types of Viscosity

There are two primary types of viscosity, and they are explained briefly as follows.

Kinematic Viscosity 

Following are some essential key points related to kinematic viscosity:

• Kinematic viscosity is the ratio between dynamic viscosity and fluid density.

• The formula of kinematic viscosity is as follows:

υ=

Here υ is the kinematic viscosity η is the absolute or dynamic viscosity, and ρ is the density of the fluid.

• When the temperature decreases, kinematic viscosity also decreases.

Dynamic or Absolute Viscosity

Following are some essential key points related to dynamic or absolute viscosity:

• Dynamic viscosity is the ratio of shear stress to the shear strain of motion.

• Dynamics viscosity helps in the interaction of the molecules dealing with mechanical stress.

Conclusion:

I hope you enjoyed reading the article and have an extensive overview of the fluid-flowing properties. Thanks for reading.