 Myself, Mr. Sachin Deshmukh, I am working as assistant professor in Vulture and Strip Technologies, Civil Engineering Department. Today, we are going to learn about velocity potential and steam functions. These are very important part of the flow dynamics in which we can define the flow is rotational or irrotational. After the learning of this concept, you are able to understand what is the velocity potential, what is the steam function and from this what we are going to achieve. Just now I told we are going to define the flow is rotational or irrotational. Now what is the rotational and irrotational? When the fluid particles are rotated about its mass center and this satisfies the Laplace equation or not that we are going to see. But before that, some important definitions that we are going to see. In this, we are going to see the definition, what is the concept, the relevant formulas and what is the relationship between velocity and potential. I will just give you the orthogonal, you can say that multiplication you are getting when you are going to multiply these two, the curves of these two, you are getting the orthogonal. That is minus one that I will tell you when the concept will see. Now for this we have to see what is the path line, what is the steam line, what is the steam streak line, what is the steam tube, what is the equipotential lines. Like this we will just go through this. First of all we will see the path line, a line traced by an individual fluid particle. Just see these are the stations, colored points and through this the color you can say die is sent and these are the particles moving from these points. The line traced by these points, it is called as a path line. Then the steam line, a line in the fluid whose tangent, see here these are the fluids flows and the tangent from this is parallel to the instantaneous velocity vector at the given instant time t. So the tangent indicates the velocity at that point. The family of streamlines at the time t are the solutions of dx by du is equal to du by dy is equal to dz by dw with the directions, where u, v, w are the functions of x, y, z direction and the velocity components in the respective direction. Steak line consists of all fluid particles in a flow that have previously passed through the common points which we have seen just now. Such a line can be produced by continuously injecting marked fluid. See here this one. The examples are the smoke in the air or die in the water when the velocity is increased. Stream tube is the surface formed instantaneously by all the streamlines. These are the streamlines, these are the streamlines and this is a stream tube. You can say our section is taken in the pipe. This is a stream tube and these are the streamlines. Now we will shift to what is the velocity potential and what is the stream function. Again I will tell you these two are the scalar functions and from this we are defining the flow is irrotational or rotational. See what the definition says. Velocity potential is defined as a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. It is denoted by phi. Mathematically we can write it is a function of phi is a function of x, y, z when it is unsteady and phi is function of x, y, z when t is not there with not respect to time then it is a steady flow. Now with the definition that is a negative derivative of that is u is a negative derivative of this x direction then v y direction and w is the z direction where u v w are the components of the velocity in x, y, z directions. The negative sign signifies that velocity component decreases with increase in the values of x, y, z direction as there is a negative sign. Now the derivation we will see for an incompressible steady flow the continuous equation continuity equation is given by that is d by dx plus dv by dy plus dw by dz is equal to 0 and by substituting the values of this u, v and w we get this particular equation that is d square phi upon dx square plus d square phi upon dy square and d square phi upon dz square is equal to 0. This equation is known as the Laplace equation related to that we must know the equations for the rotational velocities. The rotational components with x and y and z direction are these I will just tell you about the x direction that is one half into bracket dw upon dy minus dv upon dz related to this x direction you can go for finding out y direction and z direction also okay y direction not z direction okay. So you can find out this components and put the values in that particular equation okay that is you can find out x, y, z direction okay that is it is a rotational velocity in x direction, rotational velocity y direction, rotational velocity in z direction. You can only just simplify it okay put the values and simplify it put the values and simplify it one finally you will get this equation and when this rotational velocity in x, y and z direction is equal to 0 this is called as when it is 0 it is called as irrotational flow thus if velocity potential satisfies the Laplace equation it represent the possible steady incompressible and irrotational flow okay the steps you can follow okay like this okay put the values and just take the contents with respect to the rotational velocities and put the values for the x, y and z direction and find out. Similarly the stream function what is the stream function it is a function of space and time particularly for the 2D fluid is such that its partial derivative with respect to any direction gives the velocity component at right angles you just remain this word right angle to this direction it is denoted as 5 okay so in case of two dimensional flows stream functions we can write that is x, y with respect to and t for unsteady flow and only x, y when t is not there for steady flow 5 and psi okay that you have to keep in mind velocity potential and stream function respectively from definition we can write here it is not a negative derivative it is directly now that is u is equal to d psi upon dy and for v is d psi upon dx and we have to put these values in the equations okay for the two dimensional flow it is the equation d by dx plus dy is equal to 0 similar the equations substituting the values of u and v we are getting here d square psi upon dx dy because in the two directions minus the square of psi upon dx dy is equal to 0 here the existence of psi means a possible case of fluid flow here it is the existence of psi that is means it is a possible case of fluid flow similarly for rotational component okay for rotational component in the z direction it is the equation it is half into bracket dv upon dx minus dy and substituting the values of u and v we are getting here it is wz is equal to 1 half that is minus 1 half into bracket d square psi upon dx square plus d square psi upon dy square this equation is a Poisson's equation this equation is a Poisson's equation but the previous equation in the velocity component that we have seen the Laplace equation okay so again for any rotational flow when the velocity at z direction the rotational velocity with respect to z direction is 0 and again this is Laplace equation with respect to psi now we'll see what are the properties of the stream function on any streamline size constant if the flow is continuous the flow around any path the fluid is 0 the rate of change of psi with resistance in arbitrary direction is proportional to the component of the velocity normal to that direction just remain these three important properties now what is the relation between velocity potential and stream function if two points lie on the same line same stream line then there is there will there there being no flow across a streamline and difference between psi 1 and psi 2 is 0 means stream lines gives size equal to constant similarly phi is constant represent a case of velocity potential is same at that every point which is known as equipotential line these two sets of curve stream lines and equipotential line intersect each other orthogonal just I told you in the starting at all points in the intersection these are the four objective type of questions just go through this take a pause and just go through these questions very simple questions are there just now we have just studied it the answers of these questions are so velocity potential is a function of you can say space and time stream line is defined as for the only 2d flow stream function gives a velocity component at right angles to a particular direction then when velocity potential I exist the flow is irrotational okay they are very simple questions but very interesting questions okay these are some reference books you just go through it if you find any difficulties regarding this you can contact me thank you