 Hello students, myself, Mr. Siddeshwar B. Tuljapure, associate professor, Department of Mechanical Engineering, Walchand Institute of Technology, Solapur. So today, we are going to deal with the topic from the section, it is fluid dynamics and the name of the topic is Bernoulli's theorem, learning outcome. At the end of this session, the students will be able to explain the Bernoulli's theorem. Let us see the contents of this session. The first one, it is the assumptions of the Bernoulli's theorem, that is firstly assumptions we are going to cover, then the statement of Bernoulli's theorem is there, then we are going for expressing the energy in heads and then equation of the Bernoulli's theorem we are going to write. Then the flow through the horizontal pipe of varying cross section area, then we are going to deal with the flow through an inclined pipe of varying cross sectional area, then lastly it is the references. Now firstly, we look for the assumptions of the Bernoulli's theorem. In case of the assumptions, we can have the two parts, that is one it is associated with the fluid and the second one it is associated with the flow. In case of the fluid, the fluid should be ideal. Now the ideal fluid is the fluid of which the viscosity is equal to it is 0. Now all of us know that the fluids which are existing that is the fluids are having the viscosity it is and fluids are having viscosity because of which they are called as real fluids. If the fluids are not having any viscosity or zero viscosity is there, then the fluid will be called as ideal fluid and as you know the ideal conditions does not exist. The fluid should be ideal is going to remain ideal only that is we are not going to satisfy in case of the experiments are in the real world situation. Every fluid which is existing in the world is having to some extent either it is lesser or larger etcetera the value of the viscosity it is. Then the second one is the flow should be steady, the flow should be incompressible and flow should be irrotational. Now flow should be steady. In case of the steady flow, we are going to have the number of properties of the fluid flow that is velocity, pressure, discharge, etcetera at a point constant with reference to the time. In case of the second one that is the incompressible flow, so generally fluids are incompressible the flows associated with the liquids these are incompressible, third one that is the flow should be irrotational. In case of the fluid flow the fluid particles if they are not rotating about their own mass centers or axis the flow is called as irrotational flow. Now the statement of the Bernoulli's theorem is if these assumptions are satisfied then the statement is the total energy at any point in the fluid flow remains constant. Now we are having the different forms of energies, there are three different forms of energies one it is the pressure energy, second one it is the kinetic energy and the third one it is datum or the potential energy. So these three forms we are having and in case of the fluid flow the summation of all these three is called as total energy and that should remain constant like that the statement of Bernoulli's theorem is and for this one we require the assumptions to be satisfied. Now expressing the energy in the form of the heads it is. Now pressure energy can be expressed in the form of the pressure head with the help of the formula we can calculate pressure head and the formula is p upon it is rho g where p is the pressure in Newton per meter square and rho it is the density of the fluid then it is the g it is the gravitational acceleration and then the kinetic head is there in case of the kinetic head it is v square by 2g where v is the velocity of the flow and g it is the gravitational acceleration and then datum and the potential head or datum and potential head it is z only that is in terms of the meters it is the height from the reference line. Now in case of the Bernoulli's theorem if the equation it is to be written it can be written as the p by rho g plus v square by 2g plus z is equal to this constant. Now I can think of the units of the pressure head that is p by rho g then it is the kinetic head that is v square by 2g and then it is the datum or the potential head it is z. Let us see the answer of this one. So the pressure head kinetic head and the datum head these are nothing but these are the energies which are represented per unit weight of the fluid. So we are knowing that the energy it is having a unit equal to joule. So joule means it is Newton per Newton meter then per unit weight of the fluid we are having that is Newton meter divided by Newton it will be Newton Newton will get cancelled and we will get the energy in terms of the meters of the fluid it is and thus unit of all the heads is meters also you should remember that we are having the addition of the kinetic head pressure head and the datum head. So the datum head already we are knowing that the unit of that unit is meter. So meter, meter, meter only the addition is possible. So we are going to have the pressure head in terms of the meters then kinetic head in terms of the meters and then the datum head or the potential head also in terms of the meters. Now let us see a case of fluid flow through a horizontal pipe of varying cross section area. Now the pipe we can observe the axis of that when it is horizontal and because of which we have called this pipe as a horizontal pipe and then if the pipe is horizontal we are going to have the left left hand side and the right hand side sections if we consider the datum head it will be same for both this side that is left hand side and the right hand side that is we can say that the datum is kept constant. If it is different it is not constant so left hand side and the right hand side we are having. So we are going to call it as a datum is kept constant and datum energy will be same at left hand side and right hand side. Now conversion of energy from one form to another. Now at left hand side section we can observe that the cross section area it is lesser hence the velocity it is more according to the continuity equation it is q is equal to a into v and the discharge is constant if the area is lesser the velocity will be more. So if the velocity is more then kinetic energy it will be more as compared to the pressure energy. Now this comparison it has been done with reference to the left hand side and the right hand side. So at right hand side we can observe that the cross section area it is more so the velocity it is lesser so according to that continuity equation and kinetic energy will be lesser as compared to the pressure energy. So this difference of pressure energy and kinetic energy we are having on the left hand side and on the right hand side and the datum head we are kept constant. Now let us take a case of fluid flow through an inclined pipe of varying cross section area. The cross section area it is on the left hand side it is lesser as in the earlier case on the right hand side it is more as in the earlier case only. So only thing is that in case of the present situation we have made the center line of datum as inclined one instead of keeping it horizontal. So in case of this we are going to have the datum energy on the left hand side and on the right hand side it is going to be different. So the left hand side center point it is at a higher distance from the reference line and in case of the right hand side we are having the center point near to the reference line. So the difference in the datum energy will be there. So if the comparison of the energies needs to be done in case of that one. So the datum energy at LHS it will be more, datum energy at RHS it will be lesser. So as area is lesser similar to the earlier case we will have the velocity more at left hand side and kinetic energy will be more and kinetic energy will be lesser on the right hand side and pressure energy it will be lesser in case of the left hand side and pressure energy will be more on the right hand side. The pressure and the kinetic energy these are similar to the that of the for the pipe which is horizontal in the earlier case and now it has been made inclined and the datum energy difference we are going to have. So these are the references which are used thank you.