 to fluid static session 2. I am Prof. Vivek Sathe of WIT, Solapur. Now before starting with the second session, once again we will see what are the outcomes. At the end of this session, Learnall will be able to identify properties of fluid like pressure is isotropic, pressure is uniform on a horizontal plane and fluid exerts pressure at right angles. See these are very important outcomes and if you understand these outcomes in detail then you will have say compared in solving so many numerical problems in fluid mechanics. Okay, now let us go back to the first session. The question I asked you about whether the pressure is scalar or vector. Many students have confusion and they say that pressure is a vector quantity. The simple reason is they say that pressure is force upon area and as force is a vector and this is a scalar area is a scalar they say that it is a vector quantity. Now basically if I ask students what is the difference between a scalar and a vector many students tell the bihalter definition that scalar is a quantity which has only magnitude and there is no direction. So only magnitude is called as a scalar and only direction and magnitude is called as a vector. Then I ask a simple question if I take an electric current for example suppose this is an electric current coming through this is an electric current going from this and suppose the angle between this is 90 degrees this is i1, i2 and i3 suppose i1, i2 and i3 if I say i is equal to i1 plus i2 plus i3 and if I change the angle from 90 to 45 degrees will there be any change in the output then the students say no still i1 is equal to i2 plus i3 means current has got direction as well as magnitude but it is not a vector means only direction and magnitude is not the necessary condition for any physical parameter to be a vector then what is it so it is that it must add as per the vector law of addition means whenever I add this vector to this particular vector and if the addition takes place according to the triangle law or law of parallel forces then only it is called as a vector but regarding the pressure whether it is a scalar or a vector I will give a simple example pressure is basically not force upon area many times students make mistake it is force upon area it is thrust upon area what it is it is thrust upon area what is thrust suppose I have this wall and I apply this force like this at an angle some theta then this has got two components one component is along the plane and second is perpendicular the component which is perpendicular that is the thrust component is responsible for giving me the pressure and component which is along the tangent is responsible for giving me the shearing so when we talk about the pressure in fluid statics we are interested in this thrust and that is why I asked in the last session about the question that when this balloon is having the holes and the piston is pressed inside water will ooze out normally radially so if you just trace all the locus you will find that all the points are assumed to be coming from the center of the curvature means it is in the radial direction going out and this makes it clear that pressure is nothing but thrust upon area then what is thrust so this component is nothing but it is f dot n cap so when I take around the normal direction and divide it by area what I get this is the dot product dot product is a scalar quantity it is a scalar product rather so scalar upon scalar becomes a scalar and hence pressure is a scalar quantity that is why it adds like a pressure that is if I add there is no addition like a vector quantity it adds like a scalar quantity I think this much is sufficient for you about the pressure scalar or vector now we will go to the next important parameter in this session that is some properties of fluid now see first important property is pressure is isotropic now first of all there are two terms we use in mechanics one is called as isotropy and second is called as anisotropy what is called as anisotropy so isotropic means the quantity is same in all directions now you can just see here if I take a container in which there is a small element that I consider now this is a small differential element for analysis from the mathematical point of view we take dx, dy and dz it becomes a small volume element that we call as the dv which is equal to dx, dy and dz and in the limiting case dx, dy, dz tends to 0 and when they tends to 0 it becomes a point it becomes a point we will see exactly transformation from this to this exactly how it happens now see what happens here when I see the pressure acting on all the edges of this particular element if the pressure on this axis then the pressure along this axis and the pressure along say this axis if they are not balanced suppose this is my x, y and z axis this is x, this is y and this is z if I use the right hand if I use the right hand rule this is say x, y and z so if net x component is present then what will happen fluid will have acceleration to the right side if net y component is present fluid will have acceleration in the y direction if net z component is present so as my fluid particle is at rest because I am interested in fluid static it is clear that this force and this force is same this and this is same so in a nutshell what I can say that this mass elemental volume if I compress it compress it compress it it will become a point mass and I will show that the pressure is compressive so the pressure is always compressive there is no concept of tensile pressure tensile stress is there but pressure is always compressive and in static pressure when we show it on the Mohr circle it will be a point it will not be a circle because there is no shear there is no movement of the fluid taking place so this is the first point fluid pressure is isopropic another important thing is pressure is uniform on a horizontal plane now this is an important property and students have confusion regarding the same pressure along the horizontal line for two cases suppose this body is moving with uniform velocity with uniform velocity now what is the meaning of uniform velocity there is no acceleration uniform velocity means acceleration is 0 when acceleration is 0 by Newton's law we can say that there is no external force acting on the system or when the body is moving in a vertical direction and we want to measure the pressure in the horizontal direction it remains same now let us see how it works suppose I take the cylindrical element here where the cross sectional area is A pressure here is P1 pressure here is P2 and cross sectional area is ds here point is A and B force is P1 into ds and B force is P2 into ds now what happens if this is both are equal then only the particle is in equilibrium otherwise not if P1 ds is greater than P2 ds particle will move to the left sorry to the right if P2 ds is greater than P1 ds particle will move to the left but if particle is in static equilibrium what I get is equal to P2 so because of this horizontal layer has got same static pressure I am not talking about the dynamic pressure it is a static pressure similarly assume that this is moving with uniform velocity when a body moves in uniform velocity it is as if it is in the frame of reference static condition so in that situation also there is no change in the pressure along the horizontal line so these are the isobaric lines these are isobaric lines similarly if I have an accelerated motion in the vertical direction if I have acceleration in the vertical direction still my pressure in the horizontal direction is same how it is clear because whatever the pressure is coming on the top will be given by the reaction by the normal surface but as far as the horizontal layer is concerned it is not going to be affected by the acceleration in the vertical direction but acceleration in the horizontal direction will change the pressure that we see in the kinematics in the most part in the fifth session in detail so this is first point then there is a hydro static paradox hydro static paradox basically paradox means when we expect something and exactly reverse happens now see in which there is a liquid then there is some small pipe like this in which there is another liquid and some intricate geometry like this where there is some liquid but one thing I keep constant that is the height suppose head is constant then the pressure here pressure here and pressure here at this point this point this point is same so pressure along the horizontal line is same provided you are using same liquid suppose I take one liquid here another liquid here then pressure is not same so whenever I draw a line like this and if I am sure that when I pass from this point below this up to this point and the fluid is same fluid then automatically I get the same pressure everywhere this point and this point because pressure is independent shape and size it is independent it is very important of shape and size it depends only on the height so how much height of the liquid you have on the container that will give the pressure I am not asking you about what is the force exerted here force is different because force will be this area this area so force may be different but pressure is going to be same so I think now from this you have got an idea that as pressure is same pressure is a scalar quantity and when we compress the liquid it will ooze out in the normal direction and it has been made clear to you now that force upon area though it is a definition of a pressure it is a thrust upon area and thrust is a scalar quantity which is the force in the perpendicular direction and because of that we get pressure as a scalar quantity so for any reference I suggest you to go for fluid mechanics by white by TMS publication and for this test today's session we will stop over here thank you