welcome to lesson 2c equation of fluid statics in this lesson we'll derive the equation of fluid statics and then discuss a simplified case for incompressible fluid here's the derivation we consider an infinitesimal fluid element with these dimensions as we sketch here we'll always define gravity as down we can think of this element in terms of a free body diagram just like an emec class since we're talking about hydrostatics there's no acceleration the fluid is at rest and therefore sigma f equals zero in hydrostatics there can be no viscous or shear forces because as we've said previously a fluid at rest cannot resist the shear stress so only normal forces can act on this body we split sigma f into body forces and surface forces we'll develop expressions for both of these and then set this equation equal to zero let's consider body forces first let's take this as the center of the element the only body force we have here is gravity the vector gravity force is mg or rho gv or since the volume is dx dy dz rho dx dy dzg a side comment here about my notation v with a line through it means volume and v without a line through it means speed or the magnitude of velocity in the textbook we use a different font to distinguish these two capital v's since g acts in the negative z direction we can write g as negative g times k where k is the unit vector in the z direction so the sum of all the body forces is negative rho g d x d y d z times k so we have the first one of these completed now let's look at surface forces as i mentioned the only surface forces we have are normal in particular only pressure forces let's let the pressure be p naught at the center of the body when i say body that means this little fluid element this is hard to draw in 3d but let's look at the center of each of these faces here's the center of the front face right face and the top face i'll use dashed lines for the left face since it's hidden back face and the bottom face for the pressure forces let's consider the average pressure on each of the six faces since this element goes to zero we're really talking about pressure at a point but we need to have a finite fluid element in order to have some change across the faces let's consider the pressure acting on the front face and on the back face again i use dashed lines when this is hidden recall from a previous lesson that pressure acts inward and normal at any surface in general pressure is a function of x y z and t in the cartesian coordinate system in other words pressure is a function of space and time thus we have to use partial derivatives the greek symbol del not total derivatives the more common d in many places in this lecture we're going to use truncated taylor series expansions consider a point d s away from the center of our fluid element ds can be in any direction so if we're considering the point here we use a taylor series expansion as follows p at this new location is equal to the pressure at the center plus del p del s d s plus if you remember taylor series expansions 1 over 2 factorial del squared p del s squared times d s squared plus higher order term since d s goes to 0 this term is much smaller than this term so we'll ignore higher order terms this is now a truncated taylor series expansion now let's go back to our fluid element i'll erase this to get rid of some clutter let's take the x direction front and back faces and think about what is the pressure at this point in the center of the front face the pressure at the front face will be p naught plus del p del x since we're in the x direction times the distance from the center to that face well since this is dx that distance is half of dx so it's dx over 2 plus those higher order terms that we're ignoring similarly on the back face the pressure is p naught plus del p del x times negative dx over 2 since we're going in the negative x direction let's sum up the two forces in this x direction by the way those are the only two forces acting on the front and back surfaces since there's no shear stresses or shear forces now let's add those up we'll sum up all the surface forces in the x direction i'll call this sigma f surface comma x careful with our signs on the front face the pressure is acting in the negative x direction so we have to put in a negative sign while on the back face the pressure is acting in the positive x direction so we write it this way for the front face negative p naught plus del p dou x dx over 2 that's the pressure the area of that face is d y d z force forces pressure times area so this is the force on the front face for the back face we have no negative sign but we did have a negative sign here the area is also d y d z that's the area of the back face so pressure times area again is force so this is the force on the back face well these two terms cancel and these two terms add up and thus we have negative sigma f surface in the x direction is negative del p dou x d x d y d z going back to our drawing we now do the same thing for forces due to pressure in the y direction and in the z directions keeping in mind that pressure always acts inward and normal i'm not going to write all this out we do the same kind of truncated taylor series analysis as we did in the x direction i'll just do one of these on the right face the distance is d y over 2 and the partial derivative is del p del y so this is the force acting on the right face you can see that the math turns out to be the same except for changing x to y and then to z so let's go back here and say similarly for the y and z directions sigma f surface y is minus del p del y d x d y d z and sigma f surface z is negative del p del z d x d y d z these are three components of the surface force which is a vector we have the x y and z directions so now our hydrostatic equation needs to be generated by doing a vector summation of all the forces and setting it equal to zero from above we split it into body and surface forces again this is a vector equation but let's consider the x direction or the unit vector i component in the x direction there's no body force so that term is zero and in the x direction the surface force was given up here negative del p dou x d x d y d z some of these has to equal zero and we simplify this to del p del x equals 0. this is very significant it tells us that pressure does not vary in the x direction in hydrostatics we're assuming of course that gravity acts in the minus z direction similarly in the y direction zero minus del p del y d x d y d z must equal zero so del p del y equals zero we conclude that p does not vary in the y direction either in hydrostatics now let's consider the z direction now we have both the body force and a surface force in the z direction we already solve for the body force and the surface force so in the z direction we write the body force and the surface force and they must sum to zero the volume of the fluid element dxdydz cancels in both terms therefore del p del z is equal to negative rho g i'll call that equation one unlike the x and y directions pressure does vary in the z direction and hydrostatics the bottom line from this derivation can be stated as follows in fluid statics in a continuous fluid in other words no gap or not we're not talking about two different fluids here one continuous fluid p does not vary horizontally but v does vary vertically this is an extremely important concept in hydrostatics furthermore because of this negative sign p decreases as you go up or p increases as you go down that one's easier to remember for example when you go swimming you get more pressure on your ears as you go down this agrees with our previous equation for hydrostatics which was p below equal p above plus rho g delta z which you should recall from a previous lesson this agrees with our derivation now so kind of a summary in hydrostatics by the way hydrostatics typically refers to liquids whereas the more general fluid statics refers to liquids or gases so for fluid statics remember p is normally a function of x y z and t but if this is fluid at rest that nothing's changing with time we just showed that p is not a function of x or y so p is a function of z only in fluid statics so our equation 1 which was del p del z is minus rho g can be simplified since we can use d total derivative instead of del partial derivative you need partial derivatives when a variable is a function of more than one independent variable but here all we have is z so we can rewrite one as dp dz equal minus rho g i'll call that equation 2 really the same as equation 1 except with total derivatives so this is our equation of fluid statics or hydrostatics dpdz is minus rho g suppose we have some location z1 where this is the z direction and some other height z2 we can find pressure difference by integrating we integrate to get p2 if we know p1 and we know how rho and g vary with z we separate the variables and write dp is minus rho g dz which we can integrate from p1 to p2 and from z1 to z2 on the right this is the general case if either of these variables row or g is not constant you have to do an integration now g doesn't vary much with elevation so we'll always consider g a constant but rho can vary with elevation simplest case is incompressible where rho is a constant and g is constant unless delta z is huge so for the incompressible case we integrate p2 minus p1 is equal to the rho and the g can both come outside the integral since they're constants and this integral is just z2 minus z1 so for an incompressible fluid in hydrostatics we simply have this expression this holds for any incompressible fluid at rest keep in mind if you're doing homework problems for example if density is not a constant you must integrate we can rewrite this as our simple equation from previously namely without worrying about the negative signs we just rewrite this as p below equal p above plus rho g delta z i like this form because you never get confused with negative signs and it clearly shows that p increases with depth i'll end by saying this we can solve any hydrostatics problem with this equation thank you for watching this video please subscribe to my youtube channel for more videos