 In the last segment what we did is we came up with an equation for the surface forces on a fluid element due to pressure and we're deriving a hydrostatic equation. What we're going to do now we're going to take a look at the other forces that could be on that chunk of fluid. So there are a couple of other forces that can exist on the chunk of fluid. First of all we have body forces and one of the most common ones is gravity or the gravitational body force but we could have others. There could be electromagnetic or other forms of body force and what we'll do we'll look at the gravitational body force here because that's the most common one within fluid mechanics and I'll write out the gravitational differential force element as being that and then that is just rho times g, g being the gravity vector multiplied by the volume of our fluid element. And remember before we were doing force per volume so I will write that as a small f and that would then just be rho times the gravity vector g. Now the last force we're going to look at is one due to viscous shear stress and we'll see more of this later on in the course but for now just for the sake of completeness we will include it and if you recall from an earlier lecture where we talked about Newtonian flow this in a way the term that I'll show you is similar to the mu du dy which was giving us the shear stress within a fluid element and for three dimensional fluid flow now what we can do we can write out an equation and I will put it as Vs for viscous shear so that is the term for viscous shear and viscosity is our proportionality constant and we will express that as del squared V the vector form of V so with that we have the other forces that can be acting on our fluid elements so what we're going to do we're going to combine those all together now if you recall earlier this is what we had for the pressure we had the gradient of pressure was the surface force and then we have this new force here and this one here for the body force and the viscous shear let's put those all together and we will sum forces on the left hand side I will put rho a and this is force per unit volume recall so the left hand side is ma divided by volume gives us rho a and on the right hand side this will be equal to minus the gradient of the pressure field plus rho times the gravity vector plus this viscous shear stress or viscous shear force that we talked about so we can rewrite this now and this is the most general form of fluid static equation what we'll be doing actually it would not be fluid static because we have the velocity term here and so you could have velocity field but we'll make some simplifications in order to get rid of that that I'll talk about shortly but in this equation so what we can say is that if you know your acceleration vector so there and if you know the velocity vector which is there you can determine the pressure distribution in the fluid so that's essentially what that tells us we're now going to consider a number of special cases of this equation and these are the ones that we typically use when we're dealing with hydrostatics so special cases the first one would be if the flow was at rest or constant velocity so if the flow is at rest the acceleration vector a is going to be zero and del squared v is going to be equal to zero as well so if we look back at our equation what we end up with is just grad p is equal to rho times g and this would be the special case or the hydrostatic which we'll look at more extensively in the next segment so that's the hydrostatic condition another special case that can exist is if you have rigid body translation or rotation that would be where the fluid is moving but it's all moving as a solid body movement so there's no relative motion within the fluid all of the fluid particles are moving together for example you put all the fluid in a bucket and you move it for that what happens is we get grad p is rho times g minus a that would be the form of the equation for that type of scenario and the last one that can exist this is if you do have fluid motion and velocity but let's say the fluid motion is what we call irrotational so irrotational motion is characterized by del cross v equals to zero and with that what would happen is our grad squared v term would disappear and consequently we have another simplification of the equation looking back it's very similar to the solid body rotation one but with that one we would determine the pressure because we have fluid motion pressure would be determined using an equation that we'll look at later in the course which is called Bernoulli's equation so pressure via Bernoulli equation that provides the pressure in the equation that we have so those are simplifications that you can have in this section however we're mainly going to look at the hydrostatic condition and the rigid body translation rotation we'll look at the other ones later on in the course