The Viscous forces are the V in VPIGS, the forces that we need to think about when we're looking at fluids. Imagine for a moment that we've got a fluid in the middle here between two plates. And one of these plates is moving up here at the top, at some velocity capital U, and it will pull the fluid along with it at that same velocity capital U. So that this piece of fluid right here is going that fast. Down here at the bottom plate, it's not moving, so the fluid that's right adjacent to it will have no velocity, so it'll only be going this fast. And we'll wind up with a linear velocity distribution between these plates, so that the stuff at the top is going fast and slower and slower. We know this from inspection, from watching fluids behave, and also to a little bit of an extent from intuition. Now we can think of the gap between these two plates as some height H, the thickness of this layer of fluid in between the two plates. There's a surface area of the plates that's area A. They're located in this coordinate frame of reference with Y going up and X going across as usual. And this top plate pulls the fluid along at the same velocity. This bottom plate holds the fluid stationary. The force required to make this top plate move goes up with a variety of things, and we know this from observation and from measurements. It goes up with higher velocity, so that the faster we want to go, the harder we're going to have to pull on that top plate. It goes up with a smaller gap. The thinner the gap in between these two plates, the harder it is going to be to pull the top one along. It also goes up if we have a fluid that is thicker. Let's just say thicker for the moment with quotation marks. Different fluids will be easier or harder to deform this way. And it also goes up as you'd expect with the surface area. So the bigger the area of the plates, the harder you're going to have to pull to slide them along. So if we write that as an equation, the force will be equal to, well, there's the area times the velocity, that high U velocity, goes down with the thickness of the, or the height of the gap, so divided by H, and then times something to do with the thickness of the fluid. Let's just use the Greek letter mu right there. If we rearrange that, we'll get the force over the area. That's going to turn out to be a stress. We'll be equal to that mu that we don't know what it is yet, but has something to do with the, how thick the fluid is, times the ratio of the velocity and the gap size. Refine that a little bit more. That's a shear stress. This will hang on to and we'll call it viscosity. And that U over H for this linear variation in velocity, that's a velocity gradient. That's the derivative of the local value of the x velocity in the y direction. So what this tells us is that the stress that we expect in here, the shear stress that we expect, is going to depend on this constant viscosity and the gradient of the velocity. So there's the velocity gradient. That's our viscosity. And this is the shear stress. And that relates that stress to this velocity gradient by way of this constant, which turns out to be very close to a constant, the viscosity. The units for velocity gradient, while velocity is meters per second, and the distance delta y is in meters. So that's meters per second per meter. The shear stress will be in units of force, so Newton's divided by area, meters squared. That'll give us those units. And by looking at those and at this formula here and requiring it to be dimensionally consistent, we'll wind up with units for viscosity of Newton seconds per square meter. So this viscosity here, this mu is referred to as the dynamic viscosity, or just viscosity. It has the symbol mu in a Greek letter, and it has units Newton seconds per square meter. So just remember that's Greek letter mu. Fluid also has density, as you've learned in thermodynamics and elsewhere. And that's the amount of mass, kilograms per cubic meter of fluid. And it's always represented with rho, the Greek letter rho, spelled out RHO. Very often we're interested in comparing viscous forces with inertial forces, so the ratio of the viscosity and the mass are going to turn out to be important. And therefore we define a kinematic viscosity, nu equal to just the ratio of the dynamic viscosity divided by the density, and we call that nu. The units for that are meter squared per second. Later on we'll see some more complicated arrangements for shear stresses and velocity gradients in three dimensions. But the fundamental idea applies when you deform a fluid, the higher the velocity gradient, the more the velocity changes over a short distance, the higher the viscous stresses are going to be, and the more important they're going to be in the flow. So when we go to make that assessment about VPEGs, VPIGS, we're going to usually find that viscous forces are important, either if the viscosity is large, or if the gradients are large, or both.