 Consider two-dimensional incompressible flow driven by a plate moving at velocity v parallel to a stationary plate as shown below. Determine an expression for the velocity profile in the fluid stream. So the first thing we can do here is recognize that our conservation of mass allows us to say for incompressible flow that the partial derivative of u with respect to x plus the partial derivative of v with respect to y plus the partial derivative of w with respect to z must be zero. Furthermore, I recognize that there's going to be no y component of velocity nor any z component of velocity. There is only an x component of velocity, meaning I'm left with del u del x is zero. That means that the x component of velocity must only change in the y direction. That's useful because when I start looking at my conservation of momentum equations for this constant viscosity incompressible flow I can simplify many of the terms. First of all, I recognize that I only care about the x direction because there's only an x component of velocity. So I can get rid of the y direction and z direction entirely. Next, I go down the line and consider whether or not each term is important. For gravitational acceleration, I don't have any gravitational acceleration appearing in the x direction. I don't have any change in pressure in the x direction. I don't have any u-term changing with respect to x because I know it must only change with respect to y. I don't have any u-term changing with respect to z, u doesn't change with respect to time, u doesn't change with respect to x, u doesn't change with respect to z, and the y component of velocity is zero. That means that everything in this equation is zero except for this term here, the second partial derivative of u with respect to y. Then the two constants, viscosity and density, are only relevant if there's a term inside the parentheses that are themselves still present in the equation. Density disappears, but viscosity does not. That means I'm left with the viscosity, excuse me, let me get rid of that pen. That means I'm left with the viscosity times the second derivative of the x component of velocity with respect to the y direction is equal to zero. If I divide both sides of the equation by viscosity, I'm left with just that second derivative term. If I integrate once, I have a constant appearing. If I integrate a second time, I have u is equal to the first constant times y plus the second constant. For this setup, I have boundary conditions appearing on the bottom and the top. On the bottom, that is on the side of the fixed plate, at the point of the fixed plate itself, I assume that I have no movement. That is a result of the no-slip assumption. I'm assuming that there is no viscosity at the infinitesimally thin interface between the fluid and the body. So at the intersection of the fluid and the fixed plate, I'm assuming that there is no velocity. Furthermore, for the same reasons, at the intersection between the fluid and the moving plate, I'm assuming that the velocity of the fluid is the same as the plate. So at a position of zero, if I'm defining my y-axis as starting at the fixed plate and going up towards the moving plate, at a position of zero, u is zero. At a position of h, the height, my velocity is the velocity of the moving plate, which is called v. Plugging in these two boundary conditions into that function, I solve for c1 and c2. That means I'm left with my u-velocity as a function of y is the velocity of the moving plate times y over h.