 Consider two-dimensional incompressible flow between two stationary plates. Determine an expression for the velocity profile in the fluid stream. So here I'm assuming that the flow is being driven by a pressure difference. A difference in pressure from the left and right sides of this setup results in some movement of the fluid. Like with the previous example, the best place to start is with the conservation of mass. Since I have incompressible flow, the conservation of mass simplifies to del u del x plus del v del y plus del w del z is zero. I recognize that the flow is only being driven from left to right, so there must be no y nor z component of velocity. That means that del u del x must be zero, which means that u is only going to be changing in the y direction. So just like in the previous example, u is only a function of y. When I look at my conservation of momentum options, I can eliminate the y and z components of momentum, leaving me with just one equation, and I step down the line again, eliminating terms if they aren't relevant. Well, there goes the gravity term. I can't neglect the pressure with respect to x term this time, because that's what's driving my flow. But I recognize that u doesn't change with respect to x, it doesn't change with respect to z, it doesn't change with respect to time, it doesn't change with respect to x, it doesn't change with respect to z, and there is no y component of velocity. Therefore, everything simplifies except for the pressure gradient, the viscosity, and the second derivative of u with respect to y. Bringing the pressure gradient to the right-hand side of the equation, and multiplying by 1 over viscosity, I'm left with the second derivative of u with respect to y is equal to 1 over mu times del p del x. Integrating one time, I'm left with that same quantity times y plus a constant. Integrating a second time, I'm left with u as a function of y is equal to 1 over 2 times mu times the pressure gradient times y squared plus our first constant times y plus a second constant. For this setup, at a position of h, as a result of the no-slip assumption, I'm assuming that there is no velocity. Furthermore, at a position of negative h, I'm once again assuming that there's no velocity. I'm saying the fluid isn't moving at the interface between the fluid and the fixed plates. Plugging those two boundary conditions into my equation and solving, I come up with my two constants. Constant 1 is 0, constant 2 is what's left, which is negative del p del x times height squared over 2 mu. Plugging that into my equation and factoring out 1 over 2 mu times del p del x gives me 1 over 2 mu times del p del x times the quantity y squared minus h squared. I can make this a little bit more convenient to work with on a daily basis by recognizing that generally speaking, I refer to a maximum or average velocity across a velocity profile. So it's going to be useful for me to keep track of a maximum velocity. I'm defining the maximum velocity as occurring at a y position of 0, which means when I solve that equation at a y position of 0, I'm left with u max of negative h squared over 2 times mu times the pressure gradient driving the flow. When I make that substitution into my original equation, what I'm left with is u as a function of y is u max times the quantity 1 minus y squared over h squared, where u max is negative h squared over 2 times mu times del p del x. If you're wondering why there's a negative out front, it's because the velocity is developed in the opposite direction as a positive pressure gradient. So if you think about if there was a high pressure over here and a low pressure over here, my pressure gradient is positive to the left, which means it's negative to the right. The high pressure on the left will drive fluid flow to the right. Therefore, the velocity is in the opposite direction as a positive pressure gradient. So u max will be positive for a pressure gradient that is negative. That's why there's a negative in this equation.