12th
So you’ve got this viscous fluid…
and this inclined plane. We align some coordinate axes with the plane so that the x direction is downward along the plane, the y direction in perpendicular to the plane, and the z direction is determined by the x and y. The plane goes on forever in all directions because this is MATH and we do what we want. There’s a layer of the viscous fluid on the plane. The layer also goes on forever in all directions, but it has a finite thickness, h. The external force acting on the fluid is gravity. The fluid is Newtonian, which means you can’t compress it.
That simplifying assumption gives us this usable form a Navier-Stokes equation. We just let our density, rho be 1. The f in that equation is our “external forces” from gravity term. The rest of the terms deal with internal forces within the fluid.
Okay. Next. This part of the problem wants me to show that there is a solution with the velocity field*, u, being dependent only on y (i.e. on whether we’re near the top of the fluid layer or near the bottom), and with there being no motion in the x or z directions. Physically, this means that (infinitesimally thin) planar sheets of the fluid at each height above the plane are moving downhill together. If you know which plane a point is in, you know everything about it’s velocity, and everything there is to know is how fast it’s moving downhill. Oh, and it’s time-independent. The fluid doesn’t speed up or slow down. This is how I’d expect an infinite layer of jam to behave on a perfectly uniform infinite plane, for sure.
I think my professor has suggested that we do this by describing what such a solution would be like and then showing that a solution with those properties can satisfy all of our conditions. I’m a little uncomfortable about this part. I guess to really “show” it, we’d have to make a sort of backwards implication chain, where if each next step were true, it would have forced the previous step to be true, and arrive in this way at something we know we can force to be true. I think she may be a little bit carefree about the “show” part, though.
That’s enough of this problem for now. But there may be more homework blogging as the night progresses! We’ll have to see how things unfold.
*a velocity field just means a velocity for each point (x,y,z) in the fluid. It’s a vector field. Imagine being inside a 3-D grid with a little arrow at each point showing the direction the fluid is moving in at that point. The length of the arrow says how fast it’s moving in that direction
