1. Stoke’s first problem (BSL example)
Semi infinite medium: i.e. fluid extends from 0 to infinity in y direction. In the x and z directions, it extends from (minus infinity) to (plus infinity) y x
Sudden movement of plate at a constant velocity V0. The development of flow with respect to time is to be quantified
Qualitative: As time increases, you expect the ‘region of influence’ to increase; i.e. more and more fluid will be ‘aware’ of the plate movement.
Assume that if the velocity of fluid is 1% (or more) of the plate velocity V0, then we can say that the fluid is ‘aware’ of the plate movement.
Compression and expansion may move at the speed of sound in the fluid. Transfer of signal of shear stress may not.
This is fairly straight forward and the solution is given in detail in BSL (and perhaps in many other books). Note that the concept of ‘combination of variables’ is applied here to solve the PDE. This is a useful technique particularly when one of the BC involved is at ‘infinity’.

2. Stokes First pbm 1 No velocity in the y or z direction
No pressure gradient
Vx does not depend on x or z
No gravity in x direction
We take the NS equation for V-x. We ignore V-y and V-z equations since they are identically zero (except that either y or z equation relates pressure vs gravity, but it doesn’t affect our solution). We simplify it to a PDE which is first order in time and second order in y
Kinematic viscosity 1