1. CEE 262A H YDRODYNAMICS Lecture 12 Steady solutions to the Navier-Stokes equation

2. Exact solutions to the Navier Stokes Equations Real flow problems are too complicated for us to be able to solve the NS equations subject to appropriate BCs – We must simplify matters considerably! Our method: 1.Reduce real flow to a much simpler, ideal flow 2.Solve model problem exactly 3.Extract important dynamical features of solutions 4.Apply lessons learned to real flow

3. Generally, the model flow illustrates 1 of 3 possible two-way force balances (2 terms involved): Force 1Force 2Examples InertiaFrictionEkman layer Stokes Layer (Boundary layers) InertiaPressure (buoyancy) Waves Geostrophy Bernoulli Stratified flows PressureFrictionPipe flow Porous media Creeping flows Three way force balances (3 terms) are much harder to deal with

4. Poiseuille – Couette Flows Consider the non-rotating, constant density flow between two infinite parallel plates, one of which moves the other of which is fixed: U0U0 x 3 =0 Fixed plate: Moving plate: x 3 = H Constant pressure gradient Because the plates are infinite, except pressure everything

5. So, continuity tells us that Thus, most generally, u 3 =fn(x 1,x 2 ). But since u 3 = 0 on the top (or bottom plates), it must be 0 everywhere. Additionally since there is no pressure gradient or plate motion in the x 2 direction, u 2 = 0. Thus, the only non-zero flow component is u 1 (x 3 )

6. Steady (by assumption) No rotation (by assumption) Now we turn to the Navier Stokes equation to see what can be eliminated:

7. (a) u 1 =0 at x 3 =0 B=0 (b) u 1 =U 0 at x 3 =H So what is left is Subject to the boundary conditions that Thus if we integrate twice with respect to x 3 And use the boundary conditions We find that

8. where If we divide both sides by U 0 we can rewrite our velocity distribution in a convenient non-dimensional form: Pressure-friction balance

9. The parameter P represents the relative importance of the imposed pressure gradient and the moving surface -10 -5 10 5 3 -3 1 0 P=-10: dp/dx<0 P=+10: dp/dx>0 Flow

10. We can calculate the flow rate Thus, Q = 0 when P = 3, whereas Q > 0 when P 3 P = 3 – wind driven flow in a lake Wind

11. Two special cases 1. Plane Couette Flow:  Stress on plane parallel to plates x 3 =0 x 3 =2b Resultant force is in x 1 direction and constant in x 3.

12. 2. Plane Poiseuille Flow: U 0 =0 whence: and Tangential Stress = Shear stress varies linearly with depth x3x3 for Wall stress opposes p.g. Darcy's law: velocity ~ pressure gradient