Conservation of momentum also known as: Cauchy’s equation

1. Conservation of momentum also known as: Cauchy’s equation 4 equations, 12 unknowns; need to relate flow field and stress tensor Relation between stress and strain rate Navier-Stokes Equation(s)

2. Assume linear, frictionless motion under homogeneous conditions. The horizontal momentum balance is then: Continuity:

3. linear, partial differential equation (hyperbolic) Assume motion in the x direction only. Continuity becomes: And the momentum balance is then: And the continuity equation is:

4. The solution is d’Alembert’s solution, which can be studied with the sinusoidal wave form (also studied by Euler, Bernoulli and Lagrange): WAVE equation

5. Let’s now consider friction –flows along a single component of the coordinate system -- ∂ /∂y = 0, and laminar (low Re): The momentum balance can then be written as: Continuity assures that: z y x if the flow is steady:

6. Two options: 1) Flow driven by no pressure gradient; 2) Flow driven by pressure gradient  1) flow driven by no pressure gradient (e.g. wind blowing on the water’s surface; flow produced by a horizontally moving lid) Let’s solve this differential equation; integrating once: z H Integrating again:

7. Need boundary conditions  z H Constant Shear COUETTE FLOW Same for horizontally sheared flow

8. 2) flow driven by pressure gradient (e.g. river flow; flow through stationary flat walls) Let’s solve this differential equation; integrating once: Integrating again:

9. PoiseuilleFlow