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

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

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

6.  Let’s solve this differential equation; integrating once:
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)  z H
Let’s solve this differential equation; integrating once:
Integrating again:

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

8. Let’s solve this differential equation; integrating once:
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. Poiseuille Flow