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)

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

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:

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):

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:

 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:

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

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:

Poiseuille Flow