1. Navier-Stokes

2. Viscosity  A static fluid cannot support a shear.  A moving fluid with viscosity can have shear. Dynamic viscosity  Kinematic viscosity y vxvx F

3. Pipe  Steady flow in a pipe involves viscosity.  The forces must be balanced. a z l

4. Poiseuille’s Law  The flow in the pipe can be integrated. Zero velocity at the wallZero velocity at the wall  Another integration gives the total current. Poiseuille’s LawPoiseuille’s Law

5. Strain Rate Tensor  Rate of strain measures the amount of deformation in response to a stress. Forms symmetric tensorForms symmetric tensor Based on the velocity gradientBased on the velocity gradient

6. Stress and Strain  There is a general relation between stress and strain Constants a, b include viscosityConstants a, b include viscosity  An incompressible fluid has no velocity divergence.

7. Navier-Stokes Equation  The stress and strain relations can be combined with the equation of motion.  Reduces to Euler for no viscosity.

8. Bernoulli Rederived  Make assumptions about flow to approximate fluid motion. Incompressible Inviscid Irrotational Force from gravity  Apply to Navier-Stokes  The result is Bernoulli’s equation.