Navier-Stokes
Viscosity A static fluid cannot support a shear. A moving fluid with viscosity can have shear. Dynamic viscosity Kinematic viscosity y vxvx F
Pipe Steady flow in a pipe involves viscosity. The forces must be balanced. a z l
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
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
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.
Navier-Stokes Equation The stress and strain relations can be combined with the equation of motion. Reduces to Euler for no viscosity.
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.