Navier-Stokes

Pressure Force  Each volume element in a fluid is subject to force due to pressure. Assume a rectangular boxAssume a rectangular box Pressure force density is the gradient of pressurePressure force density is the gradient of pressure VV p xx yy zz

Equation of Motion  A fluid element may be subject to an external force. Write as a force densityWrite as a force density Assume uniform over small element.Assume uniform over small element.  The equation of motion uses pressure and external force. Write form as force densityWrite form as force density Use stress tensor instead of pressure forceUse stress tensor instead of pressure force  This is Cauchy’s equation.

Euler’s Equation  Divide by the density. Motion in units of force density per unit mass.Motion in units of force density per unit mass.  The time derivative can be expanded to give a partial differential equation. Pressure or stress tensorPressure or stress tensor  This is Euler’s equation of motion for a fluid.

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

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. next