Navier-Stokes

Eulerian View In the Lagrangian view each body is described at each point in space. Difficult for a fluid with many particles In the Eulerian view the points in space are described. Bulk properties of density and velocity

Streamlines A streamline follows the tangents to fluid velocity. Lagrangian view Dashed lines at left Stream tube follows an area A streakline (blue) shows the current position of a particle starting at a fixed point. A pathline (red) tracks an individual particle. Wikimedia image

Fluid Change A change in a property like pressure depends on the view. In the Lagrangian view the total time derivative depends on position and time. The Eulerian view uses just the partial derivative with time. Points in space are fixed

Jacobian Tensor A general coordinate transformation can be expressed as a tensor. Partial derivatives between two systems Jacobian NN real matrix Inverse for nonsingular Jacobians Cartesian coordinate transformations have an additional symmetry. Not generally true for other transformations

Volume Element An infinitessimal volume element is defined by coordinates. dV = dx1dx2dx3 Transform a volume element from other coordinates. components from the transformation The Jacobian determinant is the ratio of the volume elements. x3 x2 x1

Compressibility A change in pressure on a fluid can cause deformation. Compressibility measures the relationship between volume change and pressure. Usually expressed as a bulk modulus B Ideal liquids are incompressible. V p

Volume Change Consider a fixed amount of fluid in a volume dV. Cubic, Cartesian geometry Dimensions dx, dy, dz The change in dV is related to the divergence. Incompressible fluids - no velocity divergence

Balance Equations The equation of motion for an arbitrary density in a volume is a balance equation. Current J through the sides of the volume Source s inside the volume Additional balance equations describe conservation of mass, momentum and energy. No sources for conserved quantities

Mass Conservation A mass element must remain constant in time. Conservation of mass Combine with divergence relationship. Write in terms of a point in space.

Pressure Force Each volume element in a fluid is subject to force due to pressure. Assume a rectangular box Pressure force density is the gradient of pressure dV dz dy p dx

Equation of Motion A fluid element may be subject to an external force. Write as a force density Assume uniform over small element. The equation of motion uses pressure and external force. Write form as force density Use 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. The time derivative can be expanded to give a partial differential equation. Pressure or stress tensor This is Euler’s equation of motion for a fluid.

Momentum Conservation The momentum is found for a small volume. Euler equation with force density Mass is constant Momentum is not generally constant. Effect of pressure The total momentum change is found by integration. Gauss’ law

Energy Conservation The kinetic energy is related to the momentum. Right side is energy density Some change in energy is related to pressure and volume. Total time derivative Volume change related to velocity divergence

Work Supplied The work supplied by expansion depends on pressure. Potential energy associated with change in volume This potential energy change goes into the energy conservation equation.

Bernoulli’s Equation Gravity is an external force. Gradient of potential No time dependence The result is Bernoulli’s equation. Steady flow no time change Integrate to a constant

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

Stress and Strain There is a general relation between stress and strain Constants 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.