23-02-2013, 09:45 AM
Navier-Stokes
Navier.ppt (Size: 1.56 MB / Downloads: 138)
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.
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 real matrix
Inverse for nonsingular Jacobians
Cartesian coordinate transformations have an additional symmetry.
Not generally true for other transformations
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.
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.