Alternate Navier Stokes Flashcards
What is the 2D Laplacian in polar coordinates?
Under what conditions can we still use index notation in alternative coordinate bases?
If the alternative coordinate bases are orthogonal.
What is Haufen-Poiseuille flow?
A circular pipe where flow is driven by a pressure drop. Velocity is steady, uni-directional & only depends on the radial coordinate.
- What is circular Couette flow?
- What are the four assumptions that go into it?
- Which assumption is incorrect?
Using the wrong assumptions for circular Couette flow, show there is an inconsistancy between the azimuthal & radial equations.
By removing the incorrect assumption, calculate the pressure distribution in circular Couette flow.
What fact must you remember about the stress tensor for Newtonian fluids outside of Cartesian coordinates?
Note: in an exam you will be given equation sheets for the rate of strain tensor.
Suppose only the inner cylinder rotates in circular Couette flow. Determine the traction on the inner cylinder.
Given a length & velocity scale, define a time scale, dynamic pressure & viscous pressure.
By considering dimensionless variables, derive the Reynolds number in the Navier Stokes equations.
What equations do we get when we set Reynolds number to zero (slow viscous flow)?
The linear Stokes equations.
What equations do we get when we set Reynolds number to infinity (high fluid flow)?
The Euler equations.
Note : internal pressure has been scaled.
Note : These are 1st order, so we only have one BC. Say goodbye to no-slip.
What is the anzatz we derive from dimensional arguments for self-similar solutions?
Derive the ODE that needs to be solved for Rayleigh’s jerked plate. Assume no pressure gradients and parallel flow.
Solve the ODE that dictates Rayleigh’s jerked plate.
Derive the form of the velocity needed to solve for Jeffery-Hamel flow.
Derive the non-linear ODE that is solved numerically for Jeffery-Hamel flow.
By considering how the fluid must move through the point, dervive an intergal equation for Jeffery-Hamel flow.
