Fluid Mechanics

• State Kelvin's circulation theorem. What does it imply about the flux of vorticity across a surface moving with the flow?
• Prove that if a surface is initially a vortex sheet and moves with isentropic flow, then it remains a vortex sheet.
• What quantities do you need ofr the Reynolds number? Write the homogeneous, incompressible Navier-Stokes equations and explain how you arrive at an expression involving Reynolds number.
• Imagine a semi infinite flat plate (i.e. with leading edge) in a uniform flow . What characteristic length do we use for the Reynolds number? Draw what the velocity profile might look like downstream. In what region above the plate do you expect the viscous effect to be important? Conclude that the boundary layer thickness varies as the square root of the distance from the leading edge.
• State the uniform boundedness principle.
• Explain the difference between the Euler equations and the Navier-Stokes equations in the limit as the viscosity goes to zero.
• How does one obtain the no-slip boundary condition for the Navier-Stokes equations?

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