Posted: July 12th, 2023

The Navier-Stokes equations for incompressible flows

1. (10) The Navier-Stokes equations for incompressible flows written in the following form∂up 1 2= −∇+ |u| + u × ω + ν∇2 u∂tρ 2show various forces that change the velocity at a point in space.(a) The second term on the right hand side can be viewed as vorticity force. State a possiblecondition, not counting the trivial cases (u = 0 or ω = 0), under which the vorticityforce vanishes.(b) The third term on the right hand side is viscous force. State three possible conditions,not counting the trivial cases (ν = 0 or u = 0), under which the viscous force vanishes.Note that the viscous force can be written in terms of vorticity.22. (30) A “Rayleigh” plate is suddenly stopped at time t = to whenu=U21− √π√y2 νto2exp−η dη .0yUx(a) Write down the governing equation, initial and boundary conditions for the problem.(b) After examining the equations given in 2a, your colleague, Ms. Jane Math, quicklywrites down the solution without actually solving the differential equation. Explain howshe does this, and verify her solution by showing that it satisfies the equation and theinitial and boundary conditions.33. (30) Ideal fluid, i.e., irrotational and incompressible moves in a simply connected region Vbounded by a closed surface S.(a) Show that the kinetic energy1K= ρ2u2 dV,(1)∂φdS.∂n(2)Vcan therefore be written in the form1K= ρ2φS(b) Now consider an ideal flow in the gap a < r < b between two infinitely long cylinders,which are fixed. The irrotational flow between them isu=Γeθ ,2πr(3)where Γ is a constant. Show that the kinetic energy of the flow is zero from Eqn. (2).(c) The result in 3b, which was obtained using Eqn. (2), is evidently absurd. Explain thefallacy.(d) Compute the kinetic energy by correctly applying Eqn. (2). Compare this result to thatobtained directly from Eqn. (1).44. (30) Suppose that there is, in y ≥ 0, the irrotational flowu = −αx,v = αy,where α is a positive constant, and let there be a plane rigid boundary at y = 0. Suppose, inaddition, there are two line vortices, one of strength −Γ at z = z1 (t) and the other of strengthΓ at z = z2 (t) as shown below.yz1z2x(a) Write down the instantaneous complex potential for the flow.(b) Show that the vortices may remain at rest (by showing the flow velocity is zero sincethe vortices are moving with the fluid) atz1 = d(−1 + i),z2 = d(1 + i),where d2 = Γ/8πα.(c) Describe, in words, how the steady state in 4b can be achieved.

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