Fluid Dynamics I

PROBLEM SET 7

Due December 14, 1998

1. Consider the uniform slow motion with speed U of a viscous fluid past a spherical bubble of radius a. Do this by modifying the Stokes flow analysis for a rigid sphere as follows. The no slip condition is to be replaced on r = a by the condition that both u_r and the tangential stress s_rq vanish. (This latter condition applies since there is no fluid within the bubble to support this stress.) Show in particular that

and that the drag on the bubble is D = 4pmU a.

2. Construct a set of solutions of the Stokes equations in three dimensions, having the form

where e_ijk = ±1 for ijk and even (odd) permulation of 123 and is otherwise = 0, and W_k is a constant vector. What is the corresponding pressure field, if we assume that p must vanish at infinity? Use the form to find the flow field generated by a rigid sphere of radius a spinning with angular velocity W_k in a very viscous fluid, such that u = 0 at infinity. Show that the torque exerted by sphere on fluid is 8pWa³m. (Solutions of the above form, together this those given in class, forma complete set of Stokes flows in three dimensions.)

3. Prove that Stokes flow past a given, rigid body is unique, as follows. Show if p₁,u₁ and p₂,u₂ are two solutions of

satisfying u_i = -U_i on the body and

as r® ¥, then the two solutions must agree. (Hint: Consider the integral of ¶/ ¶x_i (w_j¶w_j /¶x_i) over the region exterior to the body, where w = u₁-u₂.)

4. Oseen's equations are sometimes proposed as a model of the Navier-Stokes, equations, in the study of steady viscous flow past a body. Oseen's equations, for a flow with velocity (U,0,0) at infinity, are

(a) Show that in this model the vorticity is a function of y,z alone.

(b) For the Oseen model, and for a flat plate aligned with the flow, carry out Prandtl's simplifications for deriving the boundary-layer equations in two dimensions, given that the boundary condition of no slip is retained at the body. That is find the form of the boundary layer on a flat plate of length L aligned with the flow at infinity, according to Oseen's model, and show that in the boundary layer the the x-component of velocity, u, satisfies

What are the boundary conditions on u for the flat-plate problem? Find the solution, by assuming that u is a function of yÖ{[U/(nx)}], for 0 < x < L.

(c) Compute the drag coefficient of the plate (drag divided by rU² L, and remember there are two sides), in the Oseen model.

5. What are the boundary-layer equations for the boundary-layer on the front portion of a circular cylinder orf radius a, when the free stream velocity is (-U,0,0) (see figure 1)? (Use cylindrical polar coordinates). What is the role of the pressure in the problem? Be sure to include the effect of the pressure as an explicit function in your momentum equation, the latter being determined by the potential flow past a circular cylinder studied previously. Show that, by defining x = aq,[`y] = (r-a)ÖR, where R = Ua/n is the Reynolds number, in the derivation of the boundary-layer equations, the equations are equivalent to a boundary layer on a flat plate aligned with the free stream, in rectangular coordinates, but with pressure a given function of x.

6. For a cylindrical jet emerging from a hole in a plane wall, we have a problem analogous to the 2D jet considered in class (see figure 2). Consider only the boundary-layer limit. (a) Show that

and hence that the momentum M is a constant, where

(b) Letting (u_z,u_r ) = (1/r)(y_r, -y_z ) where y(0,z) = 0 show that we must have

y = z f(h) ,   h = r²/z² . Determine the equation for f and thus show that the boundary-layer limit has the form

where h₀ is a constant. Express h₀ in terms of M, the momentum flux of the jet defined above.

7.consider the Prandtl boundary-layer equations with U(x) = 1/x, so p(x)/r = p_¥-1/(2x²). Verify that the similarity solution has the form

y = f(h), h = y/x . Find the equation for f. Show that there is no continuously differentiable solution of the equation which satisfies f(0) = f^¢(0) = 0 and f¢® 1 ,f^¢¢® 0 as h® ¥. (Hint: Obtain an equation for g = f¢.)