MecE 630 Problem Set 1

Directions: The problem set is due in class on Tuesday Sept. 28. If you collaborate with other students,(i) solutions must be written up independently; (ii) the names of those that you’ve collaborated with mustbe given. All problems carry equal weight.

Problem 1. Show, using index notation, that ∇ · ~ω = 0. Here ~ω denotes the vorticity vector.

Problem 2. An irrotational vortex satisfies uθ = C/r and ur = 0 so that ωz = 0/r (here C is a constant).Letting Γ denote the circulation, show that Γ = 2πC and Γ = 0, respectively, along any path that doesand does not enclose the origin.

Problem 3. The velocity in a certain flow is given by u = 2y, v = cos t. Is the flow steady or unsteady?Obtain the equation for the streamlines passing through the point (1, 1) at t = 0, π/2 and π. Obtain theequation of the pathline of a particle whose position at t = 0 is (1, 1).

Problem 4. The velocity components in an unsteady plane flow are given by u = x/(1 + t) andv = 2y/(2 + t). Describe the pathlines and the streamlines.

Problem 5. An elastic sphere of radius a pulsates in an inviscid, incompressible fluid such that a = a(t).By considering the flux of fluid across a spherical surface of radius r > a, show that the fluid’s radialvelocity, ur, is given by

ur =(ar

)2 dadt.

Problem 6. A rectangular tank filled with water is placed on wheels and is given a constant horizontalacceleration, a. Show that, at steady state, the angle made by the free surface with the horizontal is givenby tan θ = a/g, where g is gravitational acceleration. Neglect friction.

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