PROBLEM SET 3: DUE OCTOBER 13, 2014 FLUID MECHANICS I 1. Compute, using the Blasius formula, the force exerted by a simple source at a point c = beiθ on a circular cylinder of radius a < b. In what direction is the cylinder pulled? 2. Consider the two-dimensional steady flow formed by combining a uniform stream of speed U in the positive x direction, a source of strength m > 0 at (x, y) = (−a, 0), and a sink of strength m at (x, y) = (+a, 0) where a > 0. The pressure far upstream of the origin is p∞ . a) Write down the velocity potential and the stream function for this flow field. b) What are the coordinates of the stagnation points? c) Determine the pressure in this flow field along the y-axis. d) There is a closed streamline in this flow that defines a Rankine body. Obtain a transcendental algebraic equation for this streamline for the half-width h of this body. 3. This problem will study flow past a slender axisymmetric body whose surface is given (in cylindrical polar coordinates), by r = R(z), 0 ≤ z ≤ L. Here R(z) is continuous, and positive except at 0, L where it vanishes. By “slender” we mean that max0≤x≤L R << L. The body is placed in the uniform flow (uz , ur , uθ ) = (U, 0, 0). We are interested in the steady, axisymmetric potential flow past the body. It can be shown that such a body perturbs the free stream by only a small amount, so that in particular, uz ≈ U everywhere. On the other hand the flow must be tangent at the body, which implies φr (z, R(z)) ≈ U dR/dz, 0 < z < L. We look for a representation of φ as a distribution of sources with strength f (z). Thus Z L f (ζ) 1 p φ(z, r) = − dζ. 4π 0 (z − ζ)2 + r2 (a) Compute ∂φ that ∂r , and investigate the resulting integral as r → 0, 0 < z < L. Argue R ∂φ 1 f (z) +∞ the dominant contribution comes near ζ = z, and hence show that ∂r ≈ 4π r −∞ (1 + s2 )−3/2 ds for r << L. (b) From the above tangency condition, deduce that f (z) ≈ dA/dz where A(z) = πR2 is the cross-sectional area of the body. (c) By expanding the above expression for φ for large z, r, show that in the neighborhood of infinity Z L 1 z φ≈ A(ζ)dζ, z 2 + r2 → ∞. 4π (z 2 + r2 )3/2 0 1 2 FLUID MECHANICS I 4. Consider a Navier-Stokes fluid of constant ρ, µ, no body forces. Consider a motion in a fixed bouned domain V with no-slip condition on its rigid boundary. Show that Z Z 2 ρ|u| /2dV, Φ = µ (∇ × u)2 dV. dE/dt = −Φ, E = V V This shows that for such a fluid kinetic energy is converted into heat at a rate Φ(t). This last function of time gives the net viscous dissipation for the fluid contained in V . (Hint: ∇ × (∇ × u) = ∇(∇ · u) − ∇2 u.) 5. A stationary ideal two-dimensional vortex with clockwise circulation Γ is located at point (0, a), above a flat plate. The plate coincides with the x-axis. A uniform stream U directed along the x-axis flows over the vortex. Sketch the flow pattern and show that it represents the flow over an oval-shaped body when Γ/πa > U . Calculate the total upward force on the plate. 6. Consider steady laminar flow through the annular space formed by two coaxial tubes aligned with the z-axis. The flow is along the axis of the tubes and is maintained by a pressure gradient dp/dz. Determine the axial velocity. Find the radius at which the maximum velocity is reached, the volume flow rate and the stress distribution.
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