Problem Set 1 ME 437: Incompressible Flow D. H. Kelley Due 27 January 2015. 1. Consider a solid sphere of diameter D moving at constant speed U through stagnant, incompressible fluid of density ρ and viscosity µ. The sphere experiences a drag force, and at very low speeds the drag is independent of density because inertial forces are dominated by viscous forces. Use only dimensional analysis to derive an appropriate expression for the drag force Fd in this situation. 2. Suppose that I’m floating in a boat on a lake carrying a brick. If I throw the brick overboard, will the level of the lake change, and if so, how? 3. A closed vessel full of water is rotating with angular velocity Ω about a horizontal axis, and the water is stationary in the rotating frame. Show that the surfaces of equal pressure are circular cylinders whose common axis is at height g/Ω2 above the axis of rotation. 4. Consider a scalar field φ(x, t) and a velocity field u(x, t) defined in an inertial reference frame. Now suppose that we want to work in the non-inertial frame where X ≡ x + ξ(t) T ≡t In this new frame, the velocity and scalar fields are given by U(X, t) and Φ(X, t). (a) Express Φ and U in terms of φ and u. (b) What is the rate of change following a fluid element in the (X, T ) frame? That is, what is the relationship between the material derivatives (c) Write the acceleration of a fluid element a = Du/Dt in the (X, T ) frame. 5. Consider two second-rank tensors Aij and Bij . Suppose that Aij = Aji and Bij = −Bji ; then Aij is called symmetric and Bij is antisymmetric. Show that Aij Bij = 0. Note that this is a general result regardless of tensor rank: the contraction of a symmetric and and antisymmetric tensor vanishes. 6. Let us explore some properties of the Levi-Civita symbol ijk . (a) Consider two coordinate systems E and E 0 that are related by a reflection through each axis; that ˆi = −ˆ is, e e0i for i = 1, 2, 3. Use these two coordinate systems to show that ijk is not a third-rank tensor. (b) Prove the “epsilon–delta” rule: ijk `mk = δi` δjm − δim δj` . Explain why the right-hand side is a proper tensor while ijk is a pseudotensor. (c) Show that ijk ijk = 6. 7. Not all quantities with components are tensors, and in fact not all one-component quantities are proper scalars — some are pseudoscalars. As an example, show that the scalar triple product a · b × c, where a, b, and c are proper vectors, is actually a pseudoscalar. What happens to the scalar triple product under coordinate reflections? 8. Which of these could be valid tensor equations? For those that are not, state why not. (a) aij = bik` cj` dk (b) ψ = mii + nii xii (c) Lijk = imn rjm pkn (d) x = y (e) dn = `kn q` ijk ri uj (f) ak = rijk δij + Sijk`m fi gj` (g) y3 = qij3 zij 9. Let the quantities vi , Mij , Nijk , and Pijkl be tensors. For each of the following, combine all four to write a valid tensor equation for the indicated quantity. (a) A scalar: φ = (b) A vector: ω = (c) A second-rank tensor: Fkl = (d) A third-rank tensor: Gpqr = (e) A fourth-rank tensor: Hijkl = 10. Evaluate and/or simplify the following expressions, where Mij is a second-rank tensor, φ(x) is a scalar field, and v(x) is a vector field. Use indicial notation. (a) δij δpq Miq (b) ∂ ∂xk (xi vi ) 2 φ (c) εijk ∂x∂j ∂x k (d) δpq δpq (e) ∂2 ∂xi ∂xj (xp xq ) 2