UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2007 2008 FLUIDS DYNAMICS WITH ADVANCED TOPICS Time allowed: 3 hours Attempt question ONE and FOUR other questions. Candidates must show on each answer book the type of calculator used. Do not turn over until you are told to do so by the Invigilator. Copyright of the University of East Anglia
- 2 - For Information Below are the Navier-Stokes equations, for an incompressible fluid of constant density ρ and constant kinematic viscosity ν, in which the velocity u = u r e r + u θ e θ + u z k and the pressure is p, with zero body force, expressed in cylindrical polar coordinates r, θ, z: u r t + (u )u r u ( 2 θ r = 1 p ρ r + ν 2 u r u r r 2 ) u θ 2 r 2 θ u θ t + (u )u θ + u ru θ = 1 ( p r ρr θ + ν 2 u θ u θ r + 2 ) u r 2 r 2 θ where and Also u z t + (u )u z = 1 p ρ z + ν 2 u z u = u r r + u θ r 2 = 1 r u = 1 r θ + u z z ( r ) + 1r 2 r r 2 θ + 2 2 z. 2 r (ru r) + 1 u θ r θ + u z z. In spherical polar coordinates r, θ, α F = F r e r + 1 F r θ e θ + 1 F r sin θ α e α. and u = 1 ( ) r 2 1 (u θ sin θ) u r 2 r + + 1 (u α ) r r sin θ θ r sin θ α
- 3-1 (a) An inviscid, incompressible fluid, of constant density ρ, lies in a domain Ω which is subjected to a sudden change in velocity due to the fluid impacting part of the boundary Ω of the fluid domain. The velocity field is u b immediately before impact and u a immediately after the impact. The pressure impulse P is such that in Ω u a u b = 1 P. (1) ρ (i) Show that P satisfies Laplace s equation. (ii) On Ω the unit normal drawn out of the fluid is n. Suppose the fluid impacts a rigid impermeable surface Ω s with given velocity u b and that the fluid stays in contact with Ω s after impact. From equation (1) show that the boundary condition for P is P n = ρ u b n on Ω s. (b) At the instant of impact, incompressible fluid occupies a triangle OAB whose vertices O, A, B have coordinates (0, 0), (L, 0) and (L, L/ 3), respectively, where the constant L is positive. The side AB is a vertical rigid wall which is struck by the fluid with a velocity u b = U 0 (1 y 2 /L 2 ), where U 0 is a positive constant, and it stays in contact with the wall after impact. The hypotenuse OB is a free surface on which the boundary condition is P = 0. On the bed OA, the fluid remains in contact with the impermeable bed throughout the impact. (i) You are given that a suitable form of solution for the pressure impulse is P = k(x 3 3xy 2 ), where k is a constant. Show, from the boundary condition at x = L, that the constant k = ρu 0 /(3L 2 ). (ii) Confirm that the boundary conditions on P are satisfied on the free surface, OB, and on the bed OA. (iii) Find the fluid velocity immediately after impact, at the point B. PLEASE TURN OVER
- 4-2 (a) For a velocity field v, define concisely the vorticity. [2 marks] (b) Fluid occupies the region y > 0 above a flat plate at y = 0 with no body forces. Far from the plate the velocity is zero and the pressure is p = p 0, a constant. The plate oscillates horizontally with velocity iu 0 sin(ωt), where U 0 and Ω are positive constants. (i) Assuming that the fluid velocity u = iu(y, t), show that the Navier-Stokes equations imply that for y > 0 u t = νu yy (1) and p = p 0. (2) [6 marks] (ii) Write down the boundary conditions on u(y, t) for equation (1). [2 marks] (iii) Let A and k be constants. Confirm that a solution of eq. (1) is of the form u(y, t) = A exp( ky) sin(ky Ωt) where you must show that k = (Ω/[2ν]) 1/2. (iv) Find A. (v) Sketch u as a function of y when Ωt = π/2 and when Ωt = 3π/2. [2 marks]
- 5-3 (a) Define the Reynolds Number. [3 marks] (b) At large Reynolds Number the steady two-dimensional Navier-Stokes equations reduce to the following dimensionless boundary-layer equations for fluid above a fixed impermeable surface at y = 0: uu x + vu y = U du dx + u yy (1) u x + v y = 0, (2) where, as y, u iu(x). (i) Let u = ψ y and v = ψ x. Show that equations (1) and (2) are satisfied provided ψ y ψ xy ψ x ψ yy = U du dx + ψ yyy. (3) (ii) Let U = 1. Suppose ψ = x b f(η) where x > 0 and η = yx b. Find the constant b such that this is a similarity solution of eq. (3). (iii) Hence derive a third-order ordinary differential equation for f(η) (do not try to solve it) and supply a complete set of boundary conditions on f. [6 marks] (iv) The i-component of viscous stress on the boundary at y = 0, due to the flow, is t 1 = c u y, where c is a positive constant. Show that t 1 decreases as x increases. [3 marks] PLEASE TURN OVER
- 6-4 In spherical polar coordinates r, θ, α an axisymmetric flow has a velocity field u which can be expressed in terms of components as u = u r e r + u θ e θ, where θ is the angle measured from the positive z-axis and where e r, e θ, e α are unit vectors. Suppose the Stokes stream function ψ = ψ(r, θ) is such that u r = 1 ψ r 2 sin θ θ and u θ = 1 ψ r sin θ r. (i) Show that, expressed this way, the velocity is consistent with the fluid being incompressible. (ii) Let U be constant and let ψ = 1 2 Ur2 sin 2 θ. Show that the corresponding velocity is U cos(θ)e r U sin(θ)e θ and that this is a flow of constant speed U directed parallel to the z-axis. (iii) For flow at small Reynolds number you are given that the Navier-Stokes equations imply that ψ satisfies [ 2 r + sin θ 2 r 2 ( 1 θ sin θ )] 2 ψ = 0. ( ) θ By seeking all solutions of equation ( ) of the form ψ = r λ sin 2 θ, where λ is a constant, show that ψ(r, θ) = U(Ar 1 + Br + Cr 2 + Dr 4 ) sin 2 θ, where A, B, C, D are constants. (iv) Consider the slow viscous flow past a fixed impermeable sphere, S, centre O, and radius a. As r the velocity field has speed U parallel to the z-axis. Show first that D = 0 and C = 1 2 U. (v) Establish suitable boundary conditions on the surface of S for ψ. Hence find A and B.
- 7-5 Viscous fluid lubricates the gap between a moving steel plate at z = 0 and a fixed curved copper surface z = h(x) where h(x) > 0 is a differentiable function for x : 0 x 1. The plate has velocity U 0 i, where the constant U 0 > 0. At x = 0 and at x = 1 the pressure p is zero. The flow is two-dimensional and such that the pressure p = p(x), and the velocity u = u(x, z)i + w(x, z)k are governed by 0 = 1 dp ρ dx + νu zz (1) and u x + w z = 0. (2) (i) Show that u(x, z) = 1 dp 2µ dx ( z 2 hz ) ( + U 0 1 z ). h (ii) From equation (2) and the boundary conditions on u, show that the volume flux Q = h u dz through the gap, is independent of x. 0 (iii) Deduce that x p(x) = 6U 0 µ 0 1 x dx 12µQ h2 0 1 h 3 dx. [3 marks] (iv) Hence show that Q = U 0 2 1 0 h 2 dx 1. [3 marks] 0 h 3 dx (v) Establish that where the pressure has a local maximum, at x = x m, the corresponding gap width h(x m ) is given by h(x m ) = 2Q U 0. [3 marks] (vi) Suppose h(x) = 1 λx, where the constant λ : 0 < λ < 1. Find an expression for x m in terms of λ. [3 marks] PLEASE TURN OVER
- 8-6 An inviscid fluid flows steadily, without gravity, between two plane impermeable walls at y = 0 and y = 1. The undisturbed fluid velocity is U(y)i and the pressure is zero. This flow is now disturbed by small perturbations, such that the velocity and pressure are ( ) u(x, y, t) = i U(y) + û(y) exp(i[kx ωt]) + j ˆv(y) exp(i[kx ωt]) p(x, y, t) = ˆp(y) exp(i[kx ωt]), where the real part of each right-hand side is to be understood. (i) Using the fact that the fluid is incompressible, from Euler s equations derive three differential equations, linear in û, ˆv and ˆp. (Gravity is absent and you may neglect the products of small terms û 2, ˆv 2, ˆvˆv etc). [6 marks] (ii) Eliminate û, ˆp to show that ˆv(y) satisfies the following ordinary differential equation in which a prime is a y-derivative: d 2ˆv ( ) ku dy + 2 ω Uk k2 ˆv = 0, (1) subject to ˆv = 0 at y = 0 and y = 1. [6 marks] (iii) Suppose k is real and ω = ω R + iω I. Multiply equation (1) by the complex conjugate of ˆv, and integrate to show that 1 ω I k 0 U (y) ˆv 2 dy = 0. ω Uk 2 (iv) Deduce Rayleigh s Inflection Point Theorem: a necessary condition for this flow to be unstable is that U (y) must vanish for at least one y : 0 < y < 1. END OF PAPER