Theoretical Fluid Dynamics - Solutions for Assignment 8 | MENG 463, Assignments of Mechanical Engineering

Download Theoretical Fluid Dynamics - Solutions for Assignment 8 | MENG 463 and more Assignments Mechanical Engineering in PDF only on Docsity! MENG 463 / ENAS 704 Spring 2009 Problem Set 8 — Solutions Each of the situations described below is potentially unstable. Here, we will consider the physics of each instability. For each case, explain: (a) What are the destabilizing effects? What are the stabilizing effects? (b) Describe the base flow state you would perturb about in a stability analysis. (c) What do you expect the flow to look like after it goes unstable? 1. The Taylor–Couette instability: fluid is confined to lie between two concentric rotating cylin- ders with radii R1 and R2 and rotation rates Ω1 and Ω2. Solution: (a) The key here is to note that we will have an unstable stratification of angular momentum if Ω1R21 > Ω2R 2 2, where R2 and Ω2 refer to the outer cylinder. Fluid will be spun faster by the inner cylinder, and due to centrifugal forces will want to spread outwards. As it goes radially outwards, however, it will slow down, and want to come back towards the inner cylinder. Viscosity plays a stabilizing role if we include it in the analysis. (b) The base state is steady flow with circular streamlines. (c) The Taylor–Couette flow produces a pattern not so unlike the Rayleigh–Bénard case: the basic instability gives us a pattern of toroidal vortices, analogous to convection rolls. 2. The Rayleigh–Taylor instability: a fluid of density ρ1 lies above a fluid of density ρ2, with ρ1 > ρ2. Solution: (a) Since ρ1 > ρ2, the density stratification is destabilizing: the heavy fluid lies above the light fluid. Viscosity and potentially surface tension, if the two fluids are immiscible, are stabilizing influences. (b) The relevant base state is one of no flow. (c) The heavy fluid wants to fall down, but the light fluid is in the way, and the whole system must remain incompressible. We therefore have plumes of rising light fluid and falling heavy fluid. 3. The Saffman–Taylor instability: a fluid of viscosity µ1 is injected into a fluid of viscosity µ2 from a point source, with µ1 < µ2. Consider the fluids to be confined to a narrow gap between two parallel plates (called a Hele–Shaw cell). Solution: (a) The viscosity difference plays a destabilizing role here, as well as the inertia of the injected fluid; surface tension stabilizes the system. (b) The base state is one of expanding circle of fluid 1 with perfect azimuthal symmetry. (c) The azimuthal symmetry is broken, and “fingers” of fluid 1 extend into fluid 2. This is also known as a “fingering” instability.