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In real life, even a very small viscosity has a profound effect, leading to completely different flows from the ideal ones. We’ll get an idea why later on. 46 CHAPTER 3. NON-DIMENSIONALISATION 2. Verify that the units of E are [M][L]−1 [T]−2 . 3. Roughly how big is the steady-state displacement? 4. If the drag force is switched on suddenly at t = 0, over what timescale does the beam initially respond? 5. What is the timescale for free oscillations? We will return to versions of this model at several places later in the book.
If T (x, t) is the temperature in an incompressible fluid which is moving with velocity u, explain why the heat flux is ρcT u − k∇T. 1). [Note that in the derivation on page 40 we used incompressibility to say that the density in the material volume remains constant. If the fluid is compressible, we have to worry about what we mean by the specific heat, because the density changes. That is, we have to think carefully about the thermodynamics of the problem. Fortunately, for most liquids the density 12 At large Reynolds number the flow is turbulent and so unsteady; the drag coefficient must be interpreted as a time average.
NON-DIMENSIONALISATION Explain (in terms of a force balance) why appropriate conditions at the free surface y = h are σij nj = 0, where σij is the stress tensor. Take coordinates x downhill along the plane and y normal to it. Show that there is a solution u = (u(y), 0, 0) and that the free surface conditions reduce to p = 0, ∂u/∂y = 0. Find u(y) and verify the dimensional analysis h for Q = 0 u(y) dy. 2 on page 250). 13. Dimensional analysis of Poiseuille flow. In a Poiseuille flow down a pipe, a Newtonian viscous fluid is forced down a circular tube of crosssectional area A (or radius a) and length L by a pressure drop ∆P .