fluidsCh67 - Chapter 6 Stokes Flows One of the earlier...

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Chapter 6 Stokes Flows One of the earlier approximations to the Navier-Stokes equations goes back to Stokes himself, who studied the limit of very small Reynolds number. This has applications to very viscous flows, suspensions and bubbles, and the recently important field of micro-fluid-dynamics. Highlights include Stokes’ paradox, far-field effects, kinematic reversibility, and the role of vorticity. (Fig. 6.1). 6.1 Stokes’ equation We start from the Navier-Stokes equations, and use a velocity U and length L as scaling quantities. Notations: u i = U u * i x j = L x * j Then t u i + u j j u i = - 1 ρ i p + ν∂ 2 jj u i (6.1) becomes U∂ t u * i + U 2 /Lu * j * j u * i = 1 ρL * i p + νU/L 2 * 2 jj u * i (6.2) Since we expect the viscous term to be significant, we set its coefficient equal to 1 by multiplication by L 2 /Uν , yielding L 2 ν t u * i + Re L u * j * j u * i = - L ρUν * i p + * 2 jj u * i (6.3) 145
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146 CHAPTER 6. STOKES FLOWS G.I. Taylor’s movie: Creeping flows Vanishing Reynolds number (large viscosity or small scale or low velocity) or convective terms vanish because of geometry (e.g. Poiseuille/Couette flow at moderate Re). Kinematic reversibility (but stresses change sign). No inertia, no convective terms. Balls, drops and suspensions. Long range effects: 2 everywhere. Vorticity essential (from B.C.). Flow around a sphere: exact solution Stokes’ paradox: flow around a cylinder Stokes drag Slender bodies Lubrication Propulsion Hele-Shaw cell Figure 6.1: A wonderful movie by G.I. Taylor: low Re flows
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6.1. STOKES’ EQUATION 147 Figure 6.2: Mind-map relative to scaling of pressure In the limit of very small Re L , the nonlinear terms drop out. Mathemati- cally, this makes the equations linear in u i , hence more easily solvable. Also, the viscous time scale is L 2 ν . Flow disturbances associated with times larger than this are in effect quasi-stationary, and the field is in equilibrium with the current conditions imposed at the boundary (examples in the movie). Furthermore, we see that ρU 2 is no longer the correct scaling for pressure. In this instance, the pressure field scales as μU/L , which may be representative of the shear stress applied at a boundary. See Fig. 6.2 The resulting equation is very simple: i p = μ∂ 2 jj u i (6.4) or p = μ 2 u (6.5) or again (using incompressibility, so that ∇ × ω = -∇ 2 u ) p = - μ ∇ × ω (6.6) This should be contrasted to Crocco’s result for inviscid flow. Since we assume incompressibility ( ∇ · u = 0), it follows that 2 p = 0 . (6.7)
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148 CHAPTER 6. STOKES FLOWS Figure 6.3: Mind-map: Pressure and vorticity in Stokes flow: Laplace equa- tion and far-field effects Similarly, since ∇ × ∇ p = 0 for any smooth scalar function, we also have 2 ω = 0 . (6.8) Laplace equation everywhere! Note that in the case of (vector) vorticity, it applies to each component. For vorticity, the stretching and advection terms are negligible in Stokes flow. As boundary conditions vary (slowly enough), diffusive equilibrium with the current boundary values is reached ‘instantly’ (i.e. much faster).
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