4.7 Dispersion in an oscillatory shear flow

Transcription

1 Lecture notes in Fluid ynamics.63j/.0j) by Chiang C. Mei, MIT, Spring, dispersion.tex March, 007 [Refs]:. Aris:. Fung, Y. C. Biomechanics 4.7 ispersion in an oscillatory shear flow Relevant to the convective diffusion of salt and/or pollutants in a tidal channel, and chemicals in a blood vessel, Let us examine the Taylor dispersion in an oscillating flow in a pipe. Let the velocity profile be given, u = U s r) + R [ U w r)e iωt], 0 < r < a. 4.7.) The transport equation for the concentration of a solvent is recalled C t + uc) C = x x + r C )) r r r 4.7.) Assume the pipe to be so small that diffusion affects the whole radius in one period or so, i.e., τ o π ω a 4.7.3) We shall be interested in longitudinal diffusion across L much greater than a. Let U o be the scale of U and x = Lx, r = ar, u = U o u, t = a t, Ω = ωa 4.7.4) Equation 4.7.) is nomalized to C + Ua a u C ) = a C t L x L x + r C ) 4.7.5) r r r Let the Péclét number be of order unity Pe = Ua/ = Oa/L) 0, 4.7.5) becomes C + ǫpe u C ) = ǫ C t x x + ) r C 4.7.6) r r r the boundary conditons For brevity we op the primes from now on. C r = 0, r = 0, 4.7.7) u = U s + RU we iωt 4.7.8)

2 4.7. Multiple scale analysis-homogenization For convenience let us repeat the perturbation arguments of the last section. There are three time scales : diffusion time across a, convection time across L, and diffusion time across L. Their ratios are : a : L : L U o = : ǫ : ǫ, 4.7.9) the smallest time scale being comparable to the oscillation period. Upon introducing the multiple time coordinates t 0 = t, t = ǫt, t = ǫ t 4.7.0) and the multiple scale expansions. C = C 0 + ǫc + ǫ C ) where C i = C i x,r,t 0,t,t ), then the perturbation problems are Oǫ 0 ): C 0 = r C ) 0 t r r r the boundary conditions: : Oǫ): Oǫ ): C 0 r C 0 + C + Pe uc 0) t t 0 x C r C 0 + C + C + Pe uc ) t t t 0 x 4.7.) = 0, r = 0, ) = r r C ) r r 4.7.4) = 0, r = 0, ) = C 0 x + r C ) r r r 4.7.6) C = 0, r = 0, ) r Ignoring the transient that dies out quickly and focusing attention to the long-time evolution, i.e., t = O), the solution at Oǫ 0 ) is Strictly speaking the solution is C 0 = C 00 x,t 0,t,t ) + C 0 = C 0 x,t,t ), 4.7.8) C 0n x,t,t )e k n ) t 0 J 0 k nr) 0 where k n is the n th root of J 0ka) = 0. The series terms die out quickly in t 0 and t, leaving the limit C 00 which is independent of t 0. r. E. Qian,993)

3 3 then At Oǫ), let the known velocity be u = U s y) + R ) U w y)e iωt 0 C 0 + C + Pe { U s + R [ Ur)e ]} iωt C 0 0 t t 0 x = r C ) r r r enoting the period average by overbars, f = Ω π t0 +π/ω t 0 f dt 0 and taking the period average, C 0 C 0 + PeU s t x = r C ) r r r 4.7.9) 4.7.0) 4.7.) C = 0, r = 0, 4.7.) r Let us now integrate or average ) across the pipe, and get where angle brackets denote averaging over the cross section. C 0 t + Pe U s C 0 x = ) Now subtract 4.7.3) from 4.7.0) h = π 0 πrh C + Pe { Ũ s + R [ U w e ]} iωt C 0 0 t 0 x = r C ) r r r 4.7.4) where Ũ = U s y) U s 4.7.5) is the velocity nonuniformity Now C is governed by a linear equation, we can assume the solution to be proportional to the forcing and composed of a steady part and a time harmonic part, i.e., then C = Pe C 0 x r { Bs r) + R [ B w r)e iωt 0]} d r db s ) 4.7.6) = Ũr) 4.7.7)

4 4 and the boundary conditions r d r db w ) db s = 0 and db w After solving for B s,b w we go to Oǫ ), i.e., 4.7.6) : C 0 t + C t + C t 0 + iωb w = U w r) 4.7.8) = 0, r = 0, ) + Pe { U s + Ũs + R [ { U w e 0]} iωt Bs + R [ B w y)e iωt 0 = C 0 x + r C ) r r r ]} C 0 x ) which is a linear PE for C. From4.7.6) and 4.7.3) we find It follows that C = Pe C 0 t x U s { B s r) + R [ ]} B w r)e iωt ) C 0 t + C t 0 + Pe { Ũ s + R [ { U w e 0]} iωt Bs + R [ B w r)e iωt 0 = C 0 x + r C ) r r r ]} C 0 x Taking the time average over a period, C 0 + Pe {Ũs B s + } t R [U wbw] C 0 x = C 0 x + r C ) r r r 4.7.3) ) C = 0 r = 0, ) r Averaging ) across the pipe, we get C 0 t = E C 0 x ) E = Pe { ŨsB s + } R U wbw )

5 which is the effective diffusion coefficient or the dispersion coefficient. The first part is of molecular origin; the second part is due to fluid shear. Finally we add 4.7.3) and ) to get: + ǫ + ǫ ) C 0 + ǫpe U s C 0 t 0 t t x = ǫ E C ) x This describes the convective diffusion of the area averaged concentration, which is certainly of practical value. After the perturbation analysis is complete, there is no need to use multiple scales; we may now write C 0 + Pe U s C 0 t x = C 0 ǫe ) x in dimensionless form, or, C 0 t + U s C 0 x = C 0 E ) x in physical form. This equation governs the convective diffusion of the cross-sectional average, after the initial transient is smoothed out Aris solution for a cicular pipe R. Aris 960, Proc Roy. Soc. Lond. 59, pp ) has worked ouyt the solanwers for a flow forced by a periodic pressure gradient in a circular pipe of radius a. The analysis is carried out by using Bessel functions; only the solution is cited here. For the pressure gradient p [ ρ x = P 8 First the velocity profile is found. enoting a n cos nπt T + b n sin nπt )] T ) U s = Pa 8ν, Pe = U s a 4.7.4) α n = nπa νt, β n = nπa T, 4.7.4) The dispersion corfficient is found to be E = + Pe 48 + Pe 8 Lα n,β n )a n + b n) ) enoting σ = α n β n = ν, y = β n, )

6 6 Figure 4.7.: The function L σ y), Aris, 960. we can rewrite Lx,y) = Lσy,y) = L σ y) so that L is a function of y and the Schmitt number σ, as plotted in Figure Homework: Find the dispersion coefficient E in the oscillatory flow in a circular pipe and carry out the necesary numerical calculations. Homework mini research) : Find the dispersion coefficient E in the oscillatory flow in a blood vessel elastic wall.