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TECHNICAL PAPERS

Numerical Simulations of Peristaltic Mixing

[+] Author and Article Information
Saurabh Kumar, Ho Jun Kim

Aerospace Engineering Department,  Old Dominion University, Norfolk, VA 23529-0247

Ali Beskok1

Aerospace Engineering Department,  Old Dominion University, Norfolk, VA 23529-0247abeskok@odu.edu

1

Corresponding author.

J. Fluids Eng 129(11), 1361-1371 (Jun 06, 2007) (11 pages) doi:10.1115/1.2786480 History: Received December 28, 2005; Revised June 06, 2007

Abstract

Numerical simulations of two-dimensional flow and species transport in a peristaltically driven closed mixer are performed as a function of the Reynolds number $(Re⩽6288)$ and the normalized traveling wave amplitude $(ε⩽0.3)$ at low to moderate Schmidt number $(Sc⩽10)$ conditions. The mixer consists of a rectangular box with a traveling wave motion induced on its bottom surface. Flow and species mixing are produced by the surface motion. The numerical algorithm, based on an arbitrary Lagrangian–Eulerian spectral element formulation, is verified using the asymptotic solutions for small wave amplitude cases. Kinematics of large-deformation conditions are studied as a function of the Reynolds number. Species mixing is simulated at various $Re$ and $Sc$ conditions. Mixing index inverse $(M−1)$ is utilized to characterize the mixing efficiency, where $M−1∝exp(Pe−αt)$ is observed as the long-time behavior. Simulation data are utilized to determine the exponent $α$ at various $Re$ and $Sc$ conditions. For all simulations, $0.28⩽α⩽0.35$, typical of partially chaotic flows, have been observed. The effect of flow kinematics and species diffusion on mixing is interpreted.

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Figures

Figure 1

Top: Schematic view of the peristaltic mixer with its relevant length scales, and seventh-order spectral element discretization for ε=a∕h=0.3. The thick lines show the element boundaries, while the thin lines correspond to the internal degrees of freedom used for each element. Bottom: Zoomed view of the right bottom corner of the domain that exhibits exponential variations in the x direction, as given by Eq. 4.

Figure 2

Velocity profiles for ε=0.01 and Re=63 at various times during a cycle. (a), (b), (c), and (d) show variation of u(x,0.5,t), v(x,0.5,t), u(L∕2,y,t), and v(L∕2,y,t), respectively. Symbols show the numerical solutions and the solid lines represent the leading-order asymptotic solution. Squares, triangles, and circles represent numerical snapshots at t=0, T∕4, and T∕2, respectively.

Figure 3

Velocity profiles for ε=0.01 and Re=6288 at various times during a cycle. (a), (b), (c), and (d) show variation of u(x,0.5,t), v(x,0.5,t), u(L∕2,y,t), and v(L∕2,y,t), respectively. Symbols show the numerical solutions and the solid lines represent the leading-order asymptotic solution. Squares, triangles, and circles represent numerical snapshots at t=0, T∕4, and T∕2, respectively.

Figure 4

Velocity profiles for ε=0.3 and Re=63 at various times during a cycle. (a), (b), (c), and (d) show variation of u(x,0.5,t), v(x,0.5,t), u(L∕2,y,t), and v(L∕2,y,t), respectively. Symbols show the numerical solutions and the solid lines represent the leading-order asymptotic solution. Squares, triangles, and circles represent numerical snapshots at t=0, T∕4, and T∕2, respectively.

Figure 5

Velocity profiles for ε=0.3 and Re=400 at various times during a cycle. (a), (b), (c), and (d) show variation of u(x,0.5,t), v(x,0.5,t), u(L∕2,y,t), and v(L∕2,y,t), respectively. Symbols show the numerical solutions and the solid lines represent the leading-order asymptotic solution. Squares, triangles, and circles represent numerical snapshots at t=0,T∕4, and T∕2, respectively.

Figure 6

Time evolution of species concentration contours at Re=6000 and Pe=6000 conditions for the ε=0.3 case

Figure 7

Mixing index inverse variation as a function of the number of time periods (T) for three different flow conditions at Pe=4000. Time periods for all three cases are different from each other.

Figure 8

Mixing index inverse variation as a function of the number of time period (T) for Re=400 flow at Sc=1 and Sc=10 conditions

Figure 9

Mixing index inverse variation as a function of the number of time period (T) for Re=4000 and Re=6000 conditions

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