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Research Papers: Techniques and Procedures

Spectral Method for Analyzing Motions of Ellis Fluid Over Corrugated Boundaries

[+] Author and Article Information
M. Fazel Bakhsheshi

Department of Mechanical and Materials Engineering, The University of Western Ontario, London, ON, N6A 5B9, Canadamfazelba@uwo.ca

J. M. Floryan

Department of Mechanical and Materials Engineering, The University of Western Ontario, London, ON, N6A 5B9, Canadamfloryan@eng.uwo.ca

P. N. Kaloni1

Department of Mechanical and Materials Engineering, The University of Western Ontario, London, ON, N6A 5B9, Canadapkaloni@uwo.ca

1

Corresponding author.

J. Fluids Eng 133(2), 021401 (Feb 04, 2011) (14 pages) doi:10.1115/1.4003356 History: Received April 13, 2010; Revised December 22, 2010; Published February 04, 2011; Online February 04, 2011

A spectral method for solving the steady flow of a shear-thinning Ellis fluid is discussed for the case of a planar channel with corrugated boundaries. Polynomial approximations are employed for the velocity and viscosity distributions in the regions around singularities. The proposed algorithm employs a fixed computational domain with the physical domain of interest submerged inside the computational domain. The flow boundary conditions are imposed using the concept of immersed boundary conditions. The method, thus, eliminates the need for grid generation. The algorithm relies on Fourier expansions in the flow direction and Chebyshev expansions in the transverse direction. Various tests confirm spectral accuracy of the algorithm.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 11

Variations of the Chebyshev norm ‖DΦ(n)‖ω as a function of the Fourier mode number for the model configuration defined by Eq. 71 with the corrugation wave number α=1 and with the selected values of the corrugation amplitude S for the flow Reynolds number Re=50 and properties of the fluid described by non-Newtonian parameters β=2 and λ=0.1. Computations have been carried out using NM=22 Fourier modes and NT=81 Chebyshev polynomials.

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Figure 1

Sketch of the model flow problem. Yl and Yb denote locations of extremities of the flow domain.

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Figure 2

Distributions of the reference velocity u0(y) (a) and the reference viscosity μ0(y) (b) for selected values of the non-Newtonian parameters λ and β

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Figure 3

Variations of the mass flow rate Q0 (dashed line) and pressure loss ∂P0/∂x (solid line) as functions of the non-Newtonian parameters λ and β

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Figure 4

Distributions of the reference velocity u0(y) and its first derivative du0/dy for fluid characterized by the non-Newtonian parameters β=2 and λ=0.1

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Figure 5

Variations of the second and third derivatives of the reference velocity (a) and the reference viscosity and its first derivative (b) as functions of y. Solid and dashed lines illustrate values before and after smoothing of the singularity. The presented data are for the non-Newtonian parameters β=2 and λ=0.1.

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Figure 6

Variations of the modified viscosity as a function of ŷ before (dashed line) and after (solid line) smoothing of the singularity

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Figure 7

Matrix structure for the corrugation defined by Eq. 71 obtained with NM=10 Fourier modes and NT=61 Chebyshev polynomials used in the discretization. The nonzero elements are marked in black.

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Figure 8

Variations of the pressure correction error ‖AP‖∞ as a function of the sizes of the smoothing zones ε1, ε2, and ε3 for the fluid characterized by the non-Newtonian parameters β=2 and λ=0.1, flow with the Reynolds number Re=50, and the model configuration defined by Eq. 71 with the corrugation amplitude S=0.04 and the corrugation wave number α=2. Computations have been carried out with NT=81 Chebyshev polynomials and NM=22 Fourier modes.

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Figure 9

Variations of the maximum difference between the computed and the reference values of the x-velocity component as a function of the number of Chebyshev polynomials NT used in the computations. All computations have been carried out using NM=22 Fourier modes. The reference solution has been obtained with machine accuracy, i.e., using NT=81 Chebyshev polynomials. The presented results have been obtained for the model geometry defined by Eq. 71 with the corrugation wave number α=2, the selected values of the corrugation amplitude S shown in the figure, the flow Reynolds number Re=50, and the fluid characterized by the non-Newtonian parameters β=2 and λ=0.1.

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Figure 10

Variations of the real part of DΦ(n) as a function of y for higher modes (n>20) in the region close to the lower wall for the model configuration defined by Eq. 71 with the corrugation wave number α=5 and the corrugation amplitude S=0.05 for the flow Reynolds number Re=50 and properties of the fluid described by the non-Newtonian parameters β=2 and λ=0.1 obtained using NM=25 Fourier modes and NT=81 Chebyshev polynomials. Formation of boundary layers is clearly visible for each amplitude function.

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Figure 12

Variation of the norms ‖uL(x)‖∞ and ‖vL(x)‖∞ as a function of the total number of Fourier modes NM used in the calculation for the model geometry described by Eq. 71 with the corrugation wave number α=1 and with different corrugation amplitudes S. Calculations were carried out for the flow Reynolds number Re=50 and the material constant λ=0.1 using NT=81 Chebyshev polynomials.

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Figure 13

Variations of the x- and y-components of the velocity vector evaluated at the lower wall for the model configuration defined by Eq. 71 with the corrugation amplitude S=0.05 and the corrugation wave number α=4 evaluated using NM=22 Fourier modes and NT=81 Chebyshev polynomials for the flow Reynolds number Re=50 and properties of the fluid described by the non-Newtonian parameters β=2 and λ=0.1. The reader may note that uL and vL provide a measure of error in the enforcement of the flow boundary conditions at the corrugated wall.

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Figure 14

Fourier spectra of distributions of the x- and y-components of the velocity vector evaluated at the lower wall for the model configuration defined by Eq. 71 with the corrugation amplitude S=0.05 and the corrugation wave number α=1 evaluated using NM=20 Fourier modes and NT=81 Chebyshev polynomials for the flow Reynolds number Re=50 and properties of the fluid described by the non-Newtonian parameters β=2 and λ=0.1. The reader may note the absence of the first 20 modes, which is consistent with the construction of the IBC method.

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Figure 15

Fourier spectra of distributions of the x-component of the velocity vector, i.e., uL, evaluated at the lower wall for the model configuration defined by Eq. 71 with the corrugation amplitude S=0.04 and the corrugation wavelength λx=2π/3 evaluated using NT=81 Chebyshev polynomials for the flow Reynolds number Re=50 and properties of the fluid described by the non-Newtonian parameters β=2 and λ=0.1. Computations were carried out with the wave numbers α=3,1.5,1 using NM=10,20,30 Fourier modes in cases A, B, and C, respectively.

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Figure 16

Variation of the ‖uL(x)‖∞ norm as a function of the corrugation amplitude S for the corrugation described by Eq. 71 with selected values of the corrugation wave number α for the flow Reynolds number Re=50 and the material parameters β=2 and λ=0.1. Computations have been carried out using NM=15 (solid lines) and 20 (dashed lines) Fourier modes and NT=81 Chebyshev polynomials.

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Figure 17

Variation of the ‖uL(x)‖∞ norm as a function of the corrugation wave number α for the corrugation described by Eq. 71 with the corrugation amplitude S=0.05 for the flow Reynolds number Re=50 and the material parameters β=2 and λ=0.1. Computations have been carried out using NM=15 (solid lines) and 20 (dashed lines) Fourier modes and NT=81 Chebyshev polynomials.

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Figure 18

Variation of the ‖uL(x)‖∞ norm as a function of the flow Reynolds number Re for the corrugation described by Eq. 71 with the corrugation amplitude S=0.05 and the corrugation wave number α=2 for the material parameters β=2 and λ=0.1. Computations have been carried out using NM=22 Fourier modes and NT=81 Chebyshev polynomials

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Figure 19

Variation of the ‖uL(x)‖∞ norm as a function of the material constant λ for the corrugation described by Eq. 71 with selected values of the corrugation amplitude S and the corrugation wave number α for the flow Reynolds number Re=50 and the material parameter β=2. Computations have been carried out using NM=15,20 Fourier modes and NT=81 Chebyshev polynomials.

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Figure 20

Flow pattern in the channel with geometry defined by Eq. 74 with α=1 for the flow Reynolds number Re=50 and properties of the fluid described by the non-Newtonian parameters β=2 and λ=0.1.

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Figure 21

Variations of the pressure correction factor as a function of the corrugation amplitude S and the corrugation wave number α for the model configuration described by Eq. 71 for the flow Reynolds number Re=10 and the material parameters λ=0.1 and β=2. All results have been obtained with NM=20 Fourier modes and NT=81 Chebyshev polynomials.

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Figure 22

Variations of the pressure correction factor as a function of the material constant λ for the corrugation defined by Eq. 71 with the wave number α=2 and the amplitude S=0.02 for selected values of the flow Reynolds number Re and the second material constant β=2. All results have been obtained with NM=20 Fourier modes and NT=81 Chebyshev polynomials.

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