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Pulsating flow of a couple stress fluid in a channel with permeable walls

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Abstract

The paper is concerned with an analytical study of the oscillatory flow of a couple stress fluid in a channel, bounded by two permeable walls. The couple stress fluid is considered to be injected into the medium through one of the walls with a given velocity and to be sucked off by the other wall with an equal velocity. The problem is solved by using a perturbation technique. Analytical expressions for the velocity and volumetric flow rate are derived for the oscillatory flow of the couple stress fluid flowing in the channel. By using the method of parametric variation, distribution of the velocity of the couple stress fluid, change in velocity profiles at different instants of time, change in volumetric flow rate with change in frequency and cross-flow Reynolds number are computed, by considering an illustrative example. The study reveals that both the velocity and the volumetric flow rate are quite sensitive to the couple stress parameter, the frequency of oscillation and also to the cross-flow Reynolds number. The study will be immensely useful in resolving different problems associated with oil industries.

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Abbreviations

(x ∗,y ∗,z ∗):

Space coordinates

t ∗ :

Time

p ∗ :

Pressure

q :

Velocity vector (u ∗,v ∗,w ∗)

V :

Velocity of injection/suction

ω :

Angular velocity

R :

Cross-flow Reynolds number

ρ :

Fluid density

μ :

Dynamic viscosity

ν :

Kinematic viscosity, μ/ρ

\(\tau_{ij}, \tau_{ij}^{A}\) :

Symmetric and anti-symmetric parts of the stress tensor

F i :

External force

M ij :

Couple stress tensor

η,η′:

Physical constants depicting the couple stress behaviour of the fluid

μ ij :

The deviatoric part of M ij

W i :

Vorticity vector

d ij :

Symmetric part of the velocity gradient, \(=\frac {1}{2}[q_{i,j}+q_{j,i}]\)

l i :

Body moment

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Acknowledgements

The authors are highly thankful to the reviewers of the original manuscript for their esteemed comments, based upon which the present revised version has been prepared. One of the authors (J.C. Misra) is thankful to the Alexander von Humboldt Foundation, Germany for their partial support during his stay at the Leibnitz University, Hanover where a part of the work was carried out and to Prof. J. Wallaschek of the Institute of Dynamics and Vibration for some valuable discussion. Prof. Misra also wants to express his deep sense of gratitude to Prof. (Dr.) Manoj Ranjan Nayek, President of the Siksha O Anusandhan University, Bhubaneswar, India for providing a congenial research environment in the University and also for kind encouragement in his current research activities.

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Authors and Affiliations

  1. Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur, 721302, India

    S. D. Adhikary

  2. Institute of Technical Education and Research, Siksha O Anusandhan University, Bhubaneswar, 751030, India

    J. C. Misra

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  1. S. D. Adhikary
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Correspondence to J. C. Misra.

Additional information

J.C. Misra was formerly a Professor at the Department of Mathematics, Indian Institute of Technology Kharagpur, India.

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Adhikary, S.D., Misra, J.C. Pulsating flow of a couple stress fluid in a channel with permeable walls. Forsch Ingenieurwes 77, 49–57 (2013). https://doi.org/10.1007/s10010-013-0160-8

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