Pulsating flow of a couple stress fluid in a channel with permeable walls
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O R I G I NA L A R B E I T E N / O R I G I NA L S
Pulsating flow of a couple stress fluid in a channel with permeable walls S.D. Adhikary · J.C. Misra
Received: 23 February 2013 / Published online: 12 April 2013 © Springer-Verlag Berlin Heidelberg 2013
Velocity vector (u∗ , v ∗ , w ∗ ) Velocity of injection/suction Angular velocity Cross-flow Reynolds number Fluid density Dynamic viscosity Kinematic viscosity, μ/ρ Symmetric and anti-symmetric parts of the stress tensor External force Couple stress tensor Physical constants depicting the couple stress behaviour of the fluid The deviatoric part of Mij Vorticity vector Symmetric part of the velocity gradient, = 12 [qi,j + qj,i ] Body moment
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.
q V ω R ρ μ ν τij , τijA
Nomenclature (x ∗ , y ∗ , z∗ ) Space coordinates t∗ Time p∗ Pressure
1 Introduction
J.C. Misra was formerly a Professor at the Department of Mathematics, Indian Institute of Technology Kharagpur, India. S.D. Adhikary Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur 721302, India J.C. Misra () Institute of Technical Education and Research, Siksha O Anusandhan University, Bhubaneswar 751030, India e-mail: [email protected]
Fi Mij η, η μij Wi dij li
The fact that consideration of Newtonian fluids is quite inadequate to determine the flow characteristics of real fluids emphasize the necessity of different studies concerned with non-Newtonian fluids. In the case of most fluids used in engineering and industries, the relation between stress and strain rate is far from being linear and therefore, it is not possible to study the flow behaviour and other problems associated with it under the purview of the classical Newtonian theory. Similarly, the flow characteristics of blood and most other physiological fluids in many situations can not be accurately determined by using the theory of Newtonian fluids. In view of the above, several non-Newtonian models have been de
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