Oscillatory Darcy Flow of Non-Newtonian Casson Fluid with Temperature-Dependent Viscosity in a Porous Channel
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RESEARCH ARTICLE-CHEMICAL ENGINEERING
Oscillatory Darcy Flow of Non-Newtonian Casson Fluid with Temperature-Dependent Viscosity in a Porous Channel Z. Abbas1 · S. Rafiq1 · M. Sheikh2 · Shaban Aly3 Received: 6 August 2019 / Accepted: 10 February 2020 © King Fahd University of Petroleum & Minerals 2020
Abstract The present investigation aims to analyze the impact of temperature-dependent viscosity (TDV) on oscillatory Darcy flow of Casson fluid within a vertical channel in the presence of suction/injection and slip boundary condition which is applied at the cold wall. The governing equations of the present flow are transformed into dimensionless partial differential equations by using a set of non-dimensional variables. Analytical solution is obtained for the problem under consideration. The influence of distinct physical parameters on velocity profiles as well as on temperature profiles is demonstrated graphically. Here, it is remarkable to see that in the existence of TDV, the velocity of the fluid is lower for the case of Casson (non-Newtonian) fluid as compared to the Newtonian fluid. However, the velocity profiles for both types of fluids increase with increasing values of TDV parameter . Keywords Casson fluid · Oscillatory flow · Variable viscosity · Slip condition · Porous channel
List of symbols
μ(θ )
u p y v0 T0 , T1 u p αs μB
λ B0 k Cp Gr δ γ qw Nu T β t ρ g θ σe β∗ π πc h μ0 K Ha
Axial velocity (ms−1 ) Fluid pressure (kgm−1 s−2 ) Spatial coordinates (m) Constant horizontal velocity (ms−1 ) Reference fluid temperatures (K) Non-dimensional velocity (−) Dimensionless pressure (−) Stagnation speed (ms−1 ) Plastic dynamic viscosity of the non-Newtonian fluid (kgm−1 s−1 ) Product of the component of deformation rate with ei j itself, [π ei j ei j ] Yield stress of the fluid (kg m−1 s−2 ) py μ(T ) Temperature-dependent viscosity [ μ0 μ(θ )]
B
M. Sheikh [email protected]
1
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
2
Department of Mathematics, University of Sialkot, Sialkot 51310, Pakistan
3
Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Kingdom of Saudi Arabia
Dimensionless temperature-dependent viscosity, e−θ Non-dimensional pressure gradient (–) Magnetic field intensity (m−1 A) Thermal conductivity (Wm−1 K−1 ) Specific heat (Jkg−1 K−1 ) Grashof number (–) Thermal radiation parameter (–) Slip parameter (–) Heat flux (kg s−3 ) Nusselt number (–) Fluid temperature (K) Casson fluid parameter (–) Time (s) Fluid density (kg m−3 ) Acceleration due to gravity (ms−2 ) Dimensionless temperature Electric conductivity (s m−1 ) Volumetric expansion (K−1 ) Component of the deformation rate (–) Critical value of the product (–) Width of channel (m) Characteristic viscosity (kg m−1 s−1 ) Viscosity variation parameter (–) Porous permeability (m2 ) Hartmann number (–)
123
Arabian Journal for Science and Engineering
S Pr Da α ω τw Cf
Suction/injection parameter (–) Prandtl number (–) Darcy number (
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