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Combined effects of suction/injection and exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow: a semi-analytical approach

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Abstract

A two-dimensional mathematical model is developed and solved semi-analytically in order to theoretically examine the impact of suction/injection and an exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow through a coaxial cylinder. The walls of the cylinders are porous so as to enable the superimposition of the radial flow. The solution of the governing momentum and continuity equations are derived using a two-step process, the Laplace transformation in conjunction with the Riemann Sum-Approximation (RSA). For accuracy check, the steady state solution is computed and numerical values obtained using the Riemann-Sum Approximation (RSA) is compared with the already established results. It is found out that for an increasing time, a growing pressure gradient enhances the flow formation for both suction and injection, although the effect on the azimuthal velocity profile is subtle when suction is applied on porous walls. Moreover, the skin frictions on the walls can be minimized by imposing a decaying pressure gradient for suction/injection, however the behaviour is seen clearly when fluid particles are injected through the porous cavity.

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Abbreviations

a :

Radius of the inner cylinder (m)

b :

Radius of the outer cylinder (m)

D :

Diameter of non-circular geometry

D n :

Dean number

p :

Static pressure (Kg/ms2)

R :

Dimensionless radial distance

R c :

Radius of curvature

Re :

Reynolds number (suction/injection parameter)

s :

Laplace parameter

t :

Dimensionless time (s)

U 0 :

Reference velocity (m/s)

ν r :

Radial velocity (m/s)

ν :

Circumferential velocity (m/s)

V :

Dimensionless velocity

δ :

Coefficient of time-dependent pressure gradient

λ :

Radii ratio (b/a)

ρ :

Fluid density (kg/m3)

τ :

Skin friction

μ :

Dynamic viscosity of the fluid (Kg/ms)

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  1. Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

    Basant K. Jha & Dauda Gambo

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  1. Basant K. Jha
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  2. Dauda Gambo
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Correspondence to Dauda Gambo.

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Jha, B.K., Gambo, D. Combined effects of suction/injection and exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow: a semi-analytical approach. Int J Geomath 11, 28 (2020). https://doi.org/10.1007/s13137-020-00164-w

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