Combined effects of suction/injection and exponentially decaying/growing time-dependent pressure gradient on unsteady De
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ORIGINAL PAPER
Combined effects of suction/injection and exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow: a semi-analytical approach Basant K. Jha1 · Dauda Gambo1 Received: 21 January 2020 / Accepted: 22 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Suction/injection · Dean flow · Unsteady · Riemann-sum approximation (RSA) · Time-dependent pressure gradient Mathematics Subject Classification 76D05 · 76A02 · 76S05
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Dauda Gambo [email protected] Basant K. Jha [email protected]
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Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria 0123456789().: V,-vol
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GEM - International Journal on Geomathematics
(2020) 11:28
List of symbols a b D Dn p R Rc Re s t U0 νr ν V
Radius of the inner cylinder (m) Radius of the outer cylinder (m) Diameter of non-circular geometry Dean number Static pressure (Kg/ms2 ) Dimensionless radial distance Radius of curvature Reynolds number (suction/injection parameter) Laplace parameter Dimensionless time (s) Reference velocity (m/s) Radial velocity (m/s) Circumferential velocity (m/s) Dimensionless velocity
Greek letters δ λ ρ τ μ
Coefficient of time-dependent pressure gradient Radii ratio (b/a) Fluid density (kg/m3 ) Skin friction Dynamic viscosity of the fluid (Kg/ms)
1 Introduction The study of unsteady two-dimensional viscous incompressible flows in horizontal permeable/non-permeable and non-rotating concentric cylinders have been the interest of many investigators in the past decades. This can be ascribed to the widespread application in biomedical engineering, biofluid mechanics where the circumferential flow is seen in most of the apparatuses transporting fluids, nuclear engineering, electrical and mechanical engineering. The analysis of steady laminar flow in a curved
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