Effect of an oscillating time-dependent pressure gradient on Dean flow: transient solution

Basant K Jha,Dauda Gambo

doi:10.1186/s43088-020-00066-8

Abstract

BackgroundNavier-Stokes and continuity equations are utilized to simulate fully developed laminar Dean flow with an oscillating time-dependent pressure gradient. These equations are solved analytically with the appropriate boundary and initial conditions in terms of Laplace domain and inverted to time domain using a numerical inversion technique known as Riemann-Sum Approximation (RSA). The flow is assumed to be triggered by the applied circumferential pressure gradient (azimuthal pressure gradient) and the oscillating time-dependent pressure gradient. The influence of the various flow parameters on the flow formation are depicted graphically. Comparisons with previously established result has been made as a limit case when the frequency of the oscillation is taken as 0 (ω = 0). ResultsIt was revealed that maintaining the frequency of oscillation, the velocity and skin frictions can be made increasing functions of time. An increasing frequency of the oscillating time-dependent pressure gradient and relatively a small amount of time is desirable for a decreasing velocity and skin frictions. The fluid vorticity decreases with further distance towards the outer cylinder as time passes.ConclusionFindings confirm that increasing the frequency of oscillation weakens the fluid velocity and the drag on both walls of the cylinders.

Highlights

Navier-Stokes and continuity equations are utilized to simulate fully developed laminar Dean flow with an oscillating time-dependent pressure gradient
1 Background Research work on unsteady fully developed laminar flow attributed to circumferential pressure gradient and oscillating pressure gradient has remained very active in the past decade due its increasing applications in hemodynamics, biofluid mechanics, and engineering
3 Results In an attempt to understand the influence of time, frequency of oscillating time-dependent pressure gradient and the annular gap on the velocity, skin frictions and vorticity, a MATLAB program has been written to compute and generate line graphs and numerical values for the velocity, skin friction and vorticity

Summary

Introduction

Navier-Stokes and continuity equations are utilized to simulate fully developed laminar Dean flow with an oscillating time-dependent pressure gradient. These equations are solved analytically with the appropriate boundary and initial conditions in terms of Laplace domain and inverted to time domain using a numerical inversion technique known as Riemann-Sum Approximation (RSA). In context of blood flow in the human arterial system, the fully developed flow of a viscous and incompressible fluid with imposed periodic pressure gradient in a circular pipe was examined by Womersley [5]. Uchida [6], on the other hand, gave the exact solution for the steady motion of a viscous and incompressible fluid in a circular pipe driven by pressure gradient oscillating about a non-zero frequency.

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