Transient Dean flow in an annulus: a semi-analytical approach

Abstract

The mathematical model of a fully developed laminar transient flow formation between the gaps of two horizontally stationary concentric tubes forming concentric annulus due to the impact of sudden application of azimuthal pressure gradient is analysed. Analytical and numerical solutions of the momentum equations are obtained by the aid of two-step process. The first step is by solving the governing partial differential equation analytically by using Laplace transform technique in the Laplace domain. Using Riemann-sum approximation of Laplace inversion, the velocity and skin friction are then inverted to time domain. Expressions for steady-state velocity and skin friction are obtained for the validations of the method employed. In the course of numerical computations, it is observed that increase in dimensionless time (T) leads to increase in velocity as well as skin friction at the surfaces of tubes. It is also found that at large values of dimensionless time (T), the velocity and skin friction reach steady state.

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

“…14- (17) are to be transformed in order to determine the velocity, skin frictions and vorticity in time domain. Due to the intricate nature of the closed-form solutions, a numerical inversing procedure known as Riemann-Sum Approximation (RSA) employed by Jha and Odengle [20], Yusuf and Gambo [21], and Jha and Yahaya [22,23] which is remarkable for its precision has been utilized in transforming Eq. (14)- (17) to time domain as follows:…”

“…In their work, they utilized a numerical inversing technique known as Riemann-Sum Approximation approach (RSA) in transforming the Laplace domain solution to time domain and concluded that velocity of the fluid is an increasing function of time. Other related literatures that adopted this method of solution include the work of Jha and Odengle [20], Yusuf and Gambo [21], and Jha and Yahaya [22].…”

“…Recently, Jha and Yahaya [22,23] adopting the same method of solution as Jha and Yusuf [19], scrutinized unsteady Dean flow of a viscous and incompressible fluid with constant pressure gradient. In their work, they considered the flow in a horizontal concentric cylinder and later extended the work to the case when the walls of the cylinder are porous in order to superimpose the radial flow.…”

“…In their work, they considered the flow in a horizontal concentric cylinder and later extended the work to the case when the walls of the cylinder are porous in order to superimpose the radial flow. They obtained that an increasing time is desirable for an optimum velocity and skin friction (see Jha and Yahaya [22]). In addition, Jha and Yahaya [23] reported that the fluid velocity and skin friction are increasing functions of injection and time.…”

“…However, other facets like the behavior of fluid in the annular gap with an increasing time at fixed frequency and amplitude were not elucidated. The aim of this article is to extend the work of Jha and Yahaya [22] by considering an oscillating timedependent pressure gradient in addition to the azimuthal pressure gradient. The governing momentum equations are solved analytically in Laplace domain, and the Laplace domain solution is transformed to time domain using a numerical inversing technique known as Riemann-Sum Approximation (RSA).…”