Transient free convective flow in an annular porous medium: A semi-analytical approach

This article deliberated the effects of heat generation/absorption on transient natural convective flow in infinite vertical concentric cylinder filled with a porous material. The natural convective flow is as a result of constant heating at the inner cylinder while the inner surface of the outer cylinder is thermally insulated. A combination of Laplace transform technique and Riemann-sum approximation approach has been used to transform and invert the governing equation from the Laplace domain to the time domain respectively. The numerical values obtained from the Riemann-sum approximation excellently agree with the steady state solution at large time. The obtained results are represented graphically and the effects of the governing parameters on the velocity field, temperature field, mass flow rate, heat transfer as well as the skin-friction on both surfaces of the annulus are studied in detail.
In the course of numerical computations, it is found that the fluid velocity and temperature both increases with time, while the fluid velocity and skin friction on both walls are seen to be proportional to Darcy number but inversely proportional to the viscosity ratio. Furthermore, it is found that the mass flow rate can be controlled by increasing/decreasing the heat generation/absorption parameter.

Impact of heat generation/absorption on transient natural convective flow in an annulus filled with porous material subject to isothermal and adiabatic boundaries

The mathematical model responsible for fully developed laminar transient flow formation between the gaps of two stationary concentric porous tubes due to the imposition of azimuthal pressure gradie...

Transient Dean flow in a channel with suction/injection: A semi-analytical approach:

This paper investigates time-dependent Dean Flow of an incompressible fluid under homogeneous slip, non-homogeneous slip, and no-slip boundary conditions. The fluid flow is due to the sudden application of an azimuthal pressure gradient. The general solutions of the governing momentum equations are obtained using a two-step approach called Laplace transformation and Riemann-sum approximation method of Laplace inversion. The velocity and skin friction are determined exactly in the Laplace domain and inverted back to time domain using a numerical approach known as Riemann-sum approximation. The steady-state solutions for the velocity and the skin friction are obtained for the validation of the method employed. Graphs are plotted for analysis and numerical values are tabulated for comparison of the Riemann-sum approximation and the exact solution at large values of time. From the analysis, it is observed that the velocity profile of the fluid is higher at the wall with the highest slip coefficients. Finally, the influence of the dimensionless time ( T ) and the slip coefficients is also discussed with the aid of graphical illustrations.

Unsteady Dean flow formation in an annulus with partial slippage: A riemann-sum approximation approach

Received: 23 June 2018 Accepted: 4 September 2018 The transient flow formation due to sudden application of constant circumferential pressure gradient in an annulus partially filled with porous material has been considered. An exact solution of the governing equations has been obtained in Laplace domain using Laplace transform technique. Using suitable transformations, the equations are converted into standard Bessel equations whose solutions are available in the literature. The solutions to the Bessel equations are then inverted to the time domain using Riemann-sum approximation approach. The implicit finite difference scheme has also been employed in solving the governing equations in other to establish the accuracy of the Riemann-sum approximation approach for both transient and steady state solutions. The obtained solutions are then analysed for various dimensionless parameters entering the unified problem. There is a clear indication that large value of Da enhances the permeability of the porous region. Also, thin clear fluid region boosts τ1 but has no significant influence on τλ. While thin porous region enhances skin friction at both surfaces.

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Transient pressure driven flow in an annulus partially filled with porous material: Azimuthal pressure gradient

This study is devoted to investigate the unsteady fully developed time-dependent Couette flow in a composite channel partially filled with porous material. The Brinkman-extended Darcy model is used to simulate momentum transfer in the porous medium. The fluid and porous regions are interlinked by equating the velocity and by considering shear stress jump conditions in the interface. The solutions of the governing equations are obtained using a Laplace transform technique. However, the Riemann-sum approximation method is used to invert from Laplace domain to time domain. The solution obtained is validated by presenting comparisons with closed form solution obtained for steady flow which has been derived separately and also by implicit finite difference method. During the course of numerical comparison, an excellent agreement was found between steady-state solution obtained exactly and unsteady solution obtained by implicit finite difference method or Riemann-sum approximation method at large values of time. The effect of various flow parameters entering into the problem is discussed with the aid of line graphs. The results obtained here may be further used to verify the validity of obtained numerical solutions for more complicated time-dependent Couette flow in composite channel.

Unsteady Couette Flow in a Composite Channel Partially Filled with Porous Material: A Semi-analytical Approach

This study investigates the start-up flow in a horizontal annulus partially filled with clear fluid and partially with a fluid-saturated porous material (composite channel) as a result of the sudden application of constant pressure gradient in horizontal direction. Momentum transfer in the porous medium is simulated using the Brinkman-extended Darcy model. The fluid and porous regions are linked together by equating their velocities and incorporating the shear stress jump conditions at the interface. With the application of Laplace transform technique, the governing partial differential equations are transformed into ordinary differential equations which are solved in the Laplace domain to obtain exact solutions. Inversion of the exact solution in Laplace domain to time domain is done using the Riemann-sum approximation approach. Validation of the Riemann-sum approximation method is achieved by comparing numerical values obtained with those of exact solution obtained for steady state flow and transient solution obtained by implicit finite difference method. At large values of time, transient solution obtained using Riemann-sum approximation approach and implicit finite difference method coincide with closed form solution obtain exactly for steady state flow showing excellent agreement between these methods. Line graphs are used to discuss the effect of the different flow parameters involved in the flow formation.

J. O. Odengle

Papers

Unsteady Couette Flow in a Composite Channel Partially Filled with Porous Material: A Semi-analytical Approach

A Semi-analytical Solution for Start-Up Flow in an Annulus Partially Filled with Porous Material