Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell

1969 ◽  
Vol 39 (3) ◽  
pp. 477-495 ◽  
Author(s):  
R. A. Wooding
Keyword(s):  
Porous Medium ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Saturated Porous Medium ◽  
Mean Amplitude ◽  
Wave Number ◽  
Two Fluids ◽  
Averaged Equations ◽  
The Mean ◽  
Hele Shaw Cell

Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

Thermal Instability of a Fluid-Saturated Porous Medium Bounded by Thin Fluid Layers

1987 ◽  
Vol 109 (3) ◽  
pp. 677-682 ◽  
Author(s):  
G. Pillatsis ◽  
M. E. Taslim ◽  
U. Narusawa
Keyword(s):  
Porous Medium ◽  
Porous Layer ◽  
Thermal Instability ◽  
Numerical Solutions ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
Wave Number ◽  
Thermal Conductivity Ratio ◽  
Fluid Layers

A linear stability analysis is performed for a horizontal Darcy porous layer of depth 2dm sandwiched between two fluid layers of depth d (each) with the top and bottom boundaries being dynamically free and kept at fixed temperatures. The Beavers–Joseph condition is employed as one of the interfacial boundary conditions between the fluid and the porous layer. The critical Rayleigh number and the horizontal wave number for the onset of convective motion depend on the following four nondimensional parameters: dˆ ( = dm/d, the depth ratio), δ ( = K/dm with K being the permeability of the porous medium), α (the proportionality constant in the Beavers–Joseph condition), and k/km (the thermal conductivity ratio). In order to analyze the effect of these parameters on the stability condition, a set of numerical solutions is obtained in terms of a convergent series for the respective layers, for the case in which the thickness of the porous layer is much greater than that of the fluid layer. A comparison of this study with the previously obtained exact solution for the case of constant heat flux boundaries is made to illustrate quantitative effects of the interfacial and the top/bottom boundaries on the thermal instability of a combined system of porous and fluid layers.

scholarly journals Mixed Convection Flow over an Unsteady Stretching Surface in a Porous Medium with Heat Source

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Syed Muhammad Imran ◽  
Saleem Asghar ◽  
Muhammad Mushtaq
Keyword(s):  
Porous Medium ◽  
Heat Source ◽  
Mixed Convection ◽  
Numerical Solutions ◽  
Saturated Porous Medium ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Continuous Increase ◽  
Unsteady Mixed Convection ◽  
Unsteady Stretching Surface

This paper deals with the analysis of an unsteady mixed convection flow of a fluid saturated porous medium adjacent to heated/cooled semi-infinite stretching vertical sheet in the presence of heat source. The unsteadiness in the flow is caused by continuous stretching of the sheet and continuous increase in the surface temperature. We present the analytical and numerical solutions of the problem. The effects of emerging parameters on field quantities are examined and discussed.

Oscillatory forcing of flow through porous media. Part 2. Unsteady flow

2002 ◽  
Vol 465 ◽  
pp. 237-260 ◽  
Author(s):  
D. R. GRAHAM ◽  
J. J. L. HIGDON
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Pressure Gradient ◽  
Flow Rate ◽  
Flow Through Porous Media ◽  
Mean Flow ◽  
Two Fluids ◽  
Mean Pressure ◽  
Flow Through ◽  
The Mean

Numerical computations are employed to study the phenomenon of oscillatory forcing of flow through porous media. The Galerkin finite element method is used to solve the time-dependent Navier–Stokes equations to determine the unsteady velocity field and the mean flow rate subject to the combined action of a mean pressure gradient and an oscillatory body force. With strong forcing in the form of sinusoidal oscillations, the mean flow rate may be reduced to 40% of its unforced steady-state value. The effectiveness of the oscillatory forcing is a strong function of the dimensionless forcing level, which is inversely proportional to the square of the fluid viscosity. For a porous medium occupied by two fluids with disparate viscosities, oscillatory forcing may be used to reduce the flow rate of the less viscous fluid, with negligible effect on the more viscous fluid. The temporal waveform of the oscillatory forcing function has a significant impact on the effectiveness of this technique. A spike/plateau waveform is found to be much more efficient than a simple sinusoidal profile. With strong forcing, the spike waveform can induce a mean axial flow in the absence of a mean pressure gradient. In the presence of a mean pressure gradient, the spike waveform may be employed to reverse the direction of flow and drive a fluid against the direction of the mean pressure gradient. Owing to the viscosity dependence of the dimensionless forcing level, this mechanism may be employed as an oscillatory filter to separate two fluids of different viscosities, driving them in opposite directions in the porous medium. Possible applications of these mechanisms in enhanced oil recovery processes are discussed.

scholarly journals Stability and nonlinear regimes of flow over a saturated porous medium

2013 ◽  
Vol 20 (4) ◽  
pp. 543-547 ◽  
Author(s):  
T. P. Lyubimova ◽  
D. T. Baydina ◽  
D. V. Lyubimov
Keyword(s):  
Porous Medium ◽  
Saturated Porous Medium ◽  
Base Flow ◽  
Nonlinear Flow ◽  
Wave Number ◽  
Neutral Stability ◽  
Steady Plane ◽  
Stability Of Water ◽  
Parameter Values ◽  
Nonlinear Regimes

Abstract. The paper deals with the investigation of stability and nonlinear regimes of flow over the saturated porous medium applied to the problem of stability of water flow over the bottom covered with vegetation. It is shown that the velocity profile of steady plane-parallel flow has two inflection points, which results in instability of this flow. The neutral stability curves, the dependencies of critical Reynolds number and the wave number of most dangerous perturbations on the ratio of porous layer thickness to the total thickness are obtained. The nonlinear flow regimes are investigated numerically by finite difference method. It is found that at supercritical parameter values waves travelling in the direction of the base flow take place.

Thermal Optimization of a Circular-Sectored Finned Tube Using a Porous Medium Approach

2002 ◽  
Vol 124 (6) ◽  
pp. 1026-1033 ◽  
Author(s):  
Sung Jin Kim ◽  
Jae Wook Yoo ◽  
Seok Pil Jang
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Fluid Flow ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Effective Conductivity ◽  
Saturated Porous Medium ◽  
Finned Tube ◽  
Uniform Heat Flux ◽  
Total Thermal Resistance

The present work investigates the heat transfer characteristics of a laminar fully developed forced convection in a circular-sectored finned tube with axially uniform heat flux and peripherally uniform wall temperature. The tubes with circular-sectored fins are modeled as a fluid-saturated porous medium. Using the Brinkman-extended Darcy model for fluid flow and the two-equation model for heat transfer, the analytical solutions for both velocity and temperature distributions are obtained and compared with the exact solution for fluid flow and the numerical solutions for conjugate heat transfer in order to validate the porous medium approach. The agreement between the solutions based on the porous medium approach and the conventional method is close within 5.3 percent. Based on the analytical solutions, parameters of engineering importance are identified to be the angle of the circular sector α and the effective conductivity ratio C, and their effects on fluid flow and heat transfer are studied. Also, the total thermal resistance is derived from the analytical solutions and minimized in order to optimize the thermal performance of a tube with circular-sectored fins.

Film Boiling Heat Transfer From a Sphere and a Horizontal Cylinder Embedded in a Liquid-Saturated Porous Medium

1988 ◽  
Vol 110 (4a) ◽  
pp. 961-967 ◽  
Author(s):  
J. Orozco ◽  
R. Stellman ◽  
M. Gutjahr
Keyword(s):  
Heat Transfer ◽  
Experimental Data ◽  
Porous Medium ◽  
Theoretical Model ◽  
Layer Thickness ◽  
Numerical Solutions ◽  
Saturated Porous Medium ◽  
Horizontal Cylinder ◽  
Film Boiling ◽  
Vapor Layer

This paper analyses both theoretically and experimentally the problem of film boiling from a body embedded in a liquid-saturated porous medium. Two body geometries are investigated thoroughly: a horizontal cylinder and a sphere. The theoretical model relies on the Brinkman-extended flow model to describe the flow field inside the thin vapor layer occupying the neighborhood near the heated surface. The theoretical model also includes an improved formulation of the effective conductivity in the vicinity of the heater as a function of the vapor layer thickness and the geometry of the porous medium material. Solutions are obtained for the vapor layer thickness and the local Nusselt number as a function of angular position. Numerical solutions are also obtained for the overall heat transfer rates from the surface to the fluid for a given vapor superheat. Experimental data for a 12.70 mm stainless steel cylindrical heater embedded in a 3-mm glass particle porous medium were obtained under steady—state operation. The experimental data obtained are compared with the theoretical analysis. The comparison shows that there is a good agreement between theory and experiments. The theoretical model is also compared with the experimental data obtained by other investigators for a spherical geometry. Excellent results are obtained in such comparison.

Wave Motions in a Fluid Saturated Porous Medium

1973 ◽  
Vol 40 (4) ◽  
pp. 873-878 ◽  
Author(s):  
L. Hsieh ◽  
C. H. Yew
Keyword(s):  
Porous Medium ◽  
Incompressible Fluid ◽  
Wave Motion ◽  
Numerical Solutions ◽  
Saturated Porous Medium ◽  
Frequency Equation ◽  
Governing Equations ◽  
Strain Relationship ◽  
Fluid Saturated Porous Medium ◽  
Rotational Waves

Wave motion in a fluid saturated porous medium is studied analytically in this paper. By assuming that the porous medium is homogeneous, isotropic, elastic, and permeable and by treating the interporous fluid as an incompressible fluid, linearized governing equations for wave motion in a fluid saturated porous medium were derived using the concept of superimposed continua, and the modified Lubinski’s stress-strain relationship. Frequency equations for dilatational and rotational waves were obtained. Numerical solutions of the frequency equation are presented and discussed.

Analysis of Fluid Flow in Composite Channels: Comparision of Analytical and Numerical Approaches

1999 ◽  
Author(s):  
A. V. Kuznetsov ◽  
Ming Xiong
Keyword(s):  
Porous Medium ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Saturated Porous Medium ◽  
Single Domain ◽  
Fluid Viscosity ◽  
Clear Fluid ◽  
Composite Channels ◽  
Model Fluid ◽  
Forced Convective Flow

Abstract In this work, comparisons between the numerical solutions for forced convective flow in composite channels, partly occupied by a homogeneous (clear) fluid and partly by a fluid saturated porous medium, and the exact solutions for the same problems are carried out. This is done to establish limitations of the single-domain approach utilized to obtain the numerical solutions. These limitations result from the assumptions which are implicitly invoked once the single-domain approach is utilized to model fluid flow in the interface region between the porous medium and a clear fluid. It is shown that the single-domain approach results in correct matching of the shear stress at the porous/fluid interface only if the adjustable coefficient in the representation for the excess stress equals zero and also if the effective viscosity of the porous medium equals to the fluid viscosity.

Sign in / Sign up

Export Citation Format

Share Document