Abstract
An analysis of double diffusive convection induced by a uniformly heated and salted horizontal wavy surface in a porous medium is presented. The wavy surface is first transformed into a smooth surface via a suitable coordinate transformation and the transformed nonsimilar coupled nonlinear parabolic equations are solved using the Keller box method. The local and average Nusselt and Sherwood numbers are given as functions of the streamwise coordinate and the effects of various physical parameters are discussed in detail. The effects of the Lewis number, buoyancy ratio, and wavy geometry on the dynamics of the flow are studied. It was found, among other observations, that the combined effect of Dufour and Soret parameters is to reduce both heat and mass transfer.
MSC: 34B15, 65N30, 76M20.
Keywords:
double diffusive convection; nonsimilar solutions; porous medium; Keller box method1 Introduction
The study of doublediffusive convection has received considerable attention during the last several decades because of its occurrence in a wide range of natural and technological settings. Doublediffusive convection is an important fluid dynamic phenomenon that involves motions driven by two different density gradients diffusing at different rates (Mojtabi and CharrierMojtabi [1]). A common example of double diffusive convection is seen in oceanography, where heat and salt concentrations exist with different gradients and diffuse at differing rates. Double diffusive convection manifests in the form of “saltfingers” (see Stern [2,3]) which are observable in laboratory settings. The input of cold freshwater from an iceberg can affect both of these variables. Doublediffusive convection has also been cited as being important in the modeling of solar ponds (Akbarzadeh and Manins [4]) and magma chambers (Huppert and Sparks [5], Fernando and Brandt [6]). Doublediffusive free convection is also seen in seawind formations, where upward convection is also modified by Coriolis forces. This is of particular interest in oceans where the Earth’s rotation plays a dominant role in many of the motions observed.
In engineering applications, double diffusive convection is commonly visualized in the formation of microstructures during the cooling of molten metals, and fluid flows around shrouded heatdissipation fins. Typical technological motivations for the study of doublediffusive convection range from such diverse fields as the migration of moisture through air contained in fibrous insulations, grain storage systems, the dispersion of contaminants through watersaturated soil, crystal growth, solidification of binary mixtures, and the underground disposal of nuclear wastes. Other important applications can be found the fields of geophysical sciences and electrochemistry. A comprehensive review of the literature concerning natural convection in fluidsaturated porous media may be found in the books by Ingham and Pop [7,8], Nield and Bejan [9], Vafai [10,11], and Vadasz [12].
Most free convection studies mainly focus on the cases where the thermal boundary
conditions allow the use of similarity transformations to reduce the governing boundary
layer equations to a system of ordinary differential equations which can be handled
either analytically or numerically. The existence of selfsimilar solutions point
to the fact that, in general, the heated surface must have a plane geometry. In reality,
the heated surfaces need not always be planar. Surfaces are sometimes deliberately
roughened to achieve enhanced heat transport. Heat transfer devices like flatplate
solar collectors and flatplate condensers in refrigerators possess a nonuniform surface.
Further, in cavity wall insulating systems and grain storage systems, one can witness
pronounced surface roughness. Yao [13] and Moulic and Yao [14,15] were the first to include the effects of surface nonuniformities on the free convection
thermal boundary layer flow of a Newtonian fluid. Rees and Pop [16,17] investigated parallel thermal boundary layers due to natural convection induced by
a vertical surface with sinusoidal undulations embedded in a in porous medium. Subsequent
studies by Rees and Pop [18,19] considered the effect of surface waves on the convection due to either horizontal
or vertical wavy surfaces embedded in a porous medium. In the case of horizontal wavy
surface, Rees and Pop [18] found that the amplitude of the surface waves must be within an
Cheng [2023], in a series of papers investigated free convection due to vertical/inclined wavy surfaces in porous media. Cheng [20] studied natural convection, heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium. He showed that the average Nusselt and Sherwood numbers for a sinusoidal wavy surface are constantly smaller than the corresponding results for a flat plate. This work was generalized in Cheng [21] to a nonDarcy model for natural convection heat and mass transfer from a vertical wavy surface in a porous media. Cheng [22] considered a nonNewtonian power law liquid to study combined heat and mass transfer in free convection flow due to a vertical wavy surface in a porous medium with thermal and mass stratifications. It was found that an increase in the powerlaw index, the thermal stratification parameter, or the concentration stratification parameter leads to a smaller fluctuation of the local Nusselt and Sherwood numbers with the streamwise coordinate. Double diffusive natural convection along an inclined wavy surface in a porous medium was investigated by Cheng [23] and showed that the inclination angle enhanced the total heat and mass transfer rates.
Pop and Na [24] studied natural convection due to a frustum of a wavy cone in a porous medium and discussed the effects of geometry of a wavy cone on the heat transfer. Cheng [25] studied the natural convection heat and mass transfer near a wavy cone with constant wall temperature and concentration in a porous medium. Using a cubic spline method of solution, he showed that the buoyancy ratio leads to higher heat and mass transports while the diffusivity ratio (Lewis number) reduces heat while increasing mass transports. Cheng [26] obtained nonsimilar solutions for double diffusive convection near a frustum of a wavy cone in porous media and showed that the average Nusselt and Sherwood numbers for the wavy cone are smaller than the corresponding smooth cone. Doublediffusive natural convection along a vertical truncated wavy cone in nonNewtonian fluid saturated porous media with thermal and mass stratification has been investigated by Cheng [27]. In this study, he showed that the powerlaw index leads to smaller fluctuations in the local Nusselt and Sherwood numbers and increasing the thermal and concentration stratification parameters reduce the heat and mass transfer rates.
An energy flux is often generated by both temperature and solute gradients leading to Dufour or diffusionthermo effect and Soret or thermaldiffusion effects. Both effects have been extensively studied in gases, while the Soret effect has been studied both theoretically and experimentally in liquids; see Mortimer and Eyring [28]. It is generally accepted that the Dufour and the Soret effects are small compared to the effects described by Fourier and Fick’s laws (Mojtabi and CharrierMojtabi [1]) and can therefore be neglected in many heat and masstransfer processes. However, it has been shown in a number of studies that there are exceptions in areas such as in geosciences where Dufour and Soret effects are significant and cannot be ignored; see for instance Kafoussias and Williams [29], Awad et al.[30], and the references therein.
Mortimer and Eyring [28] used an elementary transition state approach to obtain a simple model for Soret and Dufour effects in thermodynamically ideal mixtures of substances with molecules of nearly equal size. In their model, the flow of heat in the Dufour effect was identified as the transport of the enthalpy change of activation as molecules diffuse. The results were found to fit the Onsager reciprocal relationship, Onsager [31]. Shariful et al.[32] investigated the Dufour and Soret effects on steady combined freeforced convective and mass transfer flow past a semiinfinite vertical flat plate of hydrogenair mixtures. They used the fourthorder RungeKutta method to solve the governing equations of motion. Their study showed that the Dufour and Soret effects should not be neglected. Shateyi et al.[33] investigated the effects of diffusionthermo and thermaldiffusion on MHD fluid flow over a permeable vertical plate in the presence of radiation and hall current. Recently, Narayana et al.[34] and Malashetty and Biradar [35] have investigated the cross diffusion effects on the onset of double diffusive convection in a binary Maxwell fluid saturated porous medium. Other related recent studies on the effects of Soret and Dufour parameters include those by Makinde [3638].
In this paper, we extend the work by Rees and Pop [18] to include the solute diffusion in natural convection induced by a horizontal wavy surface profile in a porous medium. The wall temperature and concentrations are assumed to be constants and the amplitude of the wave is assumed to be small enough so as not to engender any regions of reverse flow (these issues are dealt in detail by Rees and Pop [18]). Assuming the Rayleigh number Ra to be very large, the boundary layer approximation is invoked leading to a set of nonsimilar parabolic partial differential equations whose solution is obtained using the Keller box method (see, Keller [39], Keller and Cebeci [40]).
2 Mathematical formulation
Consider the problem of doublediffusive convection in a fluid around a horizontal surface with transverse sinusoidal undulations embedded in a porous medium. Figure 1 shows the schematic sketch of the problem. The wavy surface profile is described by
where
Figure 1. Schematic diagram of the physical problem.
The temperature and the solute concentrations at the surface are assumed to be constants
subject to boundary conditions
Here, g is the acceleration due to gravity, K is permeability of the porous medium, ν is the kinematic viscosity, β is the coefficient of thermal expansion,
The governing equations (2)(5) take the form:
The dimensionless parameters appearing in the above set of equations are the Rayleigh
number Ra, the buoyancy ratio N, Dufour number
The boundary conditions in (6) take the following nondimensional form:
Introducing the stream function
the continuity equation (8) is satisfied automatically and Eqs. (9)(11) can be written in the following form:
with the boundary conditions
The effect of the wavy surface and the usual boundary layer scalings are incorporated into the governing equations (15)(17) using the transformations,
along with the substitutions
Equations (19) transform the wavy surface to a smooth surface in the physical configuration.
Substituting (19) and (20) into Eqs. (15)(18) and letting
subject to the boundary conditions
The associated local Nusselt and Sherwood numbers
The mean Nusselt and Sherwood numbers from the leading edge to the free stream position x are given by
3 Solution procedure
The nonsimilar parabolic partial differential equations that govern the dynamics of the fluid flow were solved using the Keller box implicit finite difference method. This section gives details of the Keller box scheme used to solve the coupled nonlinear parabolic equations (21)(23). We present the implementation of the Keller box method for a problem that involves coupling, nonlinearity, variable coefficients. The method consists of four main steps; decomposition, discretization, linearization, and the solution of the linearization difference equation.
The decomposition of Eqs. (21)(23) into firstorder equations is achieved by setting
where
In the discretization and formation of finite difference equations, the computational domain (ξη plane) is divided into a finite number of mesh points that form the corners of the rectangle shown in Figure 2(a). The net points are defined as
where
Figure 2. Computational domain.
A finer grid that points toward the surface (
Using the boxscheme approximations (35), Eqs. (27)(32) yield the following nonlinear finite difference equations:
Further manipulation yields the following equations:
where
The boundary conditions take the form
The difference equations (42)(47) are imposed at each grid points
We use Newton’s method to linearize the nonlinear finite difference equations (42)(47)
which possesses quadratic nonlinearity. Assuming that the nth iterates of the variables are known the iterate (
We further define the following variables for brevity.
Substituting Eqs. (49)(50) into Eqs. (42)(47) and retaining only linear terms in δ, the linearized finite difference equations can be written as:
where
From the boundary conditions, we get the following difference equations:
The linearized difference system of Eqs. (51)(57) can be solved by any standard method like elimination or iterative methods. Since the linearized system has a block tridiagonal structure, we employ a blockelimination method. The block tridiagonal system in its matrix form is written as follows:
where
We first factorize the coefficient matrix as follows:
where I is an identity matrix of order 6. The procedure involves two sweeps, i.e., a forward sweep and a backward sweep. In the forward sweep, the unknowns
Forward sweep
Initial matrices:
Using the above, the successive matrices are obtained using the following formulae:
Backward sweep
The required solution of the linearized system can be written as
After obtaining the solution of the linearized system, the corrections are added
to the solutions at the nth iteration and the procedure is repeated until convergence is achieved. This gives
the solution at a given
4 Results and discussion
The problem of double diffusive convection induced by a horizontal wavy surface in
a porous medium in presence of cross diffusion effect has been analyzed. The problem
is governed by a system of coupled nonlinear parabolic partial differential equations
(21)(24). The Keller box method was used to find the numerical solutions of the problem.
The amplitude of the wavy surface was assumed to be small and the wave phase was considered
to be zero. The computations were carried out for the following range of parameter
values;
Figure 3. Effect of cross diffusion on velocity, temperature and concentration profiles for
different values of steam wise locationξwith
Figure 3 shows the cross diffusion effects on the velocity, temperature, and concentration
profiles for different values ξ. Due to the wavy nature of the surface, the dynamics of the fluid flow at different
stream wise locations will be different. We notice the existence of locations where
the velocity of the fluid attain extreme values over every undulation of the wavy
surface. The profiles are shown for two selected stream wise locations, namely,
The variation of the slip velocity
Figure 4. Variation of slip velocity
The heat transfer results are shown in Figures 5(a) and (b) for different values of the wave amplitude α. The local Nusselt number
Figure 5. Variation of local and mean Nusselt numbers withξfor different values of wave amplitudeαwith
Figures 5(c) and (d) highlight the mass transfer results for different values of α. Analogous to the heat transfer case, the local Sherwood number
Figure 6 shows the effect of the Lewis number Le and the buoyancy ratio N on the slip velocity. The Lewis number reduces the slip velocity while the buoyancy ratio has an enhancing effect as expected. Increasing aiding buoyancy generates faster moving fluid flow as is evident in Figure 6(b).
Figure 6. Variation of slip velocity
Figure 7 shows the effect of buoyancy ratio N on the local and mean Nusselt and Sherwood numbers. It is evident that the buoyancy
ratio enhances both the local and mean heat and mass transfer coefficients. The buoyancy
ratio enhances the local and the mean Nusselt numbers. As observed earlier,
Figure 7. Variation of local and mean heat and mass transfer rates withξfor different values of buoyancy numberNwith
Figure 8 shows the effect of the Lewis number Le on the local and mean Nusselt and Sherwood numbers. The Lewis number tends to reduce
the local and average heat transfer rates. Further,
Figure 8. Variation of local and mean heat and mass transfer rates withξfor different values of Lewis numberLewith
Figure 9. Cross diffusion effect on the slip velocity
The cross diffusion effect on the local and mean heat and mass transfer are depicted in Figure 10. These figures suggest that the presence of cross diffusion effects reduces both heat and mass transfer rates. The cross diffusion is more pronounced in case of heat transfer while the variation is infinitesimal in case of mass transfer. Figures 11(a) and (b) show the effect of the Dufour and Soret numbers on the slip velocity. Both parameters tend to accelerate the fluid in the boundary layer thereby increasing the slip velocity.
Figure 10. Cross diffusion effect on local and mean heat and mass transfer rates for varyingξwith
Figure 11. Variation of slip velocity
The effects of Dufour number on
Figure 12. Variation of local and mean Nusselt and Sherwood numbers withξfor different values of Dufour number
The effects of the Soret number on
Figure 13. Variation of local and mean Nusselt and Sherwood numbers withξfor different values of Soret numberSrwith
5 Conclusions
The paper analyzed double diffusive convection induced by a horizontal wavy surface embedded in a porous medium taking the cross diffusion effect into account. The governing parabolic partial differential equations have been solved using the Keller box method. The convergence rate depends on the parametric values chosen. Computations show that the Lewis number enhances mass transfer while reducing heat transfer. The buoyancy ratio enhances both the heat and mass transfer rates. The surface geometry also plays a crucial role in controlling heat and mass transfer rates. The effect of the Dufour number is to reduce heat transfer and enhance mass transfer. The effect of the Soret number is the exact opposite of the Dufour effect. We observe reduced heat and mass transfer in the presence of cross diffusion terms.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MN carried out the numerical computations and drafted the manuscript. PS, SSM, and PGS participated in the design of the study and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the National Research Foundation (NRF) and the University of KwaZuluNatal for financial support.
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