A New Exact Solution for the Flow of a Fluid through Porous Media for a Variety of Boundary Conditions

Abstract

The viscous fluid flow past a semi-infinite porous solid, which is proportionally sheared at one boundary with the possibility of the fluid slipping according to Navier’s slip or second order slip, is considered here. Such an assumption takes into consideration several of the boundary conditions used in the literature, and is a generalization of them. Upon introducing a similarity transformation, the governing equations for the problem under consideration reduces to a system of nonlinear partial differential equations. Interestingly, we were able to obtain an exact analytical solution for the velocity, though the equation is nonlinear. The flow through the porous solid is assumed to obey the Brinkman equation, and is considered relevant to several applications.

1. Introduction

The flow of a fluid through a porous medium has numerous applications in industries dealing with polymer extrusion process, glass blowing, metallurgical processes, and geophysical and allied areas (see [1]). A variety of equations have been used to describe the flow of a fluid through a porous medium as it is one of the important key factors in maintaining the temperature in the medium. These equations due to [2,3,4,5] and others, are merely approximations to the appropriate balance laws. A variety of ideas have been suggested to model the flow of mixtures, and one such approach is that which follows from the seminal works of Darcy and Brinkman and has been given a formal structure by [6,7]. Several specific problems have been solved using such an approach (see [8,9,10,11,12,13,14,15,16,17,18,19]). Here, we study the flow of a fluid through a porous media that is governed by the Brinkman equation (see [20,21,22,23,24,25,26,27,28,29,30]) for a discussion of the status of the Brinkman equation within the context of mixture theory). The fact that we are able to obtain an analytical solution to the problem makes the study all the more interesting. Despite the fact that advanced computing facilities are available to obtain the numerical solution, investigators around the world are much more interested in providing the analytical solution due to their accuracy, relevance, and convenient to analyze physical process, in comparison to numerical solutions. The analytical solution can provide a better assessment of consistency and parameter estimates. Many authors have investigated the fluid flow through porous media and provided analytical solution (see [31,32,33,34,35,36,37]). The novelty of this study is our use of a variety of boundary conditions that subsumes those that have been considered earlier, in addition to new conditions concerning slip and proportional shearing at the boundary.

In this study, we consider the flow of a fluid through a semi-infinite porous media with one boundary subject to the slipping or adherence of the fluid, the solid being proportionally sheared, and the fluid being injected at the boundary (see Figure 1). We are able to obtain an analytical solution by introducing a similarity variable that greatly simplifies the governing equation. The effects of the boundary conditions on the flow through the porous media are determined.

2. Theoretical Model

Two dimensional laminar, steady, incompressible fluid flow through a porous media is considered. The x-axis is taken along the stretching of the sheet in the direction of the motion, and y-axis is perpendicular to the slit. In order to confine the fluid flow in the region y > 0, two forces of equal strength are applied along the x-axis. u and v denote the axial as well as transverse velocities in the flow field. Figure 1 depicts the physical flow problem subjected to the boundary conditions.

We considered the flow of the classical incompressible Navier-Stokes [38] fluid through a porous half-space. We assumed that the equations governing the flow are those given by the Brinkman equation for flow through porous media which assumes that the fluid is incompressible and, hence, the conservation of mass reduces to

$$\nabla \cdot \overrightarrow{q}=0,$$

(1)

and the conservation of linear momentum that takes the form

$$\rho \left[\frac{1}{\varphi }{\overrightarrow{q}}_t+\frac{1}{\varphi }\left(\overrightarrow{q}\cdot \nabla \right)\overrightarrow{q}\right]=-\nabla p+{\mu }_{eff}{\nabla }^2\overrightarrow{q}-\frac{\mu }{K}\overrightarrow{q},$$

(2)

where ${\mu }_{eff}$ represents the effective viscosity of the fluid(see [39,40]) provides the definition of the other parameters. The Brinkman equation can be shown to be obtained as a systematic approximation using mixture theory by assuming special structures for the interaction forces between the porous solid and fluid, and assuming the porous solid is rigid (see [41] for a detailed derivation).The transformed governing equations for the conservation of mass and the balance of linear momentum are given as

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(3)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{{\varphi }^2}{{\rho }_f}\frac{\partial p}{\partial x}+{\varphi }^2{\nu }_{eff}\frac{{\partial }^{2}u}{\partial y^2}-\frac{\nu {\varphi }^2}{K}u,$$

(4)

where $\nu =\frac{\mu }{{\rho }_f}$ and ${\nu }_{eff}=\frac{{\mu }_{eff}}{{\rho }_f}$. The Forchheimer term in the interaction is neglected as it produces little impact on the fluid flow in a porous medium governed by the Brinkman equation (see [42]). Also, the pressure gradient is neglected, and the time factor is zero for the steady case.

The governing boundary conditions are (see [43,44])

$$\begin{array}{}\left(5\mathrm{a}\right)& u\left(x,y\right)=d\alpha x+A\frac{\partial u}{\partial y}+B\frac{{\partial }^{2}u}{\partial y^2},& & \mathrm{v}={\mathrm{v}}_c,& & \mathrm{at}& & y=0,\left(5\mathrm{b}\right)& u\left(x,y\right)\to 0,& & & & \mathrm{as}& & y\to \infty .\end{array}$$

Here, d is the parameter of proportional shearing at the boundary, with $d\ne 0$ and d = 0 corresponding to the boundary at y = 0 and being either proportionally sheared or being fixed. The constants A and B represent the first- and second-order slip coefficients, respectively. Also, the mass transpiration parameter, ${\mathrm{v}}_c$, represents suction or injection depending on ${\mathrm{v}}_c>0$ or ${\mathrm{v}}_c<0$, respectively.

In order to carry out the analysis, the physical stream functions in terms of similarity variables $f$ and $\eta$ are introduced as follows:

$$\psi =\sqrt{\alpha {\nu }_{eff}}xf(\eta ),$$

(6)

where

$$\eta =\frac{1}{\varphi }\sqrt{\frac{\alpha }{{\nu }_{eff}}}y.$$

(7)

In terms of physical stream function $\psi$, the axial and transverse velocities can be rewritten as follows:

$$u=\frac{\partial \psi }{\partial y},\mathrm{v}=-\frac{\partial \psi }{\partial x}.$$

(8)

The Equation (8) satisfies the continuity equation. Upon substitution of Equations (6) and (7) into Equation (4), we obtain

$$\Lambda \frac{{\partial }^3\psi }{\partial y^3}+\frac{\partial (\psi ,\frac{\partial \psi }{\partial y})}{\partial (x,y)}-K_1\frac{\partial \psi }{\partial y}=0.$$

(9)

Here, the second term indicates the Jacobian and subject. The appropriate boundary conditions are (see [43,44]) as follows:

$$\frac{\partial \psi }{\partial y}=d\alpha x+A\frac{\partial u}{\partial y}+B\frac{{\partial }^{2}u}{\partial y^2},\frac{\partial \psi }{\partial x}=v_c\mathrm{at}\mathrm{y}=0,$$

(10a)

$$\frac{\partial \psi }{\partial y}=0,\mathrm{as}\mathrm{y}\to \infty .$$

(10b)

Here, $\Lambda =\frac{{\mu }_{eff}}{\mu }$ is the Brinkman number or viscosity ratio. Using Equations (9) and (10) with Equation (6), the following transformed equation with constant coefficient is derived:

$$\Lambda f_{\eta \eta \eta }+f{f}_{\eta \eta }-f_{\eta }^2-K_1{f}_{\eta }=0.$$

(11)

The governing boundary conditions for Equation (11) are given as

$$f(0)=V_c,f_{\eta }(0)=d+{\Gamma }_1{f}_{\eta \eta }\left(0\right)+{\Gamma }_2{f}_{\eta \eta \eta }\left(0\right),\mathrm{at}\eta =0,$$

(12)

$$f_{\eta }(\infty )\to 0\mathrm{as}\eta \to \infty ,$$

(13)

where ${\Gamma }_1=A\sqrt{\frac{\alpha }{\nu }}>0$ and ${\Gamma }_2=B\frac{\alpha }{\nu }<0$ are the first- and second-order slip parameters, and $K_1=\frac{\nu {\varphi }^2}{\alpha K}$ is the reciprocal of Darcy number $Da=\frac{l^2}{K}$, with $l=\varphi \sqrt{\frac{\nu }{\alpha }}$ and $V_c=\frac{{\mathrm{v}}_c}{\sqrt{\alpha \rho }}$ as the mass suction/injection parameters. The subscript denotes the derivative with respect to $\eta$.

3. The Analytical Solution

The flow problem considered is the generalization of the classical works of [43,44,45,46,47,48]. In our problem, the viscous flow with first- and second-order velocity slips over a porous half-space that is stretched or contracted at the boundary (see Figure 1), and the flow being governed by the Darcy–Brinkman model is considered. One obtains nonlinear partial differentiation from Equations (3) and (4), which are mapped into systems of the nonlinear ordinary differential Equation (11), with a constant coefficient by means of similarity transformation subjected to the imposed boundary (12)–(13).The analytical solution for the velocity distribution is determined.

The exact analytical solution of Equation (11) subjected to the governing boundary conditions Equations (12) and (13) is derived. The condition in Equation (13) suggests choosing the equation of the form

$$f\left(\eta \right)=A_1+B_1\exp (-\beta \eta ),$$

(14)

where $\beta >0$ is to be determined later. Also, $A_1$ and $B_1$ are constants that are to be determined by using Equation (12):

$$A_1=V_c+d\left(\frac{1}{\beta +{\Gamma }_1{\beta }^2-{\Gamma }_2{\beta }^3}\right)\mathrm{and}{B}_1=-d\left(\frac{1}{\beta +{\Gamma }_1{\beta }^2-{\Gamma }_2{\beta }^3}\right).$$

(15)

It follows from Equation (11), (14), and (15) that

$$\Lambda {\Gamma }_2{\beta }^4-\left(\Lambda {\Gamma }_1+V_c{\Gamma }_2\right){\beta }^3+\left(V_c{\Gamma }_1-\Lambda -K_1{\Gamma }_2\right){\beta }^2+\left(V_c+K_1{\Gamma }_1\right)\beta -\left(d+K_1\right)=0.$$

(16)

Here, $\beta >0$ is one of the real roots (see [49,50]).

By using the transformation variable $\xi =\beta +\frac{a_3}{4}$, Equation (16) transforms into

$${\xi }^4+p{\xi }^2+q\xi +r=0,$$

(17)

where $p=a_2-\frac{3}{8}{a}_3^2$, $q=\left(A_1-\frac{1}{2}{A}_2{A}_3+\frac{1}{8}{A}_3^3\right)$, and $r=a_0-\frac{1}{4}{a}_1{a}_3+\frac{1}{16}{a}_2{a}_3^2-\frac{3}{256}{a}_3^4$, and $a_3=-\frac{\Lambda {\Gamma }_1+V_c{\Gamma }_2}{\Lambda {\Gamma }_2}$, $a_2=\frac{V_c{\Gamma }_1-\Lambda -K_1{\Gamma }_2}{\Lambda {\Gamma }_2}$, $a_1=\frac{V_c+K_1{\Gamma }_1}{\Lambda {\Gamma }_2}$, $a_0=\frac{d+K_1}{\Lambda {\Gamma }_2}$.

The four corresponding roots of the algebraic Equation (17) are

$${\beta }_1=\frac{\sqrt{C}}{2}+\frac{1}{2}\sqrt{D_1}-\frac{a_3}{4},$$

(18a)

$${\beta }_2=\frac{\sqrt{C}}{2}-\frac{1}{2}\sqrt{D_1}-\frac{a_3}{4},$$

(18b)

$${\beta }_3=-\frac{\sqrt{C}}{2}+\frac{1}{2}\sqrt{D_1}-\frac{a_3}{4},$$

(18c)

$${\beta }_4=-\frac{\sqrt{C}}{2}-\frac{1}{2}\sqrt{D_1}-\frac{a_3}{4},$$

(18d)

where

$$D_1=D-\frac{2q}{C},$$

$$\begin{array}{l}C=-\frac{2p}{3}+2^{1/3}\left(p^2+12r\right){\left[3{\left(2p^3+27q^2-72pr+\sqrt{-4{\left(p^2+12r\right)}^3+{\left(2p^3+27q^2-72pr\right)}^2}\right)}^{1/3}\right]}^{-1}\\ +{\left(2^{1/3}3\right)}^{-1}{\left(2p^3+27q^2-72pr+\sqrt{-4{\left(p^2+12r\right)}^3+{\left(2p^3+27q^2-72pr\right)}^2}\right)}^{1/3}\end{array}$$

and

$$\begin{array}{l}D=-\frac{4p}{3}-2^{1/3}\left(p^2+12r\right){\left[3{\left(2p^3+27q^2-72pr-\sqrt{-4{\left(p^2+12r\right)}^3+{\left(2p^3+27q^2-72pr\right)}^2}\right)}^{1/3}\right]}^{-1}\\ -{\left(2^{1/3}3\right)}^{-1}{\left(2p^3+27q^2-72pr+\sqrt{-4{\left(p^2+12r\right)}^3+{\left(2p^3+27q^2-72pr\right)}^2}\right)}^{1/3}\end{array}$$

Equation (18) gives the complete solution of Equation (17). However, it should be noted that there is only one feasible solution for the Equation (17) when ${\Gamma }_2<0$, and based on the flow field Equation (16) has feasible solutions for $\beta >0$.

4. Results and Discussion

In this paper, we were able to establish the impact of various physical parameters on the velocity distribution. The solutions obtained are in good agreement with that of the classical works when suitably restricted to specific conditions. The main emphasis of this study is the effect of boundary conditions on the flow through porous media. There are several possibilities at the boundaries, namely, the fluid meeting the no-slip adherence condition, the Navier slip condition, the second-order slip condition, as well as the possibility of blowing of the fluid.

The effects of physical parameters such as the mass transpiration parameter ($V_C$), first-order Navier slip (${\Gamma }_1$), second-order slip (${\Gamma }_2$),Brinkman ratio ($\Lambda$), and proportional shearing parameter ($K_1$)are discussed graphically. As the velocity distribution is an exponential function with a negative argument, it decreases with the increase in $\eta$. Since $\beta$ is a function of mass transpiration parameter ($V_C$), first-order Navier slip (${\Gamma }_1$), second-order slip (${\Gamma }_2$), Brinkman ratio ($\Lambda$), and proportional shearing parameter ($K_1$)both axial as well as transverse velocities are forced to decrease exponentially.

The solution domain of $\beta$ in Equation (16)has only complex roots when $D_1<0$, only real roots when $D_1>0$,and real repeated roots when $D_1=0$. Figure 2a–d depicts the solution behavior of $\beta$ verses $V_C$ for various values of $D_1$ and ${\Gamma }_1$. Figure 3a–d depicts the solution domain of $\beta$ versus $V_C$ for different values of $K_1$. In fact, by choosing ${\Gamma }_2=0$, Equation (16) reduces to a cubic equation and with the proper choice of ${\Gamma }_1$, $\Lambda$, and $K_1$, and the results are reduced to those obtained by [43,51,52].

Figure 4a–c demonstrates the impact of first-order velocity slip with proportional shearing on the solution domain of β versus V_c. The presence of a larger slip drags the separation curve towards the slit. The viscous fluid flow in a permeable medium with slip in a stretching boundary is quite different from that of a contracting boundary.

Figure 5a,b depicts the axial velocity profiles for various values of first-order slip parameters, and for the fixed values of other physical parameters. This plot clearly demonstrates that the increasing Navier’s slip results in the reduction of the velocity boundary. In comparison to the lower branch solution, the boundary layer thickness decreases in the case of upper branch solution. Furthermore, under the given slip parameter and mass suction parameter, one can see the increasing velocity boundary thickness with the increase in proportional shearing parameter. Also, the reduction in mass suction leads to the decrease in velocity boundary for other physical parameters fixed. However, for the case of mass injection, the velocity boundary increases with increasing values of slip parameter. Thus, the flow geometry and the rate of change of velocity boundary layer thickness are significantly influenced by the slip parameter.

Figure 6a–c demonstrates the effect of first- and second-order slip parameters on the axial, as well as transverse velocity profiles respectively, in the accelerating boundary. From the plots, it is clear that the increasing values of first-order slip for fixed values of various physical parameters results in the velocity boundary profiles decreasing, whereas the decreasing value of second-order slip for fixed values of other physical parameters results in the decreasing velocity boundary profiles.

Figure 7a,b demonstrates the effect of Brinkman ratio on the axial and transverse velocity profiles. From the plots, it can be seen that increasing values of Λ while keeping other physical parameters fixed results in enhanced boundary layer thickness, whereas in Figure 7c,d, the different values of V_C with fixed Brinkman ratio result in exactly the opposite.

Figure 8a,b depicts the effect of various physical parameters on the shear stress profile. In all of these plots, there are crossover points for the shear stress profiles, and the combined effects on the porous solid can be observed. The increase in the values of ${\Gamma }_1$, ${\Gamma }_2$, and K₁ results in increasing shear at the wall boundary. In the case of mass injection, the shear wall boundary decreases faster for a smaller value of second-order slip parameter. Interestingly, however, the increasing value of Brinkman ratio leads to decreasing shear wall boundary as seen in Figure 8a.

5. Concluding Remarks

In conclusion, the viscous fluid flow past a porous solid wherein the flow is governed by the Brinkman equation with first- and second-order slip in the presence of mass transpiration was solved, and the exact analytical solution for the governing nonlinear partial differential equation was obtained. The solution was analyzed for the effect of slip parameters, the mass transpiration parameter, the Brinkman ratio, and the extent of shearing or contraction. In the case of the boundary contracting, the solution branches (Figure 5a,b), whereas in the case of the boundary stretching there is only one branch of the solution, and depending on the mass transpiration parameter, the solution is branched (Figure 5a,b).