ARTICLE

J. Cent. South Univ. (2019) 26: 1271-1282

DOI: https://doi.org/10.1007/s11771-019-4086-7

Variable mass and thermal properties in three-dimensional viscous flow: Application of Darcy law

Iffat JABEEN^{1, 2}, Muhammad FAROOQ¹, Nazir A. MIR¹

1. Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan;

2. Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: This article concentrates on the properties of three-dimensional magneto-hydrodynamic flow of a viscous fluid saturated with Darcy porous medium deformed by a nonlinear variable thickened surface. Analysis of flow is disclosed in the neighborhood of stagnation point. Features of heat transport are characterized with Newtonian heating and variable thermal conductivity. Mass transport is carried out with first order chemical reaction and variable mass diffusivity. Resulting governing equations are transformed by implementation of appropriate transformations. Analytical convergent series solutions are computed via homotopic technique. Physical aspects of numerous parameters are discussed through graphical data. Drag force coefficient, Sherwood and Nusselt numbers are illustrated through graphs corresponding to various pertinent parameters. Graphical discussion reveals that conjugate and constructive chemical reaction parameters enhance the temperature and concentration distributions, respectively.

Key words: Newtonian heating; chemical reaction; variable mass diffusivity; Darcy law

Cite this article as: Iffat JABEEN, Muhammad FAROOQ, Nazir A. MIR. Variable mass and thermal properties in three-dimensional viscous flow: Application of Darcy law [J]. Journal of Central South University, 2019, 26(5): 1271–1282. DOI: https://doi.org/10.1007/s11771-019-4086-7.

1 Introduction

Magneto-dydrodynamic (MHD) flow has great importance and application in many fields of science and engineering, including plasma’s magnetic behavior in fusion reactors, petroleum industries, MHD generators, electric motors, jet printers, electromagnetic casting, crystal growth, design cooling systems, blood flow measurements, pumps, electromagnetic casting, ship propulsion, and flow meters etc. In view of these applications MUHAMMAD et al [1] have accounted ohmic heating and viscous dissipation effects to analyze three-dimensional MHD flow. KHAN et al [2] have explained homogeneous-heterogeneous reactions to analyze flow of Carreau fluid by implementing magnetic field. HAYAT et al [3] have utilized cylindrical coordinated to analyze endoscopic reactions in MHD Ree-Eyring fluid. HUSSAIN et al [4] presented physical features of convective heat transfer in MHD tangent hyperbolic fluid with viscous dissipation. HAYAT et al [5] considered combined effects of MHD and Soret and Dufour in the fluid flow deformed by stretchable surface. YOUSOFV et al [6] have discussed the 3D MHD nanofluid flow with mixed convection. GHASHEMI et al [7] also investigated MHD nanofluid flow in a porous medium with natural convection. NAN et al [8] studied effect of convection on magnetohydrodynamic fluid flow filled in a cavity.

The study of flow through porous medium has great importance in many fields because of its applications in different branches of engineering, e.g., biomedical engineering and polymer industries such as wire manufacturing, glass and fiber. The basic law which governs the fluid flow through Porous medium is Darcy’s law. Porous medium flow of fluid is an important subject. KHALID et al [9] have investigated behavior of wall couple stress in micropolar fluid in the presence of porous medium. HAYAT et al [10] tackled features of Darcian regime in the flow through curved channel. HUSSAIN et al [11] has applied peristaltic theory in pertaining flow through porous medium. CHEN et al [12] used Galerkin method to solve two-dimensional natural convective flows with porous medium. KHAN et al [13] examined the dual behavior of solutions for the fluid flow embedded in porous medium. MEGAHED [14] has discussed viscoelastic non- Newtonian fluid flow in a porous medium over a stretching sheet with heat generation and velocity slip. ASIAEI et al [15] addressed the entropy generation effects in mixed convection nanofluid with porous channel. EMAMI et al [16] described the Cu-water nanofluid flow through inclined porous channel with convective effects. SIAVASHI et al [17] illustrated the convective flow of nanofluid saturated in porous medium considering two-phase model. SIAVASHI et al [18] discussed the heat transfer phenomena through a porous medium in a flow of nanofluid.

Inherent property of material thermal conductivity has key role in the optimization and redesigning of cooling/heating schedules, i.e., in the design and optimization for defining safe heating and cooling schedules. Thermal conductivity plays an important role in numerous research areas of electronics and building insulation and other related fields. High thermal conductivity materials such as aluminum, silver and copper are used in electronics and turbines while materials with low thermal conductance, e.g., polystyrene and alumina, are used in building construction and furnaces. HAYAT et al [19] described the features of variable fluid features in chemically reactive and stratified fluid flow with Cattaneo-Christov theory. NAWAZ et al [20] exposed the variation in stagnant Casson fluid flow by implementing variable thermal conductivity. BILAL et al [21] demonstrated three-dimensional Williamson fluid flows by considering heat absorption/generation and thermal conductivity. HAYAT et al [22] studied variable thermal conductivity with the effect of Cattaneo-Christov heat flux. WAQAS et al [23] used Cattaneo- Christov model to analyze stagnation point and temperature-dependent conductivity in the Powell-Eyring fluid. MAJID et al [24] described the impact of forced convection on flow of nanofluid in the presence of Al₂O₃ nanoparticles with entropy generation. SARI et al [25] discussed heat transfer phenomena in a nanofluid flow between nonparallel walls. MAGHSOUDI et al [26] disclosed features of mixed convection in nanofluid flow filled in porous cavity. SIAVASHI et al [27] disclosed the heat transfer features in flow over multi-layer porous medium.

In this paper we have discussed the features of stagnation point in three-dimensional flow by nonlinear stretching sheet having variable thickness. In literature, authors discussed the stagnation point in two-dimensional flow, and three-dimensional flow. But not a single paper is available to explore the features of fluids like three-dimensional flow, stagnation point, variable sheet thickness and Newtonian heating.

Thus our main objective is to fulfill this void in this attempt. Further, it should be noted that homotopy analysis method is not valid only for single but also it works more than one independent variable. For example, we can use it for partial differential equations. First order chemical reaction is accounted to explore the mass transfer process. Variable fluid properties are implemented to study the heat and mass transport processes. Series solutions are computed through homotopic technique [28–34]. Graphical exhibition of temperature, fluid velocity and concentration distribution are demonstrated and analyzed through physical parameters. Drag force, Sherwood and Nusselt numbers are also disclosed graphically.

2 Problem statement

Assume a steady, incompressible and three- dimensional viscous fluid flow deformed by impermeable nonlinear stretchable surface having variable thickness in the region of stagnation point. Applied magnetic field of variable magnitude is directed along the z-direction in the electrically conducting field. Magnetic field of small intensity is applied in order to neglect the polarization and induced magnetic field. Thermal conductivity and mass diffusivity are assumed to vary linearly with temperature and concentration respectively. Darcy law for porous medium is implemented to compute flow analysis. Newtonian heating and mass diffusivity are considered at the surface by ignoring internal thermal and mass resistance of the material. First order chemical reaction is incorporated to analyze the features of mass transportation. Using the velocity field (V=[u(x, y, z), v(x, y, z), w(x, y, z)], see Figure 1), temperature field (T=T(x, y, z)) and concentration field (c=c(x, y, z)) the governing equations take the form:

Figure 1 Flow geometry

                           (1)

(2)

    (3)

  (4)

     (5)

Implementing boundary layer approximation (i.e., u=o(1), x=o(1), v=o(1), y=o(1), w=o(δ), w= o(δ)),

                          (7)

               (8)

               (9)

      (10)

   (11)

with the boundary conditions:

at                      (12)

                (13)

here u, v and w denote velocity components along the x, y and z directions, respectively; u_∞ represents ambient velocity along x-axis; v_∞ represents ambient velocity along y-axis; u₀ is the reference velocity; u_w(x) and v_w(x) are stretching velocities along x-axis and y-axis, respectively; v is the fluid kinematic viscosity; represents variable heat transfer coefficient; represents variable magnetic field; is the variable thermal conductivity; ρ is the density; represents variable permeability of porous medium; T is the temperature of fluid; T_∞ is the ambient temperature; is the variable chemical reaction coefficient and represents variable mass transfer coefficient.

Considering (14)

                       (14)

               (15)

continuity Eq. (1) is automatically satisfied while Eqs. (2)–(5) become

  (16)

            (17)

    (18)

  (19)

The corresponding boundary conditions:

           (20)

           (21)

             (22)

here prime denotes differentiation with respect to η and  represents the plate surface. Again by implementing F(η)= f(η–α^*)=f(ξ), G(η)=g(η–α^*)=g(ξ), Θ(η)=θ(η–α^*)=θ(ξ), and f(η)=f(η–α^*)=φ(ξ) we have the alternating forms of the governing equations as follows:

             (23)

             (24)

       (25)

 (26)

with boundary conditions:

               (27)

In the above equations, P represents porosity parameter; Pr represents Prandtl number; M denotes magnetic parameter; β₁, β₂ and β₃ represent ratio velocity parameters; n represents power index; ε represents variable thermal conductivity parameter; ε₁ represents variable mass diffusivity parameter; γ represents chemical reaction parameter. These quantities of interest are expressed as:

         (28)

in which

Skin friction coefficient and local Nusselt and Sherwood number are

                        (29)

where the shear stress τ_w,and the heat flux q_wand the mass flux j_m are given by

                  (30)

In dimensionless variables one can write

                      (31)

where is the local Reynolds number.

3 Homotopic solutions

Homotopic technique was first initiated by LIAO in 1992 [16]. This method is used to construct the solutions for especially non-linear problems. It facilitates to adopt the initial guesses and linear operators for the composition of series solutions. In the present situation, the initial guesses and auxiliary linear operators satisfy the following definitions:

                     (32)

         (33)

        (34)

            (35)

The operators have the properties given below:

           (36)

           (37)

               (38)

               (39)

where A_i (i=1–10) are the arbitrary constants.

3.1 Zeroth-order problems

                (40)

                 (41)

           (42)

         (43)

,

         (44)

Defining non-linear operators as

           (45)

          (46)

        (47)

 (48)

where is embedding parameter and andare the non-zero auxiliary parameters.

3.2 m-th-order problems

          (49)

          (50)

           (51)

          (52)

        (53)

                       (54)

                 (55)

                (56)

                (57)

   (58)

           (59)

              (60)

                         (61)

Obviously for q=0, and q=1, one may write

              (62)

              (63)

              (64)

              (65)

and with variation of q from 0 to 1, f(ξ, q), g(ξ, q), θ(ξ, q) and φ(ξ, q) vary from the initial solutions f₀(ξ), g₀(ξ), θ₀(ξ) and φ₀(ξ), to the final solutions f(ξ), g(ξ), θ(ξ) and φ(ξ), respectively. Using Taylor series for q=1, we have

                    (66)

                    (67)

                    (68)

                    (69)

The general solutions (f_m, g_m, θ_m and φ_m) of Eqs. (45)–(48) in terms of special solutions (f^*_m, g^*_m, θ^*_m and φ^*_m) are

            (70)

            (71)

                (72)

               (73)

3.3 Convergence analysis

Homotopic analysis method (HAM) facilitates us to adjust and confine the convergence region of the homotopic series solutions. Region of convergence region is that part of range which is parallel to Therefore, we have sketched the -curves in Figure 2. The allowable ranges of the auxiliary parametersandare and

Figure 2 -curve for f″(ξ), g″(ξ), θ′(ξ), f′(ξ)

4 Discussion

The object of this passage is to illustrate the behavior of distinct parameters on the fluid velocity, temperature field and fluid concentration. Figure 3 shows the variation of β₁ on velocity distribution. It is discovered that velocity profile increases for higher values of velocity ratio parameter β₁. In fact, it is due to the reason that by increasing ratio parameter β₁, fluid velocity enhances for β₁>1, and β₁<1. In both cases velocity distribution enhances either due to domination of stretching velocity or free stream velocity. It is identified that boundary layer thickness has opposite behavior for both cases. There is no boundary layer for β₁=1, because fluid and plate move with the same behavior. Figure 4 illustrates the variation of porosity parameter P on the horizontal velocity profile. It is found that the velocity field is a reducing function of porosity parameter P. Physically, increment in porosity P permeability decays which is liable for reduction in velocity distribution. Figure 5 depicts the effect of n on f′(ξ). It is demonstrated that higher value of n results in reduction of velocity distribution.Figure 6 depicts that velocity ratio parameter β₃ decreases along y-axis then velocity profile f′(ξ) decreases. Figure 7 shows that velocity ratio parameter along y-axis increases then horizontal velocity decreases. Analysis of velocity distribution corresponding to magnetic parameter is depicted in Figure 8. Velocity components demonstrate decreasing behavior for larger magnetic parameter M. Higher values of magnetic parameter M correspond to more intensity of Lorentz force which produces for fluid motion. Thus, velocity field decays. Figure 9 describes the effect of velocity ratio parameter β₃ on g′(ξ). It is illustrated that vertical velocity profile reduces due to reduced stretchable surface that stretchable surface offers less movement to a fluid hence velocity profile reduces. Figure 10 illustrates the variation of velocity ratio parameter β₂ on the vertical velocity component. It is analyzed that velocity profile shows reverse behavior near the plate and then increasing behavior away from the plate. The maximum velocity occurs at the surface of the plate if free stream velocity is less than the stretching velocity. Further, the maximum velocity can be located away from the surface if free stream velocity is higher than the stretching velocity. Figure 11 illustrates the performance of magnetic parameter M on velocity field g′(ξ) in y-direction. Velocity distribution shows reverse behavior near the surface of the plate then grows up for higher magnetic parameter M. Physically, it justifies that higher magnetic field produces higher intensity of Lorentz force which is resistive force but in the case of higher stream velocity than the stretching velocity Lorentz force assists the flow; therefore, velocity profile enhances. Features of Prandtl number Pr are illustrated in Figure 12 corresponding to temperature distribution. Higher Prandtl number Pr number is responsible for the lower temperature distribution. Physically, enlarged Prandtl number Pr is related to lower thermal conductivity which is responsible for the decrement in fluid temperature distribution. It is also demonstrated that the temperature distribution is higher at the surface of the plate for lower values of Prandtl number Pr and thickness of thermal boundary layer decays. Attributes of chemical reaction γ on temperature distribution are illustrated in Figure 13. Higher chemical reaction chemical reaction γ is responsible for the higher temperature distribution. Physically, higher chemical reaction chemical reaction γ corresponds to higher heat transfer rate which is responsible for the increment in temperature field. It is also demonstrated that the temperature distribution is higher at the surface of the plate for larger chemical reaction chemical reaction γ, and thermal boundary layer thickness also increases. Variable thermal conductivity parameter ε is shown in Figure 14. It is checked that the impact of variable thermal conductivity parameter enhances temperature profile. By increasing variable thermal conductivity parameter ε, more heat transfers to the fluid, so temperature increases. Figure 15 depicts that characteristics of concentration distribution enhances with the increment of variable mass diffusivity parameter ε₁. It is because larger mass diffusivity results in more mass transfer. Hence fluid concentration distribution increases. Figure 16 shows that with the increase in chemical reaction γ, concentration profile φ(ξ) also increases because chemical reaction depends on heating factor coefficient. So concentration profile increases with the increase of constructive chemical reaction. Figure 17 discusses the behavior of magnetic parameter M and porosity P on drag force (skin friction). It is noticed that drag force decays with increase in M and P. Influence of Prandtl number Pr and variable thermal conductivity parameter ε on Nusselt number is exposed in Figure 18. Clearly, Nusselt number increases for higher Pr and decreases for larger ε. Figure 19 discloses variation of n and mass diffusivity parameter β₁ on Sherwood number. It is found that Sherwood number increases with higher variation in n and decays for β₁.Figure 20 discusses the behavior of magnetic parameter M and porosity parameter P on skin friction coefficient. It is noticed that skin friction coefficient increases with increase in M and P along y-axis. Figures 21 and 22 show the contour behaviors of n and γ corresponding to velocity and temperature distributions respectively.

Figure 3 Effect of β₁ on f ′

Figure 4 Effect of P on f ′

Figure 5 Response of n on f ′

Figure 6 Response of β₃ on f ′

Figure 7 Response of β₁ on g′

Figure 8 Response of M on f ′

Figure 9 Effect of β₃ on g′

Figure 10 Effect of β₂ on g′

Figure 11 Response of M on θ(ξ)

Figure 12 Response of Pr on θ(ξ)

Figure 13 Response of γ on θ(ξ)

Figure 14 Response of ε on θ(ξ)

Figure 15 Response of ε₁ on f(ξ)

Figure 16 Response of γ on f(ξ)

Figure 17 Effect of P and M on c_f

Figure 18 Effect of Pr and ε on Nu

Figure 19 Effect of n and β₁ on Sh

Figure 20 Effect of P and M on c_g

Figure 21 Contour plot for n

Figure 22 Contour plot for γ

Table 1 represents the comparison of f″(0) and g″(0), with the work of KHAN et al [35]. It is concluded that both the results are in good agreement.

5 Conclusions

We have explored the characteristics of Newtonian heating with variable mass diffusivity in viscous flow. Flow is three-dimensional magnetohydrodynamic over a nonlinear stretching sheet. Chemical reaction and mass diffusivity played part for mass transport in the presence of stagnation point while adopted medium is porous. The following observation are worth mentioning.

Table 1 Numerical values of f″(0) and g″(0) for different values of n and β₁ in limiting case

1) Velocity profile demonstrates opposite behavior for higher values of porosity parameter P and velocity ratio parameter β₁.

2) Strength of variable thermal conductivity parameter ε results in the enhancement of temperature profile.

3) Temperature profile shows opposite behavior for increasing values of thermal radiation parameter R and Prandtl number Pr and temperature profile increases for increasing the Biot number γ.

4) Concentration distribution is dominant for higher solutal Biot number.

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(Edited by YANG Hua)

中文导读

三维黏性流体流动的传质和传热特性：Darcy定律的应用

摘要：应用非线性增厚表面的Darcy多孔介质变形，对饱和三维黏性磁流体的动力学特性进行了研究，对驻点附近的流动进行了分析。应用牛顿加热和可变导热系数表征了热传输特性。并采用一阶化学反应和变质量扩散率进行质量输运。通过适当的变换，将得到的控制方程进行转换(实现控制方程的转换)。用同伦技术计算了解析收敛级数解。根据图形数据讨论了参数的物理性质。用图示对阻力系数、Sherwood数和Nusselt数等相关参数进行说明。结果表明，共轭和构造的化学反应参数增强了温度和浓度的分布。

关键词：牛顿加热；化学反应；变质量扩散率；Darcy定律

Received date: 2018-08-02; Accepted date: 2019-01-26

Corresponding author: Iffat JABEEN, PhD; E-mail: iffatjabeen@yahoo.com; ORCID: 0000-0003-3278-3460