Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate in a Non-Darcy Porous Medium with Convective Boundary Condition

1 Introduction

In reality, most of the fluids are non-Newtonian, which means that their viscosity is dependent on the shear rate (shear thinning or shear thickening). In contrast to Newtonian fluids, non-Newtonian fluids display a nonlinear relation between shear stress and shear rate. But, the model of a micropolar fluid developed by Eringen [1] exhibits some microscopic effects arising from the local structure and micro motion of the fluid elements. It provides the basis for the mathematical model of non-Newtonian fluids which can be used to analyze the behavior of exotic lubricants, polymers, liquid crystals, animal bloods, colloidal or suspension solutions, etc. The detailed review of theory and applications of micropolar fluids can be found in the books by Lukaszewicz [2] and Eremeyev et al. [3]. In view of an increasing importance of mixed convection in many transport processes, the topic of mixed convection in boundary layer flow along vertical, inclined, and horizontal flat plates has been extensively analyzed and some experimental/theoretical studies [4–7] on these flow geometries in micropolar fluid have been reported.

Transport of heat through a porous medium has been the subject of many studies due to the increasing need for a better understanding of the associated transport processes. Most of the earlier studies [8-10] in porous media are based on the Darcy model which assumes proportionality between the velocity and pressure gradient. The model, however, is valid only for slow flows through porous media with low permeability, but at higher flow rates or in highly porous media the inertial effects become significant. For example, in petroleum reservoirs the high flow can occur in various scenarios such as near the well bore (perforations), hydraulically fractured wells, condensates reservoirs (low viscosity crude reservoirs), high flow potential wells and gravel packs. Forchheimer [11] proposed a quadratic extension to the Darcian model in order to more precisely simulate the inertial effect in porous media. A detailed review of convective heat transfer characteristics of different fluids in the porous medium as well as non-Darcy porous medium can be found in the book by Nield and Bejan [12] and also see the citations therein. Later, several authors to mention few [13, 14] examined the heat and mass transfer characteristics of micropolar and viscous fluid flows along a vertical plate in the non-Darcy porous medium.

The effect of buoyancy forces by taking the linear variations in temperature and density have been investigated by several researchers. But the relationship between temperature and density may become nonlinear due to different physical characteristics like temperature variation, inertia, radiation, or presence of different densities and heat released by viscous dissipation. Particularly, the flow field and heat transfer characteristics are more influenced by nonlinear density and temperature variations in the buoyancy force term (for details, see Barrow and Rao [15], Vajravelu and Sastri [16]) when the temperature difference between the surface of the plate and the ambient fluid becomes significantly large. Therefore, the concept of nonlinear convection in a fluid medium is of great importance in a variety of disciplines such as astrophysics, geophysics, geothermal and engineering applications. Moreover, the engineering applications are applied in design of thermal system, cooling transpiration, combustion, cooling of electric components, areas of reactor safety, drying of the surfaces, turbine blades, solar collectors, etc. In such special cases, the temperature-dependent relation is nonlinear and has unavoidable importance. Partha [17] studied the effect of nonlinear convection in a non-Darcy porous medium and concluded that, with the increase of nonlinear temperature-concentration, the heat and mass transfer are more in the Darcy porous medium as compared with the non-Darcy porous medium. The effects of nonlinear convection and thermophoresis in a non-Darcy porous medium are discussed by Kameswaran et al. [18], and concluded that the temperature and concentration boundary layer thickness decreases with increasing values of nonlinear temperature and concentration parameters. Recently, Sachin Shaw et al. [19] investigated the effect of nonlinear thermal convection in nanofluid flow over a stretching sheet.

Heat transfer analysis with the convective boundary condition attracted the interest of many researchers, since this condition is more realistic and general representation in engineering and industrial processes such as transpiration cooling process, material drying, etc. Aziz [20] obtained a similarity solution for the Blasius flow of a viscous fluid under convective boundary condition. Yacob and Ishak [21] investigated stagnation point flow towards a stretching/shrinking sheet immersed in a micropolar fluid with a convective surface boundary condition. In recent times, the influence of homogeneous-heterogeneous reactions on convective heat flow of a micropolar fluid along a vertical plate in porous medium under convective surface boundary is discussed by Ramreddy et al. [22] (also refer the references given therein).

The objective of the article is to analyze the effects of nonlinear variation of density with temperature and concentration on mixed convection in a micropolar fluid saturated non-Darcy porous medium over an inclined plate by considering convective boundary condition. The governing nonlinear system of partial differential equations is transformed to set of ordinary nonlinear differential equations by local non-similarity procedure and then the successive linearization method is used to solve the resulting boundary value problem. Typical results for the velocity, temperature, microrotation and concentration distributions are presented for various governing parameters. Also, the local skin-friction, wall couple stress, as well as the heat and mass transfer rates are illustrated for representative values of the major parameters.

2 Mathematical Formulation

Consider the steady laminar mixed convective flow of an incompressible micropolar fluid along a semi-infinite inclined flat plate in a non-Darcy porous medium, with an acute angle Ω to the vertical, as depicted in Fig. 1. The coordinate system is such that x measures the distance along the plate and y measures the distance normally into the fluid. The velocity of the outer flow is of the form u_∞, the free stream temperature and concentration are T_∞ and C_∞, respectively. The plate is either heated or cooled from a flow field of temperature T_f to the left by convection with T_f > T_∞ relating to a heated surface (assisting flow) and T_f < T_∞ relating to a cooled surface (opposing flow), respectively. On the wall the solutal concentration is taken to be constant and is given by C_w. The porous medium is taken to be uniform with a constant permeability and porosity, and is saturated with a fluid which is in local thermodynamic equilibrium with the solid matrix. In addition, a Forchheimer model is considered. Further, the temperature difference between the surface of the plate and the ambient fluid assumed to be significantly larger, so that the nonlinear density and temperature variations in the buoyancy force term exert a strong influence on the flow field. Further, we follow the work of many recent authors by assuming that γ=μ+κ2j[23, 24].

Fig. 1

Physical model and coordinate system.

By employing nonlinear Boussinesq approximation (Ref. [15, 16]) and making use of the standard boundary layer approximations, the governing equations for the micropolar fluid are given by

where u and v are the Darcy velocity components in x and y directions respectively, ω is the component of micro rotation whose direction of rotation lies in the xy-plane, T is the temperature, C is the concentration, g* is the acceleration due to gravity, ρ is the density, µ is the dynamic coefficient of viscosity, b is the empirical constant, ε is the porosity, K_p is the permeability, κ is the vortex viscosity, j is the micro-inertia density, γ is the spin-gradient viscosity, α is the thermal diffusivity and D is the solutal diffusivity of the medium, Ω is inclination of angle. Here β₀ and β₁ are the coefficients of thermal expansion of the first and second orders, respectively whereas β₂ and β₃ are the coefficients of solutal expansion of the first and the second orders, respectively.

The boundary conditions are

where, the subscripts w and ∞ indicate the conditions at the wall and at the outer edge of the boundary layer, respectively, h_f is the convective heat transfer coefficient and k is the thermal conductivity of the fluid.

Introducing the following dimensionless variables:

where u_∞ is free stream velocity and Re = (u_∞L)/v is the global Reynold’s number.

In view of the continuity equation (1), we introduce the stream function ψ by

Using (8) and (9) into (2)–(5), we get the following momentum, angular momentum, energy and concentration equations

where the primes indicate partial differentiation with respect to η alone. In usual definitions, ∆ = κ/µ is the micropolar or material parameter [23, 24], Gr = [g*β₀(T_f -T_∞)L³]/ν² is the thermal Grashof number, v is the kinematic viscosity, Ri = Gr /Re² is the mixed convection parameter, 𝓑 = [β₂(C_w - C_∞)]/[β₀(T_f - T_∞)] is the Buoyancy ratio, Da = Kp/L² is the Darcy number, Fs = b/L is the Forchheimer number, 𝓙 = L²/(j Re) is the micro-inertia density, Pr = ν/α is the Prandtl number, Ω is angle of inclination and Sc = ν/D is the Schmidt number. Here α₁ = [ β₁(T_f - T_∞)]/β₀ and α₂ = [β₃(C_w - C_∞)]/β₂ are the nonlinear temperature and concentration parameters, respectively.

The boundary conditions (6) become

where Bi = [h_fL/(k Re^1/2)] is the Biot number. It is a ratio of the internal thermal resistance of the plate to the boundary layer thermal resistance of the hot fluid at the bottom of the surface.

3 Skin friction, Wall couple stress, Heat and Mass transfer coefficients

The wall shear stress and the wall couple stress are:

And the heat and mass transfers from the plate, respectively are given by

The non-dimensional skin friction Cf=2τwρu∞2, wall couple stress Mw=mwρu∞2x,the local Nusselt number Nux=qwxk(Tf−T∞) and local Sherwood number Shx=qmxD(Cw−C∞),are given by

where Rex=u∞xνis the local Reynold’s number.

4 Solution of the Problem

We now obtain approximate solutions to Eqs. (10)–(13) along with the boundary conditions (14) in two steps: (i) First, we use, the local non-similarity procedure (of Sparrow and Yu [25], and Minkowycz and Sparrow [26]) to convert the set of partial differential equations (10)–(13) along with the boundary conditions (14) into set of ordinary differential equations, (ii) Next, the resulting boundary value problem is solved using a Successive Linearisation Method (SLM).

According to local similarity and non-similarity procedure (see [25, 26] for more details), the system of partial differential equations considered here is first converted to a system of ordinary nonlinear differential equations by introducing new unknown functions of ξ derivatives.

In the first level of truncation, the terms accompanied by ξ∂∂ξ are assumed to be very small. This is particularly true when ξ ≪ 1. We neglect terms containing the ξ derivatives in Eq. (10)-(14). Thus we get local similarity equations are

The corresponding boundary conditions are

For the second level of truncation, we introduce U=∂f∂ξ,V=∂g∂ξ,H=∂θ∂ξ,K=∂ϕ∂ξ to recover the neglected terms at the first level of truncation. Thus the governing equations at the second level reduces to

The corresponding boundary conditions are

At the third level of truncation, we differentiate Eqns. (24)-(28) with respect to ξ and neglect the terms ∂U∂ξ,∂V∂ξ,∂H∂ξ,∂K∂ξ to get the following system of equations.

The corresponding boundary conditions are

The coupled nonlinear differential equations (24) - (27) and (29)–(32) together with the boundary conditions (28), (33) and (34) are solved using one of the non-perturbation methods named as Successive Linearization Method (see [27–29]), it utilizes first the successive linearization and then the Chebyshev spectral collocation scheme.

Using successive linearization, the nonlinear boundary layer equations will reduce to a system of linear differential equations. For this, let us take the following Q(η) = [f (η), g(η), θ(η) , ϕ(η),U(η), V(η), H(η), K(η)] and assume that the independent vector Q(η) can be expressed as

where Q_i(η), (i = 1, 2,3....), are unknown vectors and Q_n(η) are the approximations which is obtained by recursively solving the linear part of the equation system that results from substituting (35) in (24)–(34).

The initial guess Q₀(η) is chosen such that they satisfy the boundary conditions (28), (33) and (34). The subsequent solutions Q_i(η), i ≥ 1 are obtained by successively solving the linearised form of the equations which are obtained by substituting Eq. (35) in the governing equations and neglecting the nonlinear terms. The linearised equations to be solved are

The boundary conditions reduce to

Here the coefficient parameters a_t,i-1, b_{t, i-1}, c_t,_i-1, d_{t, i-1}, e_{t, i-1}, o_t,i-1, p_{t, i-1}, q_{t, i-1} and r_t,i-1 depend on the initial guesses Q₀(η) and on their derivatives.

Once each solution for Q_i(η), i ≥ 1 has been obtained, the approximate solutions for Q(η) are then obtained as Q(η)≈∑m=0MQm(η) where M is the order of SLM approximation.

The linearized equations (36-44) are solved using the Chebyshev spectral collocation method (Canuto et al. [30]). The unknown functions are approximated by the Chebyshev interpolating polynomials in such a way that they are collocated at the Gauss-Lobatto points defined as

where N is the number of collocation points used. The physical region [0, ∞) is transformed into the region [-1,1] using the domain truncation technique in which the problem is solved on the interval [0, L] instead of [0, ∞). This leads to the mapping

and the function Q(η) is approximated at the collocation points by

where L is a scaling parameter used to invoke the boundary condition at infinity, T_m is the m^th Chebyshev polynomial defined by

The derivatives of the variables at the collocation points are represented as

where Z is the order of differentiation and D = (2/L)𝓓 with 𝓓 being the Chebyshev spectral differentiation matrix. Substituting Eqs.(45)-(49) into linearized form of equations leads to the matrix equation.

In Eq. (50), A_i-1 is a (8N + 8)×(8N + 8) square matrix and X_i and R_i-1are (8N + 1)×1 column vectors defined by

After modifying the matrix system (50) to incorporate boundary conditions, the solution is obtained as

5 Results and Discussion

In order to assess the validity and accuracy of the present analysis, the results of the principle of local similarity for Eqs. (10)–(13) have been compared with the special case of Lloyd and Sparrow [31] in the absence of micropolar parameter ∆, buoyancy ratio 𝓑, nonlinear convection parameters α₁ and α₂ with ε = 1, Da →∞, ξ = 1, Bi →, Ω = 0 as exhibited in Table 1. It shows an excellent agreement with existing results.

In the present study, we have adopted the following default parameter values for the numerical computations: 𝓑 = 1.0, Re = 200, Da = 0.1, ε = 0.5, Pr = 0.71, Sc = 0.22, Ri = 2. The value of dimensionless micro-inertia density 𝓙 = 1.0 is chosen so as to satisfy the thermodynamic restrictions on the material parameters given by Eringen [1]. These values are used throughout the computations, unless otherwise indicated.

The dimensionless velocity, microrotation, temperature and concentration profiles have been computed for different values of the fluid parameters and presented graphically in Figs. 2–9. The effects of nonlinear temperature α₁, nonlinear concentration α₂, non-Darcy parameter Fs, micropolar parameter ∆, mixed convection parameter Ri, Biot number Bi and angle of inclination Ω have been discussed.

The effect of nonlinear temperature α₁on the velocity, microrotation, temperature and the concentration profiles are shown in Figs. 2(a)–2(d). The results indicate that the velocity distribution increases with an increasing value of α₁ and the value 𝓑 = 1 implies that the thermal and concentration buoyancy forces are of the same order of magnitude. Physically, α₁> 0 implies that T_f > T_∞; hence, there will be a supply of heat to the flow region from the wall. Similarly α₁ < 0 implies that T_f < T_∞, and in such a case there will be a transfer of heat from the fluid to the wall. Again, this increase in velocity with positive values of α₁ is more prominent in the presence of the mixed convection parameter. The temperature and concentration boundary layer thicknesses decrease with the rise of α₁.

Fig. 2

Effect of nonlinear temperature parameter on the (a) velocity, (b) microrotation, (c) temperature and (d) concentration for ∆ = 1.0, Fs = 0.5, α₂ = 5, Ω = π/6, Bi = 0.5, ξ = 0.5.

Figures 3(a)–3(d) depict the influence of the nonlinear concentration parameter α₂ for a fixed value of α₁ = 5 on the behaviour of velocity, microrotation, temperature and concentration. The initial velocity is zero at the surface of the plate and rises gradually away from the plate to the free stream satisfying the boundary conditions as given in Fig. 3(a). However, the rise of nonlinear concentration changes the sign of microrotation as shown in Fig. 3(b) from negative to positive within the boundary layer. In the absence, as well as presence of nonlinear concentration parameter α₂ the magnitude of the temperature and concentration decreases with an increase of α₂ which is presented in Figs. 3(c)–3(d). We also note that the impact of α₂ on the temperature and concentration distributions is more elegant, as compared with that of α₁.

Fig. 3

Effect of nonlinear concentration parameters on the (a) velocity, (b) microrotation, (c) temperature and (d) concentration for∆ = 1.0, Fs = 0.5, α₁ = 5, Ω = π/6, Bi = 0.5, ξ = 0.5.

Figures 4(a)-4(d) display the velocity, microrotation component, temperature and concentration distribution of fluid flow for different values of the Biot number. It is noteworthy from Figs. 4(a)–4(b) that as the Biot number increases the velocity profile increase and the microrotation changes direction from decreasing to increasing values within the boundary layer. Fig. 4(c) demonstrates the effect of the Biot number on the temperature profile and serves a dual result (i.e., for convective boundary condition and isothermal condition). Since the specified convective boundary condition is changing into wall condition, when the Biot number is tends to infinity and it is proven by Fig. 4(c). As Bi increases from thermally thin case (Bi < 0.1) to thermally thick case (Bi > 0.1) the temperature distribution is increased at the plate. The effect of the Biot number on the concentration profile is displayed in Fig. 4(d) and it depicts that the concentration profile reduces within the boundary layer when the Biot number increases from least to large value.

Fig. 4

Effect of Biot number on the (a) velocity, (b) microrotation, (c) temperature and (d) concentration for ∆ = 1.0, Fs = 0.5, α₁= 5, α₂ = 5, Ω = π/6, ξ = 0.5.

Figures 5(a)–5(d) illustrate the variation of the velocity distribution, microrotation, temperature and concentration for different values of the inclination of angle (0° ≤ Ω ≤ 90°). Moreover, the non-similarity equations for the limiting cases of the vertical and horizontal plates are recovered from the transformed equations by setting Ω = 0° and Ω = 90°, respectively. The influence of inclination of the angle on the velocity profile is displayed in Fig. 5(a). Due to the reduction in the thermal buoyancy effect in Eq. (2) caused by an increase in angle Ω, reduces the velocity distribution within the momentum boundary layer as shown in Fig. 5(a). In other words, an increase in the angle of inclination leads to reduce the velocity distribution within the boundary layer region. Also, we can observe from Fig. 5(a) that the maximum buoyancy force for the temperature and concentration difference occurs for Ω = 0 (vertical plate). When the position of the flat plate is changed from vertical to horizontal, we observe that the microrotation is increasing near the plate and far away from the plate it is showing a reverse trend within the boundary layer, as shown in Fig. 5(b). It is noticed from Fig. 5(c) and Fig. 5(d) that the temperature and concentration enhances with increasing values of inclination of angle. In particular, when the surface is vertical the smallest temperature and concentration distributions are observed, whereas they become largest for the horizontal surface.

Fig. 5

Effect of inclination of angle on the (a) velocity, (b) micro-rotation, (c) temperature and (d) concentration for ∆ = 1.0, Fs = 0.5, α₁ = 5, α₂ = 5, Bi = 0.5, ξ = 0.5.

Figures 6(a)–6(b) shows the effects of the nonlinear temperature and concentration on the non-dimensional local heat and mass transfer rates against stream wise coordinate. We observe that both heat and mass transfer rates are increasing with α₁ when α₂ is fixed. The effect of α₂ on Nusselt and Sherwood numbers is showing the same behaviour with that of α₁. The effect of varying the Forchheimer number Fs and Micropolar parameter ∆ on non-dimensional local heat and mass transfer rates are presented in Figs. 7(a)-7(b). The results indicate that as Fs increases, the local heat and mass transfer rates decrease for a fixed value of micropolar parameter. Hence, the inertial effect in micropolar fluid saturated non-Darcy porous medium reduces the heat and mass transfer coefficients. Also, it can be observed from this figure that, for a fixed value of Fs, the heat and mass transfer coefficients are reducing with the increasing values of micropolar parameter and it is obvious that the rate of heat and mass transfers in the micropolar fluid (∆ ≠ 0) are lower compared to that of the Newtonian fluid (∆ = 0). Note that similar observation has been pointed out by Srinivasacharya and RamReddy [13], and RamReddy and Pradeepa [22] in the case of linear variation of density with temperature and concentration.

Fig. 6

The effect of Nonlinear temperature and concentration parameters on (a) local Nusselt number, (b) local Sherwood number for ∆ = 0.5, Fs = 0.5, Ω = π/6, Bi = 0.5.

Fig. 7

The effect of Forchheimer and Coupling numbers on (a) local Nusselt number, (b) local Sherwood number for α₁ = 5, α₂ = 5, Ω = π/e, Bi = 0.5.

The effect of Biot number Bi and the variation of mixed convection parameter Ri on the local heat and mass transfer rates are depicted in Figs. 8(a)-8(b). It is found that the local heat and mass transfer rates are increasing when the flow direction is changed from opposing to aiding, and with the enhancement of the Biot number there is a considerable increment in heat transfer rate. The local mass transfer rate is decreasing in opposing flow, whereas; it is increased in the case of assisting flow with the rise of Biot number. The influence of the angle of inclination on heat and mass transfer rates as a function of dimensionless stream wise coordinate are shown in Figs. 9(a)-9(b). The results are displayed that the local Nusselt number and Sherwood number reduce gradually when the plate is rotated from vertical to horizontal. One can notice that the effect of angle of inclination is more on mass transfer rate as compared with that of heat transfer rate.

Fig. 8

The effect of Biot number and Mixed convection paramete on (a) local Nusselt number, (b) local Sherwood number for ∆ = 0.5, Fs = 0.5, α₁ = 5, α₂ = 5, Ω = π/6.

Fig. 9

The influence of angle of inclination on (a) heat transfer rate, (b) mass transfer rate for a fixed values of ∆ = 0.5, Fs = 0.5, α₁= 5, α₂= 5, Bi = 0.5..

The proportional quantities of skin friction and the gradient of microrotation (wall couple stress) are computed for the enhanced values of ∆, Fs, Ri, α₁, α₂, Ω, Bi and the results are presented in Table 2. It is observed that an enhancement in the mixed convection parameter causes an increase in the skin friction and decrease in the wall couple stress, whereas, with the increasing of angle of inclination they show the opposite trend. Rising in the micropolar parameter tends to enhance the skin friction and the reverse change is noticed in wall couple stress. The rate of skin friction is enhanced by nonlinear temperature and concentration, but there is a small decrement in the wall couple stress. The nominal effect on the wall couple stress and considerable increment in skin friction is encountered for high enough values of Biot number. Furthermore, the skin friction parameter increases and the wall couple stress decreases as Forchheimer number increases.

6 Conclusions

The problem of nonlinear convection flow of a micropolar fluid along an inclined plate in a non-Darcy porous medium under convective boundary condition has been investigated. Using local non-similarity technique, the governing partial differential equations have been transformed into a system of coupled nonlinear ordinary differential equations, which are solved numerically by using the Chebyshev spectral collocation method together with successive linearization named as a successive linearization method. The effects of various parameters on the velocity, microrotation, temperature, concentration, heat and mass transfer, skin friction and wall couple stress have been analyzed and some important results are given. It is observed that the tangential velocity increases with increasing values of the nonlinear temperature and concentration. The temperature and concentration boundary layer thickness decreases with increasing values of a₁ and α₂. The local heat and mass transfer rates reduce with the angle of inclination Ω increases. The numerical results indicate that the micropolar parameter diminishes the wall couple stress and enhances skin friction. As the Biot number increases, there is a gradual increment in skin friction and nominal effect on wall couple stress.