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

Nonlinear Engineering 2017; 6(2): 139–151

Ch. RamReddy*, P. Naveen, and D. Srinivasacharya Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate in a Non-Darcy Porous Medium with Convective Boundary Condition

DOI 10.1515/nleng-2016-0073 rate (shear thinning or shear thickening). In contrast to Received August 6, 2016; accepted February 19, 2017. Newtonian fluids, non-Newtonian fluids display anon- linear relation between shear stress and shear rate. But, Abstract: The objective of the present study is to investi- the model of a micropolar fluid developed by Eringen [1] gate the effect of nonlinear variation of density with tem- exhibits some microscopic effects arising from the lo- perature and concentration on the mixed convective flow cal structure and micro motion of the fluid elements. It of a micropolar fluid over an inclined flat plate in anon- provides the basis for the mathematical model of non- Darcy porous medium in the presence of the convective Newtonian fluids which can be used to analyze the behav- boundary condition. In order to analyze all the essential ior of exotic lubricants, polymers, liquid crystals, animal features, the governing non-dimensional partial differen- bloods, colloidal or suspension solutions, etc. The detailed tial equations are transformed into a system of ordinary review of theory and applications of micropolar fluids can differential equations using a local non-similarity proce- be found in the books by Lukaszewicz [2] and Eremeyev et dure and then the resulting boundary value problem is al. [3]. In view of an increasing importance of mixed con- solved using a successive linearisation method (SLM). By vection in many transport processes, the topic of mixed insisting the comparison between vertical, horizontal and convection in boundary layer flow along vertical, inclined, inclined plates, the physical quantities of the flow and and horizontal flat plates has been extensively analyzed its characteristics are exhibited graphically and quanti- and some experimental/theoretical studies [4–7] on these tatively with various parameters. An increase in the mi- flow geometries in micropolar fluid have been reported. cropolar parameter and non-Darcy parameter tend to in- Transport of heat through a porous medium has been crease the skin friction and the reverse change is observed the subject of many studies due to the increasing need in wall couple stress, mass and heat transfer rates. The in- for a better understanding of the associated transport pro- fluence of the nonlinear concentration parameter is more cesses. Most of the earlier studies [8-10] in porous me- prominent on all the physical characteristics of the present dia are based on the Darcy model which assumes pro- model, compared with that of nonlinear temperature pa- portionality between the velocity and pressure gradient. rameter. The model, however, is valid only for slow flows through Keywords: Micropolar Fluid, Non-Darcy Porous Medium, porous media with low permeability, but at higher flow Nonlinear Convection, Convective Boundary Condition, rates or in highly porous media the inertial effects become Successive Linearization Method significant. For example, in petroleum reservoirs the high flow can occur in various scenarios such as near the well bore (perforations), hydraulically fractured wells, conden- sates reservoirs (low viscosity crude reservoirs), high flow 1 Introduction potential wells and gravel packs. Forchheimer [11] pro- posed a quadratic extension to the Darcian model in order In reality, most of the fluids are non-Newtonian, which to more precisely simulate the inertial effect in porous me- means that their viscosity is dependent on the shear dia. 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. *Corresponding Author: Ch. RamReddy: Department of Mathe- Later, several authors to mention few [13, 14] examined the matics, National Institute of Technology Warangal-506004, India, E-mail: [email protected]; [email protected] heat and mass transfer characteristics of micropolar and P. Naveen, D. Srinivasacharya: Department of Mathematics, National Institute of Technology Warangal-506004, India 140 | Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate viscous fluid flows along a vertical plate in the non-Darcy boundary is discussed by Ramreddy et al. [22] (also refer porous medium. the references given therein). The effect of buoyancy forces by taking the linear vari- The objective of the article is to analyze the effects of ations in temperature and density have been investigated nonlinear variation of density with temperature and con- by several researchers. But the relationship between tem- centration on mixed convection in a micropolar fluid sat- perature and density may become nonlinear due to differ- urated non-Darcy porous medium over an inclined plate ent physical characteristics like temperature variation, in- by considering convective boundary condition. The gov- ertia, radiation, or presence of different densities and heat erning nonlinear system of partial differential equations is released by viscous dissipation. Particularly, the flow field transformed to set of ordinary nonlinear differential equa- and heat transfer characteristics are more influenced by tions by local non-similarity procedure and then the suc- nonlinear density and temperature variations in the buoy- cessive linearization method is used to solve the result- ancy force term (for details, see Barrow and Rao [15], Va- ing boundary value problem. Typical results for the veloc- jravelu and Sastri [16]) when the temperature difference ity, temperature, microrotation and concentration distri- between the surface of the plate and the ambient fluid be- butions are presented for various governing parameters. comes significantly large. Therefore, the concept of nonlin- Also, the local skin-friction, wall couple stress, as well as ear convection in a fluid medium is of great importance in the heat and mass transfer rates are illustrated for repre- a variety of disciplines such as astrophysics, geophysics, sentative values of the major parameters. geothermal and engineering applications. Moreover, the engineering applications are applied in design of thermal system, cooling transpiration, combustion, cooling of 2 Mathematical Formulation electric components, areas of reactor safety, drying of the surfaces, turbine blades, solar collectors, etc. In such spe- Consider the steady laminar mixed convective flow of an cial cases, the temperature-dependent relation is nonlin- incompressible micropolar fluid along a semi-infinite in- ear and has unavoidable importance. Partha [17] studied clined flat plate in a non-Darcy porous medium, withan the effect of nonlinear convection in a non-Darcy porous acute angle Ω to the vertical, as depicted in Fig. 1. The medium and concluded that, with the increase of nonlin- coordinate system is such that x measures the distance ear temperature-concentration, the heat and mass trans- along the plate and y measures the distance normally into fer are more in the Darcy porous medium as compared the fluid. The velocity of the outer flow is of theform u , with the non-Darcy porous medium. The effects of nonlin- ∞ the free stream temperature and concentration are T and ear convection and thermophoresis in a non-Darcy porous ∞ C , respectively. The plate is either heated or cooled from a medium are discussed by Kameswaran et al. [18], and con- ∞ flow field of temperature T to the left by convection with cluded that the temperature and concentration boundary f T > T relating to a heated surface (assisting flow) and layer thickness decreases with increasing values of nonlin- f ∞ T < T relating to a cooled surface (opposing flow), re- ear temperature and concentration parameters. Recently, f ∞ spectively. On the wall the solutal concentration is taken Sachin Shaw et al. [19] investigated the effect of nonlin- to be constant and is given by C . The porous medium ear thermal convection in nanofluid flow over a stretching w is taken to be uniform with a constant permeability and sheet. porosity, and is saturated with a fluid which is in local ther- Heat transfer analysis with the convective boundary modynamic equilibrium with the solid matrix. In addition, condition attracted the interest of many researchers, since a Forchheimer model is considered. Further, the temper- this condition is more realistic and general representation ature difference between the surface of the plate andthe in engineering and industrial processes such as transpi- ambient fluid assumed to be significantly larger, so that ration cooling process, material drying, etc. Aziz [20] ob- the nonlinear density and temperature variations in the tained a similarity solution for the Blasius flow of a vis- buoyancy force term exert a strong influence on the flow cous fluid under convective boundary condition. Yacob field. Further, we follow the work of many recent authors and Ishak [21] investigated stagnation point flow towards a by assuming that γ = μ + κ j [23, 24]. stretching/shrinking sheet immersed in a micropolar fluid 2 By employing nonlinear Boussinesq approximation with a convective surface boundary condition. In recent (Ref. [15, 16]) and making use of the standard boundary times, the influence of homogeneous-heterogeneous reac- layer approximations, the governing equations for the mi- tions on convective heat flow of a micropolar fluid along a vertical plate in porous medium under convective surface Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate | 141

The boundary conditions are

∂T u =0, v =0, ω =0,−k = h (T −T), C = C at y =0 ∂y f f w (6)

u = u∞, ω =0, T = T∞, C = C∞ as y → ∞ (7) where, the subscripts w and ∞ indicate the conditions at the wall and at the outer edge of the boundary layer, re-

spectively, hf is the convective heat transfer coefficient and k is the thermal conductivity of the fluid. Introducing the following dimensionless variables: x y Re 1/2 Re 1/2 ψ(ξ , η) ξ = , η = , f(ξ , η)= , L L ξ ξ Lu∞ Lω ξ 1/2 T − T g(ξ , η)= , θ(ξ , η)= ∞ , u∞ Re Tf − T∞ C − C∞ Fig. 1: Physical model and coordinate system. ϕ(ξ , η)= (8) Cw − C∞

where u∞ is free stream velocity and Re =(u∞L)/ν is the cropolar fluid are given by global Reynold’s number. ∂u ∂v + = 0 (1) In view of the continuity equation (1), we introduce the ∂x ∂y stream function ψ by ρ ∂u ∂u 1 ∂2u ∂ω μ ∂ψ ∂ψ u = and v =− (9) 2 u + v = (μ + κ) 2 + κ + (u∞ − u) ε ∂x ∂y ε ∂y ∂y KP ∂y ∂x ρb 2 2 * 2 + (u − u )+ρg β (T − T∞) + β (T − T∞) Using (8) and (9) into (2)–(5), we get the following momen- K ∞ 0 1 P tum, angular momentum, energy and concentration equa- 2 +β2 (C − C∞) + β3 (C − C∞) cosΩ (2) tions

1 1 2 (1+Δ) f + ff + Δg ρj ∂ω ∂ω γ ∂ ω 1 ∂u ε 2ε2 u + v = 2 − κ 2ω + (3) ε ∂x ∂y ∂y ε ∂y + Riξ θ(1 + α θ)+Bϕ(1 + α ϕ) cos Ω 1 2 2 ∂T ∂T ∂ T 1 Fs ξ ∂f ∂f u + v = α (4) + ξ 1−f + ξ 1−f 2 = f − f ∂x ∂y ∂y2 Da Re Da ε2 ∂ξ ∂ξ (10) ∂C ∂C ∂2C u + v = D (5) ∂x ∂y ∂y2 Δ 1 1 where u and v are the Darcy velocity components in x and 1+ g + (f g + fg)−ΔJξ 2g + f 2 2ε ε y directions respectively, ω is the component of micro ro- ξ ∂g ∂f tation whose direction of rotation lies in the xy-plane, T = f − g (11) is the temperature, C is the concentration, g* is the accel- ε ∂ξ ∂ξ eration due to gravity, ρ is the density, μ is the dynamic 1 1 ∂θ ∂f coefficient of viscosity, b is the empirical constant, ε is the θ + fθ = ξ f − θ (12) Pr 2 ∂ξ ∂ξ porosity, Kp is the permeability, κ is the vortex viscosity, j is the micro-inertia density, γ is the spin-gradient viscosity, 1 1 ∂ϕ ∂f ϕ + fϕ = ξ f − ϕ (13) α is the thermal diffusivity and D is the solutal diffusivity Sc 2 ∂ξ ∂ξ of the medium, Ω is inclination of angle. Here β0 and β1 where the primes indicate partial differentiation with re- are the coefficients of thermal expansion of the first and spect to η alone. In usual definitions, Δ = κ/μ is the mi- second orders, respectively whereas β2 and β3 are the co- cropolar or material parameter [23, 24], Gr =[g*β (T − efficients of solutal expansion of the first and thesecond 0 f orders, respectively. 142 | Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate

3 2 T∞)L ]/ν is the thermal Grashof number, ν is the kine- 4 Solution of the Problem matic viscosity, Ri = Gr/Re2 is the mixed convection parameter, B =[β (C − C )]/[β (T − T )] is the Buoy- 2 w ∞ 0 f ∞ We now obtain approximate solutions to Eqs. (10)–(13) ancy ratio, Da = K /L2 is the Darcy number, Fs = b/L p along with the boundary conditions (14) in two steps: (i) is the Forchheimer number, J = L2/(j Re) is the micro- First, we use, the local non-similarity procedure (of Spar- inertia density, Pr = ν/α is the Prandtl number, Ω is angle row and Yu [25], and Minkowycz and Sparrow [26]) to con- of inclination and Sc = ν/D is the Schmidt number. Here vert the set of partial differential equations (10)–(13) along α =[β (T − T )]/β and α =[β (C − C )]/β are the 1 1 f ∞ 0 2 3 w ∞ 2 with the boundary conditions (14) into set of ordinary dif- nonlinear temperature and concentration parameters, re- ferential equations, (ii) Next, the resulting boundary value spectively. problem is solved using a Successive Linearisation Method The boundary conditions (6) become (SLM). ∂f According to local similarity and non-similarity pro- f (ξ ,0)=0, f(ξ ,0)=−2ξ , ∂ξ η=0 cedure (see [25, 26] for more details), the system of partial differential equations considered here is first converted to g(ξ ,0)=0, θ (ξ ,0) =−Biξ1/2 [1−θ (ξ ,0)] , ϕ(ξ ,0)=1 a system of ordinary nonlinear differential equations by in- and f (ξ ,∞)=1, g(ξ ,∞)=0, θ(ξ ,∞)=0, ϕ(ξ ,∞)=0. troducing new unknown functions of ξ derivatives. (14) In the first level of truncation, the terms accompa- nied by ξ ∂ are assumed to be very small. This is partic- where Bi = h L/(kRe1/2) is the Biot number. It is a ratio ∂ξ f ularly true when ξ 1. We neglect terms containing the of the internal thermal resistance of the plate to the bound- ξ derivatives in Eq. (10)–(14). Thus we get local similarity ary layer thermal resistance of the hot fluid at the bottom equations are of the surface. 1 1 (1+Δ) f + ff + Δg+ Riξ θ(1 + α θ) ε 2ε2 1 1 +Bϕ(1 + α2ϕ) cos Ω+ ξ 1−f 3 Skin friction, Wall couple stress, Da Re Fs + ξ 1−f 2 = 0 (19) Heat and Mass transfer Da Δ 1 1 coefficients 1+ g + (f g + fg)−ΔJξ 2g + f =0 2 2ε ε The wall shear stress and the wall couple stress are: (20) 1 1 ∂u ∂ω θ + fθ = 0 (21) τw = (μ + κ) + κω and mw = γ (15) Pr 2 ∂y y=0 ∂y y=0 1 1 ϕ + fϕ = 0 (22) And the heat and mass transfers from the plate, respec- Sc 2 tively are given by The corresponding boundary conditions are ∂T ∂C f (ξ ,0)=0; f(ξ ,0)=0; g(ξ ,0)=0; qw =−k and qm =−D (16) ∂y ∂y y=0 y=0 θ (ξ ,0) =−Biξ1/2 [1−θ (ξ ,0)] ; ϕ(ξ ,0)=1 2τw and f (ξ ,∞)=1, g(ξ ,∞)=0, θ(ξ ,∞)=0, ϕ(ξ ,∞)=0. The non-dimensional skin friction Cf = 2 , wall couple ρu∞ mw qw x (23) stress Mw = 2 , the local Nusselt number Nux = ρu∞ x k(Tf −T∞) qm x For the second level of truncation, we introduce U = and local Sherwood number Shx = D(C −C ) ,aregivenby w ∞ ∂f ∂g ∂θ ∂ϕ ∂ξ , V = ∂ξ , H = ∂ξ , K = ∂ξ to recover the neglected 1/2 2+Δ terms at the first level of truncation. Thus the governing Cf Rex =2(1+Δ) f (ξ ,0), Mw Rex = g (ξ ,0), 2J equations at the second level reduces to (17) 1 1 (1+Δ) f + 2 ff + Δg+ Riξ θ(1 + α1θ) Nu Sh ε 2ε x =−θ (ξ ,0), x =−ϕ (ξ , 0) (18) 1 Fs Re1/2 Re1/2 +Bϕ(1 + α ϕ) cos Ω+ ξ 1−f + ξ 1−f 2 x x 2 Da Re Da u∞ x where Rex = is the local Reynold’s number. ξ ν = f U − f U (24) ε2 Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate | 143

Δ 1 1 1+ g + (f g + fg)−ΔJξ 2g + f The coupled nonlinear differential equations (24) - (27) 2 2ε ε and (29)–(32) together with the boundary conditions ξ = f V − Ug (25) (28), (33) and (34) are solved using one of the non- ε perturbation methods named as Successive Linearization Method (see [27–29]), it utilizes first the successive lin- 1 1 θ + fθ = ξ f H − Uθ (26) earization and then the Chebyshev spectral collocation Pr 2 scheme. 1 1 Using successive linearization, the nonlinear bound- ϕ + fϕ = ξ f K − Uϕ (27) Sc 2 ary layer equations will reduce to a system of linear differ- The corresponding boundary conditions are ential 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 as- ( ) ( ) ( ) f (ξ ,0)=0; f(ξ ,0)=−2ξU(ξ , 0); g(ξ ,0)=0; sume that the independent vector Q(η) can be expressed θ (ξ ,0) =−Biξ1/2 [1−θ (ξ ,0)] ; ϕ(ξ ,0)=1 as and f (ξ ,∞)=1, g(ξ ,∞)=0, θ(ξ ,∞)=0, ϕ(ξ ,∞)=0. i−1 (28) Q(η)=Qi(η)+ Qn(η)(35) n=0 At the third level of truncation, we differentiate where Qi(η), (i = 1, 2, 3....), are unknown vectors and Eqns. (24)–(28) with respect to ξ and neglect the terms Qn(η) are the approximations which is obtained by recur- ∂U , ∂V , ∂H , ∂K to get the following system of equations. ∂ξ ∂ξ ∂ξ ∂ξ sively solving the linear part of the equation system that

1 3 1 results from substituting (35) in (24)–(34). (1+Δ) U + 2 Uf + 2 U f + ΔV The initial guess Q (η) is chosen such that they sat- ε 2ε 2ε 0 1 1 Fs isfy the boundary conditions (28), (33) and (34). The sub- + 1−f − ξU+ 1−f 2 Da Re Da Re Da sequent solutions Qi(η), i ≥ 1 are obtained by successively 2Fs 1 − ξ + f U + Riξ H(1 + 2α θH) solving the linearised form of the equations which are ob- Da ε2 1 tained by substituting Eq. (35) in the governing equations +BK(1 + 2α2ϕK) cos Ω + Ri θ(1 + α1θ) and neglecting the nonlinear terms. The linearised equations to be solved are B ξ 2 + ϕ(1 + α2ϕ) cos Ω = 2 U − U U (29) ε a1,i−1fi + a2,i−1fi + a3,i−1fi + a4,i−1fi + a5,i−1gi + a θ + a ϕ + a U + a U = r (36) Δ 1 3 1 6,i−1 i 7,i−1 i 8,i−1 i 9,i−1 i 1,i−1 1+ V + (U g + fV )+ Ug − Vf 2 2ε 2ε 2ε 1 1 ξ b f + b f + b f + b g + b g − ΔJ 2g + f − ΔJξ 2V + U = U V − UV 1,i−1 i 2,i−1 i 3,i−1 i 4,i−1 i 5,i−1 i ε ε ε + b6,i−1gi + b7,i−1Ui + b8,i−1Vi = r2,i−1 (37) (30)

c1,i−1fi + c2,i−1fi + c3,i−1θi + c4,i−1θi + c5,i−1Ui 1 3 1 H + Uθ + fH− f H = ξ U H − UH (31) + c H = r (38) Pr 2 2 6,i−1 i 3,i−1

1 3 1 d f + d f + d ϕ + d ϕ + d U K + Uϕ + fK− f K = ξ U K − UK (32) 1,i−1 i 2,i−1 i 3,i−1 i 4,i−1 i 5,i−1 i Sc 2 2 + d6,i−1Ki = r4,i−1 (39) The corresponding boundary conditions are e f + e f + e f + e θ + e ϕ U(ξ ,0)=0; U(ξ ,0)=0; V(ξ ,0)=0; 1,i−1 i 2,i−1 i 3,i−1 i 4,i−1 5,i−1 i + e U + e U + e U + e U + e V 1 1 −1 1 −1 6,i−1 i 7,i−1 i 8,i−1 i 9,i−1 i 10,i−1 i H (ξ ,0) = Biξ 2 H (ξ ,0) + Bi ξ 2 θ (ξ ,0) − Bi ξ 2 ; 2 2 + e11,i−1Hi + e12,i−1Ki = r5,i−1 (40) K(ξ ,0)=0 (33) o1,i−1fi + o2,i−1fi + o3,i−1fi + o4,i−1gi + o5,i−1gi U (ξ ,∞)=0; V(ξ ,∞)=0; H (ξ ,∞) =0; K(ξ ,∞)=0 + o6,i−1Ui ++o7,i−1Ui + o8,i−1Ui + o9,i−1Vi (34) + o10,i−1Vi + o11,i−1Vi = r6,i−1 (41) 144 | Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate

p1,i−1fi + p2,i−1fi + p3,i−1θi + p4,i−1Ui + p5,i−1Ui where Z is the order of differentiation and D =(2/L)D with D being the Chebyshev spectral differentiation matrix. + p6,i−1Hi + p7,i−1Hi + p8,i−1Hi = r7,i−1 (42) Substituting Eqs.(45)–(49) into linearized form of equations leads to the matrix equation. q1,i−1fi + q2,i−1fi + q3,i−1ϕi + q4,i−1Ui + q5,i−1Ui + q6,i−1K + q7,i−1K + q8,i−1Ki = r8,i−1 (43) i i Ai−1Xi = Ri−1 (50) The boundary conditions reduce to In Eq. (50), Ai−1 is a (8N + 8)×(8N + 8) square matrix and ⎫ fi(0) = fi (0) = fi (∞) = 0, gi(0) = gi(∞) = 0, ⎪ Xi and Ri−1 are (8N + 1)×1 column vectors defined by ⎪ θ (0) − Bi ξ1/2θ (0) = 0, θ (∞) = 0, ⎪ i i i ⎬⎪ A = [Apq] , p, q = 1, 2, ....8, ϕi(0) = ϕi(∞) = 0, Ui(0) = Ui (0) = Ui (∞) = 0, i−1 ⎪ T Vi(0) = Vi(∞) = 0, ⎪ Xi = [Fi Gi Θi Φi Ui Vi Hi Ki] , 1 1 −1 ⎪ H (0) − Bi ξ 2 H (0) − Bi ξ 2 θ (0) = 0, ⎪ i i 2 i ⎭⎪ R = r r r r r r r r T H (∞) = 0, K (0) = K (∞) = 0 i−1 1,i−1 2,i−1 3,i−1 4,i−1 5,i−1 6,i−1 7,i−1 8,i−1 i i i (51) (44) After modifying the matrix system (50) to incorporate Here the coefficient parameters at, i−1, bt, i−1, boundary conditions, the solution is obtained as ct, i−1, dt, i−1, et, i−1, ot, i−1, pt, i−1, qt, i−1 and rt,i−1 de- pend on the initial guesses Q0(η) and on their derivatives. −1 Xi = Ai−1 Ri−1 (52) Once each solution for Qi(η), i ≥ 1 has been obtained, the approximate solutions for Q(η) are then obtained as M Q(η)≈ m=0 Qm(η) where M is the order of SLM approximation. 5 Results and Discussion The linearized equations (36–44) are solved using the Chebyshev spectral collocation method (Canuto et In order to assess the validity and accuracy of the present al. [30]). The unknown functions are approximated by the analysis, the results of the principle of local similarity for Chebyshev interpolating polynomials in such a way that Eqs. (10)–(13) have been compared with the special case they are collocated at the Gauss-Lobatto points defined as of Lloyd and Sparrow [31] in the absence of micropolar pa- πn rameter Δ, buoyancy ratio B, nonlinear convection param- τn =cos , n = 0, 1, 2, ..., N (45) N eters α1 and α2 with ε =1,Da → ∞, ξ =1,Bi → ∞, Ω =0 where N is the number of collocation points used. The as exhibited in Table 1. It shows an excellent agreement physical region [0, ∞) is transformed into the region [-1, 1] with existing results. using the domain truncation technique in which the prob- In the present study, we have adopted the following lem is solved on the interval [0, L] instead of [0, ∞). This default parameter values for the numerical computations: leads to the mapping B =1.0,Re = 200, Da =0.1,ε =0.5,Pr= η τ +1 0.71, Sc =0.22, Ri = 2. The value of dimensionless = ,−1≤τ ≤ 1 (46) J L 2 micro-inertia density = 1.0 is chosen so as to satisfy the thermodynamic restrictions on the material parame- and the function Q(η) is approximated at the collocation ters given by Eringen [1]. These values are used throughout points by the computations, unless otherwise indicated. N The dimensionless velocity, microrotation, tempera- Q (τ)= Q (τm)Tm(τ ), k = 0, 1, 2, ..., N (47) i i k ture and concentration profiles have been computed for m=0 different values of the fluid parameters and presented where L is a scaling parameter used to invoke the bound- graphically in Figs. 2–9. The effects of nonlinear tempera- ary condition at infinity, T is the mth Chebyshev polyno- m ture α , nonlinear concentration α , non-Darcy parameter mial defined by 1 2 Fs, micropolar parameter Δ, mixed convection parameter −1 Tm(τ) = cos[m cos τ] (48) Ri, Biot number Bi and angle of inclination Ω have been discussed. The derivatives of the variables at the collocation points The effect of nonlinear temperature α on the velocity, are represented as 1 microrotation, temperature and the concentration profiles Z N d Z are shown in Figs. 2(a)–2(d). The results indicate that the Q (τ)= D T (τm), l = 0, 1, 2, ..., N (49) dηZ i ml i velocity distribution increases with an increasing value of m=0 Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate | 145

1/2 Table 1: Comparison of Nux Rex for mixed convection flow along a vertical flat plate in Newtonian fluid (Lloyd and Sparrow [31]).

Pr=0.72 Pr=10 Pr=100 Ri Lloyd and Present Lloyd and Present Lloyd and Present Sparrow [31] Sparrow [31] Sparrow [31] 0.0 0.2956 0.2956 0.7281 0.7281 1.5720 1.5718 0.01 0.2979 0.2979 0.7313 0.7312 1.5750 1.5754 0.04 0.3044 0.3043 0.7404 0.7403 1.5850 1.5861 0.1 0.3158 0.3156 0.7574 0.7572 1.6050 1.6069 0.4 0.3561 0.3559 0.8259 0.8254 1.6910 1.6996 1.0 0.4058 0.4053 0.9212 0.9207 1.8260 1.8500

α1 and the value B = 1 implies that the thermal and con- perature distribution is increased at the plate. The effect of centration buoyancy forces are of the same order of magni- the Biot number on the concentration profile is displayed tude. Physically, α1 > 0 implies that Tf > T∞; hence, there in Fig. 4(d) and it depicts that the concentration profile re- will be a supply of heat to the flow region from the wall. duces within the boundary layer when the Biot number in-

Similarly α1 < 0 implies that Tf < T∞, and in such a case creases from least to large value. there will be a transfer of heat from the fluid to the wall. Figures 5(a)–5(d) illustrate the variation of the veloc-

Again, this increase in velocity with positive values of α1 ity distribution, microrotation, temperature and concen- is more prominent in the presence of the mixed convection tration for different values of the inclination of angle◦ (0 ≤ parameter. The temperature and concentration boundary Ω ≤90◦). Moreover, the non-similarity equations for the layer thicknesses decrease with the rise of α1. limiting cases of the vertical and horizontal plates are re- Figures 3(a)–3(d) depict the influence of the nonlin- covered from the transformed equations by setting Ω =0◦ ◦ ear concentration parameter α2 for a fixed value of α1 =5 and Ω =90 , respectively. The influence of inclination of on the behaviour of velocity, microrotation, temperature the angle on the velocity profile is displayed in Fig. 5(a). and concentration. The initial velocity is zero at the sur- Due to the reduction in the thermal buoyancy effect in face of the plate and rises gradually away from the plate Eq. (2) caused by an increase in angle Ω, reduces the ve- to the free stream satisfying the boundary conditions as locity distribution within the momentum boundary layer given in Fig. 3(a). However, the rise of nonlinear concentra- as shown in Fig. 5(a). In other words, an increase in the tion changes the sign of microrotation as shown in Fig. 3(b) angle of inclination leads to reduce the velocity distribu- from negative to positive within the boundary layer. In the tion within the boundary layer region. Also, we can ob- absence, as well as presence of nonlinear concentration serve from Fig. 5(a) that the maximum buoyancy force for parameter α2 the magnitude of the temperature and con- the temperature and concentration difference occurs for centration decreases with an increase of α2 which is pre- Ω = 0 (vertical plate). When the position of the flat plate is sented in Figs. 3(c)–3(d). We also note that the impact of changed from vertical to horizontal, we observe that the

α2on the temperature and concentration distributions is microrotation is increasing near the plate and far away more elegant, as compared with that of α1. from the plate it is showing a reverse trend within the Figures 4(a)–4(d) display the velocity, microrotation boundary layer, as shown in Fig. 5(b). It is noticed from component, temperature and concentration distribution Fig. 5(c) and Fig. 5(d) that the temperature and concentra- of fluid flow for different values of the Biot number. Itis tion enhances with increasing values of inclination of an- noteworthy from Figs. 4(a)–4(b) that as the Biot number gle. In particular, when the surface is vertical the small- increases the velocity profile increase and the microrota- est temperature and concentration distributions are ob- tion changes direction from decreasing to increasing val- served, whereas they become largest for the horizontal sur- ues within the boundary layer. Fig. 4(c) demonstrates the face. effect of the Biot number on the temperature profile and Figures 6(a)–6(b) shows the effects of the nonlinear serves a dual result (i.e., for convective boundary condi- temperature and concentration on the non-dimensional tion and isothermal condition). Since the specified con- local heat and mass transfer rates against stream wise co- vective boundary condition is changing into wall condi- ordinate. We observe that both heat and mass transfer tion, when the Biot number is tends to infinity and it is rates are increasing with α1 when α2 is fixed. The effect proven by Fig. 4(c). As Bi increases from thermally thin of α2 on Nusselt and Sherwood numbers is showing the case (Bi < 0.1) to thermally thick case (Bi > 0.1) the tem- same behaviour with that of α1. The effect of varying the 146 | Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate

(a) (a)

(b) (b)

(d) (d)

Fig. 2: Effect of nonlinear temperature parameter on the (a) velocity, Fig. 3: Effect of nonlinear concentration parameters on the (a)ve- (b) microrotation, (c) temperature and (d) concentration for Δ = locity, (b) microrotation, (c) temperature and (d) concentration 1.0, Fs =0.5,α2 =5,Ω = π/6, Bi =0.5,ξ =0.5. forΔ =1.0,Fs =0.5,α1 =5,Ω = π/6, Bi =0.5,ξ =0.5. Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate | 147

(a) (a)

(b) (b)

(d) (d)

Fig. 4: Effect of Biot number on the (a) velocity, (b) microrotation, Fig. 5: Effect of inclination of angle on the (a) velocity, (b) micro- (c) temperature and (d) concentration for Δ =1.0,Fs =0.5,α1 = rotation, (c) temperature and (d) concentration for Δ =1.0,Fs = 5, α2 =5,Ω = π/6, ξ =0.5. 0.5, α1 =5,α2 =5,Bi =0.5,ξ =0.5. 148 | Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate

Forchheimer number Fs and Micropolar parameter Δ on with the enhancement of the Biot number there is a con- non-dimensional local heat and mass transfer rates are siderable increment in heat transfer rate. The local mass presented in Figs. 7(a)–7(b). The results indicate that as Fs transfer rate is decreasing in opposing flow, whereas; it increases, the local heat and mass transfer rates decrease is increased in the case of assisting flow with the rise of for a fixed value of micropolar parameter. Hence, the iner- Biot number. The influence of the angle of inclination on tial effect in micropolar fluid saturated non-Darcy porous heat and mass transfer rates as a function of dimension- medium reduces the heat and mass transfer coefficients. less stream wise coordinate are shown in Figs. 9(a)–9(b). Also, it can be observed from this figure that, for a fixed The results are displayed that the local Nusselt number value of Fs, the heat and mass transfer coefficients are re- and Sherwood number reduce gradually when the plate is ducing with the increasing values of micropolar parame- rotated from vertical to horizontal. One can notice that the ter and it is obvious that the rate of heat and mass trans- effect of angle of inclination is more on mass transfer rate fers in the micropolar fluid (Δ ≠ 0) are lower compared as compared with that of heat transfer rate. to that of the Newtonian fluid (Δ = 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.

(a)

(b)

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/6, Bi =0.5.

(b) The proportional quantities of skin friction Fig. 6: The effect of Nonlinear temperature and concentration pa- and the gradient of microrotation (wall couple rameters on (a) local Nusselt number, (b) local Sherwood number stress) are computed for the enhanced values of for Δ =0.5,Fs =0.5,Ω = π/6, Bi =0.5. Δ, Fs, Ri, α1, α2, Ω, Bi and the results are presented in Table 2. It is observed that an enhancement in the mixed The effect of Biot number Biand the variation of mixed convection parameter causes an increase in the skin fric- convection parameter Ri on the local heat and mass trans- tion and decrease in the wall couple stress, whereas, with fer rates are depicted in Figs. 8(a)–8(b). It is found that the increasing of angle of inclination they show the op- the local heat and mass transfer rates are increasing when posite trend. Rising in the micropolar parameter tends to the flow direction is changed from opposing to aiding, and enhance the skin friction and the reverse change is no- Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate | 149

(a) (a)

(b) (b)

Fig. 8: The effect of Biot number and Mixed convection parameter Fig. 9: The influence of angle of inclination on (a) heat transfer rate, on (a) local Nusselt number, (b) local Sherwood number for Δ = (b) mass transfer rate for a fixed values of Δ =0.5,Fs =0.5,α1 = 0.5, Fs =0.5,α1 =5,α2 =5,Ω = π/6. 5, α2 =5,Bi =0.5.. ticed 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 Table 2: Effect of skin friction and wall couple stress for various nominal effect on the wall couple stress and considerable values of Δ, Fs, Ri, α1, α2, Ω, Bi. increment in skin friction is encountered for high enough Δ FsRiα α Ω Bi C Re1/2 −M Re values of Biot number. Furthermore, the skin friction pa- 1 2 f x w x 0 0.5 2.0 5.0 5.0 30◦ 0.5 5.566797 0 rameter increases and the wall couple stress decreases as 2 0.5 2.0 5.0 5.0 30◦ 0.5 8.715269 1.347554 Forchheimer number increases. 4 0.5 2.0 5.0 5.0 30◦ 0.5 10.537821 2.149497 1.0 0 2.0 5.0 5.0 30◦ 0.5 6.454313 0.783937 1.0 1 2.0 5.0 5.0 30◦ 0.5 8.301189 0.805063 1.0 2 2.0 5.0 5.0 30◦ 0.5 9.789750 0.819623 6 Conclusions 1.0 0.5 1 5.0 5.0 30◦ 0.5 5.664402 0.697881 1.0 0.5 2 5.0 5.0 30◦ 0.5 7.446391 0.796626 1.0 0.5 3 5.0 5.0 30◦ 0.5 9.081631 0.879884 The problem of nonlinear convection flow of a micropo- 1.0 0.5 2.0 0 5.0 30◦ 0.5 7.127987 0.783326 lar fluid along an inclined plate in a non-Darcy porous 1.0 0.5 2.0 3 5.0 30◦ 0.5 7.320324 0.791391 ◦ medium under convective boundary condition has been 1.0 0.5 2.0 5 5.0 30 0.5 7.446391 0.796626 1.0 0.5 2.0 5.0 2 30◦ 0.5 6.002507 0.718517 investigated. Using local non-similarity technique, the ◦ 1.0 0.5 2.0 5.0 4 30 0.5 6.974598 0.771772 governing partial differential equations have been trans- 1.0 0.5 2.0 5.0 6 30◦ 0.5 7.909810 0.820479 formed into a system of coupled nonlinear ordinary dif- 1.0 0.5 2.0 5.0 5.0 45◦ 0.5 6.812794 0.762614 ◦ ferential equations, which are solved numerically by us- 1.0 0.5 2.0 5.0 5.0 60 0.5 5.952493 0.714534 1.0 0.5 2.0 5.0 5.0 75◦ 0.5 4.887985 0.651403 ing the Chebyshev spectral collocation method together 1.0 0.5 2.0 5.0 5.0 30◦ 0.2 7.121206 0.782108 with successive linearization named as a successive lin- 1.0 0.5 2.0 5.0 5.0 30◦ 2.0 8.304417 0.832472 earization method. The effects of various parameters on 1.0 0.5 2.0 5.0 5.0 30◦ 10 8.991506 0.859538 the velocity, microrotation, temperature, concentration, 150 | Ch. RamReddy et al., Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate heat and mass transfer, skin friction and wall couple stress α Thermal diffusivity have been analyzed and some important results are given. α1 Nonlinear temperature parameter It is observed that the tangential velocity increases with α2 Nonlinear concentration parameter increasing values of the nonlinear temperature and con- β0, β1 Coefficients of thermal expansion of first and second orders centration. The temperature and concentration boundary β2, β3 Coefficients of solutal expansion of layer thickness decreases with increasing values of α1 and first and the second order α2. The local heat and mass transfer rates reduce with the γ Spin-gradient viscosity angle of inclination Ω increases. The numerical results in- ε Porosity dicate that the micropolar parameter diminishes the wall η Similarity variable Ω Inclination of angle couple stress and enhances skin friction. As the Biot num- θ Dimensionless temperature ber increases, there is a gradual increment in skin friction ϕ Dimensionless concentration and nominal effect on wall couple stress. κ Vortex viscosity μ Dynamic viscosity Acknowledgement: This work was supported by of Coun- ν Kinematic viscosity cil of Scientific and Industrial Research (CSIR), New Delhi, ρ Density of the fluid τw Wall shear stress India with Research grant No. 25(0246)/15/EMR-II. ψ Stream function ω Component of micro rotation Rex Local Reynolds number Nomenclature w Wall condition Bi Biot number Re Reynolds number B Buoyancy ratio ∞ Ambient condition C Concentration Ri Mixed convection parameter C Concentration Cw Wall concentration Sc Schmidt number Cf Skin friction coefficient T Temperature C∞ Ambient concentration D Solutal diffusivity Shx Local Sherwood number Da Darcy number T Temperature f Reduced stream function Differentiation with respect to η Fs Forchheimer number g* Gravitational acceleration g Dimensionless micro rotation Gr Thermal Grashof number References hf Convective heat transfer coefficient j Micro-inertia density [1] Eringen AC. 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