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 charac- teristics 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 ther- mal 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.