Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet
Introduction
The flow properties of certain fluids like polymeric fluids, muds, colloidal fluids, animal blood, fluids containing additives, ferrofluids etc. cannot be explained properly by the Navier–Stokes equations of Newtonian and non-Newtonian fluids theory. Because of this, many constitutive models have been suggested by several researchers. Among these models, the microfluid model proposed by Eringen [1] has attracted considerable attentions. That mathematical model deals with a class of fluids having certain microscopic characters arising from the local structure and micro-motions of the fluid elements. Eringen [2] also developed a subclass of microfluids, called micropolar fluids.
In last few decades, the research interest in micropolar fluid theory has significantly increased due to its enormous applications in many industrial processes. The pioneering work of Eringen [2] was extended in boundary layer theory by Peddieson and McNitt [3]. Peddieson [4] applied the micropolar fluid model in turbulent flow also. The self-similar solution of boundary layer flow of a micropolar fluid over a semi-infinite flat plate was obtained by Ahmadi [5]. Kümmerer [6] studied the steady two-dimensional flow of micropolar fluid using a numerical technique. Guram and Smith [7] investigated the stagnation point flow of a micropolar fluid in an infinite plate with two different boundary conditions – vanishing spin and vanishing spin gradient. The free convective boundary layer flow of a thermomicropolar fluid over a non-isothermal vertical flat plate was discussed by Jena and Mathur [8]. Gorla [9] studied the micropolar boundary layer flow near a stagnation point on a moving wall.
An important contribution in micropolar flow dynamics was made by Sankara and Watson [10], when they investigated the flow of micropolar fluids past a stretching sheet. Heruska et al. [11] extended the work of Sankara and Watson [10] by considering the mass suction or injection through the porous sheet. Hassanien and Gorla [12] explained the heat transfer in a micropolar flow over a non-isothermal stretching sheet with suction and blowing. Na and Pop [13] also considered the boundary layer flow of micropolar fluid due to continuously stretching boundary. Hassanien et al. [14] reported the numerical solution for heat transfer in a micropolar flow past a stretching sheet. The convective heat transfer for a micropolar fluid in presence of uniform magnetic field was discussed by Abo-Eldahad and Ghonaim [15]. In addition, some very important investigations in this direction were made in the articles [16], [17], [18], [19], [20], [21].
Recently, the boundary layer flow near a shrinking sheet is given significant attention due to its increasing engineering applications. The flow development around the shrinking sheet was demonstrated by Wang [22] while studying behaviour of liquid film on an unsteady stretching sheet. The existence and uniqueness of steady viscous flow due to a shrinking sheet was established by Miklavčič and Wang [23] considering the suction effects and they concluded that for some specific values of suction, dual solutions exist and also in certain range of suction, no boundary layer solution is possible. Hayat et al. [24] gave an analytic solution of magnetohydrodynamic (MHD) rotating flow of a second grade fluid over a shrinking surface using homotopy analysis method (HAM). Hayat et al. [25] also studied the MHD flow and mass transfer of an upper-convected Maxwell fluid past a porous shrinking sheet with chemically reactive species. A series solution of three-dimensional MHD and rotating flow over a porous shrinking sheet was obtained by Hayat et al. [26] using HAM. Fang [27] studied the boundary layer flow over a continuously shrinking sheet with power-law surface velocity and with mass transfer. Fang and Zhang [28] obtained a closed-form analytical solution for MHD viscous flow over a shrinking sheet subjected to applied suction through the porous sheet. The unsteady viscous flow over a continuously shrinking sheet with mass suction was investigated by Fang et al. [29]. Noor et al. [30] reported a series solution for MHD viscous flow due to a shrinking sheet by Adomian decomposition method (ADM). An analytic solution was developed for shrinking flow in a rotating frame of reference by Hayat et al. [31]. The exact analytic solution of thermal boundary layer over a shrinking sheet with mass transfer was obtained by Fang and Zhang [32]. Fang et al. [33] studied the viscous flow over a shrinking sheet taking a second order slip flow model and solved it analytically. Bhattacharyya [34] investigated the steady boundary layer flow and heat transfer over an exponentially shrinking sheet. Recently, Ishak et al. [35] described the boundary layer flow of non-Newtonian power-law fluid past a shrinking sheet with suction and Yacob and Ishak [36] discussed the micropolar fluid flow over a shrinking sheet. On the other hand, Wang [37] investigated the stagnation flow towards a shrinking sheet and obtained dual solutions for some values of the ratio of shrinking and stagnation flow rates. Wang’s [37] problem was extended by many researchers showing various aspects of shrinking sheet flow [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].
Radiative heat transfer in the boundary layer flow is very important from application point of view, because the quality of the final product is very much dependent on the rate of heat transfer of the ambient fluid particles. Elbashbeshy [48] discussed the effect of radiation on the forced convection flow of along a heated horizontal stretching surface. Ouaf [49] found an exact solution of MHD steady asymmetric flow of an electrically conducting fluid past a stretching porous sheet in presence of radiation. The effects of thermal radiation on the flow of viscoelastic fluid past a stretching sheet were explained by Khan [50] and Bataller [51]. Sajid and Hayat [52] showed the influence of thermal radiation on the boundary layer flow and heat transfer past an exponentially stretching sheet. Some other important investigations in this direction were made by Mukhopadhyay and Layek [53], Aliakbar et al. [54], Chen [55], Pal and Mondal [56], [57], Hayat et al. [58] and Nandeppanavar et al. [59]. The unsteady flow and heat transfer over a stretching sheet with thermal radiation were investigated by El-Aziz [60], Mukhopadhyay [61], Hayat et al. [62] and Hayat and Qasim [63]. Bhatacharyya and Layek [64] demonstrated the radiation effects on the steady stagnation-point flow and heat transfer towards a permeable shrinking sheet. Ali et al. [65] studied the unsteady axisymmetric boundary layer flow and heat transfer induced by a permeable shrinking sheet in presence of radiation. Recently, the effect of radiation on thermal boundary layer flow over a stretching sheet in a micropolar fluid was investigated by Ishak [66].
In present investigation, the radiation effects on micropolar flow and heat transfer over a permeable shrinking sheet are studied. It is very interesting to investigate the simultaneous effects of the thermal radiation and the microrotation on the steady flow. The nonlinear self-similar ordinary differential equations obtained here are solved numerically by shooting technique with the help of Runge–Kutta method. The complete effects of several parameters are discussed in detail.
Section snippets
Analysis of the flow problem
Consider a steady two-dimensional flow of micropolar fluid and heat transfer over a porous shrinking sheet with thermal radiation. The shrinking velocity of the sheet is Uw = −cx with c > 0 being shrinking constant. Using boundary layer approximation, the equations of motion for the micropolar fluid and heat transfer may be written in usual notation as:subject to the boundary conditions:
Numerical method for solution
The nonlinear differential equations (10), (11), (12) along with the boundary conditions (13), (14) are solved using shooting method, by converting them into an initial value problem (IVP). In this method, it is necessary to consider a suitable finite value of η → ∞, say η∞. We set the following first-order system:and,with the boundary conditions:To solve (15), (16), (17) with (18)
Results and discussion
The analysis of the results obtained by the applied numerical scheme explores the condition for which the steady flow of micropolar fluids is possible. According to Miklavčič and Wang [23] and Fang and Zhang [28], the steady two-dimensional flow of Newtonian fluids (K = 0) due to a shrinking sheet with wall mass transfer occurs only when the wall mass suction parameter is greater than or equal to 2. But, for micropolar flow the case is quite different. With the increase of material parameter K,
Concluding remarks
The flow of micropolar fluid and heat transfer over a permeable shrinking sheet with thermal radiation is studied. The self-similar equations are solved numerically. The microrotation effect on the flow executes when the material parameter changes. It is observed that due to increase in material parameter the steady flow needs more amount of mass suction. In all cases dual similarity solutions for velocity and temperature exist. The skin friction coefficient, the couple stress coefficient and
Acknowledgements
Authors are thankful to reviewers for their valuable comments and suggestions which consequently lead to the improvement of the paper. One of the authors (K. Bhattacharyya) gratefully acknowledges the financial support of National Board for Higher Mathematics (NBHM), DAE, Mumbai, India for pursuing this work.
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