Abstract: An analysis is carried out to study the flow and heattransfer characteristics in the laminar boundary layer flow of visco elastic fluid over a non-isothermal stretching sheet with internal heat generation. A numerical method, Quasilinearization technique is used to study velocity and temperature profiles of the fluid. Heattransfer analysis is carried out for two types of thermal boundaryconditions namely, (i) Prescribed Surface temperature (PST) and (ii) Prescribed wall Heat Flux (PHF). The effects of various parameters such as Prandtl number, suction, visco-elasticity and temperature parameter on flow and heattransfer are presented through graphs and discussed.
The system of coupled ordinary differential Equations (2.8) to (2.12) and (3.5) to (3.7) has been solved numeri- cally using Runge-Kutta-Fehlberg fourth-fifth order me- thod. To solve these equations we adopted symbolic al- gebra software Maple which was given by Aziz . Maple uses the well known Runge-Kutta-Fehlberg fourty- fifth order (RFK45) method to generate the numerical solution of a boundary value problem. The boundaryconditions were replaced by those at 5 in accordance with standard practice in the boundary layer analysis. Numerical computation of these solutions have been carried out to study the effect of various physical parameters such as fluid particle interaction parameter , Grashof number Gr , Prandtl number Pr and Eckert number Ec are shown graphically.
flow of past a vertical semi-infinite flat plate of a nanofluid. In the similar analysis Sheikholeslami et al.  investigated heattransfer on 𝐴𝑙 2 𝑂 3 water nanofluid flows in a semi annulus enclosure using Lattice Boltzmann method. Rashidi et al.  analyzed the buoyancy effect on MHD flow over a stretching sheet of a nanofluid in the effect of thermal radiation using RK iteration scheme. Abolbashari et al.  investigated on entropy analysis for an unsteady magnetohydrodynamic flow past a stretching permeable surface in nanofluid. Sheikholeslami et al.  solved the problem for MHD natural convection heattransfer of nanofluids using Lattice Boltzmann method. In a similar way, Sheikholeslami and Ganji ( and ) studied the heat and mass transfer problems with nanofluids. Sheikholeslami et al.  studied nanofluid flow and heattransfer over a stretching porous cylinder considering thermal radiation. Sheikholeslami  studied hydrothermal behavior of nanofluid fluid between two parallel plates. In this work, one of the plates was externally heated, and the other plate, through which coolant fluid was injected, expands or contracts with time. Ferrofluid flow and heattransfer in the presence of an external variable magnetic field was studied by Sheikholeslami  using the control volume based finite element method. Control volume-based finite element method was applied by Sheikholeslami and Rashidi  for simulating Fe 3 O 4 -water nanofluid mixed
an isothermal cone due to convective boundaryconditions. Rao et al. (2015) explained the non-isothermal wedge with flow of Jeffrey’s fluid. Nasir et al. (2016) utilized the presence of the heat source with heattransfer of a couple stress fluids over an oscillator-stretching sheet. Sadia Siddiqi et al. (2017) reported the presence of thermal radiation with periodic MHD natural convection boundary layer problem obtained by the micro-polar fluid. Ram Reddy and Pradeepa (2015) presented the convective boundary condition are represented by a free convective flow along a permeable vertical plate of a micro-polar fluid. Ashmawy (2015) analyzed fully developed by the micro-polar with natural convection. Dulal Pala and Gopinath Mandal (2017) studied the micropolar with MHD effects of stretching sheet of nanofluids. Bourantas and Loukopoulos (2014) explained the MHD field in an inclined rectangular with transient, laminar and natural convection flow of a micropolar Nano fluid. Asia et al. (2016) explained the electrically showing micropolar fluid in a porous channel with contracting wall under the exploit of MHD. Hari et al. (2015) investigated the magnetic, material and viscosity parameters on natural convective flow along vertical walls in case of both asymmetric and symmetric cooling and heating of the walls. Rashad et al. (2014) have obtained a mixed convection in two – dimensional boundary layer flow of a micropolar fluid in a vertical plane with the effect of chemical reaction coupled with heat and mass transfer. Ahmad et al. (2012) investigated the laminar film flow of a micro-polar fluid with boundary layer of micro- polar fluid. Nagendra et al. (2008) investigated Peristaltic motion of a power-law fluid in an asymmetric vertical channel. Isaac Lare Animasaun (2016) analyzed the horizontal linearly stretchable melting surface with mixed convection of micropolar fluid. Aparna et al. (2017) explained the flow of fluid with slow rotation in permeable sphere in micropolar fluid. Mishra et al. (2015) investigated the concentration of a double stratified micro-polar fluid in the presence of a magnetic field
In the year of 1959, a model presented in the flow of viscoelastic fluid by Casson which was known as a Cas- son fluid model. Casson fluid exhibits a yield stress. It is well known that Casson fluid is a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear, i.e., if a shear stress less than the yield stress is applied to the fluid it behaves like a solid, whereas if a shear stress greater than yield stress is applied it starts to move. Fre- drickson  investigated the steady flow behavior of a Casson fluid in a tube. M. Nakamura et al. , studied the flow of a non-Newtonian fluid through an axisymmetric stenosis numerically. Mustafa et al.  studied and solved analytically using homotopy analysis method (HAM) for the problem unsteady boundary layer flow with heattransfer of a Casson fluid over a moving flat plate with a parallel free stream and the concept of MHD flow of the Casson fluid model over an exponentially shrinking sheet has been presented by Nadeem et al. . An exact solution of the steady boundary layer flow of Casson fluid over a stretching or shrinking sheet was studied by Bhattacharyya et al. , and analytical solution has been given by Krishnendu Bhattacharyya et al.  for the problem MHD boundary layer flow of Casson fluid over stretching/shrinking sheet with wall mass transfer whereas Swati Mukhopadhyay  studied Casson fluidflow and heattransfer over a nonlinearly stretching surface. On the other hand Peri K. Kameswaran et al.  investigated and presented Dual solutions of Casson fluidflow over a stretching or shrinking sheet. Rizwan Ul Haq et al.  studied the flow of Casson nanofluid over an exponential shrinking sheet with convective heattransfer and MHD effects. Recently the MHD flow of a Casson nanofluid with viscous dissipation over an exponentially stretching sheet by considering convective conditions is studied by T. Hussain et al. . M. Mustafa and Junaid Ahmad Khan , discussed a model for the flow of Casson nanofluid past a nonlinearly stretching sheet considering magnetic field effects.
mechanical situations. However, the interaction of peristalsis and heattransfer has not received much attention which may become highly relevant and significant in several industrial processes. Also, thermodynamical aspects of blood may become significant in processes like oxygenation and hemodialysis [13-16] when blood is drawn out of the body. The combined effects of magnetohydrodynamics and heattransfer on the peristaltic transport have been discussed by Mekheimer, Abd elmaboud and other co-workers [17, 18]. Hayat et al.  developed the problem by considering heattransfer effect on peristalsis flow of fluid filling the porous space in an asymmetric channel. Recently Nadeem and Akram , Hayat et al.  discussed the slip and heattransfer effect on peristaltic flow in an asymmetric channel under different boundaryconditions. The aim of the present investigation is to highlight the importance of heattransfer analysis of MHD peristaltic flow in an asymmetric porous channel under the influence of slip conditions in the presence of viscous dissipation terms. The governing equations of momentum and energy have been simplified using long wavelength and low Reynolds number approximations. The exact solutions of momentum and energy equations have been obtained. The features of flow and heattransfer characteristics are analyzed by plotting graphs.
Inspired by the above literature, and in the applications of numerous areas that have been discussed, an investigation of the impact of multi-slip and solutal boundaryconditions on MHD unsteady bioconvective micropolar nanofluid restraining gyrotactic microorganism, heat and mass transfer effect over a stretching/shrinking sheet (which have not been discussed before) was carried out. The main intent of contemporary study is the analysis of the radiative MHD Micropolar nanofluid having micro-organisms. Furthermore, the article is made more fascinating by the usage of solutal and thermal boundaryconditions with radiative heat flux in the unsteady Micropolar nanofluid fluidflow over the stretching sheet. The deportment of existing parameters is demonstrated graphically through an appropriate discussion. After that, suitable similarities have been used for transformation; the governing non-linear partial differential equations are composed in a non-linear system of ordinary differential equations (ODEs). The resulting system of non-linear ODEs has been solved numerically with a proficient and authenticated variational finite element method (FEM) along with the boundaryconditions. The influences of various parameters are studied graphically. Furthermore, the graphical narration of Nusselt number and microorganism flux is accessible and the skin friction behavior and also the impact of different parameters of the flow is numerically inspected. After that, the numerical comparison of the existing results has been presented and discussed with graphs. In view of this study, transient flow with slip effects with the existence of mixed convection and chemical reaction on the sheet/disk can be observed.
In recent years, the study of non-Newtonian fluids has achieved a lot success due to their practical applications in various fields like manufacturing of foods and papers, manufacturing of plastic sheets, etc. The study of boundary layer flow over a continuous solid surface moving with a constant speed was first studied by Sakiadis  in 1961. Later Crane  extended this problem to a stretching sheet whose surface velocity varies linearly with a certain distance from a fixed point. Chang  derived a closed form solution of the non-Newtonian flow problem of Rajgoplal et al. . Char  discussed the effects of magnetic field and power law surface temperature on heat and mass transfer from a continuous flat surface. Heat and mass transfer characteristics in the presence of transverse magnetic field were obtained by Abel et al. . Raptis , Abel and Gousia  analysed the viscoelastic fluidflow and heattransfer in the presence of thermal radiation under various physical conditions.
The problem of unsteady stagnation-point flow of a viscous and incompressible fluid by considering both the stretching and shrinking sheet situations have been investigated by Fan et al. . On the other hand, Bachok et al.  discussed the effect of melting on boundary layer stagnation-point flow towards a stretching or shrinking sheet. Ahmad et al.  investigated the behaviour of the steady boundary layer flow and heattransfer of a mi- cropolar fluid near the stagnation point on a stretching vertical surface with prescribed skin friction. Lok et al.  studied the steady axisymmetric stagnation point flow of a viscous and incompressible fluid over a shrinking circular cylinder with mass transfer (suction). Bhattacharyya et al.  analyzed the effects of partial slip on the steady boundary layer stagnation-point flow of an incompressible fluid and heattransfer towards a shrinking sheet. This investigation explores the conditions of the non-existence, existence, uniqueness and duality of the solutions of self-similar equations numerically. They also studied the same case but under the condition of un- steady-state towards a stretching. Stagnation-point flow and heattransfer over an exponentially shrinking sheet was analyzed by Bhattacharyya and Vajravelu . They obtained dual solutions for the velocity and the temper- ature fields and also they observed that their boundary layers are thinner for the first solution.
This article presents the effect of nonlinear thermal radiation on boundary layer flow and heattransfer of Carreau fluid model over a nonlinear stretching sheet embedded in a porous medium in the presence of non-uniform heat source/sink and viscous dissipation with convective boundary condition. The governing partial differential equations with the corresponding boundaryconditions are reduced to a set of ordinary differential equations using similarity transformation, which is then solved numerically by the fourth-fifth order Runge–Kutta-Fehlberg integration scheme featuring a shooting technique. The influence of significant parameters such as power law index parameter, Stretching parameter, Weissenberg number, permeability parameter, temperature ratio parameter, radiation parameter, Biot number, heat source/sink parameters, Eckert number and Prandtl number on the flow and heattransfer characteristics is discussed. The obtained results shows that for shear thinning fluid the fluid velocity is depressed by the Weissenberg number while opposite behavior for the shear thickening fluid is observed. A comparison with previously published data in limiting cases is performed and they are in excellent agreement.
The system of coupled non-linear equations (6) and (7) with the boundaryconditions (8) are solved numerically using the shooting method with fourth order Runge-Kutta scheme. In order to illustrate the salient features of the model, the numerical results are presented in Figs.2-7 and compared with the existing results. The results of this comparison are given in Table.1 with those of Refs. (Skelland., 1967; Wilkinson., 1960). It can be seen from this table that excellent agreement between the results exists. The effects of Eyring–Powell fluid parameters γ and β on the velocity and temperature profiles are displayed in Figs. 2(a)–2(d) respectively. It is witnessed that
Two-dimensional, nonlinear, steady, MHD laminar boundary layer flow with heat and mass transfer of a viscous, incompressible and electrically conducting fluid over a porous surface embedded in a porous medium in the presence of a transverse magnetic field including viscous and Joules dissipation is considered for investigation. An uniform transverse magnetic field of strength B 0 is applied parallel to y-axis. Consider a polymer sheet emerging out of a slit at x = 0 , y = 0 and subsequently being stretched, as in a polymer extrusion process. Let us assume that the speed at a point in the plate is proportional to the power of its distance from the slit and the boundary layer approximations are applicable. In writing the following equations, it is assumed that the induced magnetic field, the external electric field and the electric field due to the polarization of charges are negligible. Under these conditions, the governing boundary layer equations of momentum, energy and diffusion with visc- ous and Joules dissipation are
The effect of Eckert number (𝐸𝑐) for temperature distribution is shown in Figs. 13 and 14. It is observed from the figures that the temperature profiles increases for both fluid and dust phases when the values of 𝐸𝑐 increase. Eckert number expresses the relationship between the kinetic energy in the flow and the enthalpy. It embodies the conversion of kinetic energy into internal energy by work done against the viscous fluid stresses. The greater viscous dissipative heat causes a rise in the temperature and thermal boundary layer thickness for both fluid and particle phases. It is because heat energy is stored in the liquid due to frictional heating and this is true in both cases.
Most of the existing studies on steady boundary layer flow and heattransfer with slip conditions are limited to the non-Newtonian fluid. The considered slip conditions especially are important in the non-Newtonian fluids such as polymer melts which often exhibit wall slip. This motivates us to consider the slip conditions in the present work for non-Newtonian fluids. More exactly, our aim is to investigate steady boundary layer flow and heattransfer of a Casson fluid past a stretching sheet with slip conditions. The equations of the problem are first formulated and then transformed into their dimensionless forms where the Keller box method is applied to find the exact solutions for velocity, temperature, Skin-friction and Nusselt number.
Abstract : An analysis is carried out to study the flow and heattransfer characteristics in the laminar boundary layer flow of a second order fluid over a linearly stretching sheet with internal heat generation or absorption. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation. A numerical method, quasilinearization technique is used to study velocity and temperature profiles of the fluid. Heattransfer analysis is carried out for two types of thermal boundaryconditions namely, (i) Prescribed Surface temperature (PST) and (ii) Prescribed wall Heat Flux (PHF). The effects of various parameters on flow and heattransfer are presented through graphs and discussed.
The problem of non-linear stretching sheet for differ- ent cases of fluidflow has also been analyzed by differ- ent researchers. Vajravelu  examined fluidflow over a nonlinearly stretching sheet. Cortell  has worked on viscous flow and heattransfer over a non-linearly stret- ching sheet. Cortell  further investigated on the ef- fects of viscous dissipation and radiation on the thermal boundary layer, over a non-linearly stretching sheet. Raptis et al.  studied viscous flow over a non-linear stretching sheet in the presence of a chemical reaction and magnetic field. Abbas and Hayat  addressed the radiation effects on MHD flow due to a stretching sheet in porous space. Cortell  investigated the influence of similarity solution for flow and heattransfer of a quies- cent fluid over a non-linear stretching surface. Awang and Kechil  obtained the series solution for flow over nonlinearly stretching sheet with chemical reaction and magnetic field. Cortell  investigated the influence of similarity solution for flow and heattransfer of a quies- cent fluid over a non-linear stretching surface.
Governing equations in terms of stream functions are highly nonlinear and coupled. The closed from solutions for these equations seem impossible to obtain. Evidently, long wave length approximation is appropriate and applicable in the peristaltic flows as mentioned by Barton and Raynor (1968), Radhakrishnamacharya (1982), Zien and Ostrach (1970) and Jaffrin and Shapiro (1971). Peristaltic waves propagate with long wavelengths along the boundaries of tracts having small diameter or widths (Shapiro et al. (1969)). In the assumption of long wavelength, the ratio of channel width to wavelength becomes very small of negligible order. Physically, the transverse flow quantities become small and thus negligible as compared to the flow quantities in longitudinal directions. Further, peristalsis acts as a pump providing pressure rise in the flow direction. In such a case, the inertial effects are smaller as compared to the viscous effects (Shapiro et al. (1969)). This assumption results in the small Reynolds number. These assumptions simplify the nonlinearity of the governing equations and boundaryconditions. Consequently, the highly nonlinear governing equations along with boundaryconditions are partially linearized under the long wavelength and small Reynolds number approximations.
The natural convection processes involving the combined mechanism of heat and mass transfer are encountered in many natural and industrial transport processes such as hot rolling, wire drawing, spinning of filaments, metal extrusion, crystal growing, continuous casting, glass fiber production, paper production, and polymer processing, etc. Ostrach  the initiator of the study of convection flow, made a technical note on the similarity solution of transient free convection flow past a semi infinite vertical plate by an integral method. Goody  considered a neu- tral fluid. Sakiadis  analyzed the boundary layer flow over a solid surface moving with a constant velocity. This boundary layer flow situation is quite different from the classical Blasiuss problem of boundaryflow over a semi-infinite flat plate due to entrainment of ambient fluid.
Darcy number Da, magnetic parameter M and microrotation parameter G on velocity distribution. From fig.2 it is observed that velocity decreases with the increasing values of viscosity parameter. Since, by definition viscosity is inversely proportional to the velocity and hence the result is obvious. Fig.3 depicts the effect of Darcy number on velocity profiles. Physically, Darcy number is directly proportional to the permeability which causes higher restriction to the fluidflow which in turn slows its motion. From fig.4, it is observed that velocity reduces due to the increasing values of magnetic parameter M and it is due to the fact that the presence of magnetic field produces a Lorentz force which usually resists the momentum field; whereas from fig.5, it is observed that velocity enhances with the increasing values of microrotation parameter G because for small values of G, the viscous force is predominant as a result viscosity increases and consequently velocity decreases. Figures 6-10 depict the influence of viscosity parameter 𝜃 r , Darcy number Da,
The systems of linear non-dimensional equations, with the boundaryconditions are solved by using the Laplace Transform technique. The obtained results show the effects of the various non- dimensional governing parameters, such as Casson parameter (β), aligned angle (α) Magnetic parameter (M), Porosity parameter (K), Prandtl number (Pr), thermal Grashof number (Gr), mass Grash of number (Gm), thermal Radiation parameter (R), heat absorption parameter (Q), Schmidt number (Sc), chemical reaction parameter (Kr) and time (t) on the flow of velocity, temperature & concentration. Also Skin friction coefficient, Nusselt number and Sherwood number are presented in the tabular form. From figures 1-18 for cooling (Gr>0, Gm>0) and heating (Gr<0,Gm<0) of the plate. The heating & cooling takes place by setting up free convection currents due to temperature and concentration gradient.