Abstract: An analysis is carried out to study the **flow** and **heat** **transfer** 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**. **Heat** **transfer** analysis is carried out for two types of thermal **boundary** **conditions** 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 **heat** **transfer** are presented through graphs and discussed.

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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 [24]. Maple uses the well known Runge-Kutta-Fehlberg fourty- fifth order (RFK45) method to generate the numerical solution of a **boundary** value problem. The **boundary** **conditions** 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.

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an isothermal cone due to convective **boundary** **conditions**. 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 **heat** **transfer** 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

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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 [17] investigated the steady **flow** behavior of a Casson **fluid** in a tube. M. Nakamura et al. [18], studied the **flow** of a non-Newtonian **fluid** through an axisymmetric stenosis numerically. Mustafa et al. [19] studied and solved analytically using homotopy analysis method (HAM) for the problem unsteady **boundary** layer **flow** with **heat** **transfer** 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. [20]. An exact solution of the steady **boundary** layer **flow** of Casson **fluid** over a stretching or shrinking sheet was studied by Bhattacharyya et al. [21], and analytical solution has been given by Krishnendu Bhattacharyya et al. [22] for the problem MHD **boundary** layer **flow** of Casson **fluid** over stretching/shrinking sheet with wall mass **transfer** whereas Swati Mukhopadhyay [23] studied Casson **fluid** **flow** and **heat** **transfer** over a nonlinearly stretching surface. On the other hand Peri K. Kameswaran et al. [24] investigated and presented Dual solutions of Casson **fluid** **flow** over a stretching or shrinking sheet. Rizwan Ul Haq et al. [25] studied the **flow** of Casson nanofluid over an exponential shrinking sheet with convective **heat** **transfer** 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. [26]. M. Mustafa and Junaid Ahmad Khan [27], discussed a model for the **flow** of Casson nanofluid past a nonlinearly stretching sheet considering magnetic field effects.

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mechanical situations. However, the interaction of peristalsis and **heat** **transfer** 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 **heat** **transfer** on the peristaltic transport have been discussed by Mekheimer, Abd elmaboud and other co-workers [17, 18]. Hayat et al. [19] developed the problem by considering **heat** **transfer** effect on peristalsis **flow** of **fluid** filling the porous space in an asymmetric channel. Recently Nadeem and Akram [20], Hayat et al. [21] discussed the slip and **heat** **transfer** effect on peristaltic **flow** in an asymmetric channel under different **boundary** **conditions**. The aim of the present investigation is to highlight the importance of **heat** **transfer** 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 **heat** **transfer** characteristics are analyzed by plotting graphs.

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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 **boundary** **conditions** 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 **boundary** **conditions** with radiative **heat** flux in the unsteady Micropolar nanofluid **fluid** **flow** 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 **boundary** **conditions**. 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.

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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 [1] in 1961. Later Crane [2] extended this problem to a stretching sheet whose surface velocity varies linearly with a certain distance from a fixed point. Chang [3] derived a closed form solution of the non-Newtonian **flow** problem of Rajgoplal et al. [4]. Char [5] 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. [6]. Raptis [7], Abel and Gousia [8] analysed the viscoelastic **fluid** **flow** and **heat** **transfer** in the presence of thermal radiation under various physical **conditions**.

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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. [4]. On the other hand, Bachok et al. [5] discussed the effect of melting on **boundary** layer stagnation-point **flow** towards a stretching or shrinking sheet. Ahmad et al. [6] investigated the behaviour of the steady **boundary** layer **flow** and **heat** **transfer** of a mi- cropolar **fluid** near the stagnation point on a stretching vertical surface with prescribed skin friction. Lok et al. [7] 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. [8] analyzed the effects of partial slip on the steady **boundary** layer stagnation-point **flow** of an incompressible **fluid** and **heat** **transfer** 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 **heat** **transfer** over an exponentially shrinking sheet was analyzed by Bhattacharyya and Vajravelu [9]. 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.

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This article presents the effect of nonlinear thermal radiation on **boundary** layer **flow** and **heat** **transfer** 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 **boundary** **conditions** 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 **heat** **transfer** 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.

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The system of coupled non-linear equations (6) and (7) with the **boundary** **conditions** (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

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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

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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.

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Most of the existing studies on steady **boundary** layer **flow** and **heat** **transfer** 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 **heat** **transfer** 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.

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Abstract : An analysis is carried out to study the **flow** and **heat** **transfer** 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**. **Heat** **transfer** analysis is carried out for two types of thermal **boundary** **conditions** namely, (i) Prescribed Surface temperature (PST) and (ii) Prescribed wall **Heat** Flux (PHF). The effects of various parameters on **flow** and **heat** **transfer** are presented through graphs and discussed.

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The problem of non-linear stretching sheet for differ- ent cases of **fluid** **flow** has also been analyzed by differ- ent researchers. Vajravelu [12] examined **fluid** **flow** over a nonlinearly stretching sheet. Cortell [13] has worked on viscous **flow** and **heat** **transfer** over a non-linearly stret- ching sheet. Cortell [14] 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. [15] studied viscous **flow** over a non-linear stretching sheet in the presence of a chemical reaction and magnetic field. Abbas and Hayat [16] addressed the radiation effects on MHD **flow** due to a stretching sheet in porous space. Cortell [17] investigated the influence of similarity solution for **flow** and **heat** **transfer** of a quies- cent **fluid** over a non-linear stretching surface. Awang and Kechil [18] obtained the series solution for **flow** over nonlinearly stretching sheet with chemical reaction and magnetic field. Cortell [19] investigated the influence of similarity solution for **flow** and **heat** **transfer** of a quies- cent **fluid** over a non-linear stretching surface.

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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 **boundary** **conditions**. Consequently, the highly nonlinear governing equations along with **boundary** **conditions** are partially linearized under the long wavelength and small Reynolds number approximations.

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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 [1] 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 [2] considered a neu- tral **fluid**. Sakiadis [3] 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 **boundary** **flow** over a semi-infinite flat plate due to entrainment of ambient **fluid**.

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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 **fluid** **flow** 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,

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The systems of linear non-dimensional equations, with the **boundary** **conditions** 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.

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