1 Introduction

The flow of viscous fluid in channel with porous walls has been investigated by many researchers for different physical situations. In such a system,the consideration of thermal effects and thermal stability criteria ought to be a major part of the analysis. Various applications are found in geothermal energy extraction,drying of food,nuclear waste disposal,heat and fluid exchange inside human organs,insulation of building,groundwater movement,oil and gas production,surface catalysis of chemical reactions,regenerative heat exchange and adsorption,etc. In some of these physical systems,the fluids involved belong to the wide class of non-Newtonian fluids due to their failure in obeying the classical linear viscosity model. Thus, many researchers ^[1,2,3,4,5] discussed the viscous fluid situation. The flow due to a stretching sheet has applications in continuous casting of metals,glass blowing,and spinning of synthetic fibers. Due to such applications,Crane ^[6] initiated the flow due to a linear stretching sheet. Many authors extended the work of Ref. ^[6] to both viscous and non-Newtonian flows with different boundary conditions. Gupta and Gupta ^[7] discussed heat transfer from an isothermal stretching sheet with suction/blowing effects. Chen and Char ^[8] extended the work of Ref. ^[7] to that of a non-isothermal stretching sheet. Grubka and Bobba ^[9] carried out heat transfer analysis by considering the power-law variation of surface temperature. Chiam ^[10] investigated the magnetohydrodynamic (MHD) heat transfer over a non-isothermal stretching sheet. Series solutions of MHD peristaltic flow of a Jeffrey fluid in eccentric cylinders were discussed by Ellahi et al. ^[11] . Sheikholeslami et al. ^[12] discussed the effects of the MHD for Cu-water nanofluid flow and heat transfer by the control volume finite element method (CVFEM). Recently,Hayat et al. ^[13] discussed the unsteady flow of third grade fluid over a stretching surface with heat and mass transfer. The stagnation-point flow of nanofluid towards a stretching sheet was investigated by Mustafa et al. ^[14] . Hayat et al. ^[15] discussed the effect of the magnetic field on a peristaltic flow of a second-order fluid in a symmetric channel. Thermal analysis of the flow saturating porous media over a porous stretching plate was investigated by Tamayol et al. ^[16] . Makinde and Aziz ^[17] investigated the boundary layer flow of viscous nanofluid past a stretching sheet with a convective boundary condition. Ellahi and Hameed ^[18] discussed numerical analysis of steady non-Newtonian fluids with heat transfer,MHD,and nonlinear slip effects. Ellahi ^[19] reported a review for thermodynamics,stability,applications,and techniques of differential types of fluid.

The squeezing flow between two parallel boundaries is an interesting topic of research due to its abundant applications. Examples of such flows are quite prevalent in polymer processing,compression and injection modelling. The lubrication system can be discussed through the squeezing flow. The initial work on the squeezing flow was investigated by Stefan ^[20] . Following Stefan’s work,many researchers investigated such flows through different geometries. Recently,Hayat et al. ^[21] discussed the MHD squeezing flow of second grade fluid between two parallel plates. The unsteady squeezing flow of a Jeffrey fluid between parallel disks was investigated by Qayyum et al. ^[22] . Mustafa et al. ^[23] reported heat and mass transfer characteristics squeezing flow of viscous fluid. Domairry and Aziz ^[24] developed the homotopy perturbation solution (HPS) for MHD squeezed flow between the parallel disks. The three-dimensional squeezing flow in a rotating channel of lower stretching porous wall was discussed by Munawar et al. ^[25] .

In this attempt,we extend the work of Ref. ^[25] for mixed convection flow in presence of thermal radiation. We also consider the viscous dissipation effects. Such a consideration is significant,because the viscous dissipation effects (the generation of heat due to friction caused by shear in the flow) are important when the fluid is largely viscous or flowing at a high speed. The structure of this article is as follows. The problem development is given in Section 2. In Section 3,the solution is computed by the modern technique known as the homotopy analysis method (HAM) ^{[26,27,28,29,30,31,32]} . Section 4 contains the convergence analysis. Results and discussion are presented in Section 5. 2 Problem development

Let us consider an unsteady incompressible three-dimensional mixed convection rotating flow of viscous fluid between two infinite vertical plane walls. The fluid is electrically conducting. The plane (positioned at y = 0) is stretched with a time-dependent velocity U₀(t) = in the x-direction. The plane is located at a variable distance and it squeezes the fluid with a time dependent velocity in the negative y-direction. The fluid and the channel are rotating about the y-axis with an angular velocity and the plate y = 0 sucks the flow with a velocity A magnetic field is applied along the y-axis. The viscous dissipation effects are also considered. The flow configuration and the coordinate system are shown in Fig. 1. The governing equations for the velocity and temperature fields are as follows:

Here,u,v,and w are the velocity components in the x-,y- and z-directions,respectively. ρ is the fluid density,ν is the kinematic viscosity,p is the pressure,T is the fluid temperature,g is the magnitude of acceleration due to gravity,µ is the dynamic viscosity,c_p is the specific heat at constant pressure,k is the coefficient of thermal conductivity,B₀ is the magnetic field,and γ is the characteristic parameter.

The boundary conditions are

where a is the stretching rate of the wall y = 0,and V₀ is the suction/injection velocity. Substituting into Eqs. (2)-(5),we have with the boundary conditions where the squeezing parameter ,the suction/injection parameter ,the Prandtl number ,the magnetic parameter the rotation parameter the mixed convection parameter the modified Grashaf number and the Reynolds number Note that the plane at y = h(t) moves with the velocity V_h < 0 for S_q > 0 towards the plane at y = 0. For S_q < 0,the plane at y = h(t) moves apart with respect to the plane y = 0,and S_q = 0 corresponds to the steady case or the stationary plane. The skin friction coefficient and the Nusselt number are defined as where In terms of dimensionless variables (7) in (12),we get 3 Homotopy analysis solutions 3.1 Zeroth-order deformation problems

The initial guess F₀,G₀,θ₀,and the auxiliary linear operators L_F,L_G,and L_θ are defined as follows:

with where C_i (i = 1,2,· · · ,8) are the arbitrary constants. The nonlinear operators are The zeroth-order deformation problems are where _F ,_G ,and _θ are the auxiliary parameters,and p ∈ ^[0,1] is an embedding parameter. Note that,when p changes from 0 to 1,F(η,p) varies from F₀(η) to F (η) ,G(η,p) varies from G₀(η) to G(η),and θ(η,p) varies from θ₀(η) to θ(η). When p = 0 and p = 1,one obtains In view of the Taylor series,we can write The convergence of the series solution is dependent upon _F ,_G ,and _θ . We choose _F ,_G , and _θ in such a way that the series (32)-(34) converge at p = 1 and hence 3.2 mth-order deformation problems

The mth-order deformation equations are obtained by differentiating Eqs. (25)-(27) m times with respect to p and then setting p = 0,which are

The general solution to Eqs. (38)-(40) can be expressed as in which F_m^*,G_m^*,and θ_m^* epresent the special solutions. 3.3 Convergence analysis

The series solutions (38)-(40) contain the auxiliary parameters _F ,_G ,and _θ . These parameters adjust the convergence of the obtained series solutions.Figures 2-4 show the -curves of the functions F,G,and θ for suction. The permissible values of these auxiliary parameters _F,_G ,and _θ are −1.3 6 (_F,_G ) 6 −0.3 and −1.2 6 _θ 6 −0.4. Table 1 is useful in making a guess of what order of approximations are necessary for a convergent solutions. This table shows that the 25th-order of approximations is enough for the convergent solutions for suction.

4 HAM-based MATHEMATICA package BVPh 2.0

We also use MATHEMATICA package BVPh 2.0 to solve the coupled nonlinear ODEs (8)-(10) with the boundary conditions (11). We use BVPh ^[11,15] to obtain the 15th-order of approximations for the momentum and energy equations. The minimum squared residual errors at 15th-order of approximations are 3.237 5 × 10⁻²³ ,4.936 58 × 10⁻²⁵,and 1.697 41 × 10⁻²³ for F,G,and θ,respectively. Figure 5 is sketched for the solutions of F,G,and θ. Figure 6 shows the total residual errors corresponding to different orders of approximations.

5 Results and discussion

To investigate the effects of different parameters on the flow and heat transfer,Figs. 7-14 are plotted. The velocity and temperature profiles against η are plotted for different values of the involved parameters. The effect of Sq on the velocity component F is investigated in Fig. 7. It is observed that as S_q increases,the velocity profile F increases. The effect of the suction parameter S on the velocity profile F is discussed in Fig. 8. With the increase in S,the velocity profile F increases. The permeable nature of the right disk allows the fluid particles to move closer to the boundary,which makes the boundary layer thinner. The effects of the mixed convection parameter g_s on the velocity profile F^' and F,are shown in Figs. 9 and 10,respectively. With the increase in the mixed convection parameter g_s,the magnitude of the velocity profile increases for 0 6 η 6 0.5,and it decreases for 0.5 6 η 61. The influence of g_s on the velocity component F is plotted in Fig. 10. Here,the velocity field F decreases with the increase in g_s . The effect of the squeezing parameter Sq on the temperature profile θ is observed in Fig. 11. With the increase in S_q ,the temperature profile decreases,and the thermal boundary thickness also decreases. The influence of the Prandtl number Pr on the temperature profile θ is displayed in Fig. 12. Due to the large viscous dissipation effect,the temperature profile shows a rapid increase by increasing Pr,while the thermal boundary layer thickness decreases with increasing values of Pr. The effect of the Eckert number Econ the temperature profile θ is observed in Fig. 13. The effects of Ecand Pr are similar in a quantitative sense. The viscous dissipation effects are important only when the fluid is very viscous. This phenomenon occurs at high Eckert number (Ec >> 1). The effect of the mixed convection parameter g_s on the temperature profile θ is plotted in Fig. 14. The temperature decreases with the increase in g_s. Physically, the mixed convection parameter depends on the buoyancy force. Therefore,the increase in the mixed convection parameter means an increase in the buoyancy force. An increase in the buoyancy force gives rise to the fluid flow by which the velocity increases and the temperature decreases. Figures 15-19 present the three-dimensional plot of the velocity components of u,v, and w. Figure 15 shows the effects of x and η on the velocity component u. With the increase in x,the stretching velocity increases near the left plane and is minimum at the centre of the channel. Figures 16 and 17 show the effects of t and η on the velocity components u and v. With the increase in t and η,the velocity components u and v increase near the planes but decrease near the centre of the channel. Figures 18 and Figures 19 indicate the effects of x and t on the velocity component w. With the increase in x and t,the velocity component w increases at the centre of the channel.

Tables 2 and 3 are prepared for the numerical values of the skin friction and the Nusselt number for different parameters. With the increase in the rotation parameter Ω,the suction parameter S,the magnitude of the skin friction coefficient,and the magnitude of the local Nusselt number increase, while with the increase in the squeezing parameter S_q and g_s,the magnitude of the skin friction coefficient and the magnitude of local Nusselt number at the left plane decrease. However,the behavior of these quantities on the right plane is opposite. Table 3 shows the effects of Pr,Re,and Ec on the Nusselt number at the left and right planes. With the increase in Pr and Ec,the magnitude of the Nusselt number increases at the surface of the right plane and decreases at the surface of the left plane. With the increase in Re,the magnitude of the Nusselt number decreases at the left plane and increases at the right plane. Table 4 gives a comparison of the present results with the previous published work. This table shows that the present results are in excellent agreement with previous results in a limiting case.