Heat transfer characteristics of magnetohydrodynamic two fluid oscillatory flow in an inclined channel with saturated porous medium

Abstract

This investigation aims to examine the hydromagnetic flow of two liquid flows of fully developed incompressible Newtonian fluid in an inclined channel through the porous medium accounting for radiative heat flux, Joule heating, and viscous dissipation. The walls of the channel are maintained at different temperatures. A double perturbation method is employed to derive analytical results for velocity and temperature. Graphical results are presented for various arising parameters such as Hartmann number, Grashoff number, ratio of viscosity parameter, thermal conductivity ratio, and porous parameter. Further, the results for mass flux are presented in a tabular form and discussed. Reduction in the velocity and temperature distribution is observed by enhancing Darcy dissipation and frequency parameter. As the angle of inclination increases, there is a rise in flow and heat distribution. With the strength of the magnetic field and the rise in the Reynolds number, the mass flux decreases. A comparative study is carried out with the previously published work and the results are found to be in good agreement.

Introduction

The natural environment involves multi-phase fluid flow such as rainy, snowy winds, water, air pollution, tornadoes, and volcanoes. Chemical biological metallurgical industries and the production of oil, gas, and food are a wide variety of work processes that emphasize investigating the behavior of multi-phase systems. Studying the use of immiscible liquid in multi-layer in the natural and industrial systems is significantly increasing. The first two-phase flow was initiated by Thome. ¹ Researchers have captured the attention by reducing the power required to pump oil in a pipeline by adding appropriate water.^2–4 Packham and Shail ⁵ investigated stratified laminar flow in a horizontal pipe of two immiscible liquids. Shail ⁶ considered Hartmann flow of a conducting fluid in the channel between two horizontal insulating plates of infinite extent, there is a layer of non-conducting fluid between the conducting liquid and the upper channel wall.

The inclined geometry got a lot of attention in the study of heat transfer, especially in solar collector technology. On the other hand, the study of flow and heat transfer in two-phase inclined channel has received little attention, despite the fact that the problem is of practical importance in many areas of technology. Catton ⁷ reviewed the basic research in this area.

Models pertaining to fluid-saturated media have attracted the attention of many. In areas such as energy-efficient drying process, nuclear industry, cold storage, insulation for the buildings, and geothermal energy have an interest in studying convective flow with porous media. The macroscopic velocity of a porous medium is not small, so the force of inertia is all most vanishes. Furthermore, velocity deformation causes shear stresses, which lead to a viscous force. It is important to analyze the convective flow with a particular type of coarse porous medium with generalized Darcy law that includes the Darcy resistance and both forces of inertia and viscous.

Lohrasbi and Sahai ⁸ investigated heat transfer characteristics of magnetohydrodynamic (MHD) generator channels with the effect of the slag layer. In another study, Malashetty and Leela ⁹ examined the heat transfer situation in a horizontal channel for which both phases are electrically conducting. Malashetty and Umavathi ¹⁰ reported on heat transfer of two-phase MHD flow in an inclined channel. Umavathi et al. ¹¹ studied the velocity and temperature fields of two immiscible fluids in an inclined channel containing both fluid and porous layers. Zivojin et al. ¹² reported MHD flow and heat transfer of two immiscible fluids between moving plates in the presence of external electric and inclined magnetic fields. Nikodijevic et al. ¹³ investigated the MHD Couette flow of two immiscible fluids in a parallel plate channel in the presence of an applied electric and inclined magnetic field. Umavathi et al. ¹⁴ reported on the characteristics of the flow and heat transfer of the fluid-saturated porous medium in a double passage channel. Hasnain et al. ¹⁵ studied the MHD convection flow of two immiscible liquids in an inclined channel. Anil Bhaurao Wakale et al. ¹⁶ examined the buoyancy-driven flow of two immiscible liquids in an inclined two-dimensional confined channel with differentially heated walls. Ramana Murthy et al. ¹⁷ examined the entropy generation characteristics for an inclined channel of two immiscible micropolar fluids. Recently Hasnain et al. ¹⁸ investigated the flow and heat transfer analysis of two immiscible fluids in an inclined channel embedded in a porous medium. Kalyan and Srinivas ¹⁹ reported on the unsteady MHD flow of Casson fluid over an inclined permeable stretching sheet. Pramod and Ankit ²⁰ have explored an inclined magnetic field effect on entropy production of non-miscible fluid flow. Despite a good number of investigations reported on the subject of oscillatory flows in channels and tubes, very few works have been reported in the literature pertaining to immiscible liquids with an oscillatory pressure gradient (see Padma Devi and Srinivas ²¹ and several references therein). Bharath and Srinivas ²² studied the pulsatile flow of hydromagnetic Eyring–Powell nanofluid through a vertical porous channel. Recently Padma Devi and Srinivas ²¹ studied the thermal characteristics of flows with time-dependent pressure gradients. The MHD effects on heat and mass transfer phenomena in third-grade fluid past an inclined exponentially stretching sheet fixed in a porous medium with Darcy–Forchheimer law influence was reported recently by Abbas et al. ²³ More recently Abbas et al. ²⁴ explored the thermal-diffusion and diffusion thermo effects on heat and mass transfer in third-grade fluid with Darcy–Forchheimer relation impact over an exponentially inclined stretching sheet embedded in a porous medium.

Motivated by the studies of Malashetty et al.,^25,11 the investigation of two immiscible pulsating flow in an inclined channel with a porous medium accounting the radiative heat flux, Darcy dissipation, and Joule heating is not yet explored and hence it is the objective of the study. Such investigations of multi-phase immiscible flows with multi-component mass and heat transfer are important in several scientific and engineering disciplines, which include geophysics, plasma physics, petroleum industry, etc. Closed-form expressions have been obtained for the dimensionless velocity and temperature distributions. The influence of several arising parameters of physical interest on velocity, temperature, and rate of heat transfer is presented graphically and numerical values for flow rate are given in a tabular form and discussed quantitatively.

Figure 1.

Sketch of the model.

Mathematical formulation

The linear equations of momentum and energy equation of Newtonian fluid in an inclined channel (See Figure 1) are given by

$${\rho }_i\frac{\mathrm{\partial }{\overrightarrow{U}}_i}{\mathrm{\partial }t}=-\mathrm{\nabla }P+\overrightarrow{J}\times \overrightarrow{B}+{\rho }_i\overline{g}-\frac{{\mu }_i}{k}{\overrightarrow{U}}_i+{\mu }_i{\mathrm{\nabla }}^2{\overrightarrow{U}}_i$$

(1)

$$\frac{\mathrm{\partial }{T}_i}{\mathrm{\partial }t}=k_i{\mathrm{\nabla }}^2{T}_i+\frac{J^2}{{{\sigma }_i}^2}-\mathrm{\nabla }.q_r+\mu {\mathrm{\Phi }}_i+\frac{{\mu }_i}{k}{\overrightarrow{U}}_i.{\overrightarrow{U}}_i$$

(2)

In equation (2), the terms included are Joule heating, radiative heat flux, and viscous and Darcy’s dissipation. To obtain high permeability porous medium, Brinkman extension is applied. The porous medium in the two regions is isotropic and homogenous. Further, the liquid and solid phases in the porous space are with constant thermophysical properties. The walls of the channel are impermeable, and flow is considered two-dimensional.

Consider the fully developed steady laminar flow of two immiscible incompressible liquids through an inclined channel with a porous medium. Different constant temperatures are maintained at the walls of the channel. The two regions (Region I) and (Region II) are filled with incompressible Newtonian fluids. The governing equations of the momentum and energy under Oberbeck–Boussinesq approximation are given by (Malashetty et al.^25,11 Padma and Srinivas ²¹ ) and

Region I

$$\begin{array}{rl}{\rho }_1\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }t}& =-\frac{\mathrm{\partial }p}{\mathrm{\partial }x}+{\mu }_1\left(\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{y_1}^2}\right)+{\rho }_{1}g{\beta }_1(T_1-T_0)\sin \varphi \\ & -\frac{{\mu }_1{u}_1}{k}-{\sigma }_1{B_0}^2{u}_1\\ \left({\rho }_1{c}_p\right)\frac{\mathrm{\partial }{T}_1}{\mathrm{\partial }t}& =k_1\frac{{\mathrm{\partial }}^2{T}_1}{\mathrm{\partial }{y}^2}+{\mu }_1{\left(\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }{y}_1}\right)}^2+{\sigma }_1{B_0}^2{u_1}^2\end{array}$$

(3)

$$-\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }{y}_1}+\frac{{\mu }_1}{k}{u_1}^2$$

(4)

Region II

$$\begin{array}{rl}{\rho }_2\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }t}& =-\frac{\mathrm{\partial }p}{\mathrm{\partial }x}+{\mu }_2\left(\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{y_2}^2}\right)+{\rho }_{2}g{\beta }_2(T_2-T_0)\sin \varphi \\ & -\frac{{\mu }_2{u}_2}{k}-{\sigma }_2{B_0}^2{u}_2\\ \left({\rho }_{2}cp\right)\frac{\mathrm{\partial }{T}_2}{\mathrm{\partial }t}& =k_2\frac{{\mathrm{\partial }}^2{T}_2}{\mathrm{\partial }{y_2}^2}+{\mu }_2{\left(\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{y}_2}\right)}^2+{\sigma }_2{B_0}^2{u_2}^2\end{array}$$

(5)

$$-\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }{y}_2}+\frac{{\mu }_2}{k}{u_2}^2$$

(6)

The conditions used at the boundary and interface (26), are

$$\begin{array}{r}\begin{array}{cc}& u_1=0\\ & T_1=T_{w1}\end{array}{|}_{y=-h}\end{array}$$

(7)

$$\begin{array}{r}\begin{array}{cc}& u_2=0\\ & T_2=T_{w2}\end{array}{|}_{y=h}\end{array}$$

(8)

$$\begin{array}{r}\begin{array}{cc}& u_1=u_2\\ & T_{w1}=T_{w2}\end{array}{|}_{y=0}\end{array}$$

(9)

$$\begin{array}{cc}& {\mu }_1\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }y}{|}_{y=0}={\mu }_2\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }y}{|}_{y=0}\\ & k_1\frac{\mathrm{\partial }{T}_1}{\mathrm{\partial }y}{|}_{y=0}=k_2\frac{\mathrm{\partial }{T}_2}{\mathrm{\partial }y}{|}_{y=0}\end{array}$$

(10)

Non-dimensional quantities and pressure gradients are given by

$$\begin{array}{c}{u_1}^{*}=\frac{u_i}{{\overline{u}}_1},y_1=\frac{y_i}{h_i},{\theta }_i=\frac{T_i-T_0}{T_{w2}-T_{w1}},Gr=\frac{g{\beta }_1{h_1}^3(T_{w2}-T_{w1})}{{v_1}^2}\\ Pr=\frac{{\mu }_1}{k_1}cp,\sigma =\frac{h}{\sqrt{k}},Ec=\frac{{\overline{u}}_1}{cp(T_{w2}-T_{w1})}\\ P=\frac{{h_1}^2}{{\mu }_1{\overline{u}}_1}\left(\frac{\mathrm{\partial }p}{\mathrm{\partial }x}\right),b=\frac{{\beta }_2}{{\beta }_1},h=\frac{h_1}{h_2},k=\frac{k_1}{k_2},m=\frac{{\mu }_1}{{\mu }_2}\\ n=\frac{{\rho }_2}{{\rho }_1},S=\frac{{\sigma }_2}{{\sigma }_1},t^{*}=\frac{u}{h^2}t\\ \text{ Rd = }\frac{4{\sigma }^{*}}{\mathit{k}{\mathit{k}}_{\mathit{1}}}{\mathit{T}}_1^3,{\mathit{q}}_{\mathit{r}}=\frac{4\sigma *}{3{\mathit{k}}^{*}}\frac{\mathrm{\partial }{\mathit{T}}^{{*}^4}}{\mathrm{\partial }{y}^{*}}\text{ where; }{\mathit{T}}^{{*}^4}\cong 4{\mathit{T}}_1^3{\mathit{T}}^{*}-3{\mathit{T}}_1^4\\ M=B_{0}h\sqrt{\frac{\sigma }{{\mu }_1}},Re\text{=}\frac{{\rho }_1\mathit{h}\mathit{u}}{{\mu }_1}\end{array}$$

(11)

where $Pr$ , $Gr$ , $Rd$ , $Ec$ , $M$ , and $Re$ are Prandtl Number, Grashoff number, Thermal Radiation, Eckert number, Hartmann number, and Reynolds number, respectively. Non-dimensional equations after dropping “asterisks” are as follows:

Region I

$$\begin{array}{rl}\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }t}& =-P+\left(\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{y_1}^2}\right)+\frac{Gr}{\mathrm{Re}}{\theta }_1\sin \varphi -({\sigma }^2+M^2)u_1\\ \frac{\mathrm{\partial }{\theta }_1}{\mathrm{\partial }t}& =(\frac{1}{Pr}+\frac{4Rd}{3Pr})\frac{{\mathrm{\partial }}^2{\theta }_1}{\mathrm{\partial }{y_1}^2}+Ec{\left(\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }y}\right)}^2\end{array}$$

(12)

$$+(M^2+{\sigma }^2){u_1}^{2}Ec.$$

(13)

Region II

$$\begin{array}{rl}mn\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }t}& =-mP{h}^2+\left(\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{y_2}^2}\right)+bmn{h}^2\frac{Gr}{\mathrm{Re}}{\theta }_2\sin \varphi \\ & -{\sigma }^2{u}_2{h}^2-Sm{h}^2{M}^2{u}_2\\ nk\frac{\mathrm{\partial }{\theta }_2}{\mathrm{\partial }t}& =\frac{1}{Pr}\frac{{\mathrm{\partial }}^2{\theta }_2}{\mathrm{\partial }{y_2}^2}+\frac{k}{m}Ec{\left(\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{y}_2}\right)}^2+S{h}^{2}K{M}^2{u_2}^{2}Ec\end{array}$$

(14)

$$+\frac{4Rd}{3Pr}\frac{{\mathrm{\partial }}^2{\theta }_2}{\mathrm{\partial }{y_2}^2}+\frac{{\sigma }^2{h}^{2}K{u_2}^{2}Ec}{m}$$

(15)

Non-dimensional boundary and interface conditions of velocity and temperature

$$\begin{array}{r}\begin{array}{cc}& u_1=0\\ & {\theta }_1=0\end{array}{|}_{y=-1}\end{array}$$

(16)

$$\begin{array}{r}\begin{array}{cc}& u_2=0\\ & {\theta }_2=1\end{array}{|}_{y=1}\end{array}$$

(17)

$$\begin{array}{r}\begin{array}{cc}& u_1=u_2\\ & {\theta }_1={\theta }_2\end{array}{|}_{y=0}\end{array}$$

(18)

$$\begin{array}{cc}& {\mu }_1\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }y}{|}_{y=0}={\mu }_2\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }y}{|}_{y=0}\\ & k_1\frac{\mathrm{\partial }{\theta }_1}{\mathrm{\partial }y}{|}_{y=0}=k_2\frac{\mathrm{\partial }{\theta }_2}{\mathrm{\partial }y}{|}_{y=0}\end{array}$$

(19)

velocity and temperature distributions, for the governing flow problem, can be obtained by assuming

$$\begin{array}{cc}{u}_1& =u_{11}+\epsilon u_{12}{e}^{i\omega t}\\ {\theta }_1& ={\theta }_{11}+\epsilon {\theta }_{12}{e}^{i\omega t}\\ u_2& =u_{21}+\epsilon u_{22}{e}^{i\omega t}\\ {\theta }_2& ={\theta }_{21}+\epsilon {\theta }_{22}{e}^{i\omega t}\\ -P& =P_s+\epsilon P_o{e}^{i\omega t}\end{array}$$

(20)

where $P_s$ and $P_o$ are considered to be steady and oscillating pressure gradient. substituting the equation (20) in the equations (12) to (15) we have

Region I

$$\frac{d^2{u}_{11}}{d{y_1}^2}+{\theta }_{11}B-F_1{u}_{11}=-P_s$$

(21)

$$\frac{d^2{u}_{12}}{d{y_1}^2}+{\theta }_{12}B-F_2{u}_{12}=-P_o$$

(22)

$$A\frac{d^2{\theta }_{11}}{d{y_1}^2}+Ec{\left(\frac{d{u}_{11}}{d{y}_1}\right)}^2+F_{1}Ec{u_{11}}^2=0$$

(23)

$$A\frac{d^2{\theta }_{12}}{d{y_1}^2}+2Ec\frac{d{u}_{11}}{d{y}_1}\frac{d{u}_{12}}{d{y}_1}+2F_{1}Ec{u}_{11}{u}_{12}=i\omega {\theta }_{12}$$

(24)

Region II

$$P_{s1}+\frac{d^2{u}_{21}}{d{y_2}^2}+J_1{\theta }_{21}-J_2{u}_{21}=0$$

(25)

$$J_5{u}_{22}=P_{o1}+\frac{d^2{u}_{22}}{d{y_2}^2}+J_1{\theta }_{22}-J_2{u}_{22}$$

(26)

$$A\frac{d^2{\theta }_{21}}{d{y_2}^2}+J_{6}Ec{\left(\frac{d{u}_{21}}{d{y}_2}\right)}^2+J_{7}Ec{u_{21}}^2=0$$

(27)

$$A\frac{d^2{\theta }_{22}}{d{y_2}^2}+2J_{6}Ec\frac{d{u}_{21}}{d{y}_2}\frac{d{u}_{22}}{d{y}_2}+2J_{7}Ec{u}_{21}{u}_{22}=J_{10}{\theta }_{22}$$

(28)

Double perturbation on Eckert number is applied due to the coupled and nonlinearity in the equation due to the presence of parameters Joule heating and thermal radiation.

$$\begin{array}{cc}& u_{11}=u_{111}+Ec{u}_{112}\\ & {\theta }_{11}={\theta }_{111}+Ec{\theta }_{112}\\ & u_{12}=u_{121}+Ec{u}_{122}\\ & {\theta }_{12}={\theta }_{121}+Ec{\theta }_{122}\\ & u_{21}=u_{211}+Ec{u}_{212}\\ & {\theta }_{21}={\theta }_{211}+Ec{\theta }_{212}\\ & u_{22}=u_{221}+Ec{u}_{222}\\ & {\theta }_{22}={\theta }_{221}+Ec{\theta }_{222}\end{array}$$

(29)

substituting equation (29) in equations (21) to (27) and collecting the coefficients of ${Ec}^0$ and ${Ec}^1$ we get:

Region I

$$\frac{d^2{u}_{111}}{d{y_1}^2}+{\theta }_{111}B-F_1{u}_{111}=-P_s$$

(30)

$$\frac{d^2{u}_{112}}{d{y_1}^2}+{\theta }_{112}B-F_1{u}_{112}=0$$

(31)

$$A\frac{d^2{\theta }_{111}}{d{y_1}^2}=0$$

(32)

$$A\frac{d^2{\theta }_{112}}{d{y_1}^2}+{\left(\frac{d{u}_{111}}{d{y}_1}\right)}^2+F_1{u_{111}}^2=0$$

(33)

$$\frac{d^2{u}_{121}}{d{y_1}^2}+{\theta }_{121}B-F_2{u}_{121}=-P_o$$

(34)

$$\frac{d^2{u}_{122}}{d{y_1}^2}+{\theta }_{122}B-F_2{u}_{122}=0$$

(35)

$$A\frac{d^2{\theta }_{121}}{d{y_1}^2}=i\omega {\theta }_{121}$$

(36)

$$A\frac{d^2{\theta }_{122}}{d{y_1}^2}+2\frac{d{u}_{111}}{d{y}_1}\frac{d{u}_{121}}{d{y}_1}+2F_1{u}_{111}{u}_{121}=i\omega {\theta }_{122}$$

(37)

Region II

$$P_{s1}+\frac{d^2{u}_{211}}{d{y_2}^2}+J_1{\theta }_{211}-J_2{u}_{211}=0$$

(38)

$$\frac{d^2{u}_{212}}{d{y_2}^2}+J_1{\theta }_{212}-J_2{u}_{212}=0$$

(39)

$$J_5{u}_{221}=P_{o1}+\frac{d^2{u}_{221}}{d{y_2}^2}+J_1{\theta }_{221}-J_2{u}_{221}$$

(40)

$$J_5{u}_{222}=\frac{d^2{u}_{222}}{d{y_2}^2}+J_1{\theta }_{222}-J_2{u}_{222}$$

(41)

$$A\frac{d^2{\theta }_{211}}{d{y_2}^2}=0$$

(42)

$$A\frac{d^2{\theta }_{212}}{d{y_2}^2}+J_6{\left(\frac{d{u}_{211}}{d{y}_2}\right)}^2+J_7{u_{211}}^2=0$$

(43)

$$A\frac{d^2{\theta }_{221}}{d{y_2}^2}=J_{10}{\theta }_{221}$$

(44)

$$A\frac{d^2{\theta }_{222}}{d{y_2}^2}+2J_6\frac{d{u}_{211}}{d{y}_2}\frac{d{u}_{221}}{d{y}_2}+2J_7{u}_{221}=J_{10}{\theta }_{222}$$

(45)

$$\begin{array}{cc}& u_{111}(-1)=0\\ & u_{112}(-1)=0\\ & u_{121}(-1)=0\\ & u_{122}(-1)=0\\ & u_{211}\left(1\right)=0\\ & u_{212}\left(1\right)=0\\ & u_{221}\left(1\right)=0\\ & u_{222}\left(1\right)=0\\ & u_{111}\left(0\right)=u_{211}\left(0\right)\\ & u_{112}\left(0\right)=u_{212}\left(0\right)\\ & u_{121}\left(0\right)=u_{221}\left(0\right)\\ & u_{122}\left(0\right)=u_{222}\left(0\right)\\ & u_{111}{}^{\prime }\left(0\right)=u_{211}{}^{\prime }\left(0\right)\\ & u_{112}{}^{\prime }\left(0\right)=u_{212}{}^{\prime }\left(0\right)\\ & u_{121}{}^{\prime }\left(0\right)=u_{221}{}^{\prime }\left(0\right)\\ & u_{122}{}^{\prime }\left(0\right)=u_{222}{}^{\prime }\left(0\right)\\ & {\theta }_{111}(-1)=0\\ & {\theta }_{121}(-1)=0\\ & {\theta }_{112}(-1)=0,\\ & {\theta }_{122}(-1)=0\\ & {\theta }_{211}\left(1\right)=1\\ & {\theta }_{212}\left(1\right)=0\\ & {\theta }_{221}\left(1\right)=0\\ & {\theta }_{222}\left(1\right)=0\\ & {\theta }_{111}\left(0\right)={\theta }_{211}\left(0\right)\\ & {\theta }_{112}\left(0\right)={\theta }_{212}\left(0\right)\\ & {\theta }_{121}\left(0\right)={\theta }_{221}\left(0\right)\\ & {\theta }_{122}\left(0\right)={\theta }_{222}\left(0\right)\\ & {\theta }_{111}{}^{\prime }\left(0\right)={\theta }_{211}{}^{\prime }\left(0\right)\\ & {\theta }_{112}{}^{\prime }\left(0\right)={\theta }_{212}{}^{\prime }\left(0\right)\\ & {\theta }_{121}{}^{\prime }\left(0\right)={\theta }_{221}{}^{\prime }\left(0\right)\\ & {\theta }_{122}{}^{\prime }\left(0\right)={\theta }_{222}{}^{\prime }\left(0\right)\end{array}$$

(46)

Solution of the problem

Region I

$$\begin{array}{rl}{u}_{111}\left(y\right)& =\{\begin{array}{c}{c}_3\cosh \left(\sqrt{F1}y\right)+c_4\sinh \left(\sqrt{F1}y\right)\\ +F_{10}-F_{11}\end{array}\end{array}$$

(47)

$$\begin{array}{rl}{u}_{112}\left(y\right)& =\{\begin{array}{c}{c}_7\cosh (\sqrt{F_1}y)c_8\sinh (\sqrt{F_1}y)+F_{28}{y}^2\\ +F_{29}-F_{30}{y}^4+F_{31}\cosh \left(2\sqrt{F_1}y\right)\\ +F_{32}\sinh \left(2\sqrt{F_1}y\right)+F_{33}y\cosh \left(\sqrt{F_1}y\right).\end{array}\end{array}$$

(48)

$$\begin{array}{rl}{u}_{121}\left(y\right)& =\{\begin{array}{c}{c}_{11}\cosh (\sqrt{F_2}y)+c_{12}\sinh \left(\sqrt{F_2}y\right)\\ +F35\cosh \left(\sqrt{D1}y\right)+F_{36}\sinh \left(\sqrt{D1}y\right)\\ +\frac{P_o}{F_2}\end{array}\end{array}$$

(49)

$$\begin{array}{rl}{u}_{122}\left(y\right)& =\{\begin{array}{c}{c}_{15}\cosh \left(\sqrt{F_2}y\right)+c_{16}\sinh \left(\sqrt{F_2}y\right)\\ -F_{61}\cosh \left(\sqrt{D_1}y\right)+F_{62}\sinh \left(\sqrt{D_1}y\right)\\ +F_{63}\cosh (\sqrt{F_3}y)\end{array}\end{array}$$

(50)

$${\theta }_{111}=c_1+c_{2}y$$

(51)

$$\begin{array}{rl}{\theta }_{112}\left(y\right)& =\{\begin{array}{c}\frac{F_{20}}{2}{y}^2-\frac{F_{21}}{12}{y}^4+\frac{F_{72}}{6}{y}^3\\ +F_{22}\cosh \left(2\sqrt{F_1}y\right)+F_{23}\sinh \left(\sqrt{F_1}y\right)\\ +F_{24}\cosh \left(\sqrt{F_1}y\right)+F_{25}\sinh \left(\sqrt{F_1}y\right)\\ +F_{26}y\cosh \left(\sqrt{F_1}y\right)\end{array}\end{array}$$

(52)

$${\theta }_{121}\left(y\right)=c_9\cosh \left(\sqrt{D_1}y\right)+c_{10}\sinh \left(\sqrt{D_1}y\right)$$

(53)

$$\begin{array}{rl}{\theta }_{122}\left(y\right)& =\{\begin{array}{c}{c}_{13}\cosh \left(\sqrt{D_1}y\right)+c_{14}\sinh \left(\sqrt{D_1}y\right)\\ +F_{53}\cosh \left(\sqrt{F_3}y\right)+F_{54}\cosh \left(\sqrt{F_4}y\right)\\ +F_{55}\sinh \left(\sqrt{F_3}y\right)+F_{56}\sinh \left(\sqrt{F_4}y\right)\end{array}\end{array}$$

(54)

Region II

$$\begin{array}{rl}{u}_{211}\left(y\right)=& \{\begin{array}{c}{c}_{19}\cosh \left(\sqrt{J_2}y\right)+c_{20}\sinh \left(\sqrt{J_2}y\right)\\ -J_8+J_{9}y\end{array}\end{array}$$

(55)

$$\begin{array}{rl}{u}_{212}\left(y\right)& =\{\begin{array}{c}{c}_{23}\cosh \left(\sqrt{J_2}y\right)+c_{24}\sinh \left(\sqrt{J_2}y\right)\\ +J_{30}\cosh \left(2\sqrt{J_2}y\right)+J_{31}\sinh \left(2\sqrt{J_2}y\right)\\ +J_{32}y\cosh \left(\sqrt{J_{32}}y\right)\end{array}\end{array}$$

(56)

$$\begin{array}{rl}{u}_{221}\left(y\right)& =\{\begin{array}{c}{c}_{27}\cosh \left(\sqrt{D_2}y\right)+c_{28}\sinh \left(\sqrt{D_2}y\right)\\ +J_{41}\cosh \left(\sqrt{J_{10}}y\right)+J_{42}\sinh \left(\sqrt{J_{10}}y\right)\\ +\frac{P_{o1}}{J_{10}}\end{array}\end{array}$$

(57)

$$\begin{array}{rl}{u}_{221}\left(y\right)& =\{\begin{array}{c}{c}_{31}\cosh \left(\sqrt{D_2}y\right)+c_{32}\sinh \left(\sqrt{D_2}y\right)\\ -J_{83}\cosh \left(\sqrt{J_{10}}y\right)-J_{84}\sinh \left(\sqrt{J_{10}}y\right)\\ -J_{85}\cosh \left(\sqrt{D_5}y\right)-J_{86}\cosh \left(\sqrt{D_6}y\right)\\ -J_{87}\sinh \left(\sqrt{D_5}y\right)-J_{88}\sinh \left(\sqrt{D_6}y\right)\\ -J_{89}\cosh \left(\sqrt{D_7}y\right)-J_{90}\cosh \left(\sqrt{D_2}y\right)\\ -J_{91}\sinh \left(\sqrt{D_7}y\right)-J_{92}\sinh \left(\sqrt{D_8}y\right)\\ -J_{93}\cosh \left(\sqrt{D_2}y\right)-J_{94}y\sinh \left(\sqrt{D_2}y\right)\\ -J_{95}y\cosh \left(\sqrt{J_{10}}y\right)-J_{96}y\sinh \left(\sqrt{J_{10}}y\right)\\ -J_{97}\sinh \left(\sqrt{J_{10}}y\right)-J_{98}\cosh \left(\sqrt{J_{10}}y\right)\\ -J_{99}{y}^2\sinh \left(\sqrt{D_2}y\right)-J_{100}{y}^2\cosh \left(\sqrt{D_2}y\right)\\ -J_{102}{y}^2\sinh \left(\sqrt{J_{10}}y\right)-J_{103}{y}^2\cosh \left(\sqrt{J_{10}}y\right)\\ -H_{17}\cosh \left(\sqrt{J_2}y\right)-H_{18}\sinh \left(\sqrt{J_2}y\right)\\ -H_{19}-H_{20}y\end{array}\end{array}$$

(58)

$${\theta }_{211}\left(y\right)=c_{17}+c_{18}y$$

(59)

$$\begin{array}{rl}{\theta }_{212}\left(y\right)& =\{\begin{array}{c}{J}_{21}\cosh \left(2\sqrt{J_2}y\right)+J_{22}\sinh \left(2\sqrt{J_2}y\right)\\ +J_{23}y\cosh \left(\sqrt{J_2}y\right)+J_{24}y\sinh \left(\sqrt{J_2}y\right)\\ +J_{25}\sinh \left(\sqrt{J_2}y\right)+J_{26}\cosh \left(\sqrt{J_2}y\right)\\ +J_{27}{y}^2+J_{28}{y}^4+J_{29}{y}^3+c_{21}y+c_{22}\end{array}\end{array}$$

(60)

$${\theta }_{221}\left(y\right)=c_{25}\cosh \left(\sqrt{J_{10}}y\right)+c_{26}\sinh \left(\sqrt{J_{10}}y\right)$$

(61)

$$\begin{array}{rl}{\theta }_{222}\left(y\right)& =\{\begin{array}{c}{c}_{29}\cosh \left(\sqrt{J_{10}}y\right)+c_{30}\sinh \left(\sqrt{J_{10}}y\right)\\ +J_{67}\cosh \left(\sqrt{D_5}y\right)+J_{68}\cosh \left(\sqrt{D_6}y\right)\\ +J_{69}\sinh \left(\sqrt{D_5}y\right)+J_{70}\sinh \left(\sqrt{D_6}y\right)\\ +J_{71}\cosh \left(\sqrt{D_7}y\right)+J_{72}\cosh \left(\sqrt{D_8}y\right)\\ +J_{73}\sinh \left(\sqrt{D_7}y\right)+J_{74}\sinh \left(\sqrt{D_8}y\right)\\ +J_{75}\cosh \left(\sqrt{D_2}y\right)+J_{76}\sinh \left(\sqrt{D_2}y\right)\\ +J_{77}y\cosh \left(\sqrt{J_{10}}y\right)+J_{78}y\sinh \left(\sqrt{J_{10}}y\right)\\ +J_{79}{y}^2\sinh \left(\sqrt{J_{10}}y\right)+J_{80}{y}^2\cosh \left(\sqrt{J_{10}}y\right)\\ +J_{81}y\cosh \left(\sqrt{D_2}y\right)+J_{82}y\sinh \left(\sqrt{D_2}y\right)\\ +H_{13}\cosh \left(\sqrt{J_2}y\right)+H_{14}\sinh \left(\sqrt{J_2}y\right)\\ +H_{15}+H_{16}y\end{array}\end{array}$$

(62)

Rate of heat transfer

The Nusselt number is given by

$$Nu={\left(\frac{\mathrm{\partial }{\theta }_1}{\mathrm{\partial }y}\right)}_{y=-1,1}+\epsilon e^{i\omega t}{\left(\frac{\mathrm{\partial }{\theta }_2}{\mathrm{\partial }y}\right)}_{y=-1,1}$$

(63)

Mass flux

The flow rate is given by

$$Q={\int }_{-1}^0{u}_1\left(y\right)dy+{\int }_0^1{u}_2\left(y\right)dy$$

(64)

Results and discussion

The solutions obtained for the flow variables are evaluated numerically and the results are discussed graphically (Figures 2 to 5) for different parameters and numerical values for mass flux (Table 1) are presented. The velocity falls as the Reynolds number increases, as shown in Figure 2(a). A drag force is caused due to the applied magnetic field in the direction against the flow, resulting in a decrease in fluid velocity, as shown in Figure 2(b). The effect of the Grashoff number on the velocity distribution is represented in Figure 2(c). A higher Grashof number corresponds to more buoyancy forces and more free convection since it measures the ratio of buoyancy forces to viscous forces. As a consequence, the fluid velocity rises with the increase of Grashof number, and the same is reflected in Figure 2(c). The effect of $\omega$ is shown in Figure 2(d), which shows that the velocity decreases as the frequency parameter increases (Figure 2(e)). Because of the damping effect of Darcy’s resistance, frictional resistance against convection is very high for the values of the space porosity that lowers the fluid velocity in lower and upper regions, and as a result, the velocity decreases (Figure 2(f)). The fluid’s velocity increases as $m$ increases. The viscosity of the fluid in the upper region is small when compared to the lower region (Figure 2(g)). As the magnitude of the driving forces increases, when the angle of inclination of the velocity raises. Figure 3(a) shows that the temperature distribution enhances. $\varphi$ denotes that the channel tends to be vertical, and thus the effect of buoyancy force increases due to gravity. The frequency parameter raises as the temperature decreases, as shown in Figure 3(b). The temperature falls in Figure 3(c) as $M$ increases. Temperature decreases as porosity increases, as shown in Figure 3(d). The variation of Prandtl number on the temperature distribution of the fluid is depicted in Figure 3(e). The increase in viscous diffusion enhances internal temperature due to the presence of viscous dissipation and hence there is a rise in temperature, with a rise of $Pr$ , as shown in Figure 3(e). A rise in the Eckert number shows a high kinetic energy, which causes fluid molecules to vibrate and collide more frequently. Hence higher temperature profile, Figures 3(f) and 5(c) result from the enhanced heat dissipation in the boundary layer region due to more molecular collisions. Figure 3(g) shows that as $k$ increases, the temperature distribution decreases due to a decrease in thermal conduction. Further, from Figure 3(h), for the smaller values of electrical conductivity, there is a fall in temperature distribution. Figures 4(a), (b), and 5(a), (b) show the velocity and temperature of oscillating unsteady profiles of frequency parameter and porosity. Temperature enhances with the rise $Ec$ in reveals in Figure 5(c), which can be observed. Figure 5(d) shows an increase in temperature as the Prandtl number increases. The heat transfer rate at the channel wall is observed in Figure 6. As shown in Figure 6(a) and (b), increasing thermal radiation causes a reduction in the rate of heat flow at the lower wall and an enhancement at the upper wall. With a rise in the porosity parameter, the rate of heat transfer drops at the lower wall and enhances at the upper wall as displayed in Figure 6(c) and (d). One can observe that with the strength of the electrical conductivity, the heat transfer rate at the lower wall falls (Figure 6(e)) and reverse trend at the upper wall (Figure 6(f)). Figure 6(g) and (h) indicates that the heat transfer rate enhances at the lower wall and decreases at the upper wall with the rise of the Eckert number. In Figure 7(a), for accuracy and validity, we compared the results of the present investigation with that of Malashetty et al. ²⁵ in Figure 7(a) for velocity distribution, in the absence of thermal radiation and porosity. One can notice that our results are in good agreement with that of Malashetty et al. ²⁵ Further, a comparative study is also performed in Figure 7(b) for velocity distribution, with numerical results obtained by using the Runge–Kutta fourth-order with shooting technique. It is observed that the numerical and analytical results are in excellent agreement.

Evaluation of mass flux, for various parameters, is obtained and numerical values are presented in Table 1. From Table 1(a), it can be observed that $Re$ rises and mass flux decreases. Table 1(b) shows that as the density ratio increases, consequently does the mass flux increases $\omega t$ . The mass flux increases as $m$ increases in Table 1(c). Mass flux, for various values of Hartmann Number, has been depicted in Table 1(d) . Oscillatory behavior is observed as the magnetic field strength increases. For fixed $n$ , a fall in the mass flux can be observed for a rise in $\omega$ .

Table 1.

(a) Variation of mass flux with Reynolds number $(M=1,S=6,\alpha =1,n=1.4,m=\mathrm{1.5.})$ (b) Variation of mass flux with ratio of density $(,Re=2,M=1,S=6,\alpha =1,m=1.2)$ (c) Variation of mass flux with ratio of viscosity $(Re=2,M=1,S=6,\alpha =1,n=1.4)$ (d) Variation of mass flux with Hartmann number $(Re=2,S=6,\alpha =1,n=1.4,m=1.2).$

$\omega t\mathrm{\setminus }Re$	1	3	5	7	9
0	1.901637	1.054928	0.910522	0.851040	0.8186095
$\pi /4$	1.917179	1.050969	0.902981	0.841997	0.808740
$\pi /2$	1.889508	1.017631	0.868607	0.807189	0.773693
$\pi$	1.785181	0.946703	0.803825	0.744986	0.712909
$\omega t\mathrm{\setminus }n$	1	1.1	1.2	1.3	1.4
0	1.155250	1.177734	1.200479	1.223486	1.246756
$\pi /4$	1.154927	1.177687	1.200686	1.223930	1.247423
$\pi /2$	1.121795	1.144864	1.168154	1.191672	1.215425
$\pi$	1.042589	1.065736	1.089113	1.112721	1.136566
$\omega t\mathrm{\setminus }m$	1	1.1	1.2	1.3	1.4
0	1.182185	1.216369	1.246756	1.273949	1.298434
$\pi /4$	1.182134	1.216695	1.247423	1.274927	1.299697
$\pi /2$	1.150942	1.185071	1.215425	1.242602	1.267084
$\pi$	1.075759	1.107938	1.136566	1.162204	1.185305
$\omega t\mathrm{\setminus }M$	1	1.1	1.2	1.3	1.4
0	1.246756	0.7190247	0.464682	1.273949	65.100285
$\pi /4$	1.247423	0.715842	0.460926	59.691081	334.723713
$\pi /2$	1.215425	0.693029	0.444111	51.508277	289.398438
$\pi$	1.136566	0.6456387	0.412583	44.812215	245.060826

Figure 2.

Velocity profiles (a) varying Reynolds number $M=1$ , $Gr=1$ , $\varphi =30$ , $s=6$ , $\sigma =1.4$ , $Pr=6.9$ , $Ec=0.01$ (b) varying Hartmann numebr $Re=5$ , $n=1.5$ , $Gr=1$ , $\sigma =1.4$ , $s=2$ , $\varphi =30$ , $Pr=6.9$ , $Ec=0.01$ (c) varying frequency parameter $Re=5$ , $n=1.5$ , $M=1$ , $Gr=1$ , $\varphi =30$ , $s=2$ , $\sigma =1.4$ , $Pr=6.9$ , $Ec=0.01$ (d) varying Grasoff number $Re=5$ , $n=1.5$ , $M=1$ , $\varphi =30$ , $s=2$ , $\sigma =1.4$ , $Pr=6.9$ , $Ec=0.01$ (e) varying porasity parameter $Re=5$ , $n=1.5$ , $M=1$ , $Gr=1$ , $\varphi =30$ , $s=6$ , $Pr=6.9$ , $Ec=0.01$ (f) varying Ratio of viscosity $Re=5$ , $n=1.5$ , $M=1$ , $Gr=1$ , $\varphi =30$ , $s=6$ , $\sigma =1.4$ , $Pr=6.9$ , $Ec=0.01$ (g) varying angle of inclination $Re=1$ , $n=1.4$ , $M=1$ , $Gr=1$ , $s=6$ , $\sigma =1$ , $Pr=6.9$ , $Ec=0.01$ .

Figure 3.

Temperature profiles (a)varying the inclination angle $n=1.4$ $Re=1$ , $M=1$ , $m=1.2$ , $Gr=1$ , $s=6$ , $\sigma =1.4$ , $k=1$ , (b) varying the frequency parameter, $Re=5$ , $n=1.4$ , $k=0.5$ , $M=1$ , $Gr=5$ , $m=0.5$ , $s=2$ , $\sigma =1.4$ (c) varying the Hartmann number $Re=5$ , $m=0.5$ , $k=1$ , $Gr=1$ , $s=2$ , $\sigma =1.4$ , $n=1.4$ (d) varying porasity $Re=5$ , $n=1.4$ , $M=1$ , $Gr=5$ , $m=0.5$ , $s=2$ , $Pr=21$ , $Ec=0.1$ (e) varying Prantl number $Re=5$ , $n=1.5$ , $M=1$ , $Gr=1$ , $m=0.5$ , $s=2$ , $\sigma =1.4$ , $Ec=0.1$ (f) varying Eckert number $Re=2$ , $n=1.4$ , $M=1$ , $Gr=1$ , $m=0.5$ , $s=2$ , $\sigma =2$ , $Pr=6.9$ (g) varying thermal conductivity $Re=5$ , $n=1.4$ , $M=1$ , $Gr=1$ , $m=0.5$ , $s=2$ , $\sigma =1.4$ , $Pr=6.9$ , $Ec=0.1$ (h) varying the electrical conductivity $Re=1$ , $n=1.5$ , $M=1$ , $Gr=5$ , $m=0.5$ , $\sigma =1.4$ , $Pr=21$ , $Ec=\mathrm{0.1.}$

Figure 4.

Unsteady profiles of velocity (a) varying the frequency parameter $Pr=21$ , $M=1$ , $Gr=1$ , $Rd=0.1$ , $n=1.5$ , $m=1.2$ (b) varying there porosity parameter $Pr=21$ , $M=1$ , $Gr=1$ , $Rd=0.1$ , $n=\mathrm{1.5.}$

Figure 5.

Unsteady profiles of temperature (a) varying the frequency parameter $Re=5$ , $n=1.5$ , $M==1$ , $Gr=5$ , $\varphi =30$ , $s=6$ , $\sigma =1.5$ , $Pr=21$ , $Ec=0.1$ (b) varying the porosity $Re=1$ , $n=1.5$ , $M=1$ , $Gr=5$ , $\varphi =60$ , $s=3$ , $Pr=21$ , $Ec=1$ (c) Varying Ec $Re=5$ , $n=1.5$ , $M=1$ , $Gr=1$ , $\varphi =30$ , $s=6$ , $\sigma =1$ , $Pr=21$ (d) varying Pr $Re=1$ , $n=1.5$ , $M=1$ , $Gr=1$ , $\varphi =30$ , $s=6$ , $\sigma =1$ , $Ec=1$ .

Figure 6.

Nusselt number profiles (a) varying the thermal radiation at lower region (b) varying at upper region $Re=5$ , $n=1.5$ , $M=1$ , $Gr=5$ , $\varphi =45$ , $s=2$ , $\sigma =1$ , $Pr=21$ , $Ec=1$ (c) varying the porosity parameter at lower region (d) varying the porosity at upper region $Re=5$ , $n=1.5$ , $M=1$ , $Gr=5$ , $\varphi =45$ , $s=2$ , $Pr=21$ , $Ec=1$ (e) varying electrical conductivity at lower region (f) varying the electrical conductivity at upper region $Re=5$ , $n=1.5$ , $M=1$ , $Gr=1$ , $\varphi =45$ , $\sigma =1.4$ , $Pr=21$ , $Ec=1$ (g) varying Eckert number at lower region (h) varying Eckert number at upper region $Re=1$ , $n=1.5$ , $M=1$ , $Gr=1$ , $\varphi =45$ , $S=2$ , $\sigma =1$ , $Pr=21$

Figure 7.

(a) Comparison of the results for velocity with previous and present paper $M=1$ , $s=0.1$ , $n=3.5$ , $m=0.5$ . (b) Comparison of the results for velocity analytically and numerically $m=0.5$ , $Re=2$ , $\sigma =1.5$ , $Pr=21$ , $h=1$ .

Conclusion

The flow and heat transfer problems of two immiscible fluids in the presence of a uniform magnetic field were investigated. The governing equations are nonlinear and coupled. Fluids in both regions are assumed to be Newtonian and conduct electricity. The results were presented graphically for various parameters such as inclination angle, Hartmann number, Joule heating, thermal radiation, electric conductivity, porosity, Eckert number, and Prandtl number. As an outcome of the investigation, the following conclusions have been drawn:

A rise in the magnitude of the oscillations in the unsteady velocity and temperature distributions occurs with the frequency parameter of the pressure gradient.

Velocity distribution rises with the increase of the Grashof number.

Velocity decreases with the rise of Reynolds number, Hartmann number, frequency parameter, and porosity.

As magnetic field strength and Reynolds number increase the mass flux decreases.

Temperature distribution enhances with an increase in the angle of inclination, Eckert number, and Prandtl number.

Temperature of the liquid falls with the rise of frequency parameter, porosity, Hartmann number, thermal radiation, and electric conductivity.

The results for the hydrodynamic case can be obtained when $M=0$ as a special case. Further, by taking $\sigma =0$ , $Rd=0$ our analysis reduces to that of Malashetty et al. ²⁵