# Unsteady oscillatory MHD boundary layer flow past a moving plate with mass transfer and binary chemical reaction

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## Abstract

In this work, we have studied analytically the heat and mass transfer of unsteady MHD natural convective flow past a motioning plate with binary chemical reaction. The flow surface spontaneously moves with unvaried velocity in the flow direction or opposite under the influence of magnetic field. We studied both frequency-dependent effects and “long-time” effects that would require non-practically long channels to be observed in steady flow. We also explored mathematically the important aspects of reactive fluid flow, especially the residence time flow behaviour, scale-up and scale-down procedures. From the present study, it is seen amongst others that temperature is enhanced as the fluid angular velocity rises which leads to maximum temperature in the body of the liquid. It is discovered that mean velocity decrease with a rise in the species reaction and reaction order. The graphical results representing the governing flow parameters effect were presented and discussed. The flow conditions at the wall were also investigated, presented and described.

## Keywords

Magnetohydrodynamic flows (MHD) Boundary layer Oscillating plate Binary chemical reaction## 1 Introduction

Several reacting processes contain chemical species with different chemical kinetics; such occur in toxic pollution control and geothermal. The relations between chemical reactions and species transfer are mostly very compound, and this can be noticed in the generation and utilization of species reactant at various degrees both in the contest of the flow liquid and concentration transfer, Salawu et al. [1]. The investigation into the flow of hydromagnetic fluid and energy transfer has gotten significant consideration recently because of its wide usage in technology and engineering processes which include nuclear reactors, studies of plasma, extractions of geothermal energy, MHD generator and so on, Hassan et al. [2]. MHD is the multi-disciplinary study of the fluid flow of an electrically conducting fluid in electromagnetic fields, such fluids include plasmas, liquid metals, and urine (salt water). MHD over a moving plate has several applications in in stellar and solar structures, astrophysics and geophysics, radio propagation, interstellar matter, ionosphere and so on. Flow of convective fluid has a well pronounced modern application in geothermal systems, granular and fibre and many more. Buoyancy effect is very significant in the domain where variances between air and land heat gives rise to complex flow patterns Salawu and Fatunmbi [3]. In nature, numerous transport processes occur where species and temperature transfer exists concurrently due to the combined effect of buoyancy forces as a result of chemical and thermal diffusions. Convective flows of transient oscillatory fluid assume a critical role in aerospace technology, turbo machinery and chemical engineering; such flows emerge because of unsteady boundary motion or unsteady temperature boundary. Additionally, temperature and free stream momentum can cause the flow unsteadiness. Unsteady rotating fluid flows have many possible applications and uses in nuclear mechanical engineering, dynamics of geophysical liquid and chemical processes.

Many researchers reported their findings in the area of thermal hydromagnetic fluid with buoyancy induced flows, and flows in rotating fluids as well as flow of chemical binary reactive fluid in the presence of activation energy. Some of the earliest studies in the aforementioned areas could be found in the work of Soundalgekar et al. [4], who examined the transient viscous, incompressible, rotating flow liquid over an infinite permeable wall. Bergstrom [5] reported on the oscillating flow of rotating fluid problem in a boundary layer past a vast half-plate. Other earlier works include [6, 7, 8, 9, 10, 11]. Waqas et al. [12] reported on thermally developed Falkner–Skan bioconvection flow of a magnetized nanofluid in the presence of motile gyrotactic microorganism: Buongiorno’s nanofluid model. The authors gave various aspects of involved physical quantities of the flow. Nguyen et al. [13] considered Macroscopic modeling for convection of Hybrid nanofluid with magnetic effects. From the results, it is found that radiation parameter and magnetic boosts the Nusselt number. Nguyen-Thoi et al. [14] investigated Analysis on the heat storage unit through a Y-shaped fin for solidification of NEPCM.

Not quit long, in the work of Maleque [15], the solution to an unsteady temperature and species transport of natural convective flow through permeable surface in a boundary layer was obtained using similarity solution. Maleque [16] investigated the impacts of the temperature dependent chemical reactive fluid and Arrhenius energy kinetic on the flow momentum, concentration and temperature. Awad et al. [17] solved transient rotating viscous, incompressible nonlinear coupled equations governing the flow over a stretching plate in the existence of Arrhenius kinetic and chemically binary reaction using the method of spectral relaxation. Mallikarjuna et al. [18] analyzed convective variable porosity of MHD rotating flow of mass and energy transfer in a vertical cone with chemical reaction. While Zhang et al. [19] examined radiative MHD nanofluid in a permeable medium with chemical reaction and surface variable heat flux. Abbas et al. [20] presented mass transfer in a thermal radiation of unsteady Casson stagnation fluid past an extending or contracting boundary layer surface. In the study, the Arrhenius kinetic and chemical binary reaction effect on a temperature dependent viscous fluid is considered. The work of Shafiquen et al. [21] reported on the effects of unidirectional Maxwell revolving flow of species and energy transfer over a stretching sheet. The authors presented proportionally mixed binary concentration to both parameters fitted rate \(n\), the reaction rate \(r\), activation energy \(E\), and rotation \(k\). It was noticed that the parameters causes concentration solute distribution reduction.

Other most recent work include the work of Khan et al. [22] on energy and species transport on the flow of chemical reactive axisymmetric convection MHD Maxwell fluid flow propelled by isothermal widening disks and exothermal. Mabood et al. [23] reported on mass and temperature transfer of hydromagnetic stagnation nanofluids flow in porous media with chemical reaction, viscous dissipation and radiation while Rawat et al. [24] presented MHD micropolar fluid flow of energy and chemical species transport past a nonlinear elongating plate in a non-Darcy permeable medium with variable micro inertia density, heat flux and chemical reaction. Despite the effort of the earlier scholars, there is evolving requests for determination of the binary chemical reaction of unsteady natural convection MHD boundary layer flow past a moving sheet with mass and heat transfer due to its numerous applications in technology, engineering and industry.

## 2 Problem formulation

*x*-axis in the upward vertically direction with

*y*-axis is assumed perpendicular to the flow direction. A uniform strength magnetic field \(B_{0}\) is introduced in the flow direction with the neglecting of the temperature field. Initially, the conducting fluid and sheet have equal temperature \(T_{w}\) in a fixed condition and fluid species level \(C_{w}\) at very points. The sheet begins oscillating with velocity \(U_{0}\) at time

*t*> 0 in its own plane. The flow concentration and temperature at \(C_{w}\) and \(T_{w}\) respectively raised in level as the plate oscillating. For an incompressible flow, the governing compactible continuity equation and MHD momentum flow equation in a porous medium is defined as

All the physical quantities maintain their normal denotations.

## 3 Method of solution

## 4 Discussion of results

The influence of an increment in the various parameters values \(V, M, k_{p}\), \(A, \delta , {\text{Grt}}\), \({\text{Grc}}\), \(\lambda , r\) and \(\omega\) on the behavior of the flow fluid for fixed values of \({\text{Sc}}\), \(\Pr\) and \(\epsilon\) are carried out. To obtain the impact of the terms on the flow characteristic, for practical purpose, the Schmidt number is chosen as \({\text{Sc}} = 0.62\) while the Prandtl number is taken as \(\Pr = 0.71\) at a single atmospheric pressure depicting the air at temperature 25 °C. For simplicity, \(\left| {\psi_{12} } \right| \equiv \psi_{12} ,\) where \(\psi = u, \emptyset\) and \(\theta\). The default values of the parameters are taken as follows: \(V = 0.3, M = 0.2, k_{p} = 0.5, A = 0.5, \delta = 0.2, {\text{Grt}} = 0.5, {\text{Grc}} = 0.2, \lambda = 0.1, r = 1.0\) and \(\omega = \pi t\) with unvaried value of \(\epsilon = 0.05\) and \(\Pr = 0.71\) and unless otherwise stated.

### 4.1 Temperature field

### 4.2 Concentration field

### 4.3 Velocity field

### 4.4 Skin friction

## 5 Conclusions

Temperature increases as the fluid angular velocity increases.

Maximum temperatures exist in the body of the fluid.

Concentration increases with a rise in the generative chemical reaction and vice versa.

Concentration phase reduces with an enhancement in the in generative chemical reaction.

Amplitude of concentration phase increase with increase in reactivity parameter.

Mean velocity decrease with a rise in chemical reaction generative and reaction order.

Velocity diminishes with increase in mass buoyance and thermal Grashof number for heating of the plate.

Skin-friction increases as generative/destructive chemical reaction increases.

Skin friction increases as limiting surface moves in opposite direction and decreases as the limiting surface motions in the fluid flow direction.

## Notes

### Compliance with ethical standards

### Conflict of interest

All authors have agreed and approved the manuscript, and have contributed significantly towards the article. There is no conflict of interest among the authors.

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