#### Abstract

This article addresses the hydrodynamic boundary layer flow of a chemically reactive fluid over an exponentially stretching vertical surface with transverse magnetic field in an unsteady porous medium. The flow problem is modelled as time depended dimensional partial differential equations which are transformed to dimensionless equations and solved by means of approximate analytic method. The results are illustrated graphically and numerically and compared with previously published results which shown a good agreement. Physically increasing Eckert number of a fluid amplifies the kinetic energy of the fluid, and as a novelty, the Eckert number under the influence of chemically reactive magnetic field is effective in controlling the kinematics of hydrodynamic boundary layer flow in porous medium. Interestingly, whilst the Eckert number amplifies the thermal boundary layer thickness and velocity as well as the concentration of the fluid, the presence of the magnetic field and the strength of the chemical reaction have a retarding effect on the flow. Also, the chemical reaction parameter and permeability of porous medium are effective in reducing skin friction in chemically reactive magnetic porous medium and are relevant in practice because reduced skin friction enhances the efficiency of a system. The results of the current study are useful in solar energy collector systems and materials processing.

#### 1. Introduction

Heat and mass transfer are greatly influenced by the hydrodynamic boundary layer of the flow. Recently, researchers have developed much interest in the study of hydrodynamic boundary layer flow in porous medium under the effects of a magnetic field because of the possible applications in engineering systems and industrial. In industrial and engineering operations, heat as well as mass transfer are crucial processes. They take place simultaneously in hot rolling, paper manufacturing, wire drawing, plastic manufacturing, glass fiber manufacturing, rubber sheet manufacturing, and the cooling of hot metal plates, among others. To accomplish equal cooling of the product, these processes sometime require stretching materials in a fluid media. The quality of the finished product is determined by the rate at which it is stretched and cooled. The mechanical qualities of the final product are depended on the thermal conductivity, which varies linearly with temperature [1].

Sakiadis [2] was the first to investigate fluid flow with a constant velocity on a continuous surface. Crane [3] expanded Sakiadisâ€™ work to include flows on an elastic sheet. Using a shooting approach, Chiam [4] investigated viscous fluid flow across a porous stretched sheet with varying thermal conductivity and found that the thermal conductivity amplifies the fluid temperature. Misra et al. [5] enhanced this by looking at the heat flow problem with both stretching surface temperature and stretching heat flux. Chen [6] investigated heat transfer on a stretched plate in a fluid flow, which was broadened by Ishak and colleagues [7] to incorporate magnetic field intensity on the flow. As a result of higher skin friction coefficient experienced in the flow, they discovered that the magnetic field strength tends to decay the flow. Dandpat and Chakraborthy [8] examined the effects of properties of variable fluid flow on a stretching surface and realized that the variable viscosity of the fluid decreased the velocity profiles.

In the last decade, many researchers have worked on the subject of magnetohydrodynamics (MHD) as well as flow in porous media. In fact, the usefulness of flow in porous medium under the effects of magnetic field is enormous. It is applied to investigate the underground water in agricultural engineering, filtration, and purification process in chemical engineering to investigate the movement of water, oil, or natural gas in reservoirs in the petroleum engineering, among others. Due to the applications mentioned above, several researchers studied MHD flow in a porous medium with varied configurations, which the present study also seeks to contribute. Hsiao [9, 10] studied magnetohydrodynamics convection flow of viscoelastic fluid using finite difference method and magnetohydrodynamics stagnation flow of viscoelastic fluid subject viscous dissipation effect. He concluded that the strength of the magnetic field adversely affects the heat transfer rate. Furthermore, Nadeem et al. [11] analyzed MHD fluid properties over a linearly stretching sheet embedded in porous medium using the Runge-Kutta method and concluded that the magnetic field retards the flow. Similarly, Seini and Makinde [12] examined MHD fluid flow on stretching sheet with radiation and chemical reaction effects. A nonlinear velocity term attributed to the magnetic field presence was included in the energy equation, and results revealed that the heat transfer rate decreased with the magnetic field and the radiation parameter. Ali et al. [13] extended Seini and Makinde work to include heat transfer in inclined stretching sheet with magnetic field. Similarly, Sahoo et al. [14] analyzed the MHD viscoelastic flow on a stretching plate and found that fluids with larger Prandtl numbers adversely affect the temperature due to low thermal diffusivity. Ahmed et al. [15] studied numerical/analytical solutions for MHD flow in porous medium past oscillating vertical sheet and realized that the shear stress and fluid velocity depressed as a result of increment in permeability parameter whilst the velocity and the temperature profiles decreased due to an increased in conduction-radiation parameter. Krishna el al. [16] extended Ahmed et al. [15] study to an unsteady flow on an infinite oscillating porous vertical sheet and observed the same trends. Sulemana et al. [17] applied Laplace transform method to study unsteady boundary layer flow on a vertical surface subject to heat source and magnetic field in porous medium. Also, Sulemana et al. [18] extended their study in [17] to examine unsteady convective flow on a vertical surface subject to porous medium with Newtonian heating. They concluded in both studies that the heat absorption parameter decays the temperature. Recently, Sulemana and Seini [19] studied time-dependent hydromagnetic flow on a vertical plate in porous medium and concluded that as time passed, skin friction coefficient reduced. Hussain et al. [20] studied MHD viscoinelastic fluid flow on stretching plate using the law of Fourier heat conduction and observed that the movement of fluid is defied by strength of the magnetic field. Khan et al. [21] analyzed non-Newtonian flow on a stretching/shrinking permeable surface using HAM. It was established that increased values of radiation parameter enhanced the temperature whilst the temperature fell for the rising Prandtl number. Chemical reaction effect on MHD dissipative nanofluid on slippering stretching surface embedded in porous medium is investigated by Yousef et al. [22]. They observed that in the absence of magnetic field, the nanofluid experiences a faster velocity. Saeed et al. [23] studied MHD Casson flow on stretching surface in porous medium using HAM. The influence of chemical reaction is also investigated in this study. They noticed a reduction in mass fluid diffusivity due to increasing values of chemical reaction. Homogeneous-heterogeneous reactions on three-dimensional, MHD flow on exponentially stretching/shrinking sheet is studied by Algehyne et al. [24]. In this investigation, the effects of chemical reaction and magnetic field are also considered. It was observed that the velocity along the -direction and the -direction is reduced by the stretching ratio parameter. Similarly, Rasheed et al. [25] investigated the effects of chemical reaction and hall current on unsteady MHD flow past a vertical surface employing ND-solve method in Mathematica and observed that the temperature of the fluid developed with increased in Schmidt number, Eckert number, and radiation parameter but diminished with the Prandtl number.

To authorsâ€™ best knowledge, hydrodynamic boundary layer flow of a chemically reactive fluid over an exponentially stretching vertical surface with transverse magnetic field in an unsteady porous medium is not documented. The present study attempts to investigate the problem using approximate analytic method. In the current study, the works of Ahmed et al. [15] and Krishna *et al*. [16] are extended to include the influence of chemically reactive in a magnetic porous medium under unsteady conditions over nonoscillating stretching vertical surface. The chemical reaction species is introduced in the flow because of its practical applications in diffusional operations in many engineering systems. The results obtained by the approximate analytic method used in the current study indicate that chemically reactive magnetic field can be used to influence the flow field.

#### 2. Mathematical Formulation of the Problem

Consider a magnetohydrodynamic flow on stretching vertical surface in a chemically unsteady porous reactive medium. The flow assumptions are as follows: (i)The flow is unsteady(ii)The stretching plate is taken along the -axis direction(iii)The -axis is at right angle to the surface(iv)A magnetic field of uniform strength, is applied transversely to the plate(v)The Reynolds number is assumed to be too small; hence, the induced magnetic field is negligible(vi)Also, viscous dissipation is further assumed to be too small hence neglected(vii)The ambient fluid temperature and the plate temperature are and , respectively(viii)At first, the fluid and the stretching plate are at the same temperature with concentration level at all points(ix)For time , the surface is moving continuously with an exponential velocity in the positive -direction and the temperature of the stretching plate is linearly with (x)The concentration levels closer to the plate are raised to (xi)The physical properties of the fluid like thermal conductivity and viscosity and shall be assumed constant(xii)The plate is infinitely along the -direction; hence, all the physical parameters are functions of and only(xiii)Gray fluid, nonscattering but absorbing/emitting radiation is used in the analysis

Figure 1 shows the physical fluid flow system.

By the assumptions above and extending the works of Ahmed et al. [15] and Krishna et al. [16] with nonoscillating but stretching vertical surface under the influence of chemically reactive in a magnetic porous medium, the equations of the boundary layer are as follows:

With boundary conditions similar to Krishna et al. [16],

Introduce dimensionless parameters similar to Ahmed et al. [15] as follows:

From Rosseland approximation,

Since the difference in temperature of the flow is too small and following Ahmed et al. [15] and Krishna et al. [16], expressing as a linear function of the temperature and applying Taylorâ€™s series expansion of and neglecting terms of higher order gives

Substituting (8) into (7) results in

With equations (6) and (9), equations (1)â€“(4) are transformed into the following dimensionless equations:

where and .

It is associated with dimensionless boundary conditions as follows:

#### 3. Approximate Analytic Method

The nonlinearly transformed dimensionless equations (10)â€“(12) subject to boundary conditions in (13) are solved using approximate analytic method outlined below. This method is similar to numerical/Laplace transform solution proposed by Ahmed et al. [15] but novel. Generally, these nonlinear equations must be integrated numerically, but for a case where the Reynolds number is small, approximate analytic solution can be found by applying perturbation method. By using this technique, a linear approximated solution of the nonlinear equations is found.

Now considering the dimensionless concentration model in (12), expanding in terms of Eckert number using power series gives

Considering powers of Ec,

In power series, the coefficient terms are such that ; hence, higher terms will be neglected. Also, the Eckert numbers are chosen to be less than 1 (i.e., ); hence, terms such us , will be much smaller hence neglected.

In (19),

From boundary conditions in (13), for all , put in (20) gives

Applying Laplace transform to (18) gives

But

Equation (24) is linear nonhomogeneous second-order ODE.

The general solution (24) is

By convolution theorem, in (26), the inverse Laplace transform is

Similarly, the Laplace transform of (17) gives

The general solution of (28) is

Applying convolution theorem in (30) gives the inverse Laplace transform as

The general solution of the concentration model is

Similarly, by considering the dimensionless temperature model in (11) and the dimensionless velocity model in (10), expanding in power series in terms of Eckert number, the general solutions are, respectively,

where

After obtaining the temperature field, it is possible to investigate the heat transfer coefficient rate at the stretched plate in terms of the Nu. The , , , Ec, and Pr effects on Nu can be investigated. The dimensionless Nusselt number is

The mass transfer coefficient rate at the stretching plate in terms of the Sh can be calculated from the concentration field. The , Ec, Sc, and Kc effects on Sherwood number are investigated. The Sherwood number is

After obtaining the velocity profile, it is also important to investigate variations in skin friction as a result of the physical parameters , , , , Ec, Pr, Kc, , Gr, and Gc. The skin friction is

#### 4. Results and Discussion

The physical dynamics of the problem can be studied, the controlling parameters effects on the temperature (), velocity (), and concentration () profiles are illustrated graphically using a written MATLAB codes. Numerically, the values of local Nusselt number (), skin friction coefficient (), and Sherwood number () are presented in tables.

##### 4.1. Graphical Results

###### 4.1.1. Temperature Profiles

Figure 2 illustrates the temperature profiles indicating the effects of time, . The fluid temperature decreases as time passes. Figures 3 and 4 show the effects of Prandtl number (Pr) and Eckert number (Ec) on the temperature profiles, respectively. Increase in Pr decreases the thermal boundary layer thickness of the fluid whilst increase in Ec increases the fluid temperature. Physically, increasing Pr results in decreasing the thermal conductivity of fluid and increase in viscosity of fluid; hence, the thermal boundary layer decreases. Since Ec relates the kinetic energy in the flow and the heat enthalpy, so the kinetic energy of the fluid is amplified by increasing Ec; hence, the temperature of the fluid increases.

Figures 5 and 6 show the temperature profile of the effects of radiation parameter () and heat absorption parameter (), respectively. From the graphs, temperature increases with increase in either or . Physically, the radiation parameter relates the conduction heat transfer to the thermal radiation transfer. So an increased in the radiation parameter caused the temperature within the boundary layer to rise.

###### 4.1.2. Concentration Profiles

In Figures 7â€“10, illustrate the concentration profiles of the effects of , Sc, Kc, and Ec, respectively. It is noticed that there is a decrease in the concentration of the fluid as time passes resulting in decrease in porous medium permeability. Initially, increase in Sc increases the concentration boundary layer thickness but at the twist of the flow, further increase in Sc decreases concentration as a result of the unsteadiness in the flow. This is also due to the fact that, physically, the molecular diffusivity of a fluid decreases as Sc increases causing a reduction in the concentration boundary layer thickness. However, the concentration of the fluid increases as Kc or Ec increases due to enhancement of the kinetic energy of the fluid. The fluid converged at termination point for concentration field for Kc and Ec, as shown in the diagrams.

###### 4.1.3. Velocity Profiles

Figures 11â€“14 show the profiles of the velocity of the effects of thermal Grashof number (Gr), mass Grashof number (Gc), magnetic parameter (), and Eckert number (Ec), respectively. In Figure 11, increase in Gr increases the speed of the flow as a result of the strength in the buoyancy force aiding the velocity to raise in the boundary layer. In Figure 12, however, increase in Gc decreases the velocity due to the delay in convection motion of the fluid as a result of the magnetic field influence. Also, in Figure 13, the fluid velocity diminishes as increases. In practice, this is true because magnetic field produces a Lorentz force that has the tendency to retard a free convective flow. The kinetic energy of the fluid is amplified due to increase in Eckert number (Ec) resulting in increase of the velocity of the fluid as depicted in Figure 14.

##### 4.2. Numerical Results

Tables 1â€“3 illustrate the behaviour of Nusselt number, , Sherwood number, , and skin friction coefficient, for distinct values of , *M*, Ec, Kc, Pr, Sc, , , Gr, , and Gc. In Table 1, Nusselt number, rises with increase in ,Ec, and Pr; it however decreases as and increase. Table 2 shows that the Sherwood number, , decreases as Kc increases but increases for increase in , Sc, and Ec. In Table 3, it is noticed that skin friction coefficient diminishes for increase in , , Ec, , Kc, Gr, and Gc but increases for increasing values of Pr, , Sc, and . The controlling parameters such as , , Ec, , Kc, Gr, and Gc are effective in reducing skin friction in chemically reactive magnetic porous medium and are relevant in practice. This is good in practice because reduced skin friction enhances the efficiency of a system. For example, fuel efficiency of many equipment is improved due to reduced skin friction. In Tables 4â€“6, present study results are compared with previous results of Ahmed et al. [15] and Krishna et al. [16] and there is consistency.

#### 5. Conclusion

A magnetohydrodynamic fluid flow across a stretching vertical plate in a chemically unsteady porous reactive medium has been studied. Dimensionless parameters are used to model the associated partial differential equations into dimensionless differential equations. Approximate analytic method was employed to solve the resulting dimensionless equations, and the results obtained are shown in graphs and tables. From the graphical and numerical results obtained, it was established that (1)The thermal boundary layer thickness decreases with Prandtl number as time passes but increases with the radiation parameter, Eckert number, and the heat absorption parameter(2)The concentration boundary layer thickness diminishes with time but increases with the rate of chemical reaction in the fluid, Eckert number, and Schmidt number. It is however noticed that there is a reduction in the species concentration in the fluid due to the Schmidt number at the twist of the flow(3)The velocity diminishes with increase in the mass Grashof number and magnetic parameter but increases with increase in thermal Grashof number and Eckert number(4)As time passed, reduced skin friction is achieved with increase in either the permeability of porous medium, chemical reaction parameter, Eckert number, thermal Grashof number, or the mass Grashof number

In general, the results obtained in the current study indicate that chemically reactive magnetic field can be used to influence a flow field. The controlling parameters such as , , Ec, , Kc, Gr, and Gc are effective in reducing skin friction in chemically reactive magnetic porous medium and are relevant in practice because reduced skin friction enhances the efficiency of a system. For example, fuel efficiency of many equipment is improved due to reduced skin friction. The results of the current study are useful in solar energy collector systems and material processing.

#### Nomenclature

: | and components of velocity, respectively (ms^{-1}) |

: | Velocity of the plate (ms^{-1}) |

: | Coordinate axis normal to the plate (m) |

: | Time (s) |

: | Kinematic viscosity (m^{2}â€‰s^{-1}) |

: | Thermal expansion coefficient (K^{-1}) |

: | Concentration expansion coefficient (K^{-1}) |

: | Fluid density (kgm^{-3}) |

: | Fluid temperature near the plate (K) |

: | Fluid temperature at the plate surface (K) |

: | Temperature of the free stream (K) |

: | Concentration in the fluid (Kmolm^{-3}) |

: | Concentration far away from the plate (Kmolm^{-3}) |

: | Concentration at the plate surface (Kmolm^{-3}) |

: | Acceleration parameter (-) |

: | Chemical molecular diffusivity (m^{-2}â€‰s^{-1}) |

: | Thermal diffusivity (m^{2}â€‰K^{-1}) |

: | Specific heat at constant pressure (Jkg^{-1}â€‰K^{-1}) |

: | Rate of chemical reaction (-) |

: | Permeability coefficient of the porous medium (m^{2}) |

: | Radiation heat flux (-) |

: | Heat source parameter (-) |

: | Dimensionless coordinate axis normal to the plate surface (-) |

: | Dimensionless velocity in direction (-) |

: | Dimensionless velocity in direction (-) |

: | Dimensionless temperature (-) |

: | Dimensionless time (-) |

: | Dimensionless permeability of the porous medium (-) |

: | Radiation parameter (-) |

: | Magnetic parameter (-) |

: | Heat absorption parameter (-) |

: | Constant (-) |

: | Uniform external magnetic field (Telsa) |

: | Dynamic viscosity (kgm^{-1}â€‰s^{-1}) |

: | Soret number (-) |

Sc: | Schmidt number (-) |

Pr: | Prandtl number (-) |

Gr: | Thermal Grashof number (-) |

Gc: | Mass Grashof number (-) |

: | Electrical conductivity (sm^{-1}) |

: | Dimensionless concentration in the fluid (-) |

: | Acceleration due to gravity (ms^{-2}) |

: | Thermal conductivity of the fluid (Wm^{-1}â€‰K^{-1}) |

Kc: | Dimensionless rate of chemical reaction (-) |

Ec: | Eckert number (-) |

: | Rosseland mean absorption coefficient (-) |

: | Stefan-Boltzman constant (-). |

#### Data Availability

Numerical data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.