Viscous dissipation effect on unsteady magneto-convective heat-mass transport passing in a vertical porous plate with thermal radiation

The effects of radiative and viscous dissipation on the transfer of unsteady magnetic-conductive heat-mass across a vertically porous sheet is studied in this article. The non-dimensional ODEs are solved by applying the Finite Difference Method (FDM) through the MATLAB software numerically. The fluid temperature and velocity enhance for uplifting values of the Eckert number. Enhancing values of the transpiration parameter the velocity, concentration, and temperature distributions reduce. The local skin friction enhances about 9%, and 18% due to increase the Eckert number (0.5–3.0) and Dufour number (0.5–4.0), respectively and reduces 17%, 38%, and 31% due to increase Prandtl number (0.71–7.0), magnetic force parameter (0.5–3.0), and suction parameter (0.5–3.0), respectively. Enhancing values of the Eckert number (0.5–3.0) reduces the heat transfer rate by 40%. The increasing value of the Prandtl number (0.71–7.0) and the suction parameter (0.5–3.0) increases the heat transfer rate by 27% and 92%, respectively. With an increase in the values of the Schmidt number (0.22–0.67), the mass transfer rate increased by approximately 94%. At last, the numerical results of this paper has compared with the previously published paper. We noticed that the comparison has an excellent acceptance.


Introduction
The viscosity needs the highest consideration in the study of fluid flow for all the fluid properties. In the 20th century [1], introduced a new area of fluid dynamics by considering viscosity and thus theoretical hydrodynamics and unifying hydraulics. Many researchers carried on MHD (magnetohydrodynamics) flow of the boundary layer problem or a radiating gas inside a vertical pathway. The impact of this type of viscous dissipation term on an unsteady condition was often ignored. The influence of this heat dissipation function cannot be neglected from a practical point of view because of its momentous in several flow issues. It is the bearing lubricant that provides the source of temperature rise and geodynamic heating. For much lower velocity methods the impact of viscous dissipation in the temperature profile is comparatively small. The influence of viscous dissipation cannot be neglected in the manner The prime purpose of this research is to take into account the above issues upon a vertically perforated plate considering the effects of viscous dissipation. The comparison of our numerical results with published paper is the main novelty of this paper. We have also improved this paper further by assuming the radiative and viscous dissipation through the Finite Difference Method (FDM) which is not been analyzed yet. This current work is to extend the works of [22] by considering the radiative and viscous dissipation. Non-dimensional numbers/parameters have been calculated for a wide range such as Eckert number, heat radiation, Schmidt number, magnetic force parameter, and Prandtl number explained graphically. Furthermore, mass, and heat transfer rates and the local friction coefficient are narrated with tabular structures.

Model and mathematical formulations
The unsteady 2D hydro-magnetic flow of an incompressible viscous and electrically conducting fluid along a permeable vertical flat sheet joined in a permeable medium is considered. The fluid flow direction is on the x-axis. The free stream velocity is parallel to this axis which is vertical. The vertical porous sheet is perpendicular to the y-axis. A magnetic field with uniform strength B is transversely applied to the flow direction. The permeable sheet begins passing impulsively in its self-plane with a velocity U 0 for t > 0. The fluid temperature at the sheet is increased to T w . The fluid concentration at the sheet is increased to C w . The physical model and coordinate systems are plotted on Fig. 1 [26]. The fluid is considered to have certain properties except for the effects of concentration variation with temperature and concentration. This effects are assumed only in terms of physical forces. The velocity components are the function of y and t.
We assume the applied uniform magnetic field is remarkably small. The magnetic Reynolds number has been compared to one of the studies [28]. Then the magnetic force lines are B = (0, B 0 , 0). The density of current is J = (J x , J y , J z ). The charge continuity equation is given by ∇.J = 0. The solution of this equation is J y = constant. The y-axis is considered along only the propagation direction. There is no change in the direction of this propagation along the y-axis. For this reason ∂Jy ∂y = 0. Since the plate is electrically non-conductive then this constant of integration is zero, so J y = 0 at the plate.
Considering that the boundary-layer as well as Boussinesq approximation, the equations of governing fluid flow are given as [26]: ∂T ∂t ∂C ∂t Boundary conditions are given as: where T = fluid temperature ,u = velocity components in the x − axis, v = velocity components in the y − axis, C = fluid concentration, β = volumetric expansion coefficient including temperature, T w = wall temperature, C w = wall concentration, ρ = fluid density, C ∞ = fluid concentration in the free stream, k = plate thermal conductivity, T ∞ = fluid temperature in the free stream, C s = concentration susceptibility, C p = specific heat at constant pressure, the k T = thermal diffusion ratio, υ = fluid kinematic viscosity, g = acceleration Assuming a similarity parameter σ which is given by σ = σ(t) (7) where σ is the unsteady length scale. The solution of continuity equation (1) is assumed in terms of this unsteady length scale. This solution is given by the following relation: Here v 0 < 0 displays blowing and v 0 > 0 displays suction, where v 0 is the non-dimensional normal velocity at the sheet. The radiative heat flux (q r ) [29] is given as:

∂T 4 ∂y
where K * is the coefficient of mean absorption and σ * is the constant of Stefan-Boltzmann.
We consider from Ref. [30] that the difference between the liquid temperature and the free flow temperature is quite small. Now, T 4 is being expanded in a Taylor series about T 0 . We ignored the higher-order terms yield: We used the following similarity transformations [16]: The governing equations (1)-(4) are transformed into the non-dimensional coupled ODEs by using the above equations (7)-(9) The corresponding transformed boundary conditions are given by where (Tw− T∞)Cp = Eckert number and ξ = η + v0 2 . The Significant physical quantities need to be noted, for example, local skin friction (f ′ (0)), Nusselt number (N u ) and Sherwood number (S h ) given by the following relationships:

Numerical solution
The principal aim of this study is to apply the FDM (Finite Difference Methods) for solving the couple ODEs (10)-(12) corresponding boundary conditions (13)- (14). Such methods are tested for efficiency and accuracy in solving various problems [31,32]. The solution domain location is discretized in FDMs (Finite Difference Methods). Let the grid size is Δη = h > 0 in η-direction and Δη = 1 N , Let the numerical values of f, θ and φ are F i , Θ i and Φ i at the i th node, respectively. We take: the system of ODES (10), (11), and (12) is separated in space by applying the FDM which is the principal step. We get (10), (11), and (12) from (16) and (17) into and delete the truncation errors. Then for (i = 0, 1, …, N), the final algebraic equations may take the form: Also, the boundary conditions are The system of equations 18-20 is a nonlinear system of algebraic equations in F i ,Θ i ,and Φ i . We have solved these nonlinear system of algebraic equations by including the Newton iteration method through the MATLAB software.

Results and discussions
In this numerical analysis, the viscous dissipation effect on time-dependent magneto-convective mass and heat transport through a vertical permeable sheet in presence of the radiative is discussed. The initial valued problems with the set ODEs (10)-(12) are numerically resolved using the "MATLAB ODE45 ′′ software using the limiting difference method (FDM) associated with the boundary conditions (13) [16]. It is seen that they are in good contract which is displayed in Table 9 and Table 10

Velocity profiles for several values of non-dimensional numbers/parameters
Velocity distribution for several values of the Eckert number (Ec) is shown in Figs. 2 and 6. The effect of the Eckert number is to create more skin friction coefficients in the border layer area and to improve the fluid velocity as shown in Fig. 2. Larger viscous dissipative heat enhances the temperature profile. The nature of radiative parameter (R) on the velocity profile is exhibited in Fig. 3.
The radiative parameter is defined R = 16σ * T 2 W 3k * k and shapes in the improved thermal diffusion term in Eqn. (11) i.e. 1 Pr (1 + R)θ ′′ (η). The comparative contribution of radiative with thermal conductive heat transfer is defined by the radiative parameters. When R > 1, then the radiative dominates over thermal conduction, on the other hand the thermal conduction dominates for R < 1. The thermal conduction and radiative contributions both are equal when R = 1. We considered only the case R > 1 for our current study. Fig. 3 shows that a powerful acceleration of linear velocity as R increases. Strengthening the flow results in increased thermal expansion and then decreasing velocity expansion. This momentum leads to a reduction in the boundary layer thickness. Fig. 4 illustrates the velocity        Fig. 5 shows that the rising values of the local gravitational number enhances the value of the wave velocity due to the increase in the wave force. The velocity profile formed a symmetrical form for negative and positive values of the local Grashof number. We observed that when the mode of heating is increased, the velocity also increases but a converse impact is observed for the case of cooling. Consequently, symmetric figures are obtained. It is also well established that the improvement of buoyancy parameters improves the fluid flow. The impact of Darcy number (Da) on velocity profile is revealed in Fig. 6. Fig. 6 shows that the velocity of the liquid exacerbates with increasing the value of Da. For huge values of the Darcy number, the permeability of the medium decays, so the fluid flows slowly.  Fig. 8 exhibits the influence of radiative parameter (R) on the temperature distribution. From Fig. 8 it can be observed that the surface temperature gradient decreases to increase the value of radiative parameters. The heat transfer rate reduces due to the improving values of R on the surface. The radiative parameter is liable for thickening the thermal boundary. The fluid flow gives up the heat energy from the flow zone and as a result, the system cools down. Fig. 9 shows the influence of the Prandtl number (Pr) on the temperature distribution. This is because the Rosseland approximation  increases the temperature. The thermal conductivity is inversely proportional to the Prandtl number (Pr). As the value of thermal conductivity decreases, the Prandtl number increases. From Fig. 8, it is seen that the temperature profile diminishes by reducing values of the thermal conductivity. In practically, a higher Prandtl number has relatively a lower thermal conductivity, which reduction in the thermal conductivity, and therefore the temperature decreases. For this reason, the heat transfer rate enhances as the magnitude of Pr increases. Hence, the fluid temperature diminishes. Fig. 10 presents the effect of the separate values of the Schmidt number (Sc) on the concentration profile. The Schmidt number is inversely proportionate to the molecular (species) diffusivity. The momentum level and density (species) will have the same thickness and diffusivity rate for Sc = 1. When Sc > 1 then the rate of reproduction of the momentum overcomes the rate of reproduction of the species. When Sc < 1 then the opposite behavior happened. The concentration profile in Fig. 10 shows the decrease in concentration for the improving values of the Schmidt number. The reduction associated with mass diffusivity leads to a smaller force mass transfer which reduces the density level. So, the thickness of the concentration boundary layer reduces. Therefore, the mass transfer applies interplay with the field of velocity and the distribution of species in the matter may be dominated by the Schmidt number.

Local skin friction coefficient, heat transfer rate and mass transfer rate
The authors are interested in discussing not only the field of concentration, velocity as well as temperature but also the values of the mass transfer rate, local skin friction coefficient as well as heat transfer rate. Tables 1-8 Tables 1-8 It is  revealed from Tables 1-8, that the local skin friction coefficient enhances for improving values of the local Grashof number, Eckert number, local modified Grashof number, and Dufour number. Besides, the local skin friction coefficient decreases for growing values of Prandtl number, magnetic force number, Schmidt number, and suction parameter. The heat transfer rate improves for raising values of the suction parameter as well as Prandtl number. On the other hand, the reverse trends are shown for the Dufour number and Eckert number. Also, increase the mass transfer rate to improve the suction parameters as well as the value of the Schmidt number.

Comparison
We are interested in comparing our results with previously published papers [16]. Tables 9 and 10 illustrate the comparison of the local Sherwood number and the local Nusselt number. Some limited cases are evaluated with previously published results and we find good agreement.

Conclusions
The effects of condensate dissipation on the transfer of unstable magnetic-conductive heat-mass across a vertically permeable sheet with radiative effect has been analyzed. From the above numerical results, the following conclusions may be drawn.    Table 3 Local skin friction coefficient, heat, and mass transfer rates for several values of the suction (v 0 ).  Table 4 Local skin friction coefficient, heat, and mass transfer rates for several values of the Grashof number (Gr).  Table 5 Local skin friction coefficient, heat and mass transfer rates for several values of the modified Grashof number (G m ).  Table 6 Local skin friction coefficient, heat, and mass transfer rates for several values of the Prandtl number (Pr).  Table 7 Local skin friction coefficient, heat, and mass transfer rates for several values of the Dufour number (Df). The outcome results of this study may be helpful for geodynamic heating, bearings lubricant, drawing of plastic films, polymer sheet extrusion from a dye, geothermal energy extraction, geophysical flows, etc.

Author contribution statement
Md. Hasanuzzaman, Ph. D: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.
Sathi Akter, M. Sc; Md Mosharrof Hossain, Mphil: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.
Shanta Sharin, M. Sc; Md Amzad Hossain, PhD: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.
Akio Miyara, PhD: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.

Funding statement
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement
The data that has been used is confidential.

Declaration of interest's statement
The authors declare no conflict of interest.  Table 9 Comparison of local Sherwood number (Sh) for different values of So and Df when Ec = 0 and R = 0. [