Unsteady dynamical analysis of convective hydromagnetic thermal migration of chemically reacting tiny species with dissipation and radiation in an inclined porous plate

1. Introduction

Hypothetical and investigational studies of the consequences of Soret and viscous dissipation on hydro-magnetic natural convection fluid stream of a continuously elongating surface have piqued the interest of many engineers, industrial geophysicists, and astrophysicists in recent years due to the numerous technological complex designing, industrial geological science, and cosmological applications, for example, synthetic compound production, earthenware manufacturing, sustenance preservation, regulating of fissile reactors, boosted lubricant retrieval, subsurface energy passage, attracted plasma transport, rapid plasma wind, galactic planes, and astral systems. Various researchers have extensively studied the upshots of reactive species on time-dependent natural convective hydro-magnetic transport. Due to the incredible applications of such results on fluid movement, several investigators^{1,2,3,4,5,6,7,8,9} carefully considered the outcomes of chemical reactions on MHD natural convective transport with thermal and solutal movement. Prabhakar Reddy and Peter¹⁰ offered the outputs of chemical reacting species on hydro-magnetic transport over an impetuously initiated inestimable upright sheet by inconstant temperature and solutal dispersion under the existence of Hall current employing the FDM approach. Shamshuddin et al.¹¹ explained the impact of spinning time-varying multi-physicochemical magneto-micropolar flow in absorbent surfaces using the Galerkin finite element approach. Goud et al.¹² presented their numerical study of micropolar fluid in which chemical reactions were suspended. Further, Goud et al.^13,14 also presented similar study considering Casson fluid and regular fluid in which viscous dissipation and Soret effects are included.

The free flow exists in fluid due to temperature alteration that causes density difference ensuing in resistance forces involved in the fluid. Natural convection is a situation of thermal transmission in natural transport. The warming of houses and structures with the help of heaters illustrates thermal transmission by natural convection. Radiation is the progression of energy via electromagnetic rays in nuclear reactor. Radiative convective transport is essential in multitudinous industry-base and the surface surrounding physical processes like warming and conserving grooms, vaporization from huge-exposed water tanks, cosmological transports and sun energy technology. Owing to the outstanding significance of the stated material facets, numerous studies^{15,16,17,18,19,20,21,22} explored the mathematical descriptions of natural convective transport of compress-resisting viscous fluid subjected to assorted transport geometries where thermal radiation was regarded. Shamshudinn et al.²³ inspected the upshots of gyrating periodic convective MHD radiative micropolar thermo-solutal transport via FEM. Raju et al.²⁴ discussed that variable suction provides significant prospects for enhancing the characteristics of unsteady regular convective fluid flow via FEM. Reddy and Goud²⁵ provided a thorough report on the use of nanofluid for heat transmission in stagnation point flow influenced by thermal radiation.

The thermal-diffusion effect is a physical process where less-weight, intermediate, and hefty molecules are dispersed and subjected to the temperature gradient. Regularly, this outcome is vital when numerous reacting species exist with a highly bulky temperature change. The models of this type are very convenient for isotope parting and concoction amid gaseous elements with minimal molecular weight. It often arises in production involving blistering metal rolling, iron cable designing, glass-fiber construction, paper manufacture, plastic film design, metal rotating, and metallic and synthetic compound excrescence processes. Due to such significance, various early investigations worth mentioning^{26,27,28,29,30,31,32,33,34} included thermal-diffusion and diffusion-thermal’s upshot on natural convective MHD transport. Shamshuddin and Thumma³⁵ examined thermal-diffusion and diffusion-thermal consequences on micro-polar fluid’s time-dependent hydro-magnetic natural convective movement under periodic vibrating plate velocity regarding viscous dissipation impacts.

To a higher extent, the studies stated above disregard viscous dissipation’s consequences on fluid transport. To put it another way, there are some situations where the resistance of the fluid flow can fluctuate significantly with temperature variations. On account of this, the influence of viscous dissipation is crucial for intense transport field dimensions, low temperatures, and large gravitational fields. This type of movement is utilized in plentiful high-production processes, including the aerodynamic gibbsite of plastic materials, physical conveyors, regulating or drying of fabrics, production of optical fiber, etc. As a result of these critical applications, countless researchers are becoming increasingly curious about the upshots of viscous dissipation on fluid transport. Studies^{36,37,38,39,40,41,42,43,44,45,46} measured the upshots of dissipative outcomes in magnetohydrodynamic free convection transport subjected to various circumstances. In the presence of reacting species and dissipative impact, Barik and Dash⁴⁷ examined the consequence of heat radiation on time-dependent magneto-hydrodynamic flow past a sloped porous heated sheet. Shamshudinet et al.⁴⁸ discussed the periodic dissipative conjugate thermal and solutal transport in chemically reacting micro-polar transport with skin couple stress.

The exploration of heat source or sink outcomes in fluid transport is momentous due to various physical challenges, such as fluids going through chemical processes. Prabhakar Reddy and Sademaki⁴⁹ examined the upshots of conjugate heating on time-varying natural convective hydro-magnetic Casson fluid transport over a periodic vibrating upright absorbent plate encased in a permeable medium. Problems^50,51 described the consequences of various governing factors on MHD natural convective thermal and solutal transport, including heat source. Chamkha⁵² investigated time-varying convective thermal-solutal movement past a semi-unlimited absorbent accelerating sheet with a heat sink. Hady et al.⁵³ inspected the model of natural convection transport across an upright wavy surface enclosed in MHD fluid-drenched permeable surface in the existence of inner heat source or sink outcomes.

The analysis of convective hydro-magnetic unsteady thermal diffusion of chemically reacting species and heat-dissipating fluid with radiation across an inclined permeable plate has yet to be investigated. The results of this research are essential in production involving blistering metal rolling, iron cable drawing, glass-fiber construction, paper manufacture, designing of plastic films, and metal rotating, in addition to metallic and synthetic compounds excrescence processes, etc. Hence, the novelty of this current framework aims to scrutinize the thermal diffusion and viscous dissipation on magneto-hydrodynamic heat-riveting chemical-reacting species transport through an inclined absorbent sheet in the existence of radiation with time-varying concentration and temperature. The finite difference of the semi-implicit method has been followed to find the solutions for constructing nonlinear highly coupled PDEs of the physical model. The consequential deviations and constituents of different characteristics like engineering parameters, temperature, velocity, and concentration have been broadly paid attention to observe the conceivable variations in the performance of the fluid.

2. Mathematical Conceptualization

Consider a 2D time-dependent viscous convective MHD chemical reactive heat-riveting transport of incompressible and electrically conducting fluid past a periodic wavering immeasurable slanted permeable plate at an angle $\gamma$ upright (see Fig. 1). The $y^{\prime }=0$ coordinate is viewed to be in accompaniment to the plate and the $x^{\prime }=0$ coordinate normal to it. A constant magnetic field of extent $B_0$ is engaged perpendicularly to the plate. The crosswise employed magnetic field and magnetic Reynolds number are anticipated to be extremely tiny, so the generated magnetic field is insignificant. At $t^{\prime }=0$ both fluid and the plate are at unvarying temperature $T_{\infty }^{\prime }$ and the concentration is close to the plate $C_{\infty }^{\prime }$. At points, the plate gains impulsive progression in the transport direction, i.e., along $x^{\prime }$-axis opposes the gravitational force with fixed velocity $U_0$. Further, the consistent chemical reaction of first-order diffusing particles. It is also presumed that neither applied nor polarized voltage exists, which signifies the deficiency of the electric field. It is reckoned that the plate concentration and temperature at the plate are changing additively with time. The fluid has unvarying kinematic viscosity and unchanging heat conductivity. By regarding the assumptions above collectively, and the Boussinesq’s approximation, we deduce the following PDEs and related initial and boundary conditions as follows⁴⁷ :

and where “$u^{\prime },g,\beta ,{\beta }^{•},T^{\prime },C^{\prime },\upsilon ,\mu ,\rho ,\sigma ,K^{•},C_p,Q^{\prime },T_m,k_r^{\prime },D_m,\gamma ,K^{\prime },t^{\prime }$ are the fluid velocity, acceleration due gravity, volumetric coefficient of heat expansion, volumetric coefficient of concentration expansion, fluid temperature, fluid concentration, kinematic viscosity, fluid viscosity, fluid density, electrical conductivity, thermal diffusion ratio, specific heat at constant pressure, heat source constant, fluid mean temperature, chemical reaction constant, chemical molecular diffusivity, slanted angle of the plate, permeability of the media and time”.

This study treats the fluid as grey imbuing-emitting radiation but as a non-dispersing medium. The radiative thermal flux $q_r$ with the Rosseland approximation is realized by

where ${\sigma }^{\ast }$ and $k^{\ast }$ denote, respectively, the Stefan–Boltzmann constant and absorption coefficient. Letting a slight temperature difference between the fluid temperature $T^{\prime }$ and free stream temperature $T_{\infty }^{\prime }$ is expanded as a linear function about a free transport temperature $T_{\infty }^{\prime }$ after ignoring 2nd and higher order terms to produce a result of the form Using Eqs. (5) and (6), Eq. (2) yields Introducing dimensionless quantities Using Eq. (8) into Eqs. (1), (3) and (7), the following flow descriptive dimensionless PDEs with associated boundary conditions are obtained : and

3. Numerical Procedure

The adopted numerical procedure is taken from the finite difference of semi-implicit method by Refs. 54 and 55 for a thermal reacting fluid species. As demonstrated by Makinde and Chinyoka,⁵⁶ the numerical system is stretched to energy and concentration equations by considering the intermediate implicit terms on the time level of $(m+\zeta )$ in $0\le \zeta \le \infty$. The procedure engaged by Chinyoka⁵⁷ utilizes $\zeta =1/2$ thus, this study will adopt the formulation by Salawu et al.⁵⁸ and consider accommodating high time steps. The presented computational algorithm is conjectured to be applicable for various values of time steps because it is close to fully implicit. A discretized Cartesian even grid and linear mesh of the flow model assumed at the point finite-differences are demarcated. The central differences of order two are approximated for the spatial derivatives, and the grid points are modified for the boundary constraints to be captured. Thus, the module for velocity in a semi-implicit scheme is

The forward difference formula is utilized for the derivative in Eq. (13) satisfying $\frac{\partial \alpha }{\partial \tau }=\frac{({\alpha }^{(m+1)}-{\alpha }^{(m)})}{\mathrm{\Delta }\tau }$. The description for $u^{(m+1)}$ gives The $u^{(m+1)}$ solution technique resulted in tri-diagonal matrices; this gives its merit over a whole implicit method. Therefore, the integration scheme in semi-implicit for the energy equation component resembles the velocity module, this is treated as where ${\theta }^{(m+\zeta )}$ is define The term $r_2=\zeta \mathrm{\Delta }\tau /\mathrm{\Delta }{\eta }^2$. Thus, the solution becomes an inverse tri-diagonal matrix. Likewise, the semi-implicit integral structure for the concentration model follows the order as the heat equation where is defined Here, $r_3=\zeta \mathrm{\Delta }\tau /\mathrm{\Delta }{y}^2$. Also, the method of solution changes to inverse tri-diagonal matrices. The consistency of schemes (14), (16) and (18) is confirmed for order one accuracy and order two in space. Here, $\zeta =1$ is considered to accept giant time steps, and the steady solutions still converged. Hence, the solution procedure works for diverse values of time steps. A maple code is developed to determine the solution algorithm and examined for time-based and dimension convergence. As confirmed, the solutions for $\mathrm{\Delta }\tau =1$ and $\mathrm{\Delta }\tau =5$ for 200- and 40-time steps, are similar. The numerical solution converges at the three iterations, after which no changes are seen in the computed outputs for the flow characteristics components $u,\theta$ and $\varphi$, see Table 1 for the convergence results. The computational default values and others are taken from the previously theoretical and experimental studies by scholars.

Some important physical characteristics linked to thermal and mass flow quantities include the wall-friction coefficient, rate of thermal and solutal transport which are described in dimensionless form by

3.1. Validation of the current numerical schema

Table 2 describes the approximation of the friction factor for the several affected flow parameters by published work (Barik et al.⁴⁷ and Reddy et al.⁵⁹). The results show that second-branch solutions are in excellent agreement with Reddy et al.⁵⁹ Because of the perfect alignment of the solutions in Table 2, we are persuaded that the offered code is well-designed to uncover the unknown outcomes of the current challenge.

4. Analysis of the Results

Thermo-diffusion (Soret effect) becomes important and needs to be analyzed whenever the influence of mass transfer happens. This survey investigates thermal and solutal effects for the transport of viscous fluid over an oscillating immeasurable plate. The examination of numerical outcomes in this section is discussed in both forms — quantitatively and graphically for the unstable outcomes. As a result, to fully understand the double outcomes for a limiting scenario, it is first necessary to analyze the technique, the code dependability, and the result validations. The composite nonlinear structure of Eqs. (9)–(11) with assumed boundary restrictions (12) is computed by deploying a traditional numerical approach such as the finite difference-based semi-implicit method. The effect of various other controlling factors has also been analyzed through different graphs, i.e., Tables 2–4 and Figs. 2–15.

Figure 2 exhibits the variation of velocity with magnetic parameters in three dimensions. It is perceived that velocity declines as the magnetic quantity upsurges over time while the velocity diminishes as the boundary layer rises. Figure 3 displays the instance of change of time and boundary layer. It is distinguished that as time escalates, temperature also rises. This is due to temperature being an increasing function of time. Figure 4 displays the variation of concentration with time. It is detected that dilating values of time increase species concentration.

In Fig. 5, it is noted that the fluid velocity in the plate falls with the increasing value of magnetic quantity M. The impact of a crosswise magnetic field on an electron-flowing field produces Lorentz force comparable to resistive force that competes with the fluid movement, henceforth decreasing the fluid velocity along the plate. This physically reveals that as magnetic field increases it expands the magnetic strength and it additionally upsurges the fluid particle, so velocity diminishes. The fluctuations of the permeability parameter K and the velocity profile are demonstrated in Fig. 6. It is detected that the fluid velocity progresses when the absorbent value K is enlarged. The mounting porosity parameter K means the magnitude of the holes within the porous membrane is improved, due to which the opposing force is decreased and hence the fluid velocity enhances.

Figure 7 depicts the influence of Gr on the fluid velocity profile. It is observed that the fluid velocity heightens with the growing Gr which tends to intensify the heat and mass buoyancy forces. Hence fluid velocity increases. The effect of the solutal Grashof number Gm on the velocity profile of fluid is shown in Fig. 8. It is perceived that the fluid velocity upsurges with the elevating Gm. Like Gr, Gm enhances the thermal buoyancy force.

Figure 9 demonstrates the viscous dissipation Ec outcomes on the temperature profile. It is detected that the fluid temperature increases as viscous dissipation Ec increases. Viscous dissipation involves the conversion of energy due to motion into internal energy by work done counter to the viscous fluid forces. The higher viscous dissipation enhances the lowering of the plate, meaning the heat vanishes from the plate to the fluid; henceforth, superior viscous dissipative heat boosts the fluid temperature. Also, this Eckert number is the relation on that enthalpy and kinetic energy, moreover, the escalation in Ec aids in the dissipative heat being reserved on the fluid over frictional heating, that basically inspires the temperature on the fluid. Figure 10 exhibits the upshot of the Prandtl number $Pr$ on the temperature profile. It is noted that fluid temperature shrinks with a mounting value of the Prandtl number $Pr$. The heat boundary thickness lessens as the Prandtl number rises. The lesser $Pr$ values increase thermal conductivity across the plate; hence, the heat can dissipate forth from the heated plate hurriedly for greater $Pr$. An analogous tendency is depicted in Fig. 11 as the radiation quantity N amplified. In the existence of radiation, the thermal boundary layer is constantly bladed, consequently, the fluid temperature declined. Thermal radiation has a negative physical effect on the medium’s thermal diffusibility, which declines the temperature profile. The upshot of heat absorption Q on the temperature profile is illustrated in Fig. 12. It is observed that the fluid temperature diminishes with elevating values of the heat absorption quantity Q. Heat absorption enhances heat loss through diffusion across the plate, decreasing the fluid temperature.

Figure 13 exemplifies the consequence of Sc on the concentration profiles. Substantially the intensification of Sc conveys the lessening of molecular dispersion. Therefore, the species concentration escalates for smaller quantities of Sc and decreases for larger values Sc. The upshot of Kr on the fluid concentration profile is exposed in Fig. 14. It is noted that the progression in chemical reaction parameters Kr lowers the fluid species concentration. Reactive species lower the movement of the fluid molecules, decreasing the fluid concentration. Figure 15 depicts the outcomes of the thermal-diffusion parameter on the fluid concentration. It can be witnessed that there is an increasing behavior in fluid concentration with boosting values of Sr. Physically improving Sr enhances mass buoyancy force, which yields increased species concentration, consequently boosting transport in the boundary layer.

Tables 3–5 show the estimated numerical findings of the wall friction and thermal and solutal transport rate. Table 3 clearly shows that raising the magnetic parameter yields to a drop in skin friction, but increasing the permeability parameter, Grashof number, and thermal Grashof number causes the differing effect. Table 4 demonstrates that the Nusselt number upsurges as the Eckert number and radiation parameter increase; however, the reverse pattern is observed if the heat absorption parameter and Prandtl number increase. According to Table 5, a boost in Schmidt number and chemical reaction parameter results in a rise in the Sherwood number, while escalation of the Soret number has the reverse effect.

5. Conclusion

A numerical computation is executed to study thermofluidic parametric sensitivity of thermal migration of chemically reacting hydromagnetic tiny species with time-dependent flow. The impact of radiation and dissipation on the flowing fluid in an oscillating inclined absorbent plate in the existence of a chemical reaction is examined. The adequate finite difference of the semi-implicit method is exerted to attain the arithmetical solution of constructing partial differential equations. The key findings are

• The velocity escalates with mounting values of K, Gr and Gm whereas there is an opposite trend in the velocity for the upsurge in $M.$

• A rise in enhanced fluid temperature while an inverse consequence is perceived when $Pr,N$ and Q are enlarged.

• Increasing $Sr$ escalates the concentration of fluid species while boosting Sc, Kr and has reverse results.

• Wall friction declines with upward values of K, Gr and Gm while it advances with intensifying values of M.

• Rate of thermal transfer heightens with improvement in Ec, N and $\tau$ while it lessens with enlarging $Q.$

• Mounting Sc and Kr improve Sherwood number whereas they become lower with the escalation of Sr.

ORCID

L. Joseph Sademaki  https://orcid.org/0000-0003-1357-7346

MD. Shamshuddin  https://orcid.org/0000-0002-2453-8492

S. O. Salawu  https://orcid.org/0000-0001-6951-7524

B. Prabhakar Reddy  https://orcid.org/0000-0002-1776-5064