Soret Effect on Magneto Hydrodynamic Free Convective Heat and Mass Transfer Effects Flow over an Inclined Plate Embedded in a Porous Medium

Researchers working on alternative energy sources, astrophysics, and hypersonic aerodynamics have shown a great deal of interest in investigations of heat transfer and mass transfer using MHD flow over an inclined plate.

The study of movement of electrically conducting fluids is known as Magnetohydrodynamic. Examples of MHD are salt water, liquid metals, electrolytes and plasmas. MHD is basis to many exciting fields of astrophysics and geophysics and has a wide range of applications in engineering, laboratories and society. Magneto fluid mechanics and magneto gas dynamics are other different terms used to denote MHD. Manufacturing process of petroleum industries, cooling of clear reactors, boundary layer control in aerodynamics are some of the important applications of MHD.

In the problem a solid plate submerged within a porous medium is considered. The porous medium encircling the plate has the potential to influence the movement of fluids and the transport of mass. The inclination of the plate influences the movement of the fluid. Some applications of inclined plates immersed in porous media encompass the investigation of flow patterns within soil containing buried obstacles, the refinement of designs aimed at controlling soil erosion, and the examination of fluid interactions within densely packed beds.

Heat Transfer (HT): The process by which thermal energy is transferred from a region of higher temperature to lower temperature region is called as HT. It can be functions of time and geometric coordinates, inside the area of matter. HT occurs through three primary modes: conduction, convection, and radiation. HT foretells the speed at which energy is conveyed over a surface because of difference in temperature at the surface and difference of temperature amidst varied surfaces. The quantity of HT per unit area through unit time is known as the heat flux.

HT theory finds practical applications in various fields. It finds application in computing thermal stresses in jet engine turbine blades, optimizing the layout of combustors in internal combustion engines, directing heat treatment procedures, estimating the power loads required for furnace operation, and evaluating the essential cooling water demands for electrical power plants. Mass Transfer (MT): The concept of transfer of mass from higher concentration part to region of lower concentration is termed as MT. The three modes of MT are 1. Diffusion MT, 2. Convective MT and 3. Phase change MT. Examples of MT are dissolution of salt in gravy and evaporation of naphthalene balls into atmosphere.

The HT task consistently goes with MT procedure in circumstances like evaporation, chemical reaction and condensation. Hence the study of consolidated HT and MT is useful in perception of various technical transmit activities.

The transmit of mass caused by gradient in temperature is called as Soret effect (Sr). The transferral of heat by concentration difference is termed as dufour effect (Df). In hydrology, petrology and geosciences the reaction of Sr and Df play a fundamental role.

The incorporation of Sr in MHD free convective flow over an IPIPM provides further applications and considerations across various engineering fields such as crystal growth, astrophysical fluid dynamics, chemical process industry, electrokinetic devices, geological fluid dynamics and food processing.

The reaction of Df and heat generation on MHD free convection flow of an IECVF above an IPIPM using closed analytical method has been considered by Islam et al. [1]. Taid and Ahmed [2] worked on the outcome of heat source, chemical reaction and Sr on the stable 2D MHD free convective flow across IPIPM, using perturbation method. Here, it was noticed that the velocity, concentration and SF drop off with the rise in Kr, also temperature, Nu and Sh hikes with the boost up in the value of Kr.

The consequence of Sr, hall current and rotation on an unstable MHD free convection heat and MT flow of IECVF past an IPIPM was measured by Sarma and Pandit [3]. In this scenario, the entire system rotates at a constant angular velocity perpendicular to the plate, while a uniform magnetic field is applied in the same perpendicular direction. Falodun et al. [4], worked on the reactions of negative and positive Sr and Df on unsteady mass and HT flow in the existence of viscous dissolution, heat and mass buoyancy. A spectral relaxation method, known for its robustness and accuracy, is utilized to solve unsteady nonlinear PDEs. The outcome of Sr on unstable free convection flow of incompressible viscous fluid past a permeable means with peak porosity bounded by an erect unbounded plate under the effect of heat source, chemical reaction and thermal radiation was addressed by Ibrahim et al. [5]. The study shows that ‘u’ increases with increment in the values of buoyancy parameters, ‘θ’ declines with rise in the value of Pr and ‘φ’ drops off with boost up in the values of Sc and Kr.

Dhanalakshmi et al. [6] studied the results of Sr and Df reaction of Kuvshinshiki fluid on MHD free convection flow through an erect permeable plate in the existence of chemical reaction with heat and MT by applying perturbation technique. Also by using Matlab, the results are obtained both in graphical and numerical ways for different values of flow factors.

Numerical investigations have been conducted on the steady 2D MHD free convective flow, as well as MT, occurring around a semi-infinite erect porous plate within a permeable medium. This study also encompasses the consideration of the Dufour and Soret effects which was done by Alam and Rahman [7]. The momentum, energy, and concentration equations are normalized using similarity transformations. These equations are then numerically solved using the Nachtsheim-Swigert shooting method combined with a sixth-order Runge-Kutta integration scheme.

Reddy et al. [8] had examined the impact of thermodiffusion and chemical influences on HT in MHD mixed convection flow and MT alongside an infinite vertical plate. This investigation takes into account Ohmic heating and viscous dissipation as significant factors in the analysis. The results are obtained by perturbation techniques. The existence of the chemical effect leads to a reduction in ‘u’ and ‘θ’ caused by thermal diffusion. The values of ‘u’ and ‘θ’ are greater for mercury when compared to electrolytic solution. ‘φ’ increases with rise in Sr while decreases with increment in chemical effect.

The analytical analysis of the flow of free convection and MT within a vertical channel created by two vertical parallel plates was done by Jha and Ajibade [9]. The values of ‘u’, ‘θ’ and ‘φ’ are obtained by Laplace transform technique. Df is noted for inducing anomalous conditions in ‘u’ and ‘θ’ especially for small Pr. The study also disclosed that the equilibrium state of the problem remains unaffected by Df effect.

Postelnicu [10] explored the heat and MT features of natural convection around an upstanding surface embedded in a saturated porous medium and exposed to a magnetic field. This analysis takes into consideration both Df and Sr, aiming to comprehensively understand the complex interactions within this. The governing PDE are converted into a set of coupled DEs and are solved in a numerical way. Finite difference method is used here. There is a rise in the values of Nu and Sh number by an increment in the value of magnetic parameter.

Niche et al. [11] numerically studied the transient double-diffusive natural convection occurring within a square cavity. In this setup, the erect walls of the enclosure are subjected to heating and cooling, resulting in both solutal and thermal gradients. The two flat walls of the enclosure remain impermeable and adiabatic; while the analysis takes into account the Dufour and Soret effects. The governing equations are solved by SIMPLER algorithm and control volume method. The primary objective of this study is to determine the flow regime in situations where thermal and solutal factors dominate.

Ahmed et al. [12] studied the influence of thermal diffusion and radiation on a transient free convective flow of an IECVF through an infinite upstanding porous plate within a rotating system, where the Hall current is considered. Additionally, this investigation accounts for variations in temperature and concentration at the plate, both of which fluctuate periodically with time. Under the assumption of a non-scattered medium and a non-gray nature of the fluid with emitting-absorbing characteristics, the equations are resolved using analytical methods.

Sarma et al. [13] endeavored to explore the Schmidt number (Sr) in MHD free convective flow through a porous medium. The flow is constrained by an infinitely tall erect porous plate with constant heat flux. Furthermore, a uniform magnetic field is applied perpendicular to the plate.

Hasan and Hossain [14] performed numerical studies of MHD mixed convection flow occurring through a permeable, impulsively stretched erect plate. The investigation takes into account of Sr and Df.

Mondal et al. [15] investigated the characteristics of Prandtl fluid when subjected to the effects of MHD heat and MT. The fluid flow occurs along an exponentially inclined vertical plate, and the analysis of the flow model employs boundary layer approximations.

Sheri and Modugula [16] aimed to analyze the combined heat and MT effects within an unsteady MHD flow scenario. The flow occurs over an IPIPM. In this analysis, the presence of Sr and Df as well as chemical reactions are considered.

Studying the influence of chemical reactions, heat sources and sinks, and the Joule effect on HT and MT mixed convection flow along an infinitely tall erect plate in magnetohydrodynamics (MHD) is a major research focus. Additionally, the study includes analyses of viscous dissipation and ohmic heating. This study was done by Ananthi and Sujatha [17].

Chamka [18] investigated the problem of 2D, unsteady, laminar boundary layer flow by examining an IECVF with heat absorption properties. The flow takes place along a semi-infinite vertical permeable moving plate. The study also accounts for the existence of a uniform transverse magnetic field and buoyancy effects due to thermal and concentration gradients. Rama Prasad et al. [19] investigated the effects of an aligned magnetic field and the Kuvshinski fluid model on unsteady MHD free convective flow over a moving inclined plate. The study also takes into account thermal radiation, radiation absorption with chemical effects, and mass blowing or suction.

In the current work, the emphasis is given to Soret effect. To analyze the impact of Sr and radiation parameter on the MHD free convective flow of an IECVF over a slanting plate immersed in a permeable substance is the prime aim of this paper.

Boussinesq’s approximation is a simplifying assumption utilized in fluid dynamics, specifically in the examination of buoyancy-driven flows. It is typically applied in scenarios where density fluctuations resulting from temperature differences are relatively minor. The buoyancy term emerges from density differences induced by temperature variations, giving rise to the inherent buoyant forces observed in fluids.

By the presumptions mentioned earlier and Boussinesq’s approximation, the governing equations are mentioned as follows:

Equation of Conservation of Linear Momentum is represented in Eq. (1).

Equation of Energy is presented in Eq. (2).

Equation of Mass transfer is presented in Eq. (3).

$\begin{align}  & \frac{\partial {{u}^{\prime }}}{\partial {{t}^{\prime }}}=v\frac{{{\partial }^{2}}{{u}^{\prime }}}{\partial {{y}^{\prime 2}}}+g\beta \left( T-{{T}_{\infty }} \right)\cos \alpha  \\ & +g{{\beta }_{c}}\left( C-{{C}_{\infty }} \right)\cos \alpha -\frac{\sigma {{B}_{0}}^{2}}{\rho }{{u}^{\prime }}-\frac{v}{{{K}^{*}}}{{u}^{\prime }} \\\end{align}$                      (1)

$\frac{\partial T}{\partial {{t}^{\prime }}}=\frac{K}{\rho {{C}_{p}}}\frac{{{\partial }^{2}}T}{\partial {{y}^{\prime 2}}}-\frac{1}{\rho {{C}_{p}}}\frac{\partial {{q}^{\prime }}}{\partial {{y}^{\prime }}}$                       (2)

$\frac{\partial C}{\partial {{t}^{\prime }}}={{D}_{M}}\frac{{{\partial }^{2}}C}{\partial {{y}^{\prime 2}}}-{{K}_{1}}\left( C-{{C}_{\infty }} \right)+{{D}_{1}}\frac{{{\partial }^{2}}T}{\partial {{y}^{\prime 2}}}$                 (3)

The conditions for the boundary are:

$\begin{align}  & {{t}^{\prime }}\le 0:{{u}^{\prime }}=0,T={{T}_{\infty }}, \\ & C={{C}_{\infty }}\text{ for all }{{y}^{\prime }}>0;{{t}^{\prime }}>0:{{u}^{\prime }}={{U}_{0}}, \\ & T={{T}_{\infty }}+\left( {{T}_{w}}-{{T}_{\infty }} \right)A{{t}^{\prime }}, \\ & C={{C}_{\infty }}+\left( {{C}_{w}}-{{C}_{\infty }} \right)A{{t}^{\prime }},\text{ at }{{y}^{\prime }}=0;{{u}^{\prime }}=0, \\ & T\to {{T}_{\infty }},C\to {{C}_{\infty }}\text{ as }{{y}^{\prime }}\to \infty  \\\end{align}$                          (4)

where, $A=\frac{u_0^2}{v}$. The constant $A t^{\prime}$, implies the gradual change in temperature over time. It is influenced by initial temperature difference and the rate of change of temperature. The Rosseland approximation has been accepted for radiative heat flux $q_r^{\prime}$, as follows:

$q_{r}^{\prime }=\frac{-4{{\sigma }^{*}}}{3a}\frac{\partial {{T}^{4}}}{\partial {{y}^{\prime }}}$                (5)

Here‘$\sigma *$’ is Stefan Boltzmann constant and‘a’is mean absorption coefficient. By Taylor series expansion of T⁴about T_∞ and ignoring higher order terms, the following term is obtained:

${{T}^{4}}\approx 4TT_{\infty }^{3}-3{{T}_{\infty }}^{4}$             (6)

The Eqs. (1)-(3) are normalized by introducing the dimensionless quantities mentioned below:

$\begin{align}  & u=\frac{{{u}^{\prime }}}{{{U}_{0}}},t=\frac{{{t}^{\prime }}U_{0}^{2}}{v},y=\frac{{{y}^{\prime }}{{U}_{0}}}{v},\theta =\frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}},\varphi =\frac{C-{{C}_{\infty }}}{{{C}_{w}}-{{C}_{\infty }}}, \\ & Gr=\frac{vg\beta \left( {{T}_{w}}-{{T}_{\infty }} \right)}{U_{0}^{3}},Gm=\frac{vg{{\beta }^{*}}\left( {{C}_{w}}-{{C}_{\infty }} \right)}{U_{0}^{3}},\Pr =\frac{\mu {{C}_{p}}}{K}, \\ & {{K}^{*}}=\frac{{{a}^{*}}U_{0}^{2}}{{{v}^{2}}},Sc=\frac{v}{{{D}_{M}}},M=\frac{\sigma {{B}_{0}}^{2}v}{\rho U_{0}^{2}},K=\frac{{{K}_{1}}v}{{{U}_{0}}^{2}}, \\ & N=\frac{16\sigma T_{\infty }^{3}}{3K{{a}^{*}}},Q=\frac{{{v}^{2}}{{Q}^{1}}}{KU_{0}^{2}},Sr=\frac{{{D}_{1}}\left( {{T}_{w}}-{{T}_{\infty }} \right)}{v\left( {{C}_{w}}-{{C}_{\infty }} \right)} \\\end{align}$             (7)

Eqs. (1) to (3) in nondimensional form are as follows:

$\begin{align}  & \frac{\partial u}{\partial t}=\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+Gr\theta \cos \alpha +Gm\varphi \cos \alpha  \\ & -\left[ M+\frac{1}{K} \right]u \\\end{align}$                    (8)

$\frac{\partial \theta }{\partial t}=\left( \frac{1+N}{\Pr } \right)\frac{{{\partial }^{2}}\theta }{\partial {{y}^{2}}}$                  (9)

$\frac{\partial \varphi }{\partial t}=\frac{1}{Sc}\frac{{{\partial }^{2}}\varphi }{\partial {{y}^{2}}}-K\varphi +Sr\frac{{{\partial }^{2}}\theta }{\partial {{y}^{2}}}$                 (10)

The boundary conditions (B.C.) corresponding to above equations are:

$\begin{align}  & t\le 0:u=0,\theta =0,\varphi =0\text{ for all }y>0 \\ & t>0:u=1,\theta =t,\varphi =t,\text{ } \\ & \text{at }y=0\text{ As }y\to \infty :u\to 0,\theta \to 0,\varphi \to 0 \\\end{align}$                (11)

The impact of different parameters like radiation parameter (N), Hartmann number (M), Schmidt number (Sc), thermal Grashof number (Gr), mass Grashof number (Gm), Prandtl number (Pr), permeability parameter (K), Soret number (Sr) on velocity profile (VP), temperature profile(TP), concentration profile(CP), SF, Nu and Sh are studied in this paper by maintaining other parameters as fixed.

Applying magnetic filed in perpendicular manner along the flow of the fluid will lower the velocity, hence VP drops with gain in the value of M, which is implied in Figure 2.

Figure 3 exhibits that velocity is higher for mercury (Pr = 0.025) when compared to electrolytic solution (Pr = 1). Viscosity of the fluid boosts up if the value of Prandtl number increases. As viscosity rises, the heightened internal friction opposing fluid motion becomes more evident. Elevated viscosity hinders the smooth flow of the fluid. This resulsts in a decrease in velocity. Hence the rise in the value of Pr decreases the VP.

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Figure 2. Effect of M on VP

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Figure 3. Effect of Pr on VP

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Figure 4. Effect of Soret on VP

The Soret number is a computation of the ratio between thermal diffusion and mass diffusion within a fluid. As Soret effect rises, it signifies an increased impact on thermal gradients and this can lead to enhanced buoyancy-driven flows and convection motions resulting in an increment in velocity. Thus, Figure 4 shows, the increase in Sr value, rises the fluid velocity. As Sc increases, momentum diffusivity improves. Hence from Figure 5 it is noted that, VP rises with hike in the value of Sc. Radiation creates aconsiderable growth in thermal condition of the temperature of the fluid that leads t more fluid to flow in the boundary layer because of buoyancy effect. Hence velocity in the fluid rises. Thus a surge in N effects to rise in velocity, which is depicted in Figure 6. Figure 7 shows, as time boosts up, the velocity of the flow grows.

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Figure 5. Effect of Sc on VP

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Figure 6. Effect of N on VP

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Figure 7. Effect of time on VP

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Figure 8. Effect of Gr on VP

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Figure 9. Effect of Gm on VP

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Figure 10. Effect of K on VP

Gr provides insight into the relationship between thermal buoyancy power and the velocity energy due to viscosity in the boundary layer, while Gm explains the ratio of the concentration buoyancy impact to the viscous velocity stamina. Hence VP improves due to the increase of thermal and species buoyancy forces, that is expressed in Figure 8 and Figure 9. Figure 10 implies, when the porosity of the fluid rises, fluid gets further space to flow and hence the velocity rises.

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Figure 11. Effect of K on TP

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Figure 12. Effect of Pr on TP

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Figure 13. Effect of N on TP

Figure 11 shows, the improvement in the value of K, declines TP, due to distinct interactions between the fluid and the porous medium. Figure 12 depicts that TP diminishes with upsurge of Pr value. The temperature in the boundary layer declines rapidly for greater value of Pr, since the width of the boundary layer diminishes with a rise in Pr value. N defines the specific contribution of heat conduction to thermal radiation transfer. Hence Figure 13 shows the raise in the N value leads to development in TP within the boundary layer. Figure 14 depicts that TP rises with hike in the value of time.

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Figure 14. Effect of time on TP

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Figure 15. Effect of K on CP

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Figure 16. Effect of N on CP

Increased permeability enables a greater volume of fluid to pass through the porous medium, leading to a more efficient dispersion of solutes. This enhanced fluid flow facilitates the reduction of solute concentration, resulting in reduction of CP, with increase in K, which is shown in Figure 15. Radiative HT usually results in fluid cooling, consequently impacting the distribution of concentration. The unique interactions between radiation and the fluid induce alterations in temperature gradients, affecting the mass transport of fluid components and leading to a decline in CP and is depicted in Figure 16.

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Figure 17. Effect of Pr on CP

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Figure 18. Effect of Sc on CP

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Figure 19. Effect of Sr on CP

The increment in the value of Pr signifies a prevalence of thermal diffusivity over momentum diffusivity, affecting temperature gradients and subsequently influencing mass transport, resulting in a reduction in CP, which is shown in Figure 17.

Figure 18 shows that the increment in Sc implies reduction of molecular diffusivity that lead to contraction of concentration boundary layer. Thus CP falls off for boost in Sc value. The rise in Sr intensifies the influence of thermal diffusion on concentration distribution, leading to a rise in the fluid’s concentration. Hence as Sr raise, CP boosts up, which is presented in Figure 19.

Figures 20 and 21 convey that SF decreases with gain in N and Sr. The elevation in the value of N facilitates radiative cooling, leading to a decline in the temperature and viscosity of the fluid near the surface. This results in a decrease in SF value, and is depicted in Figure 20.

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Figure 20. Effect of N on SF

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Figure 21. Effect of Sr on SF

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Figure 22. Effect of N on Nu

The hike in the value of Sr modifies concentration distribution due to thermal gradients, resulting in changes in viscosity of the fluid and consequent reduction in SF and is shown in Figure 21.

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Figure 23. Effect of Pr on Nu

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Figure 24. Effect of N on Sh

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Figure 25. Effect of Pron Sh

Radiation, as a mechanism of heat transfer, may exhibit lower efficiency in enhancing heat transfer compared to convection. At that time as the value of N increases, Nu decreases and is shown in Figure 22.

An increase in Pr indicates that the fluid possesses a lower thermal diffusivity relative to its momentum diffusivity. It results in a thicker thermal boundary layer (BL). This thicker BL facilitates more efficient heat transfer. Hence as Pr rises up, the value of Nu hikes as depicted in Figure 23.

When radiation leads to the augmentation of the convective mass transfer coefficient or induces alterations in the temperature distribution that favour mass transfer, then a rise in N results in increment of Sh value as exposed in Figure 24.

A hike in Pr implies that the fluid exhibits a reduced capability for conducting heat compared to transferring momentum. This contributes to the development of a thicker thermal BL relative to the momentum boundary layer. The increased BL thickness serves to improve mass transfer resulting in an increased Sh. This is exhibited in Figure 25.

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Figure 26. Effect of Scon Sh

An increment in Sc implies that the fluid has reduced efficiency in transporting mass in comparison with momentum. This results in a thinner mass BL relative to the momentum BL. The consequential reduction in mass BL thickness contributes to an improved mass transfer process, leading to a higher Sh, which is shown in Figure 26.

Improved permeability in porous media facilitates increased fluid flow and elevated interaction between the fluid and the medium. This interaction contributes to higher mass transfer rates, resulting in a hike in Sh value and is depicted in Figure 27.

To quantify the impacts of different parameters, the numerical calculations of SF, Nu and Sh at the boundary surface are worked out and expressed in Table 1 and Table 2.

The mentioned parameter values are considered for calculations unless specified otherwise in the table: Gr = 5, Gm = 3, Sc = 0.60, Sr = 1, N = 0.5, α = 60, ω = 0.1, K = 0.5, Pr = 0.71, M = 3, t = 5.

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Figure 27. Effect of Kon Sh

Table 1. Numerical results of SF, Nu and Sh for distinct values of parameters

Table 2. SF, Nu and Sh results for various values of parameters

Soret effect has many applications in practical life. Various particles move in different speed by the strength of temperature gradient. Hence different types of particles are detached from one another if they are blended together by applying Soret force. In the creation of optical fibre through vacuum coating processes, Soret effect is used. In the realm of optical fiber manufacturing, the Soret effect finds valuable application in precisely controlling the deposition and alignment of materials along the inner surface of a hollow preform. This preform serves as the precursor to the optical fiber. The Soret effect induces the movement of particles or molecules of the cladding material, (made of glass or dielectric materials) causing them to migrate from regions of higher temperature to lower temperature within the preform. This migration aids in the deposition and alignment of the cladding material along the preform's inner surface. As a result, this technique plays a pivotal role in ensuring that the core and cladding layers attain the exact dimensions and optical properties necessary for efficient signal transmission.

By tracking down aptamer binding, Soret effect aids in drug discovery process. The Soret effect, also referred to as thermophoresis, holds significant potential in assisting drug discovery processes, particularly in the realm of aptamer binding studies. Aptamers, which are short, single-stranded DNA or RNA molecules, possess the remarkable ability to specifically bind to target molecules, encompassing a wide range of biomolecules and potential drug candidates. In the pursuit of drug discovery, aptamers serve as invaluable tools fortargeting specific molecules of interest, such as proteins, nucleic acids, or small molecules. A fundamental aspect of this endeavor involves a deep understanding of the binding affinity and kinetics governing these interactions. In this context, the Soret effect emerges as a valuable tool for probing these binding interactions. By leveraging the Soret effect as a means to investigate thermophoretic motion, researchers can gain crucial insights into the binding interactions between aptamers and their target molecules. This approach contributes significantly to the development of novel therapeutic agents.

The consideration of the Sr effect in MHD flows over an IPIPM expands the scope of application, especially in processes where both thermal and mass transport is pivotal.

1. Biomedical Engineering: The mentioned study has potential applications in drug delivery systems, where heat transfer and controlled fluid flow are essential. The Sr effect can influence the concentration distribution of drug particles within the fluid.
2. Heat Exchangers: The considered concept becomes crucial in applications demanding precise temperature and concentration control, as in chemical processing industries.
3. Environmental Engineering: For simulating contaminant transport in porous media, the prescribed concept of the paper is applied. Grasping the impact of temperature gradients on concentration gradients is essential for accurate modelling and effective management of ground water contamination.