Effect of Double Stratification on MHD Williamson Boundary Layer Flow and Heat Transfer across a Shrinking/Stretching Sheet Immersed in a Porous Medium

Te present study aims to provide a mathematical model of the Williamson fuid fow via a permeable stretching/shrinking sheet in the MHD boundary layer in the presence of a heat source, chemical reaction, and suction. Tis study is novel because it investigates the physical efects of thermal and solutal stratifcation on convective heat and mass transport using thermal radiation. Te fow’s PDEs are numerically solved using the BVP4c approach and the pertinent similarity variables until a stable solution is found. Trough visual analysis, the efects of dimensionless factors on temperature, velocity, and concentration profles are examined. Tis encompasses the mass transfer rate, the heat transfer rate, and the coefcient of friction. Te results of the present analysis are found to be consistent with those of previously published studies. Te fndings demonstrate that enhanced temperature and concentration profles cause the Williamson, magnetic, and permeability parameters to rise in conjunction with a drop in the dimensionless velocity. In relation to temperature, the thermal stratifcation parameter exhibits the opposite tendency. Regarding the solutal stratifcation parameter, concentration profles are seen to show the opposite trend. Lastly, the current work will have important implications for the removal of dust and viruses from viscoelastic fuid in bioengineering, the medical sciences, and medical equipment.


Introduction
Non-Newtonian fuids may be classifed as pseudoplastic Williamson fuids.As its practical uses and inherent interest grow, the feld of research known as boundary layer fow in pseudoplastic fuids is growing in prominence.Te power law model, the Carreaus model, the Cross model, and the Ellis model are only some of the better examples of mathematical frameworks that may be used to explain the behaviour of pseudoplastic fuids.Williamson studied the behaviour of pseudoplastic fuids and proposed an equation-based model to defne their motion in 1929.Tere have been empirical confrmations of this idea ever since.Williamson fuid peristaltic fow in a vertical annulus was investigated for its impact on heat and mass transmission by Nadeem and Sher Akbar [1].In an asymmetric tube with porous walls, Vajravelu et al. reported the peristaltic motion of a Williamson fuid [2].Te Williamson fuid model and its many uses in studying fuid dynamics are presented.Te Williamson fuid injection into a rock fracture perturbation solution was described by Dapra and Scarpi [3].Khan and Khan [4] investigated many approaches to four fow issues using a Williamson fuid by employing the homotopy analysis method (HAM).Williamson fuid fow over a linear and exponentially stretched surface has been modelled in two dimensions by Nadeem et al. [5].Hayat et al. explored the timeindependent MHD fow of Williamson fuid across a porous plate and found a solution with a number of phases [6].Te fow of a non-Newtonian fuid through a pipe with variable permeability and an inclined plane was taken into consideration by the authors Bandi et al. [7].Sudheer Babu et al. [8] investigated the impact of chemical reactions on MHD heat and mass transfer in a Jefrey fuid fow with viscous dissipation and joule heating.Tey used a porous stretched sheet.
Te existence of many fuids, fuctuations in temperature, or both may all contribute to fuid stratifcation.Natural convection in a doubly stratifed medium is a fascinating and vital area of research owing to its wide range of practical applications in engineering.Termal sources include the condensers in power plants, whereas thermal energy storage techniques include solar ponds.Heat may also be dissipated into bodies of water like lakes, rivers, and seas using this method.Te impact of medium stratifcation on the process of removing heat from a fuid has not gotten much attention in the literature despite its signifcance.Jumah and Mujumdar [9] examined free convection heat and mass transfer away from a vertical fat plate in saturated porous media for non-Newtonian power law fuids under yield stress.Murthy et al. [10] looked at how double stratifcation afected free convection heat and mass transfer in a porous Darcian fuid-saturated medium with uniform wall heat and mass fow.Lakshmi Narayana and Murthy [11] investigated free convection heat and mass transport away from a vertical fat plate in a doubly stratifed non-Darcy porous material using the series solution technique.Cheng [12] considered the natural convection fow from a vertical wavy surface in a power-law fuid-saturated porous medium with temperature and mass stratifcation.Narayana et al. [13] studied free convection heat and mass transport in multilayer porous media saturated with a power-law fuid.
Analysis of heat along with the mass transfer on fuids over an exponentially extending surface with chemical reaction impact has a crucial role in nuclear reactors, aeronautics, drying procedures, cosmic fuid dynamics, heat exchangers, geothermal, chemical engineering, building construction, solar physics, solar collectors, and also oil recovery.Many researchers [14][15][16][17][18] are assessed over chemical reactions over a stretching sheet.Kumar et al. [19] studied the efects of Soret, DuFour, hall current, and rotation on MHD natural convective heat and mass transfer fow past an accelerated vertical plate through a porous medium.Goud et al. [20] analysed the thermal radiation and Joule heating efects on a magnetohydrodynamic Casson nanofuid fow in the presence of chemical reaction through a nonlinear inclined porous stretching sheet.Reddy et al. [21] investigated the efect of heat generation/absorption on MHD copper-water nanofuid fow over a nonlinear stretching/shrinking sheet.MHD boundary layer fow of nanofuid and heat transfer has been studied by the authors in [22][23][24][25].Unsteady MHD free convective fow past a vertical porous plate with variable suction discussed by Srinivasa Raju et al. [26].Te researchers whose studies the writers evaluated included Al-Ajmi and Mosaad [27], Hamimid et al. [28], Choukairy and Bennacer [29], Labed et al. [30], and Ram and Bhandari [31].Reddappa et al. [32], Krishnamurthy et al. [33], Krishna et al. [34], Khan et al. [35], Chetteti and Srivastav [36], Yousef et al. [37], Shaheen et al. [38], and Ananthaswamy et al. [39] studied doublestratifcation Jefrey fow, MHD free convection, and the infuence of second-order chemical processes in a porous material across an exponentially stretched sheet.
A fuid system that displays two separate layers of stratifcation, usually in relation to temperature or concentration gradients, is referred to as double stratifed.All properties of a fuid, including density, temperature, or concentration, that vary or layer with regard to depth are referred to as stratifcation.In order to create a more complex stratifed structure, double stratifcation indicates that the fuid contains two distinct gradients or variations.Heat transfer across a shrinking/stretching sheet submerged in a porous medium, and the MHD Williamson boundary layer fow is examined in the setting of double stratifcation.Tis allows for an examination of the consequences of additional complexity in the fuid system.With separate gradients for each stratifed layer, the temperature or concentration distribution is altered, while additional buoyancy efects are added.Understanding the efects of double stratifcation is essential to accurately portraying the physical processes occurring in the system.Tere may not have been enough research done on the efects of two-fold stratifcation in the context of MHD Williamson boundary layer fow with a shrinking/stretching sheet immersed in a porous medium.Te existing research may focus primarily on single stratifcation situations or fow confgurations.
Te necessity of employing various non-Newtonian fuids to store thermal energy has led to a signifcant increase in the use of Williamson fuid.Te Williamson fuid model may provide a starting point for analyzing shearthinning fuids, and it has limitations when applied to complex fuid dynamics problems involving double stratifcation, MHD, and heat transfer across porous media.A collection of partial diferential equations (PDEs) that are nonlinear express the fow mathematically.Te governing equations are converted into a set of ordinary diferential equations (ODEs) with the necessary boundary conditions using similarity transformations, which can then be numerically solved using MATLAB's built-in solver bvp4c.Te profles of velocity, temperature, and concentration have been graphically shown and tabulated, along with the efects of changing the parameters that control the fow.Te fndings show that the stratifcation parameter signifcantly afects the fow feld when wall suction is present.Tis research also includes an evaluation of the heat transfer coefcient from the perspective of industrial applications.Te fndings are expected to advance the body of knowledge and have real-world applications from earlier research.

Fluid Model
For Williamson fuid model, Cauchy stress tensor S is defned as [40].

2
International Journal of Chemical Engineering where S is extra stress tensor, μ 0 is limiting viscosity at zero shear rate and μ ∞ is limiting viscosity at infnite shear rate, Γ > 0 is a time constant, A 1 is the frst Rivlin-Erickson tensor, and _ c is defned as follows: Here, we considered the case for which μ ∞ � 0 and Γ _ c < 1. Tus, � τ can be written as By using binomial expansion, we get (5)

Formulation of the Problem
Consider the motion of a stretched sheet in a Williamson fuid that is incompressible, viscous, and electrically conductive.Te stretched sheet is enclosed in a porous material; therefore, it is assumed that the fuid is fowing through it k ′ .Assumptions about the fuid content along the sheet C w and the sheet temperature are made in a similar vein y � 0 is T w .Te fuid is fowing smoothly and has a somewhat low viscosity.Te boundary layer region, which is the thin layer adjacent to the sheet surface, appears to be the zone where the modifying infuence is least likely to occur.Within such a layer, the fuid velocity rapidly changes from its starting value to the mainstream value.Tis model was selected because it ofers a decent approximation of the behavior of various non-Newtonian fuids over a wide range of shear rates.Te origin is located at the leading edge of the sheet, and the y-axis is perpendicular to the x-axis, which is selected to be parallel to the surface of the sheet in the fow direction.B is a magnetic feld that has been applied perpendicular to the fow direction.Te induced magnetic feld will be much smaller than the applied magnetic feld, assuming that the fuid has some conductivity (Figure 1), and the magnetic Reynolds number will be much lower than unity.Let T stand for the fuid's temperature, u and v for the perpendicular velocity, and represent the x-axial velocity.
According to this hypothesis, the temperature and mass concentration of the surrounding medium are linearly stratifed and take the form of where u and v are the components of velocity in the x and y directions, respectively, υ � μ/ρ denotes kinematic fuid velocity, ρ denotes fuid density, μ is the coefcient of fuid viscosity, Γ stands for a Williamson time constant, k ′ is the permeability of the porous medium, T and T ∞ , respectively, are fuid temperature and ambient fuid temperature, Q is the heat source parameter.A uniform magnetic feld of strength B is applied in the transverse direction of the fow; due to the small magnetic Reynolds number, it is not necessary to introduce the efect of the induced magnetic feld.c p signifes the specifc heat at constant pressure, k r is the reaction rate, D is the difusion coefcient, and C ∞ is the ambient fuid concentration.Constraint set for boundary is [42] where U w � cx for the case of a stretched sheet and U w � −cx for the case of a contracting sheet, with c > 0 the amount of contraction or expansion held constant.v w speed at which mass is transferred over a wall in preparation v w > 0 for mass injection v w < 0 or mass suction.International Journal of Chemical Engineering Similarity transformations are [43] ψ where ψ is the stream function and η is the similarity variable.
Stream functions are given by u � zψ zy and From equations ( 11) and ( 12), the suitable similarity transforms are Equations ( 7)-( 9) can be simplifed by applying the aforementioned similarity modifcations.
Te following are the fow's relevant boundary conditions: is the non-Newtonian Williamson parameter, M � (σB 2 0 /ρc) is the magnetic parameter, λ 2 � υ/k ′ c is the permeability parameter, Pr � μ c p /k is the Prandtl number, e 1 � a 2 /a 1 is the thermal stratifcation parameter, c � Q/cρc p is the heat generation/absorption parameter, Sc � υ/D is the Schmidt number, e 2 � b 2 /b 1 is the solutal stratifcation parameter, K 1 � k r /c is the chemical reaction parameter, and S � v w / �� cυ √ is the wall mass parameter (with S > 0 (i.e., v w > 0) wall mass suction and S < 0 (i.e., v w < 0) wall mass injection).
Furthermore, the drag force coefcient in terms of C f , the local Nusselt number Nu, and the local Sherwood number Sh are determined by [43] by introducing the transformations ( 11) and ( 12), we have where Re x � U w x/υ is the local Reynolds number.

Methodology
Te BVP4c technique is a numerical method used to solve boundary value problems (BVPs).BVP involves fnding a solution to a diferential equation that satisfes specifed boundary conditions.Te BVP4c technique provides a robust and efcient way to solve BVPs numerically by discretizing the problem, formulating and solving the resulting system of equations, and incorporating the specifed boundary conditions (Figure 2) Adding new variables to transform higher-order differential equations into linear equations, Equations ( 14)-( 16) are transformed into the following the frst-order ODE.
with the boundary conditions: 4 International Journal of Chemical Engineering with the use of MATLAB bvp4c programming, approximative solutions are numerically calculated to show the practical importance of nondimensional quantities.

Results and Discussion
Numerical calculations have been made using the technique outlined in the preceding section for a variety of variables in order to assess the fndings like wall mass parameter (S), Williamson parameter (Wi), magnetic parameter (M), permeability parameter (λ 2 ), Prandtl number (Pr), thermal stratifcation parameter (e 1 ), heat generation/absorption parameter (c), Schmidt number (Sc), chemical reaction parameter (K 1 ), and solutal stratifcation parameter (e 2 ).
For illustrations of the results, numerical values are plotted in Figures 3-17.Table 1 compares the current fndings with the body of prior research in a few selected examples.Te current results and the body of literature showed higher agreement, we discovered.Tis demonstrates the reliability of the results as well as the precision of the numerical method we employed in this investigation.Table 2 displays the values for the skin friction coefcient, Nusselt number, and Sherwood number for a range of relevant parameter values.As magnetic parameter M, permeability parameter λ 2 , and thermal stratifcation parameter e 1 are increased, research demonstrates that an increase in the Williamson parameter Wi results in an increase in the skin friction coefcient; however, the trend is the opposite for the Sherwood number and the Nusselt number.It is evident that the skin friction coefcient, Nusselt number, and Sherwood number decrease.As increase in the wall mass parameter S, the skin friction coefcient results to decrease, whereas an opposite behaviour is noticed in Nusselt number and Sherwood number.As increase in solutal stratifcation parameter e 2 , it can be seen that the Nusselt number augments, while the Sherwood number shows the reverse pattern.Te Nusselt number increases as the Prandtl number Pr rises, according to observations, although the heat generation/absorption parameter c exhibits the reverse pattern.It is evident that when the parameters for chemical reactions K 1 and Schmidt number Sc rise, the Sherwood number amplifes.
Te results of the Williamson parameter on the concentration, temperature, and velocity profles during  6 International Journal of Chemical Engineering expansion and contraction are shown in Figures 3-5.Te graph shows that increasing Wi causes fuid velocity to drop, while increasing Wi causes temperature and concentration distributions to climb.Physically, a higher Wi decreases the fuid's velocity and enhances the fuid's heat distribution through the boundary layer by increasing the viscous forces that hold the Williamson fuid layers together.Fluid motion is impeded by the extended relaxing time.In physical studies of viscoelastic fows, the Wi is used to measure the contribution of elastic to viscous forces.Te temperature, concentration, and velocity profles for the shifting magnetic parameter during expansion and contraction are shown in Figures 6-8, respectively.As M increases, it is found that the fuid's velocity drops.However, the Lorentz force, which acts against fuid motion, increases with M, so the transport rate actually decreases with M.However, when it is kept constant, the fuid temperature and concentration distributions grow M, leading to a thicker thermal boundary layer.
In Figures 9-11, we see the profles of velocity, temperature, and concentration as a function of the permeability parameter for both the expanding and contracting cases.In both cases, the results improve with decreasing velocity λ 2 .Te temperature and concentration profles act diferently for diferent values of λ 2 .Te fuid temperature was shown to rise with an increasing porosity parameter.Te same behaviours occur when concentrating a fuid.In Figures 9-11, we see the profles of velocity, temperature, and concentration as a function of the permeability parameter for both the expanding and contracting cases.In both cases, the results improve with decreasing velocity.Te temperature and concentration profles act diferently for diferent values.Te fuid temperature was shown to rise with increasing porosity parameter.Te same behaviours occur when concentrating a fuid.Te physical efect of the porosity is to slow the fow of fuids and increase the temperature and concentration gradients across the material.In either the expanding or contracting situation, a large increase in the Prandtl number causes a signifcant decrease in fuid temperature and a thinning of the thermal boundary layer.When a sheet is stretched, this does not happen, but when a sheet is shrunk, an enormous Pr, fuid causes thermal unsteadiness at the superfcial (i.e., a negative value of Nu ), Figure 12 illustrates this efects.Figure 13 displays the temperature profle's response to increasing and decreasing the thermal stratifcation parameter.As becomes larger e 1 , the temperature profle fattens out.Physically, a high level of thermal stratifcation reduces the diference in temperature between the surface and the surrounding air.Changing the heat generation/ absorption parameter c results in a corresponding change in temperature, as seen in Figure 14.It has been determined that there is an upward trend in the temperature profle.
See  for visual representations of the fuid concentration's sensitivity to changes in the Schmidt number, the chemical reaction parameter, and the solutal stratifcation parameter.Te concentration characteristics and the concentration border thickness decrease as Sc, K 1 and e 2 augment together with the mass transfer rate.It follows that the solutal boundary layer grows as a result of both heat difusion and constructive reaction.Te solutal stratifcation parameter is the outcome of the possibility of a lower concentration of fuid near the plate than in the surrounding medium.

Conclusions
Te current work provides the Williamson fuid fow in the presence of a chemical reaction via a permeable stretching/ shrinking sheet in the MHD boundary layer.Tis work is innovative in that it uses thermal radiation to examine the physical consequences of thermal and solutal stratifcation on mass transport and convective heat.Te Williamson fuid model introduces non-Newtonian behavior, and exploring how the rheological properties infuence the fow and heat transfer can be a novel aspect.Understanding how shearthinning behavior afects the velocity and temperature profles would be valuable.Interesting discoveries, such as how these dual gradients impact heat transport close to the shrinking/stretching sheet, may result from the double stratifcation.In applications involving industrial or environmental processes, this element can be especially important.Te governing equations have been converted via dimensionless transformations into a set of nonlinear International Journal of Chemical Engineering ordinary diferential equations, which are then numerically solved using the BVP4c technique.Te efects of some regulating factors were examined and visually shown.Our numerical fndings lead to the following conclusions: (i) A decline in the dimensionless velocity and an improvement in the temperature and concentration profles lead to an increase in the permeability, magnetic, and Williamson properties (ii) Te temperature of a fuid decreases as its Prandtl number rises and vice versa (iii) By contrast, the heat generation/absorption parameter has the opposite impact on the temperature profle when it is increased (iv)

Table 2 :
Te values of the skin friction coefcient, Nusselt number, and Sherwood number for various values of Wi, M, λ 2 , Pr, e 1 , c, Sc, K 1 , e 2 , S.
Te Schmidt number, solutal stratifcation parameter, and chemical reaction parameter are all impacted by fuid concentration Concentration characteristics and border thickness decrease with increasing Sc and K 1 (v) Te temperature and local Nusselt numbers are reduced when the thermal stratifcation parameter e 1 is present (vi) Increasing values of the solutal stratifcation parameter e 2 lead to a decrease in the concentration and Sherwood numbers Condition at the free stream w: Condition at the surface.