Williamson MHD nanofluid flow with radiation effects through slender cylinder

Results in Engineering 22 (2024) 101966 Available online 14 March 2024 2590-1230/© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/bync/4.0/). Williamson MHD nanofluid flow with radiation effects through slender cylinder Saquib Ul Zaman a , Muhammad Nauman Aslam a , * , Muhammad Bilal Riaz b , c , Ali Akgul d , e , f , Azad Hussan g a Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan b IT4Innovations, VSB – Technical University of Ostrava, Ostrava, Czech Republic c Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon d Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon e Siirt University, Art and Science Faculty, Department of Mathematics, 56100, Siirt, Turkey f Near East University, Mathematics Research Centre, Department of Mathematics, Near East Boulevard, PC:99138, Nicosia, Mersin 10, Turkey g Department of Mathematics, University of Gujrat, Gujrat, Pakistan ARTICLE INFO Keywords: Williamson model Numerical solution Slender cylinder MHD Radiation Nanofluid ABSTRACT The aim of this article is to analyze the Williamson nanofluid flow with magnetohydrodynamics (MHD) and radiation effects through the slender cylinder. The mass and heat transfer are analyzed under the different assumptions of viscosity, density, and thermal conductivity. Conservation of momentum and energy are modeled to exhibit the impact of the problem. The Buongiorno model is used for this analysis. The flow of Williamson nanofluid through a slender cylinder along with magnetohydrodynamics and radiation effects with constant viscosity is not studied yet. Which is the novelty of current research work. Flow governing equations are firstly converted into ordinary differential equations and then demonstrated numerically by using MATLAB bvp4c. The effects of dimensionless numbers on the non-dimensional fields are investigated and shown in graphical and tabular form. We concluded that the velocity profile reveals the decreasing behavior for curvature and buoyancy parameter. Radiation parameter and Prandtl number boost the temperature profile. The thermophoresis parameter decreases the concentration profile while the Brownian parameter increases it. Applications of this specific study in various scientific and engineering fields that ultimately benefit humanity’s health, technology, and environment. 1. Introduction Recent researchers have paid particular attention to non-Newtonian fluids. These fluids are resplendent and popularized areas of research in fluid mechanics due to their powerful benefits in medical, engineering, mathematics, and as well as industries. Various models are studied because of non-linear behavior betwixt the stress and rate of deformation. Non-Newtonian fluids have many examples in daily life as the preservation of foods, manufacturing of plastics, performance of various lubricants, and production of paper. Viscous fluids are non-linear in behavior because of their non-linearity, it is a hellacious work to explore a single model that shows all of its properties. Examples of nonNewtonian fluids include dirt, pasta, cheese, ice cream, polymer solutions, asphalt and others. The usual properties of solutions and melts of polymer have made a significant contribution to the global polymer processing industry. Non-Newtonian fluids play a significant role in a variety of industries, including the processing of materials, engineering, food, agricultural water distribution, the chemical industry, oil reservoir engineering, biomedical engineering, and many others. Megahed & Abbas [1] analyzed the motion of non-Newtonian fluid through penetrable surfaces and also demonstrated the impact of heat generation on flow. Hageman & de Borst [2] formulated the non-Newtonian behavior of fluid flow by using finite element shape. An et al. [3] studied the industrial applications of non-Newtonian fluid flow and heat transfer analysis through the penetrable medium. Gomez-Constante & Rajagopal [4] investigated the flow of non-Newtonian fluid inside an elliptic tube and considered the under-discussion tubes perfectly circular. Khan et al. [5] demonstrated the three-dimensional fluid flow behavior via a stretchable (shrinkable) permeable sheet and solved their problem by * Corresponding author. The University of Lahore, Lahore, Pakistan. E-mail addresses: [email protected] (S.U. Zaman), [email protected] (M.N. Aslam). Contents lists available at ScienceDirect Results in Engineering journal homepage: www.sciencedirect.com/journal/results-in-engineering https://doi.org/10.1016/j.rineng.2024.101966 Received 5 September 2023; Received in revised form 30 November 2023; Accepted 28 February 2024 Results in Engineering 22 (2024) 101966 2 using the homotopy analysis method. Trivedi et al. [6] analyzed heat transfer and characteristics of non-Newtonian fluid through the cylinder. The Williamson fluid model is also an example of a non-Newtonian fluid that shows the pseudoplastic fluids properties. The fluid undergoes an instant decrease in viscosity with the enlargement in shear price. As compared to other pseudo-plastic fluids less work has been done on Williamson fluid. Vinodkumar Reddy et al. [7] discussed presence of chemical reaction and suction/injection mhd radiative flow of williamson nanofluid with cattaneo-christov model over a stretching sheet through a porous media. Amanulla et al. [8] analyzed heat and mass transfer via hydromagnetic flow in a nano williamson fluid past a vertical plate with momentum and thermal slip effects: a numerical investigation. In fluid mechanics and physics, a fine layer of fluid formed near the boundary by the fluid flow along the surface is a boundary layer flow. The fluid interaction with the boundary wall induces a no-slip boundary condition. The boundary layer has two types. One is laminar boundary layer flow and the other is turbulent boundary layer flow. Emam [9] investigated the boundary layer, incompressible and steady flow of fluid through a cylinder by considering it permeable and solved the problem analytically and numerically. Khan et al. [10] presented the formulation of Maxwell fluid boundary layer flow via a rotating stretchable cylinder by adding the magnetic influence. The approximation of boundary layer flow is used in different practical flow problems of engineering and scientific interest. In many industrial processes, improving heat transmission presents a significant challenging behavior. Low thermal conductivity materials are problematic to utilize as heat transfer boosters. The majority of solids, like metals, have three times greater heat conductivity than common materials (i.e., oil, water, glycol, ethylene, etc). It implies that solid-particle-containing fluids can improve the thermal conductivity of common materials. Scientists and engineers were inspired by this discovery to find fluids with high thermal conductivity in order to stop energy loss. Using this concept, nanofluids are created which are used often in engineering and physiological activities. To enhance the heating as well as cooling systems, hyperthermia, batteries, medication delivery systems, and other techniques may be used in such operations. Waqas et al. [11] formulated a nanofluid flow problem mathematically and attained the solution numerically by using MATLAB bvp4c. They also discussed the applications of cylindrical shape nano sized devices. They analyzed impact of various parameters against thermal and flow fields. Sucharitha et al. [12] discussed Convective flow of MHD non-Newtonian nanofluids on a porous sheet undergoing chemical reaction with double diffusion due to Cattaneo-Christov. Vajravelu et al. [13] analyzed a comparative examination of MHD non-Newtonian fluid flows over a stretching sheet including the impact of radiation and chemical reaction. Waini et al. [14] demonstrated the flow impact of hybrid nanofluid and also presented the concentration, velocity and temperature profiles for various parameters. Amanulla et al., [15] discussed numerical models of non-Newtonian nanofluid flow over a semi-infinite vertical surface with slip effects in magnetohydrodynamics. Waini et al. [16] investigated benefits of nanofluid flow over shrinkable cylinder and obtained unique solution by using bvp4c solver. Ros¸ca et al. [17] investigated the Buongiorno nanofluid model theoretically and also obtained a closed-form solution analytically. Magneto hydrodynamic (MHD) contains fluid, magnet and a field which collectively analyze the development of gadgets. MHD flow has several practical and hypothetical investigations due to their industrial applications. Because of number of manufacturing applications like electric propulsion, air craft formation and cooling of nuclear reactors. MHD is a vigorous area of interest in biological engineering. In many physical, geothermal, and industrial fields, magneto-hydrodynamic (MHD) flow has a prodigious number of experimental and theoretic investigations. The MHD flows associated with warmness transfer have won a good-sized appeal from the researchers. This is due to its applications in numerous business fields along with cooling nuclear reactors, electric-powered propulsion for space investigation, aircraft, crystal evolution in liquids, microelectronic devices etc. Meenakumari et al. [18] analyzed the two-way non-linear stretching surface with temperature-dependent conductivity and heat generation/absorption in MHD 3D flow of Powell eyring fluid. Ramamoorthy et al. [19] analyzed the heat and mass transfer in an inclined, asymmetric porous conduit with second-order slip flow of a conducting Jeffrey nanofluid. Mahfoud et al. [20] numerically analyzed fluid flow problems between coaxial cylinders under magnetic effects. They solved the problem by using the finite volume method and concluded that magnetohydrodynamic effects control the transition to asymmetry flow and transfer of heat. Amanulla et al. [21] analyzed the impact of slip on an electric conducting viscoelastic fluid flowing past an isothermal cylinder is simulated numerically. Amanulla et al. [22] analyzed non-Newtonian boundary layer flow of nanofluid via a semi-infinite vertical plate with partial slip: a computational analysis. Nourbakhsh et al. [23] demonstrated the fluid flow problem with the impact of magnetohydrodynamic and radiation over a static plate. They solved the equations of three basic principles of fluid mechanics by using the finite difference technique and calculated results for magnetohydrodynamic, Prandtl number, the parameter of radiation on velocity, and temperature of the fluid. KS [24] analyzed the fluid turbulent flow with the impact of magnetohydrodynamic behavior and thermal enhancement over a circular cylinder. Pallavarapu et al. [25] discussed radiation effects on a three-dimensional MHD Williamson fluid flow on a stretchy surface. Amanulla et al. [26] discussed Nomenclature u,w Velocity components x Velocity along x-direction r Velocity along r-direction K Thermal conductivity ρ Density Cp Specific heat T Temperature Uw Velocity at wall Tw Temperature at wall μ ∞ Infinite viscosity A1 Rivilin-Erickson tensor λ Buoyancy parameter γ Curvature parameter Le Lewis number M Magnetic parameter ψ Concentration parameter f Dimensionless velocity θ Dimensionless temperature Uw Velocity at wall Pr Prandtl number φw Concentration at wall ν Kinematic viscosity Nt Thermophoresis parameter Nb Brownian parameter R Radiation knf Thermal conductivity of nanofluid Γ Time relaxation η Similarity variable T∞ Ambient temperature S.U. Zaman et al. Results in Engineering 22 (2024) 101966 3 impact of wavy wall amplitudes on mixed convection heat transfer in a ventilated wavy cavity with a central circular cold body and a copper-water nanofluid filled. Zaman et al. [27] discussed the chemically reactive fluid flow with thermophoresis and brownian effects. Aslam et al. [28] analysed heat transfer analysis of Eyring-Powell fluid with temperature dependent viscosity. Zaman et al. [29] discussed williamson flid flow with radiation effects. Literature shows non-Newtonian nanofluid flow with radiative magnetohydrodynamic impact and radiation effects with constant viscosity is not studied yet. Which is the novelty of current research work. The developed model is highly nonlinear. The governing system of PDE’S is transformed into highly nonlinear ODE’S by utilizing proper similarity transfiguration along with the boundary conditions. The resulting system of ODE’S is pursued numerically by bvp4c. Results for concentration, temperature and velocity of fluid are presented in tabular and graphical form. 2. Geometry of the problem Two-dimension, incompressible and steady state Williamson nanofluid flow in the presence of a boundary layer approximation theory is studied. 3. Mathematical modelling τ =[ μ ∞+ μ 0− μ ∞ 1−Γ˙γ]A1,(1) In equation (1) the τ symbolizes for stress tensor of Williamson’s fluid model, A1 is the First Rivlin–Erickson tensor, Γ is the time relaxation material constant, μ 0 and μ ∞ signifies the variable viscosity and viscosity at infinite rates. Π=1 2tr(gradV +gradVT)2(2) Equation (2) is the second invariant tensor. Equation (2) involved in ˙γ= 1 2 √Π. Applying Γ˙γ<1 for shear thinning property of Williamson’s fluid and μ ∞=0, we modify equation (1) as τ =[ μ 0 1−Γ˙γ]A1(3) By binomial expansion to equation (3) and also neglecting Γ2 and higher terms, equation (3) reduces to τ = μ 0(1−Γ˙γ)A1,(4) Using all above assumption the continuity, momentum and energy equation are (rw)r+(ru)x=0(5) uux+wur= υ [1 rur+urr +Γ 2 ur 2 r+ 2 √Γururr]− σ B2 0 ρ w(6) uTx+wTr=k ρ Cp(Trr +1 rTr+ +Txx)+ τ (DBφrTr+DT T∞ Tr 2) −1 r ∂ ∂ r[rqf]+q∗=0,(7) wφr+uφx=DB(φrr +1 rφr)+DT T∞(Trr +Txx)− Rr(φ−φ∞).(8) here w and u are considered as the velocity components along the z and x direction respectively, pressure is p, ρ denotes the density, T is temperature, k is thermal conductivity, γ is curvature, kinematic viscosity is υ , cp represent specific heat using constant pressure and the Rosseland model radiative heat flux qf is provided by qf=−4 σ ∗ 1 3k∗ 1 T4 r.(9) T4 can be written as with the help of Taylor series, T4=4T3 ∞T−3T4 ∞.(10) So qf is, qf=−16 σ ∗ 1 3k∗ 1 Tr.(11) U∞ is the velocity at the surface, free stream velocity is defined as U= U∞(x l). The boundary conditions in dimensional form are [30] u(x,a)→U(x),u(x,a) = 0 for r→∞, T(x,a)→T∞,T(x,a) = Tw(x)as r→∞.}(12) To solve equations (6)–(8) we utilize appropriate similarity transformations: u=xU∞ lf ′ ( η ),w=−a r(U∞ l)f( η ), ψ =φ−φ∞ φw−φ∞ ,θ=T−T∞ Tw−T∞ , η =r2−a2 2a(U∞ υ l). ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (13) The PDE’s (6–8) are transformed into an ODE’s by using transformations (13) and can be written as (1+2γ η )f ‴ +2γf ″ +3 2(1+2γ η )1 2λf ″ 2+λ(1+2γ η )3 2f ″ f ‴ +ff ″ −f ′ 2−Mf ′ =0, (14) 1 Pr(1+2γ η )(1+Rd)θ ″ +Nb(1+2γ η )θ ′ ψ ′ +Nt(1+2γ η )θ ′ 2+2Pr(1+Rd)γθ ′ +f θ ′ −f ′ θ=0, (15) (1+2γ η ) ψ ″ +2γ ψ ′ +Le Pr(f ψ ′ −f ′ ψ )+Nt Nb([(1+2γ η )θ ″ +2γθ ′ ]− Q∘ ψ =0. (16) where curvature parameter is γ=1 a x υ U∞ √, Buoyancy parameter λ= Γ 2U∞3 x υ √, Pr = υ α is Prandtl number, magnetic parameter is M=l σ B2 0 ρ U∞ ,Nb= τ DB(φw−φ∞) υ is Brownian parameter for motion, Nt= τ DT(Tw−T∞) υ T∞ is thermophoresis parameter and Le = α DB is Lewis number. Dimensionless boundary conditions are f(0) = 1,f ′ (0) = 0,f ′ →1,as η →∞, θ(0) = 1,θ→0,as η →∞, ψ (0) = 1, ψ →0,as η →∞. ⎫ ⎬ ⎭ (17) Nusselt number, skin friction and Sherwood number are the basic physical quantities that need to be examined and are defined as Cf=2 τ w ρ u2 w ,Nux=xnw k((Tw−T∞),Shx=xqw DB(φw−φ∞),(18) where τ w= μ (ur+Γ  2 √ur2)is stress over the cylinder, nw= − k(Tr)|r=R, qw= − DB(φr)|r=R is heat flux. By using all values in equation (18), we get Nusselt number Nu x , Skin friction Cf, Sherwood number Shx given below S.U. Zaman et al. Results in Engineering 22 (2024) 101966 4 NuxRex−1 2= − θ ′ (0), 1 2CfRex−1 2=f ″ (0)+1 2λf ″ 3(0), ShxRex−1 2= − ψ (0). ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (19) Rex−1 2 is the Reynold number. 4. Numerical technique For the computational results of generalized non-linear DE’s (differential equations) (14–16), we will use the favourable numerical technique of Bvp4c. The graphical interpretations are elaborated that gives the better results of our cumulative data. The governing system is substituted as follows: f( η )=y1,f ′ =y2,f ″ =y3,f ‴ =y ′ 3,(20) θ( η )=y4,θ ′ =y5,θ ″ =y ′ 5,(21) ψ ( η )=y6, ψ ′ =y7, ψ ″ =y ′ 7.(22) Hence the first-order system is given as below: 5. Graphs and discussions We investigated the MHD nanofluid flow phenomenon through slender cylinder by adding radiation effects. Firstly, we mathematically analyzed the problem governing Navier stokes equations and simplified them by using similarity transformation. Then obtained the solution numerically by using MATLAB Bvp4c. The graphical and tabular results are presented below. Following governing parameters requirements are considered here, which are physically realistic for Williamsons nanofluid flow through a slender cylinder. 0.1≤λ≤5.3,7≤Pr ≤13,0.1≤ Nb≤16,0.1≤Rd≤0.9, Nt≥1 Fig. 1 shows the geometry of the flow problem. Fig. 2 demonstrates the influence of temperature profile due to Rd. Temperature goes down and then up by increasing Rd =0.1,0.3,0.5, 0.7,0.9. As by increasing radiation or heat, the fluid’s temperature keeps increasing until it reaches a point where the heat being added is balanced by the heat being lost or radiated away. At this point, the temperature curve levels off, and the temperature remains relatively constant. And when suddenly reduce or stop the radiation or heat input, the fluid doesn’t cool down immediately. Fig. 3 depicts the temperature curve for thermophoresis parameter.Temperature goes down and then up by increasing values of thermophoresis parameter 3,4,5,6,7.Fig. 4 demonstrates the influence of temperature profile due to Pr. Temperature goes down and then up by increasing Pr =9,10,11,12,13. Fig. 5 demonstrates the profile of temprature for Brownian parameter. Curve goes up and then down by increasing for Brownian parameter 1,4,8,12, 16.While increasing the Brownian parameter, the fluid becomes more chaotic on a microscopic level. This initial chaos can actually cause the temperature to drop. Think of it like stirring a cup of hot coffee vigorously, it cools down a bit because you’re making it mix more. Fig. 6 depicts the temperature curve for M.Curve shows positive and its Fig. 1. Physical model of the flow problem. Fig. 2. Response of radiation parameter on temperature. y/ 3(x) = (−2γγ3−3λγ(1+2xγ)(1/2)(y3)2−y1y3+(y2)2−λN(1−φ∞)(y4+Nry6)− Hay2) (1+2xγ)+2λ(1+2xγ)3/2y3(x),(23) y/ 5(x) = −2γγ5−Pr(y1y5+y2y4)− (1+2xγ)(Nby5y7+Nty2 5)+P∗y2+Q∗y4 (1+2xγ), y/ 7(x) = − 2γγ7−Le Pr(y1y7−y2y6)− (Nt/Nb)[(1+2xγ)y/ 5+2γcy5−Qoy6] (1+2xγ).(25) (24) S.U. Zaman et al. Results in Engineering 22 (2024) 101966 5 opposite response due to rise in M=1,2,3,4,5.Fig. 7 presents the velocity profile for λ. Curve of velocity goes up and then down by increasing for λ=1.8,2.4,3.9,4.6,5.3.When the buoyancy parameter increases, it means the fluid becomes less dense and flows more easily. It’s a like how hot air rises because it’s less dense than cold air. So, when we heat up Williamson’s fluid or increase the buoyancy parameter, it tends to flow faster. Fig. 8 presents the velocity profile for γ.Curve of velocity shows positive and its opposite response due to rise in γ=0.1, 3.3,6.3,8.3,10.3.Fig. 9 presents the velocity profile for M. Curve goes up and then down by increasing for M=2.3,3.5,4.7,5.6,6.2.Fig. 10 is for concentration profile against thermophoresis parameter. When the thermophoresis parameter is low, it means that temperature changes in the fluid have a relatively weak impact on how these tiny particles move around. So, the concentration of these particles in the fluid stays relatively stable, and you might see a consistent concentration curve. The effect of temperature on the movement of these particles becomes stronger. Essentially, the heat causes these particles to move away from hotter regions. Curve shows positive response due to rise in Nt=1,2, 2.5,3,3.5,4.Thermophoresis explains the mechanism of mass and heat transfer due to small particles movement. Fig. 11 demonstrates the profile of concentration for Brownian parameter. Curve goes down for Nb=0.1,0.7,1.3,2.1,2.9.This happens because when the Brownian parameter is higher, the random movements of particles within the fluid increase. As a result, these particles disperse more widely throughout the fluid rather than remaining concentrated in specific regions. This increased dispersion leads to a more uniform distribution of particles, causing the concentration curve to decrease as the concentration of particles in any specific area becomes more spread out. Fig. 12 shows variation of γ on concentration profile. Profile bending down due to increment in γ=0.11,0.14,0.16,0.18,2.Fig. 13 represents concentration profile for Le.Curve shows decreasing behavior for Le=0.1,0.5, 0.8,1.5,2.3.Higher Lewis number leads to a decreased rate of mass diffusion relative to thermal diffusion. Consequently, the concentration gradient of the particles or components in the fluid becomes less Fig. 3. Behavior of T P on temperature. Fig. 4. Effects of Pr on temperature. Fig. 5. Temperature profile for B P. Fig. 6. Temperature profile for M. S.U. Zaman et al. Results in Engineering 22 (2024) 101966 6 pronounced. This results in a smoother distribution of the particles throughout the fluid, causing the concentration curve to decrease as the concentration differences between different points in the flow reduce due to increased thermal diffusivity dominating over mass diffusivity. Fig. 14 represents concentration profile. Curve exhibits positive response for M=1,2.4,5.4,7.4,9.4. Fig. 15 represents concentration profile for Pr.Due to enhancement in Pr =7.1,8.1,9.1,10.1,11.1 the curve of temperature shows decreasing behavior. Consequently, the temperature gradient decreases as heat dissipates less efficiently compared to the dissipation of momentum within the fluid. This results in a smoother temperature distribution throughout the flow, causing the temperature curve to decrease as the temperature differences between different points in the flow decrease due to increased momentum diffusivity dominating over thermal diffusivity. Prandtl number is basically a ratio of thermal diffusivity and momentum. Fig. 16 represents results of Nusselt number for different values of Pr and curvature. Also Table 1 shows the result of Nusselt number for different values of Pr and curvature. 6. Validation of code The algorithm is given. The advantages of this method are time savings, ease of usage, and consistent results. The whole process is shown in the stages that follow [30]. (1) Developing the Williamsons nanofuid’s flow-governing mathematical model. (2) The similarity transformation is used to transform the governing partial differential equations into ordinary differential equations. Fig. 7. Velocity profile against λ. Fig. 8. Behavior of velocity for γ. Fig. 9. Impact of M on velocity. Fig. 10. Results of thermophoresis parameter on concentration profile. S.U. Zaman et al. Results in Engineering 22 (2024) 101966 7 (3) The implementation of MATLAB’s computational program like bvp4c; utilizing bvp-4c to get required numerical solution of transformed partial differential equations. (4) Numerical data will be produced by achieving the dimensionless form using the Nusselt number and shear skin relation. (5) Finally, coding the entire problem in MATLAB, producing graphical and numerical findings, and offering a conclusion. 7. Conclusion Non-Newtonian nanofluid under magnetohydrodynamics (MHD) and radiation effects through slender cylinder studied. Impact of radiation in the flow problem is significant because of its application in industrial area, engineering and physics. Williamsons nanofluid flow with magnetohydrodynamic impact and radiation effects is not studied yet. Which is the novelty of current research work. We change the governing non-linear equations into ordinary differential system by using transformation. Behavior of the distinct parameters on the curve of concentration, velocity and temperature are studied and also shown graphically. Important major points are: 1) Velocity curve shows both increasing and decreasing behavior by rising the values of λ and γ. The change in viscosity with increasing curvature can cause changes in the flow profile. It is observed that the fluid flows more rapidly near regions of higher curvature, while it flows more slowly in regions with lower curvature. 2) Temperature curve shows both increasing and decreasing behavior by increasing Prandtl number, Nb, Nt, and radiation parameter. Fig. 11. Response of Brownian parameter on concentration profile. Fig. 12. Variations of γ for concentration profile. Fig. 13. Results of L e on concentration profile. Fig. 14. Results of M on concentration profile. S.U. Zaman et al. Results in Engineering 22 (2024) 101966 8 Depending on the specific values and interactions among these parameters, the temperature curve might exhibit regions of both increasing and decreasing behavior. The net effect on the temperature curve results from the interplay and dominance of each parameter’s influence under varying conditions within the Williamson fluid flow. 3) The concentration curve shows decreasing behavior by increasing Le, Pr, Nb and increasing behavior by increasing thermophoresis. A higher Brownian diffusion coefficient in Williamson fluid helps things mix faster, leading to an increase in concentration when you add more of a substance to the fluid. CRediT authorship contribution statement Saquib Ul Zaman: Writing – original draft, Methodology. Muhammad Nauman Aslam: Formal analysis. Muhammad Bilal Riaz: Funding acquisition. Ali Akgul: Methodology, Project administration, Software. Azad Hussan: Supervision, Validation. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. References [1] A.M. Megahed, W. Abbas, Non-Newtonian cross fluid flow through a porous medium with regard to the effect of chemical reaction and thermal stratification phenomenon, Case Stud. Therm. Eng. 29 (2022) 101715. [2] T. Hageman, R. de Borst, Flow of non-Newtonian fluids in fractured porous media: isogeometric vs standard finite element discretisation, Int. J. Numer. Anal. Methods GeoMech. 43 (11) (2019) 2020–2037. [3] S. An, M. Sahimi, T. Shende, M. 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