Numerical study of thermal and solutal stratification via Williamson nanofluid model with Cattaneo–Christove heat flux

Nomenclature

BVP

boundary value problem

C

concentration

C _fx

skin friction

Ec

Eckert number

IVP

initial value problem

k

positive integer

K

thermal conductivity

k*

absorption constant

Le

Lewis number

M

magnetic parameter

MHD

magnetohydrodynamics

N ₁

solutal stratification parameter

N ₂, N ₃

temperature ratio

Nb

Brownian motion parameter

Nt

thermophoresis number

Nu _x

Nusselt number

p

pressure

Pr

Prandtl number

Rd

thermal radiation parameter

RK

Runge–Kutta

T

fluid temperature

T _∞

ambient temperature

u, v

velocity components

α

thermal diffusivity

λ

porosity constant

λ _d

Cattaneo–Christov concentration parameter

λ _t

Cattaneo–Christov temperature parameter

μ

dynamic viscosity

ν

kinematic viscosity

ρ

density

ρc _p

heat capacity

σ*

Stefan–Boltzmann constant

1 Introduction

With the rapid advancement of scientific and technological disciplines, researchers have been compelled to extend their analysis to boundary layer flow over linear stretching sheets. This area of study has wide range of industrial applications, including metal spinning, rubber sheet production, fiber glass manufacturing, and polymer processing such as wire drawing. The investigation of boundary layer flow over both linear and nonlinear stretching surfaces is particularly important because it addresses several biomedical and engineering challenges, such as cooling of electronic components, metal sheet processing, wire drawing, and petroleum extraction. Boundary layer theory, a fundamental concept in fluid mechanics, plays a crucial role in bridging theoretical solution with practical applications. Since its development in the 20th century, the theory has advanced rapidly due to its extensive applications. For instance, it has been used to calculate drag forces on objects moving through fluids, including blades, ships, airplanes, turbines, and airfoils. In doing so, it connects simplified theoretical model with real world problem, with progress often validated through experimental studies. In recent decades, significant attention has been directed toward heat and mass transfer in two-dimensional (2D) boundary layer flow over nonlinear stretching surfaces. This interest arises from its broad practical applications across engineering and industry, including extrusion, sheet manufacturing, air cooling systems, paper production, and drying technologies [1], 2]. Khan et al. [3] investigated the analytic solution of magnetized bio-convective second grade fluid on a porous media through a nonlinear flexible surface due to abounded of Cattaneo–Christov theory. Usafzai [4] derived an exact solution for a second-grade nanofluid considering slip boundary conditions and mass and heat transfer through a porous permeable medium. In order study, Khan et al. [5] conducted a numerical investigation on mixed convective flow of a Casson fluid influenced by the Cattaneo–Christov double diffusion model over an inclined, linearly flexible sheet. Furthermore, Usafzai et al. [6] investigated a comparatively analysis of hybrid nanofluid through a linear stretching surface in the presence of MHD. The numerical solution for boundary layer flow in the existence of Carreau fluid in the influence of thermal conductivity and viscous dissipation across a nonlinear flexible surface was reported by Kudenatti et al. [7]. Shaheen et al. [8] investigated the analytic solution of heat and mass transfer due to the existence of Williamson nanofluid through a permeable porous media. Nasir et al. [9] reported the nanoparticle dynamics in the influence of binary chemical process in the corresponding porous media by using a purely numerically technique. Sivakumar et al. [10] analyzed the different shape factor impact of nanofluid on mixed convection flow over a permeable medium.

Fourier [11] proposed the classical law of heat conduction, which has long served as the function of heat transfer analysis. To address the limitation of instantaneous propagation implied by Fourier’s law Cattaneo [12] introduce a relaxation time, and Christov [13] further refined this formulation by replacing the time derivative with the Oldroyd upper convective derivative, resulting in the Cattaneo–Christov heat flux law. This model extends Fourier’s law by incorporating finite heat propagation speed and thermal relaxation effects, thereby resolving the paradox of infinite thermal wave velocity and providing a more realistic description of heat transfer in complex systems. It highlights the influence of relaxation time on boundary layer thickness, fluid temperature, and heat transfer coefficients, and has been shown to yield more accurate solutions for viscoelastic and non-Newtonian fluids with complex rheological properties. Due to these advantages, the Cattaneo–Christov model has been widely applied in engineering problems such as heat exchangers, melting and evaporation processes, and multiphase flow systems, where its ability to couple conservation laws of momentum, mass, and heat with realistic heat flux prediction is particularly valuable [14]. Tharapatla et al. [15] examined the role of the Cattaneo–Christov heat flux model in describing the flow properties of a Williamson nanofluid over a porous media. Salahuddin and Awis [16] analyzed the non-isothermal flow over an exponential surface and reported that larger values of the thermal relaxation parameter lead to reductions in both the temperature distribution and the heat transfer coefficient.

Williamson fluid represents an important class of non-Newtonian fluids due to their broad range of applications in science, technology, and engineering. They have been utilized in processes involving material fabrication, nuclear and chemical industries, bioengineering, and geophysical flow. Their behavior under the influence of thermal radiation makes them particularly relevant to solar energy systems, advance manufacturing, and various heat transfer operations. Practical applications also extend to oil recovery, filtration, polymer processing, ceramic production, and petroleum extraction, where accurate modeling of their rheology is essential for optimizing performance. The pioneering contributions of Williamson [17] provide the first framework for describing pseudoplastic materials and has since inspired extensive research on their flow properties. In the present work, the Williamson nanofluid model is adopted to characterize the fluid motion. This model describes shear thinning behavior by incorporating the reduction of viscosity at higher shear rates, which is highly relevant for nanofluids containing suspended particles. Compared with other non-Newtonian formulations, such as the Maxwell or Oldroyed-B models, the Williamson model offers a balance of physical realism and mathematical simplicity, making it suitable for analytical and numerical analysis. It’s usefulness in simulating polymeric solutions, biomedical fluids, and industrial suspensions has been demonstrated in prior studies [18], 19] which further supports its selection for the current investigation. Obalalu et al. [20] adopted the Galerkin weighted residual technique to examine the combined effects of chemical reactions activation energy, and solar radiation on three-dimensional non-Newtonian fluid flow over an inclined stretching surface. Anwar et al. [21] conducted a study on the flow behavior of the Williamson fluid model over a moving surface incorporating the effects of nonlinear convective diffusion and variable thermal conductivity. Anwar [22] presented modeling and numerical simulations addressing fractional nonlinear viscoelastic. Khan et al. [23] carried out a numerical investigation of nonlinear time fractional fluid model to analyze heat transport processes in porous media.

Stratification, play a crucial rule both nature and industrial process, is very important for various operations. It arises from variations in density, concentration, temperature differences among fluids. When heat and mass transfer analysis take place at the same time while the double stratification occurs. In natural water bodies such as reservoirs and seas, thermal layering take place as stratification, while saltines change are seen in oceans, canals, ground water, rivers and dams. The process plays an important rule environmental adjustment and food production, the mixing substance have slightly difference density due to gravity. This change of density effect how liquid mixed and behave moreover, the flow of fluid near a stratified sheet is a remarkable in field of environmental science, engineering and oceanography, where layer system are common. The stratification has a small impact mass and heat transfer analysis. Heat transfer, a way to interchange energy among substance is a complex subject studied by scientist, mathematician, engineers and physicist. Investigate how heat transfer work in distinct materials and in boundary layer flow is it very important due to its large importance in many biological and industrial, such as paper making, fermentation, and bubble absorption. The primary objective off this heat transfer research is to reduce energy loss between significant industrial processes [24]. Inspected the numerically study of magnetized Maxwell nanofluid flow through a stratified linear flexible sheet with the presence of Cattaneo–Christov effect by lone et al. [25]. Hyder et al. [26] studied the influence of MHD tri hybrid nanofluid (Cu–Al₂O₃–TiO₂/H₂O) in the existence of stratification and thermal radiation over a perpendicular permeable sheet in a porous media. Lam et al. [27]. However, numerical studies on stratified on the distribution over a vertical cylindrical sheet in the existence of gravity current. Daniel et al. [28] exploring thermal stratification influence on MHD nanofluid through a linear flexible surface with the presence of variable thickness. Muzmmal et al. [29] focused on the heat and mass transfer analysis of a quadratic stratified nanofluid flow over the effect of inclined magnetic field and thermophoresis parameter.

Stratification refers to the vertical arrangement of fluid layers, which plays an important role in a variety of industrial processes, engineering applications, and atmospheric phenomena. In practical situations, where heat and mass transfer occur simultaneously, it becomes necessary to investigate the effects of dual stratification on nanofluids. Stratified fluids are commonly present in natural and engineered systems, including the release of heat into the atmosphere, thermal energy storage mechanisms such as solar ponds, and heat exchange from source like groundwater reservoirs and oceans. Thermal stratification also contributes to ecological processes, for instance by increasing oxygen concentration in the deeper layers of seas and reservoirs [30]. When the denser fluid remains beneath the lighter fluid, the system is said to exhibit stable stratification, as buoyancy forces act to resist vertical motion and suppress convective mixing. Conversely, unstable stratification occurs when a lighter fluid underlies a denser fluid, leading to buoyancy driven instabilities and enhance convective transport. These two modes of stratification play an essential role in geophysical and engineering flows, influencing processes such as thermal storage in solar ponds, mixing in reservoirs, and transport in porous media. The significance of stable and unstable stratification has been highlighted in studies on heat and mass transfer in nanofluids and porous systems, where the stability conditions directly affects entropy generation, convective flow patterns, and thermal efficiency [31], [32], [33]. Abbasi et al. [34] investigated the influence of dual stratification on the mixed convective flow of a Maxwell nanofluid in the existence of heat generation and absorption. Bilala et al. [35] conducted the numerical study of entropy optimization analysis of non-Darcian MHD Williamson nanofluid flow with the effect of thermal radiation through a stratification surface. Khan et al. [36] investigated double stratified stagnation point flow of Williamson nanofluid through a porous media with particular emphasis on entropy generation.

Studies on flow models for non-Newtonian fluids have attracted considerable attention because these fluids play an important role in many practical applications due to their unique properties. Their wide range of industrial and manufacturing uses has drawn the interest of researchers and scholars, particularly over the past few decades. Non-Newtonian fluids are employed in diverse fields such as planetary research, fusion reactors, polymer solutions, chemical production, geophysics, and bioengineering, each benefiting from their specialized behavior. Because of this demand, scientists have investigated non-Newtonian flow phenomena in various geometries to better understand and optimize their applications in industry and technology. One notable contribution was made by Powell and Eyring in 1944, who introduced a model describing how non-Newtonian fluids behave like viscous fluids under high shear stress. This model is grounded in the kinetic theory of gases, supplemented by empirical formulations to capture the complex behavior of these fluids [37]. Proposed the numerical study of non-Newtonian nanofluids of blade coating with the influence of MHD and slip boundary conditions presented by Javed et al. [38]. The effect of heat transfer in magnetohydrodynamics (MHD) nanofluid flows involving non-Newtonian fluids in the presence of polymer solution was examined by Sahreen et al. [39]. Furthermore, the numerical study of heat and mass transfer due to the presence of MHD non-Newtonian fluid flow across a titled, permeable porous media were examined by Tharapat et al. [40] Khan et al. [41] applied a well-known numerical method to analyzed MHD double diffusive convection of non-Newtonian fluid between parallel cylindrical surfaces. In additionally, Venthan et al. [42] carried out a computational analysis of MHD non-Newtonian fluid behavior, focusing on how variation in fluid properties influence by heat and mass transfer through a flexible linear surface.

MHD heat transfer is widely recognized for its significance in both scientific and industrial utilization, like metallurgical processing, fluid mechanics, and biomedical systems. It primarily aims to regulate and enhance fluid flow behavior by adjusting boundary layer parameters [43]. In this context the numerical study of MHD flow involving dissipative Williamson nanofluid through avertical porous domain, employing optimal homotopy asymptotic method (OHAM) for better solution accuracy carry out by Sohail et al. [44]. Additionally, Rehman et al. [45] investigated the effect of viscous dissipation in MHD Casson nanofluid flow over a flexible, nonlinear flexible surface, contributing to understanding energy loss in such systems. Finally, Priyadharshini et al. [46] observed the bio-convective properties of MHD Williamson nanofluid through a collateral stretching sheet, highlighting the interaction between microorganism movement and heat.

One of the primary challenges in modern science and technology lies in enhancing the heat transfer capabilities of convectional bas fluid. Efficient thermal management is critical for applications such as electronic circuit cooling, heat exchangers, and automotive thermal systems, where high performance, reduced operating temperatures, reliability, and extended lifespan are essential. Consequently, researchers have increasingly focused on improving the thermal transport properties of fluids through the incorporation of solid particles. Early attempts in this filed were made by Maxwell. [47], who introduced the concept of dispersing micro sized solid particles in conventional fluids to increase thermal conductivity. However, these efforts were unsuccessful due to sedimentation and flow obstruction. Building upon this foundation, Choi [48] later demonstrated that suspending nanoparticles particles on the nanometer scale in the base fluid significantly improves its heat transfer characteristics. The resulting suspension is termed a nanofluid. Compared to suspensions containing micro or milli sized particles, nanofluids exhibit enhanced thermal transport properties along with improved stability. This discovery sparked considerable interest within the scientific community, leading to extensive investigations into the potential applications of nanofluids across diverse engineering and industrial domains. For instance, Anwar [49] examined heat transfer characteristics in MHD convective stagnation point flow over a stretching surface in the presence of thermal radiation. Puneeth et al. [50] investigated the impact of bioconvection on the free stream flow of a pseudoplastic nanofluid past a rotating cone. Anwar et al. investigation investigated the heat transfer characteristic of MHD hybrid nanofluid with variable viscosity under the influence of thermal radiation. Yu et al. [51] explored heat transfer optimization in viscous ternary nanofluid flow over a starching or shrinking thin needle. Irfan et al. [52] conducted a numerical analysis on MHD bioconvective of Carreau nanofluids considering the effect of nonlinear thermal radiation Joule heating and gyrotactic microorganisms. Khan et al. [53] examined the numerical study of (MHD) flow using the Ellis model, investigate multiple synergistic effects. Hussain et al. [54] investigated nanoparticle radius dynamics in MHD driven dusty fluid heat transfer over linearly stretching surfaces. Shree et al. [55] conducted a numerical investigation of heat and mass transfer in electrically conducting hybrid nanofluids confined between side by side, linear stretching surface. Alharbi et al. [56] studied multiscale heat transfer enhancement in hybrid nanofluid flow through permeable porous media. Similarly, Awan et al. [57] performed a numerical study on tri-hybrid nanofluid heat and mass transfer over linearly stretching flexible cylinders. Kumar et al. [58] investigated the rheological behavior and heat transfer enhancement of Williamson nanofluids in mixed convection flow over a starching cylinder/plate. Sowmiya et al. [59] conducted a numerical investigation on the influence of a magnetic field and radiative heat flux on the rheological behavior of Williamson nanofluid in a stretching cylinder. Kanimozhi et al. [60] explored the effects of multiple slip conditions on MHD Williamson and Maxwell nanofluids over a stretching sheet saturated with a porous medium. Geetha et al. [61] reported that the influence of double stratification on MHD Williamson boundary layer flow and heat transfer over a shrinking sheet embedded in a porous medium.

In this study, we developed a comprehensive mathematical model to investigate the behavior of stratified sheet flows influenced by non-linear thermal radiation. The analysis incorporates key physical parameters, including Brownian motion, Prandtl number, Eckert number, magnetic number, and thermophoresis parameter, to examine their effects on critical thermophysical quantities. Results reveal that the skin friction coefficient decreases with an increase in the Weissenberg number, dropping from approximately 2.3 to 2.2. Furthermore, a reduction in the Cattaneo–Christove temperature parameter diminishes the temperature profile while simultaneously enhancing the concentration distribution. These finding provide deeper insights into the impact of non-linear radiative effects on stratified fluids flows. To solve the discretized governing equations, a detailed algorithm based on the shooting method combined with a fourth-order Runge–Kutta (Rk-4) scheme has been implemented. The main contributions of this work are summarized as follows:

1. Developing a numerical framework for Williamson nanofluid incorporating Cattaneo–Christov heat and mass fluxes.

2. Analyzing the dual stratification thermal and solutal effects together, which is rarely addressed in earlier works.

3. Since analytical solution to the fluid flow problems is often intractable, the model is examined numerically using the shooting method in combination with the (RK-4) method.

4. Appropriate similarity transformations are applied to reformulate the governing equations into a coupled system of nonlinear ordinary differential equations (ODEs, which are then solve numerically.

The novelty of this research lies in its comprehensive treatment of thermal and solutal stratification effects in a Williamson nanofluid flow incorporating the Cattaneo–Christov double diffusion model. While prior studies have examined heat and mass transfer in non-Newtonian fluids, MHD flows, and even Cattaneo–Christov modifications individually, the simultaneous consideration of dual stratification, nonlinear thermal radiation, and the non-Newtonian features of Williamson fluid has not been adequately addresses. By integrating these complex mechanisms into a single framework, this study advances the understanding of how key thermophysical parameters interact to shape velocity, temperature, and concentration distributions. Moreover, the work extends existing models by including Joule heating, viscous dissipation, and nonlinear radiation, thereby offering a richer and more realistic representation of stratified nanofluid systems relevant to industrial and environmental processes. The objective of this study is to formulate and analyze a mathematical model for Williamson nanofluid flow over a stratified stretching sheet under the Cattaneo–Christov heat and mass flux framework. Using similarity transformations and the shooting method, the work investigates how key physical parameters like Weissenberg number, Brownian motion, Eckert number and Lewis number affect velocity, temperature, concentration, as well as the associated skin friction. While many studies have examined non-Newtonian nanofluids, MHD flows, and stratification, most treat thermal or solutal stratification separately and often neglect Cattaneo–Christov heat and mass fluxes. The combined influence of dual stratification, nonlinear radiation, Joule heating, and viscous dissipation in Williamson nanofluids has not been thoroughly explored, leaving a gap in modeling realistic industrial and environmental processes.

2 Mathematical modeling

The behavior of the Cattaneo–Christov MHD Williamson nanofluid has been systematically investigated over a stratified starching sheet. This study focuses on the effects of Joule heating, thermal conductivity, and Ohmic dissipation, which perform a remarkable role in shaping the fluids behavior. The stretching motion of the sheet is considered along the x-axis, with its velocity defined as u = U _w (x) = bx, where b is a positive constant representing the rate of stretching. The physical flow model is shown in Figure 1, where the y-axis is oriented perpendicular to the stretching direction, and the x-axis aligns parallel to it. The ambient temperature varies linearly with the x-coordinate and is expressed as T _∞ = T ₀ + Bx. Similarly, the ambient concentration follows a linear distribution, described by C _∞ = C ₀ + Ex. The temperature and concentration at the sheet surface are defined as T _w  = T ₀ + Ax and C _w  = C ₀ + Dx, respectively, where A and B are constants representing their respective gradients. These boundary conditions together with the given norms, form the basis for deriving the governing equations of the proposed flow model as discussed in [17].

Figure 1:

Flow configuration.

3 Quantities of interest

The dimensionless skin friction coefficient, Nusselt number, and Sherwood number are critical parameters in engineering and industrial applications. Their definitions are provided below:

4 Numerical treatment

This study focuses on solving the nonlinear coupled system of ordinary differential equations (8)–(10), along with associated boundary conditions (11), using the well-established shooting method [63]. To facilitate the solution process, a set of new variables is introduced.

By transformation, equations (8)–(10) are rewritten as a set of seven first order ODEs:

The following are the corresponding boundary conditions:

To meet the specified boundary conditions, unknown initial values Z ₃(0) = P ₁, Z ₅(0) = P ₂, and Z ₇(0) = P ₃ are assumed. These initial estimates, P ₁, P ₂, and P ₃, are progressively improved using Newton’s iterative method until convergence to the acceptable level of accuracy is attained. The iterative process is terminated once the following convergence is satisfied.

Additionally, for the numerical calculations, we have taken into consideration a bounded domain [0, ∞] rather than [0, ∞). From our computational result, it can be observed that boosting η _max, no substantial fluctuations are noticed in the computational results (Figure 2).

Figure 2:

Flow chart of shooting method.

5 Result and discussion

The numerical solution for the set of (ODEs) describing the flow problem in dimensionless form is the main focus of this section. The findings are illustrated through table and graphs to enhance the clarity of the analysis. This study examines on a non-Darcian (MHD) nanofluid using the Cattaneo–Christov double diffusion model. The discussion emphasizes the dimensionless velocity, temperature and concentration profile, along with other relevant physical characteristic specified earlier.

6 Graphical results

This section covers the effect of key parameters on the flow and heat transfer properties of the modeled systems. The graphs for various dimensionless quantities illustrate how fluid motion and heat transfer are governed under different conditions. Specially, the figures show the variations in the concentration profile g(η), thermal distribution θ(η), and velocity distribution f′(η), with respect to the governing parameters. Unless otherwise specified, the following values are used for all graphical representation of velocity, temperature and concentration profile: We = 0.3, Ec = 0.4, Rd = 0.4, Le = 1.0, Ec = 0.4, M = 0.3, Pr = 0.7, Nt = 0.5, Nb = 0.5, N ₁ = 0.8 and λ _d  = 0.1.

6.1 Analysis of the velocity profile

Figure 3 demonstrate the effect of magnetic parameter M on the dimensionless velocity distribution. An increase in the magnetic number intensifies the Lorentz force, which acts in opposition to the flow. As a result, the thickness of the velocity boundary layer diminishes. The analysis of Weissenberg number We on the momentum boundary layer is observe in Figure 4 the relationship among the relaxation period and time scale of the fluid flow, can be describe that as we enhance the relaxation time lengthens, permeable for large flow resistance. Due to which, the thickness of the corresponding momentum boundary layer grows, as a result, the velocity of the fluid decrees.

Figure 3:

Effect of M on f′(η).

Figure 4:

Effect of We on f′(η).

6.2 Analysis of the temperature profile

This section provides a derailed and structured explanation of the influence of various dimensionless parameters on the temperature distribution. The analysis carefully examines these parameters, highlighting their roles and effects in shaping the thermal behavior. Each parameter is discussed systematically, offering a comprehensive understanding of their contributions and the interrelation between these parameters and the resulting temperature profile.

Figure 5 illustrate the effect of the Cattaneo–Christov parameter λ _t on the temperature distribution. It is observed that as the values of λ _t increase, the temperature profile exhibits a noticeable decline. Generally, this parameter accounts for thermal relaxation, as a result heat take time to generate through the material alternatively of spreading immediately. As the factor enlarge, the delayed heat transfer go to slows fall down the open out of thermal energy, remarkable to lower temperature throughout the system. Figure 6 analyses the changes in the dimensionless temperature distribution influence by the magnetic parameter. A gradually increase in magnetic intensity leads to a rise in the thermal boundary layer thickness. Physically, this occurs because the strengthening magnetic field intensifies the resistive force. The role of viscous dissipation is characterized by the Eckert number, which establishes a relationship among kinetic energy and change in thermal energy. The influence of Ec on the thermal boundary layer is depicted in Figure 7. It is observed that a gradual rise in the Eckert number results in a decline in the temperature field. This behavior is attributed to the dominance of enhanced energy transport mechanisms associated with higher flow kinetic energy, which outweigh the effect of viscous dissipation and lead to a reduction in the overall temperature distribution.

Figure 5:

Effect of λ _t on θ(η).

Figure 6:

Effect of M on θ(η).

Figure 7:

Effect of Ec on θ(η).

Figure 8 demonstrates the effect of the Brownian motion parameter Nb on the temperature distribution. An increase in Nb values leads to a noticeable rise in temperature profile. This behavior is attributed to the intensified random motion of nanoparticles, which enhances both the thermal and momentum boundary layer thicknesses at elevated Nb levels. Figure 9 explores the role of the solutal stratification parameter N ₁ on the temperature distribution. A steady increase in N ₁ results in a reduction of the thermal boundary layer thickness, indicating a cooling effect. Furthermore, Figure 10 examines the influence of the thermophoresis parameter Nt on the temperature distribution. As Nt increase, the temperature profile shows a corresponding rise. This trend occurs because thermophoretic forces cause hotter particles to drift away from the heated surface toward cooler regions, thereby increasing the thermal energy within the fluid domain.

Figure 8:

Effect of Nb on θ(η).

Figure 9:

Effect of N ₁ on θ(η).

Figure 10:

Effect of Nt on θ(η).

Figure 11 highlights the influence of the Prandtl number Pr on the temperature field. As Pr increases the temperature gradient becomes declined. This behavior arises because a higher Prandtl number, which represents the ratio of momentum diffusivity to thermal diffusivity, corresponding to lower thermal diffusivity. Consequence, heat spreads less efficiently, resulting in a declined temperature profile. Figure 12 shows the influence of the radiation parameter Rd on the temperature distribution. As Rd increases, the temperature profile increase. Physically, increasing Rd strengthens radiative heat transport, which adds more thermal energy to the fluid and thickens the thermal boundary layer, so the temperature profile increases.

Figure 11:

Effect of Pr on θ(η).

Figure 12:

Effect of Rd on θ(η).

6.3 Analysis of the concentration profile

Figure 13 analyzes the effect of the solute stratification number on the concentration profile. From the figure, it is clearly observed that the gradually increasing values of N ₁, the corresponding velocity distribution is decline. This occurs because solutal stratification suppresses mass diffusion, which weakens particle transport and reduces the concentration profile. Furthermore, Figure 14 illustrate the relationship between the Brownian motion parameter and the associate concentration distribution. As the ascending order values of Nb increases as a result the concentration profile declines. In generally, Brownian motion heat fluid particles within the boundary layer and moves them away from the fluid region, leading to a reduction in the concentration distribution. Finally, Figure 15 presents the influence of the thermophoresis parameter Nt on the concentration profile. The results indicate that the concentration increases with higher values of Nt. Physically, this occurs because stronger thermophoretic forces drive particles away from the heated surface toward cooler regions, thereby enhancing their concentration.

Figure 13:

Effect of N ₁ on g(η).

Figure 14:

Effect of Nb on g(η).

Figure 15:

Effect of Nt on g(η).

Figure 16 indicates that the concentration profile increases with an increase in the Prandtl number Pr. Physically a large Pr is associated with low thermal conductivity that has poor heat diffusion in comparison with momentum diffusion. This decreased diffusion of heat inhibits mixing of heat to the fluid hence resulting to the solute species being concentrated at the surface. The concentration boundary layer is therefore increased resulting in a better concentration profile. Finally, there is an incremental development in the dimensionless concentration distribution as the Cattaneo Christov concentration parameter increases gradually as observed in Figure 17. This measure is the relaxation time of the concentration diffusion. Increase in this parameter causes delay in diffusion of the solute species and it compromises mass diffusion at the surface. Consequently, solute will be concentrated towards the boundary thus creating a better concentration profile.

Figure 16:

Effect of Pr on g(η).

Figure 17:

Effect of λ _d on g(η).

7 Conclusions

In the present study, numerical investigation has been conducted on thermal and solutal stratification in Wiliamson nanofluid flow, incorporating Cattaneo–Christov heat flux, over a linearly stretching sheet.