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Comparative Analysis of Cross Diffusion Effects on MHD Casson Williamson Ternary Nanofluid Flow with Heat and Mass Transfer

This paper investigates the steady, two-dimensional magnetohydrodynamic flow of a Casson-Williamson ternary nanofluid containing TiO₂, Ag, and Cu nanoparticles over a linearly stretching surface by employing both regular perturbation and numerical methods to analyze how parameters like Brownian motion, thermophoresis, and thermal radiation influence heat and mass transfer characteristics.

Original authors: Mahadev Biradar, Amar Patil

Published 2026-09-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mahadev Biradar, Amar Patil

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern engineering, the quest for better heat management is a constant challenge. Whether cooling the tiny chips inside a smartphone or managing the intense temperatures in a solar power plant, the fluids used to carry heat away must be incredibly efficient. Scientists have long turned to "nanofluids" to solve this. These are ordinary liquids, like water, that have been infused with microscopic solid particles. These particles act like tiny heat conductors, making the fluid move thermal energy much faster than the liquid could on its own. However, many real-world fluids are not simple; they are thick, stretchy, or behave strangely under pressure, much like ketchup or blood. To understand how these complex fluids move and transfer heat, researchers combine the study of these special liquids with the effects of magnetic fields, a field known as magnetohydrodynamics. When a magnetic field is applied to a moving, electrically conductive fluid, it creates a force that can slow the fluid down or change how it flows, offering a way to control the process without any moving mechanical parts.

A recent study by researchers Mahadev Biradar and Amar Patil dives deep into a specific and highly complex version of this problem. They investigated a fluid that is a mixture of water and three different types of nanoparticles: titanium dioxide, silver, and copper. This "ternary" mixture is designed to be more effective than fluids containing just one or two types of particles. The fluid they studied is not a simple liquid; it is a hybrid model that combines the properties of two different non-Newtonian fluids, known as Casson and Williamson fluids. These models describe fluids that have a yield stress, meaning they act like a solid until a certain force is applied, and then they flow like a liquid, while also thinning out as they are stretched. The researchers wanted to see how this specific, complex fluid would behave when stretched over a surface, heated, and subjected to a magnetic field, while also accounting for how the tiny particles move due to heat and random collisions.

To solve this, the team did not rely on a single method. Instead, they built a mathematical model of the situation and solved it using two different approaches to ensure their results were accurate. The first approach was a specialized calculation technique that works well when the fluid's elasticity is very weak, providing a quick, semi-analytical answer. The second approach was a robust computer simulation that could handle the full, complex behavior of the fluid, even when the elastic effects were strong. By comparing the results of these two methods, the researchers confirmed that their models were consistent and reliable. They then used these tools to run a series of experiments, changing one factor at a time to see how the fluid reacted.

The researchers found that applying a magnetic field acts as a brake on the fluid. As the strength of the magnetic field increased, the speed of the fluid dropped significantly because the magnetic force pushed back against the motion. This slowing of the fluid also caused the layer of heat near the surface to become slightly thicker, as the moving fluid was less effective at carrying the heat away. Conversely, the behavior of the tiny particles was driven by two main forces. One force, called Brownian motion, is the result of the particles bumping into each other randomly. When this motion was stronger, the fluid mixed better, leading to a thicker layer of heat and a wider spread of particles. The other force, thermophoresis, pushes particles away from hot areas toward cooler ones. When this effect was increased, the researchers observed that more particles accumulated in the cooler regions, which also thickened the layers of heat and concentration near the surface.

The study also revealed how the fluid's own internal properties changed the outcome. The researchers found that the fluid's elasticity, which describes how much it resists stretching, played a role in reducing the drag near the surface. They also discovered that the specific mix of three nanoparticles significantly improved the fluid's ability to transfer heat compared to simpler mixtures. The thickness of the fluid's boundary layer, where the heat and particles are concentrated, was heavily influenced by the fluid's Prandtl number, a measure of how easily heat spreads compared to momentum. A higher value meant heat moved more slowly, creating a thinner, sharper temperature layer. Similarly, the Lewis number, which compares how fast heat spreads versus how fast particles spread, controlled the thickness of the concentration layer; a higher value meant the particles spread more slowly, creating a sharper drop in concentration near the surface.

Ultimately, the work by Biradar and Patil provides a clear and reliable framework for understanding these complex fluid systems. They demonstrated that while simple calculation methods work well for fluids that are only slightly elastic, computer simulations are necessary to capture the full picture when the fluid behaves more strongly. Their findings show that by carefully adjusting the magnetic field, the type of nanoparticles, and the thermal conditions, engineers can fine-tune how these advanced fluids move and transfer heat. This level of control is essential for designing better cooling systems for electronics, improving energy conversion technologies, and optimizing industrial processes where precise thermal management is critical. The study confirms that combining different types of nanoparticles with non-Newtonian fluid models offers a powerful way to enhance performance, provided the complex interactions between magnetic forces, particle movement, and fluid elasticity are fully understood.

Technical Summary: Comparative Analysis of Cross Diffusion Effects on MHD Casson Williamson Ternary Nanofluid Flow

Problem Statement
This research investigates the steady, two-dimensional magnetohydrodynamic (MHD) flow of a Casson-Williamson ternary nanofluid over a linearly stretching surface. The study addresses a gap in existing literature, which has predominantly focused on mono or hybrid nanofluids, by modeling a ternary system consisting of three distinct nanoparticles—Titanium Dioxide ( $TiO_2$ ), Silver ( $Ag$ ), and Copper ( $Cu$ )—dispersed in a water base fluid. The physical model incorporates complex non-Newtonian behaviors (yield stress and shear-thinning via the Casson-Williamson hybrid model), external magnetic fields, thermal radiation, and cross-diffusion effects (Brownian motion and thermophoresis). The primary objective is to analyze the flow, thermal, and concentration boundary layers under these combined conditions and to evaluate the efficacy of different solution techniques.

Methodology
The study employs a dual-approach methodology to solve the governing partial differential equations (PDEs):

Mathematical Formulation: The conservation equations for mass, momentum, energy, and species concentration are derived under boundary layer approximations. The ternary nanofluid's effective thermophysical properties (density, heat capacity, thermal conductivity, viscosity, and electrical conductivity) are modeled using Maxwell and Brinkman models. The system is transformed into a set of coupled non-linear ordinary differential equations (ODEs) using similarity variables.
Analytical Approach (Perturbation Method): A regular perturbation expansion is developed for small Weissenberg numbers ( $We \ll 1$ ). The solution is expanded into zeroth-order and first-order systems. The zeroth-order system represents the base flow, while the first-order system accounts for weak elastic effects. This provides a semi-analytical solution valid for weakly elastic flows.
Numerical Approach: A robust numerical solution is implemented using Python's scipy.integrate.solve_bvp routine. This method utilizes a collocation technique with adaptive mesh refinement to solve the full non-linear boundary value problem across a broad range of parameters, including larger Weissenberg numbers where perturbation methods may fail.
Validation: The numerical solver is validated against existing literature for limiting cases (e.g., pure fluid flow without nanoparticles or magnetic fields), showing excellent agreement with previous studies.

Key Contributions

Ternary Nanofluid Modeling: The paper formulates a comprehensive mathematical model for a Casson-Williamson fluid containing three distinct nanoparticle types ( $TiO_2$ , $Ag$ , $Cu$ ), offering a more realistic representation of advanced working fluids compared to single or hybrid models.
Comparative Framework: The study provides a direct comparison between a regular perturbation method and a full numerical solver. It establishes the validity range of the perturbation approach (specifically for $We \ll 1$ ) and demonstrates its utility as a rapid estimation tool and an initial guess for numerical solvers.
Parametric Analysis: The research systematically quantifies the effects of key dimensionless parameters, including the magnetic parameter ( $M$ ), Weissenberg number ( $We$ ), Brownian motion ( $N_b$ ), thermophoresis ( $N_t$ ), Prandtl number ( $Pr$ ), Lewis number ( $Le$ ), and thermal radiation ( $R_d$ ).

Results and Discussion
The analysis yields several critical findings regarding the flow dynamics and heat/mass transfer characteristics:

Magnetic Field ( $M$ ): Increasing the magnetic parameter induces a Lorentz force that opposes fluid motion, significantly reducing the velocity profile and thickening the thermal boundary layer due to weakened convection.
Elasticity ( $We$ ): Higher Weissenberg numbers reduce the velocity gradient near the wall, indicating that elastic effects reduce drag. The perturbation solution (zeroth-order) aligns closely with numerical results for $We \leq 0.5$ , with relative errors in wall gradients remaining below 1%.
Cross-Diffusion Effects:
- Brownian Motion ( $N_b$ ): An increase in $N_b$ enhances the random motion of nanoparticles, leading to improved thermal mixing and thicker thermal and concentration boundary layers.
- Thermophoresis ( $N_t$ ): Higher $N_t$ drives nanoparticles from the heated surface to the cooler ambient region, strengthening both thermal and concentration boundary layers.
Diffusivity Ratios:
- Prandtl Number ( $Pr$ ): Increasing $Pr$ (e.g., moving from water to oil-like fluids) reduces thermal diffusivity, resulting in thinner thermal boundary layers and steeper temperature gradients at the wall.
- Lewis Number ( $Le$ ): Increasing $Le$ reduces the concentration boundary layer thickness as mass diffusion becomes slower relative to heat diffusion.
Performance: The incorporation of ternary nanoparticles is shown to substantially improve heat transfer performance compared to conventional nanofluid configurations. The local Nusselt number increases with enhanced nanoparticle transport mechanisms.

Significance and Claims
The paper claims to provide a "reliable and efficient framework" for understanding complex non-Newtonian ternary nanofluid systems. It emphasizes that the hybrid analytical-numerical approach allows for the optimization of such systems in advanced thermal engineering and energy-conversion applications. The authors assert that the numerical scheme remains highly stable over a broad parameter range, whereas the perturbation solution is restricted to regimes of small Weissenberg numbers. The study concludes that these findings offer a worthy computational basis for future work in biomedical engineering, polymer processing, and thermal management, though it explicitly notes that extending the model to unsteady flow and temperature-dependent viscosity is reserved for future research.

Technical Summary: Comparative Analysis of Cross Diffusion Effects on MHD Casson Williamson Ternary Nanofluid Flow

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