Advanced Fractional Model for Predicting MHD Yield Stress Fluid Flow
Background
In modern scientific and industrial research, the dynamic behavior of non-Newtonian fluids has garnered significant attention due to their unique rheological properties and wide-ranging applications. Unlike Newtonian fluids, non-Newtonian fluids such as polymers, slurries, and biological fluids exhibit complex nonlinear relationships between shear stress and strain rate. Particularly, Casson fluids, characterized by their yield stress and nonlinear stress-strain relationships, hold significant importance in biological fluid dynamics and industrial processes. However, traditional integer-order derivative models have limitations in describing the behavior of these fluids, especially when it comes to memory effects and boundary concentration dynamics. To address this issue, fractional calculus has been introduced to more accurately capture the unique characteristics of these fluids.
This paper is co-authored by Shazia Riaz, M. S. Anwar, Ayesha Jamil, and Taseer Muhammad from the Department of Mathematics at the University of Jhang, Pakistan, and King Khalid University, Saudi Arabia. The study was published on February 16, 2025, in the journal Nonlinear Dynamics under the title “Advanced fractional model for predicting MHD yield stress fluid flow with boundary effects.”
Research Process
1. Mathematical Model Construction
The study first established a Casson fluid flow model based on fractional calculus. The model considers the effects of magnetohydrodynamics (MHD), chemical reactions, and diffusion processes on concentration profiles. Specifically, the authors used the Caputo fractional derivative to introduce non-local behavior and memory effects. The momentum and concentration equations were modified using fractional derivatives to more accurately describe the fluid’s behavior.
2. Numerical Methods
To solve these complex fractional partial differential equations (PDEs), the study employed a hybrid numerical method combining the finite difference method (FDM) and the finite element method (FEM). The finite difference method was used for the discretization of the time variable, while the finite element method was applied to the spatial variables. This hybrid approach not only improved computational efficiency but also ensured the accuracy of the numerical solutions.
3. Numerical Simulations and Results Analysis
Through numerical simulations, the study conducted a detailed analysis of the effects of various physical parameters on fluid velocity and concentration distributions. These parameters include the fractional derivative parameters α and β, the Casson parameter β′, the magnetohydrodynamic parameter M, the diffusion parameters λ7 and λ8, the Schmidt number Sca, and the chemical reaction parameter kc, among others. The simulation results revealed that an increase in the fractional derivative parameter α significantly enhances fluid velocity, while an increase in β leads to a reduction in concentration. Additionally, an increase in the Casson parameter β′ also enhances fluid mobility, whereas an increase in the magnetohydrodynamic parameter M suppresses fluid motion.
Key Findings
1. Velocity Distribution
The study found that the fractional derivative parameter α has the most significant impact on fluid velocity. As α increases, fluid velocity exhibits a clear upward trend. This phenomenon indicates that fractional derivatives can better capture the non-local behavior and memory effects of fluids. Moreover, an increase in the Casson parameter β′ also significantly enhances fluid velocity, which is attributed to the reduction in yield stress.
2. Concentration Distribution
In terms of concentration distribution, the study found that an increase in the fractional derivative parameter β leads to a reduction in concentration. This result highlights the critical role of fractional derivatives in controlling concentration diffusion and transport processes. Additionally, an increase in the Schmidt number Sca also significantly reduces concentration distribution, indicating that the ratio of momentum diffusivity to mass diffusivity has a substantial impact on concentration profiles.
3. Impact of Chemical Reactions
The study also explored the effects of chemical reactions on fluid behavior. The results showed that an increase in the homogeneous reaction parameter kc significantly enhances concentration distribution, suggesting that higher chemical reaction rates help maintain the concentration distribution of reactants. Conversely, an increase in the heterogeneous reaction parameter λ9 enhances the diffusion rate of chemical species, further accelerating the chemical reaction process.
Conclusions and Significance
By introducing fractional calculus, this study successfully constructed a mathematical model capable of accurately describing the flow behavior of Casson fluids. The model not only considers the effects of magnetohydrodynamics and chemical reactions but also introduces non-local behavior and memory effects, significantly improving the model’s predictive capabilities. The findings not only deepen the understanding of non-Newtonian fluid dynamics but also provide a new theoretical framework for analyzing complex fluid behaviors in industrial processes and biomedical applications.
Research Highlights
- Application of Fractional Calculus: For the first time, Caputo fractional derivatives were introduced into the Casson fluid flow model, significantly enhancing the model’s accuracy and applicability.
- Hybrid Numerical Method: The combination of the finite difference method and the finite element method effectively solved complex fractional PDEs, offering new insights for numerical simulations of similar problems.
- Multi-Parameter Analysis: A detailed analysis of the effects of various physical parameters on fluid velocity and concentration distributions provides important references for optimizing industrial processes and biomedical applications.
Additional Valuable Information
This study was funded by the Research and Development Department of King Khalid University, Saudi Arabia, under grant number RGP.2/113/45. The authors declare no conflicts of interest, and all data are included in the article.
Through this research, the authors not only filled theoretical gaps in the field of non-Newtonian fluid dynamics but also provided valuable practical guidance for related industrial applications.