A NUMERICAL STUDY OF MAXWELL FLUID MHD PARTICLES MOVING THROUGH A STRETCHED SHEET IN THE PRESENCE OF THERMAL RADIATION

Abstract

This study presents a numerical model to investigate heat and mass transfer in a two-flow Maxwell nanofluid over a rubber sheet under the influence of thermal radiation and an induced magnetic field. By employing similarity variables, the governing equations for the system are transformed into a dimensionless form, enabling a more efficient analysis. The Runge-Kutta-Fehlberg (RKF) method, a reliable and precise numerical approach, is then used to solve these equations. The analysis focuses on providing detailed profiles of velocity, temperature, and concentration, which are crucial to understanding the behavior of the Maxwell nanofluid under combined thermal and magnetic effects. Calculations reveal how these parameters evolve across the flow field, offering insights into how heat and mass are transported in the presence of viscoelastic properties and external forces. This model can guide improvements in applications involving advanced materials, such as rubber processing and nanofluid-based thermal management systems.

Specification

Description:FIELD OF INVENTION
This study falls within the field of fluid dynamics and magnetohydrodynamics (MHD), focusing on Maxwell fluid particles moving through a stretched sheet under thermal radiation effects. It investigates the behavior and stability of non-Newtonian fluids in industrial and engineering processes, particularly in applications involving heat transfer, electromagnetic fields, and fluid mechanics in manufacturing and material processing.
BACKGROUND OF INVENTION
The study of Maxwell fluid MHD (magnetohydrodynamic) particles moving over a stretched sheet under thermal radiation has significance in the field of non-Newtonian fluid mechanics, particularly in the context of materials and industrial processes that require precise control over heat and magnetic interactions. Maxwell fluids, as viscoelastic materials, exhibit both viscosity and elasticity, unlike traditional Newtonian fluids, which makes their flow behavior complex, especially when exposed to external forces such as magnetic fields and thermal radiation.
In numerous industrial applications, such as polymer processing, aerodynamic heating, magnetic cooling, and materials manufacturing, understanding the heat transfer characteristics and behavior of such fluids under stretching conditions is crucial for optimizing product quality and process efficiency. The presence of thermal radiation adds complexity, as it can significantly influence temperature distribution, fluid stability, and viscosity. The magnetic field (MHD) component also affects flow patterns by inducing additional force on the particles, enabling control over the fluid's motion and stability.
This study employs numerical methods to model and analyze the behavior of Maxwell fluids under these conditions, providing insights for engineers and scientists to design more efficient thermal and fluid systems, contributing to advancements in industries like electronics, aerospace, and polymer manufacturing.
The patent application number 201937037751 discloses a flow-through fluid purification device and means for accommodating a radiation source.
The patent application number 202017050990 discloses a thermal management fluid systems and methods for using them.
The patent application number 202117021067 discloses a self-supporting flexible thermal radiation shield for a superconducting magnet assembly.
The patent application number 202228003661 discloses a fluid-tight, thermally insulated tank.
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SUMMARY
This study presents a numerical investigation into the behavior of Maxwell fluid particles in a magnetohydrodynamic (MHD) setting as they move across a stretched sheet, influenced by thermal radiation. The Maxwell fluid model represents a class of non-Newtonian fluids with viscoelastic properties, which makes the analysis of such systems complex due to their combined elastic and viscous responses. The inclusion of a magnetic field (MHD effects) introduces an additional layer of control over the fluid's behavior, as the magnetic forces interact with the electrically conductive fluid, affecting its flow patterns and stability.
Thermal radiation further complicates this setup, as it influences the heat distribution within the fluid, impacting viscosity and flow consistency. By employing numerical simulation methods, this study models the governing equations of the Maxwell fluid's motion and heat transfer, capturing the intricate dynamics induced by the magnetic field and thermal radiation. The study identifies key parameters that affect fluid velocity, temperature, and stress distribution, providing valuable insights for managing heat and flow in industrial applications.
The findings can support optimized process control in fields such as polymer extrusion, cooling technologies, and materials engineering, where precise control over non-Newtonian fluid flow and heat transfer is essential for product quality and process efficiency.
DETAILED DESCRIPTION OF INVENTION
Maxwell nanofluids are complex systems integrating fluid dynamics, thermodynamics, radiation heat transfer, and nanotechnology. These fluids exhibit unique behavior influenced by variables like nanoparticle type and concentration, temperature, and mechanical stress, making them ideal for enhancing heat transfer, cooling, and related processes. Researchers are focused on characterizing and optimizing these fluids for specific industrial applications by leveraging their distinctive properties.
Khan studied fluid flow and heat transfer in Maxwell nanofluids. Sui analyzed the influence of various factors on heat and mass transfer in the boundary layer, while Goyal investigated viscoelastic nanofluids with partial slip conditions in boundary layer flow over a stretching sheet. Ishak explored magnetohydrodynamic (MHD) stagnation point flow toward a stretching sheet, with Pavlov examining MHD flow of an incompressible viscous fluid on a deformed plane surface. Hayat analyzed MHD flow over a permeable stretching sheet with slip conditions, and Reddy examined Soret and Dufour effects on MHD convective flow in porous media. Mustafa modeled Casson nanofluid flow under a magnetic field, while Ibrahim studied the effects of induced magnetic fields and nanoparticles on MHD stagnation point flow.
Additional studies include Mahapatra on MHD stagnation flow, Ishak on mixed convection boundary layers, Pal on micropolar nanofluid flow, and Das on entropy generation in unsteady magneto-nanofluid flows. Gireesha explored Hall effects in two-phase dusty nanofluid flow using the Kelvin-Voigt model, while F.M. Ali and Kumari studied boundary layer flow and mixed convection in MHD scenarios.
Our research focuses on understanding heat and mass transport in 2-D Maxwell fluids influenced by stretching, magnetic fields, and thermal radiation with nanoparticle addition. These insights have significant applications across industries like copper wire production, polymer extrusion, and fiber manufacturing.
This problem deals with a steady, two-dimensional, magnetohydrodynamic (MHD) boundary layer flow over a stretched surface with heat and mass transfer of a Maxwell fluid containing nanoparticles. The mathematical formulation involves several governing equations and their corresponding boundary conditions, which can be summarized as follows:
Governing Equations

The motion of the Maxwell fluid under the influence of both stretching velocity and magnetic field is described by the Navier-Stokes equation with added terms for viscosity and induced magnetic forces:

Describes the heat transfer within the fluid and the effects of radiation, thermal diffusivity, and convection:

Similarity Transformations
The dimensional governing equations are transformed into a set of similarity variables to reduce them into ordinary differential equations. These transformations are:

Dimensionless Governing Equations
After applying the similarity transformations, the governing equations are converted into the following dimensionless form:


These equations and boundary conditions describe the MHD flow, heat, and mass transfer process with nanoparticles and their associated physical quantities, including heat and mass fluxes at the surface.
Numerical Solution
To solve the boundary value problems (BVPs) associated with the ordinary differential equations (ODEs), the Runge-Kutta-Fehlberg (RKF45) method is employed, along with the shooting technique. The shooting technique involves converting the BVP into an initial value problem (IVP). This is done by estimating a boundary condition value, solving the resulting IVP using the RKF45 method, and iteratively adjusting the boundary condition until the solution aligns with the desired boundary behavior.
Results and Discussion
Calculations are conducted to evaluate the velocity, induced magnetic field, temperature, and concentration profiles across a range of governing parameter values. Figures illustrate the effects on these profiles, offering insights into how each parameter influences the flow characteristics and heat transfer in the system.

Figure 1 illustrates the effect of the magnetic parameter MMM on the velocity profile. As MMM increases, the thickness of the velocity boundary layer decreases. This occurs because the presence of a transverse magnetic field generates a Lorentz force, which acts as a drag force on the fluid flow. Consequently, higher values of MMM lead to a reduction in velocity, resulting in a more flattened velocity profile.
Figure 2 shows the effect of the Deborah parameter β\betaβ on flow velocity. As the Deborah parameter increases, the velocity profile rises. Physically, a higher Deborah number indicates greater elasticity and reduced viscosity in the material, which enhances the fluid velocity by lowering internal resistance.

Magnetic Parameter (M) Impact on Induced Magnetic Field and Skin Friction:
Increasing MMM strengthens the induced magnetic field initially but also increases skin friction, resulting in reduced boundary layer thickness. This effect signifies that higher magnetic fields hinder fluid motion due to increased Lorentz forces, which leads to higher resistance.

Velocity Ratio Parameter (B) Influence on Velocity:
The velocity boundary layer thickness grows as BBB increases, which boosts the velocity. This parameter influences the rate at which fluid layers slide past each other, indicating that higher values of BBB promote faster fluid flow near the surface.

Reciprocal Magnetic Prandtl Number (A) on Induced Magnetic Field:
Parameter AAA (ratio of thermal diffusivity to magnetic diffusivity) impacts how the magnetic field distributes within the fluid. Higher AAA values suggest a more conductive than viscous fluid, enabling rapid magnetic field dispersion. Conversely, lower AAA values denote stronger induced magnetic fields due to slower magnetic dissipation relative to heat.

Radiation Parameter (NrNrNr) on Temperature:
As the radiation parameter increases, the temperature profile shifts, indicating more thermal radiation from the surface. NrNrNr directly affects radiative heat transfer in the boundary layer, emphasizing the material's radiative properties.
Biot Number (Bi) and Deborah Number (β\betaβ) on Temperature:
Higher Biot numbers indicate more significant convective heating at the surface, enhancing the thermal boundary layer thickness and surface temperature.
The Deborah number β\betaβ, which relates to viscoelastic relaxation time, affects temperature dissipation. Higher β\betaβ values allow more uniform heat distribution due to rapid material response to deformation, while lower β\betaβ values may lead to localized heating.
Lewis Number (Le) and Thermophoresis Parameter (Nt) on Concentration:
As the Lewis number increases, concentration near the surface decreases, indicating enhanced mass transfer. This can result in a steeper concentration gradient.
An increase in thermophoresis parameter NtNtNt thickens the concentration boundary layer, suggesting improved particle transport due to decreased diffusion resistance, which diminishes surface concentration gradients.



DETAILED DESCRIPTION OF DIAGRAM
Fig 1: Effect of M on velocity
Fig 2: Effect of β on velocity
Fig 3: Effect of M on Induced Magnetic field.
Fig 4: Effect of B on velocity
Fig 5. Effect of A on Induced Magnetic field
Fig 6. Effect of Nr on Temperature
Fig 7. Effect of Bi on Temperature
Fig 8. Effect of β on Temperature
Fig 9. Effect of Le on Concentration
Fig 10. Effect of Nt on Concentration , Claims:1. A numerical Study of Maxwell fluid MHD particles moving through a stretched sheet in the presence of thermal radiation claims that thermal radiation increases the temperature profile within the boundary layer, as it boosts radiative heat transfer, promoting energy dissipation in the fluid near the surface.
2. A strong magnetic field (higher MMM) opposes fluid motion due to the Lorentz force, which reduces the velocity profile and results in a thicker thermal boundary layer.
3. The velocity ratio parameter BBB significantly enhances the flow velocity near the surface, indicating that fluid motion is directly affected by boundary conditions and external stretching forces.
4. The reciprocal magnetic Prandtl number AAA determines the spread of the induced magnetic field within the fluid, with higher values promoting quicker magnetic diffusion compared to heat.
5. A higher Deborah number β\betaβ, reflecting quicker material response, helps achieve a more uniform temperature distribution across the boundary layer due to faster relaxation and energy dissipation.
6. As the Biot number BiBiBi increases, the convective heating from the stretched sheet intensifies, resulting in elevated temperatures and a thicker thermal boundary layer.
7. A higher Lewis number LeLeLe, indicating a dominant mass diffusion over thermal diffusion, reduces surface concentration levels, thereby increasing the concentration gradient near the surface.
8. An increased thermophoresis parameter NtNtNt enhances the thickness of the concentration boundary layer by decreasing diffusion resistance, facilitating better particle movement within the fluid.
9. The magnetic parameter MMM affects the skin friction coefficient, with higher values decreasing boundary layer thickness and skin friction, impacting surface interactions.
10. The interplay between concentration, thermal effects, and magnetic parameters dictates overall fluid behavior, demonstrating that temperature, velocity, and concentration profiles are interconnected and sensitive to these parameters.

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