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Exact Solutions for Relative Magnetic Field Effects on Casson Hybrid Nanofluid Flow with Exponential thermal Memory

This study derives exact analytical solutions for the unsteady magnetohydrodynamic flow and heat transfer of a Casson hybrid nanofluid containing magnetite and cobalt ferrite particles in a vertical porous channel, utilizing the Caputo-Fabrizio fractional integral to model thermal memory and demonstrating how fractional parameters, nanoparticle concentration, and magnetic fields significantly influence the fluid's thermal and hydrodynamic behavior.

Original authors: Muhammad Saqib, Fisal Asiri, Asma Khalid, Amani Mubarak, Muhammad Alfarhan

Published 2026-08-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Saqib, Fisal Asiri, Asma Khalid, Amani Mubarak, Muhammad Alfarhan

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

Imagine you are trying to cool down a super-hot computer chip or keep a car engine from melting. You need a liquid that can carry heat away fast. Scientists have been using special "nanofluids"—liquids mixed with tiny, microscopic particles—to do this job better than water alone. But here's the tricky part: heat doesn't always move instantly like a light switch flipping on. Sometimes, it has a "memory," meaning the heat flow depends on what happened a split second ago. Think of it like a heavy, thick syrup; if you push it, it doesn't move immediately, it drags a bit before catching up. In the world of physics, this is called "thermal memory," and ignoring it can lead to wrong predictions about how hot or cold things get.

Now, imagine you add magnets to the mix. If you put a magnetic field near a liquid that conducts electricity, it creates a force that can slow the liquid down or speed it up, kind of like an invisible brake or accelerator. This is called Magnetohydrodynamics (MHD). Scientists are trying to figure out exactly how these magnetic forces, the "memory" of the heat, and the tiny particles all work together to control the flow and temperature of these special fluids. It's a complex puzzle, but solving it could help us design better cooling systems for everything from electronics to medical devices.

This paper dives into that puzzle by looking at a very specific type of fluid called a "Casson hybrid nanofluid." Imagine this fluid as a special smoothie made of two base liquids (water and ethylene glycol) mixed with two types of magnetic nanoparticles (magnetite and cobalt ferrite). The researchers wanted to see how this smoothie flows through a narrow, vertical channel filled with a sponge-like material (a porous medium) when magnetic fields are applied. They were particularly interested in two things: how the "thermal memory" affects the heat, and whether the magnetic field is stationary relative to the moving fluid or the stationary walls of the channel.

To solve this, the authors didn't just run a computer simulation; they used advanced math to find "exact solutions." Think of this as finding the perfect, precise recipe for the fluid's behavior rather than just guessing how it might taste. They used a mathematical tool called the Caputo-Fabrizio fractional integral to model that "thermal memory" effect, treating the heat flow as if it has a short-term memory of its past states. They also tested two different magnetic setups: one where the magnetic field moves with the fluid, and one where it stays fixed to the channel walls.

Here is what their math revealed. First, the "memory" of the heat really matters. When they increased the memory effect (represented by a parameter called $\alpha$ ), the fluid actually got cooler. It's as if the fluid's memory made it harder for heat to spread out quickly, slowing down the temperature rise. This is different from the old, classic way of thinking about heat, which assumes heat moves instantly. The study shows that when you account for this memory, the temperature profile drops.

Second, the magnetic field acts like a brake. When the magnetic field gets stronger, the fluid slows down. This is because the magnetic field creates a force (Lorentz force) that fights against the flow. Interestingly, the setup matters: when the magnetic field is fixed to the moving wall (the plate), it actually pushes the fluid faster than when it's fixed to the fluid itself. It's like the difference between a conveyor belt moving under a box versus the box trying to move on its own; the moving belt gives it a better boost.

The study also found that adding more of those tiny magnetic particles (increasing the volume fraction $\phi$ ) makes the fluid hotter and faster. The particles are so good at conducting heat that they warm up the fluid quickly, which creates a stronger "buoyancy" force—like a hot air balloon rising—pulling the fluid up the channel faster. However, if the fluid is designed to absorb heat (a parameter called $Q$ ), the temperature drops, and the fluid slows down because it loses that buoyant lift.

Finally, the researchers calculated exactly how much friction the fluid creates against the walls (skin friction) and how well it transfers heat (Nusselt number). They found that all these factors—the memory of the heat, the magnetic field strength, the type of fluid, and the sponge-like channel—change these numbers significantly. For instance, increasing the "memory" parameter increases the friction on one wall but decreases it on the other.

In short, this paper provides a precise mathematical map for how these complex, memory-holding fluids behave under magnetic influence. It proves that ignoring the "memory" of heat or the specific way the magnetic field is set up can lead to big mistakes in predicting how fast the fluid moves or how hot it gets. These exact formulas now serve as a reliable benchmark for other scientists to test their own computer models against, ensuring that future cooling systems are designed with the right physics in mind.

Technical Summary: Exact Solutions for Relative Magnetic Field Effects on Casson Hybrid Nanofluid Flow with Exponential Thermal Memory

Problem Statement
This study addresses the unsteady magnetohydrodynamic (MHD) flow and heat transfer (HT) of a Casson hybrid nanofluid (HNF) within a vertical porous channel. The fluid is modeled as a 50:50 mixture of water and ethylene glycol base fluid containing magnetite ferrite ( $Fe_3O_4$ ) and cobalt ferrite ( $CoFe_2O_4$ ) nanoparticles. A critical limitation in classical heat transfer modeling is the assumption of instantaneous thermal response via Fourier's law, which fails to capture memory effects and non-local phenomena in complex fluids. Furthermore, existing MHD models often assume a stationary magnetic field relative to the fluid, neglecting the impact of relative motion between the magnetic field and the flow. This research investigates these gaps by incorporating a fractional thermal memory model and analyzing two distinct magnetic field configurations: one fixed relative to the fluid and one fixed relative to the moving plate.

Methodology
The mathematical formulation utilizes the Casson fluid model for non-Newtonian behavior and a generalized heat conduction model based on the Caputo-Fabrizio fractional integral (CFFI) operator. The CFFI operator is employed to modify the classical Fourier law, introducing an exponential kernel to account for thermal memory effects. The governing momentum and energy equations, coupled with boundary conditions for a vertical channel with porous media, are non-dimensionalized.

To obtain exact analytical solutions, the authors employ a combination of the Laplace Transform (LT) and the Finite Fourier Sine Transform (FFST). This dual-transform approach allows for the decoupling and solution of the fractional governing equations. The study derives closed-form analytical expressions for the temperature and velocity profiles. Additionally, explicit formulas for the local Nusselt number ($Nu$) and the skin friction coefficient ( $C_f$ ) are derived for both channel walls. The thermophysical properties of the hybrid nanofluid are calculated using established correlations for density, viscosity, electrical conductivity, thermal expansion, and thermal conductivity.

Key Contributions
The paper presents several specific contributions to the field of fractional fluid dynamics and heat transfer:

Fractional Thermal Modeling: The study generalizes the standard Fourier heat conduction equation using the CFFI operator, which utilizes a non-singular exponential kernel to model thermal memory.
Relative Magnetic Field Analysis: Unlike standard models, this work explicitly compares two magnetic field configurations: a field fixed relative to the hybrid nanofluid ( $\epsilon=0$ ) and a field fixed relative to the moving plate ( $\epsilon=1$ ).
Exact Analytical Solutions: The derivation of exact solutions for both temperature and velocity profiles in a fractional MHD Casson HNF system, avoiding the need for purely numerical approximations.
Comprehensive Parameter Study: The analysis investigates the influence of the fractional parameter ( $\alpha$ ), nanoparticle volume fraction ( $\phi$ ), Casson fluid parameter ( $\beta_0$ ), magnetic parameter ( $M$ ), permeability parameter ( $K$ ), heat generation/absorption parameter ( $Q$ ), and the relative magnetic field parameter ( $\epsilon$ ).
Benchmarking: The exact solutions serve as benchmarks for validating future numerical methods developed for fractional MHD Casson HNF problems.

Results and Discussion
The graphical and numerical results reveal the following physical behaviors:

Thermal Memory ( $\alpha$ ): Increasing the fractional parameter $\alpha$ (moving away from the classical case where $\alpha=0$ ) reduces the temperature profile. This is attributed to the stronger memory effect, which makes the thermal diffusion process less efficient compared to the instantaneous response of classical Fourier conduction.
Nanoparticle Concentration ( $\phi$ ): An increase in the hybrid nanoparticle volume fraction enhances the temperature profile due to improved thermal conductivity. This, in turn, strengthens the buoyancy force, leading to an increase in the fluid velocity.
Heat Absorption ( $Q$ ): The parameter $Q$ acts as a heat sink. Increasing $Q$ reduces both the temperature and velocity profiles by extracting thermal energy and weakening the buoyancy-driven flow.
Casson Parameter ( $\beta_0$ ): Increasing $\beta_0$ reduces the apparent plastic viscosity and yield stress of the fluid, resulting in higher flow velocities.
Magnetic Field ( $M$ ): The magnetic field generates a Lorentz force that opposes the flow, significantly reducing velocity. This braking effect is more pronounced when the magnetic field is fixed relative to the moving plate ( $\epsilon=1$ ) compared to being fixed relative to the fluid ( $\epsilon=0$ ).
Permeability ( $K$ ): Higher permeability reduces the Darcy drag force, allowing for increased fluid velocity.
Relative Magnetic Field ( $\epsilon$ ): The configuration where the magnetic field is fixed relative to the moving plate consistently yields higher velocity profiles than the configuration fixed relative to the fluid.
Engineering Quantities: The Nusselt number and skin friction coefficient are shown to be highly sensitive to the fractional parameter, nanoparticle concentration, and magnetic field orientation. Specifically, increasing $\alpha$ and $\phi$ generally increases the skin friction at the left boundary while altering the shear stress distribution at the right boundary.

Significance and Claims
The authors claim that the primary significance of this work lies in the rigorous analytical treatment of unsteady MHD flow with fractional thermal memory and relative magnetic field effects. By providing exact solutions, the study offers a theoretical foundation for understanding how thermal memory and magnetic field configurations influence the hydrodynamic and thermal properties of Casson hybrid nanofluids. The results suggest that the fractional parameter acts as a significant tool for regulating thermal conduction and that the relative motion of the magnetic field plays a crucial role in determining the induced Lorentz force and subsequent flow characteristics. The study concludes that these interactions are vital for the design of effective heat transfer management systems in applications involving non-Newtonian fluids and porous media.

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