Mixed Convection Darcy-Forchheimer MHD flow and radiative heat transfer of water-ethylene glycol hybrid fluid based copper nanofluid via a permeable stretching surface with convective boundary conditions

Technical Summary: Mixed Convection Darcy-Forchheimer MHD Flow of Hybrid Nanofluid

Problem Statement
This study investigates the steady, laminar, incompressible flow and heat transfer characteristics of a hybrid nanofluid composed of copper (Cu) nanoparticles suspended in a water-ethylene glycol ($H_2O - C_2H_6O_2$) base fluid. The flow occurs over a permeable stretching surface embedded within a Darcy-Forchheimer porous medium. The physical model incorporates several complex phenomena simultaneously: magnetohydrodynamics (MHD) via an external magnetic field, mixed convection, thermal radiation, Joule heating, viscous dissipation, and internal heat generation/absorption. The boundary conditions include convective heating at the surface and mass transfer via suction or injection. The primary objective is to analyze how these combined physical parameters influence the velocity profile, temperature distribution, skin-friction coefficient, and Nusselt number.

Methodology
The governing equations, representing conservation of mass, momentum, and energy, are formulated as partial differential equations (PDEs) using boundary layer approximations. To render the system solvable, suitable similarity transformations are applied to convert the PDEs into a system of non-linear coupled ordinary differential equations (ODEs).

The resulting system of ODEs is solved analytically using the Differential Transform Method (DTM). This technique transforms the differential equations into algebraic equations, allowing for the construction of series solutions for the dimensionless velocity ($f(\eta)$) and temperature ($\theta(\eta)$) functions. The study validates its numerical results by comparing them with previously published data (Shaiq et al., Gorla and Sidawi, and Khan et al.) under limiting conditions, demonstrating excellent agreement.

Key Contributions and Physical Parameters
The paper systematically analyzes the effects of several dimensionless parameters:

• Biot Number ($Bi$): Represents convective boundary conditions.
• Brinkman Number ($Br$): Relates to viscous dissipation.
• Mixed Convection Parameter ($\lambda$): Balances forced and natural convection.
• Suction/Injection Parameter ($F_w$): Controls mass transfer at the boundary.
• Magnetic Parameter ($M$), Porosity ($\Lambda$), and Radiation Parameter ($R$).

The study defines thermophysical properties for the hybrid nanofluid, specifically calculating density, viscosity, electrical conductivity, thermal expansion, and thermal conductivity based on the volume fraction of copper nanoparticles.

Results and Discussion
The graphical analysis reveals the following trends:

• Velocity Profile:

  • The velocity profile increases with higher values of the Brinkman number and the mixed convection parameter.
  • Conversely, the velocity decreases as the suction/injection parameter increases (i.e., stronger suction or injection reduces the velocity magnitude in the boundary layer).
  • The effect of the Biot number on velocity varies depending on the injection/suction condition; generally, velocity retards with increasing Biot number for injection and impermeable cases but shows an opposite trend for suction.

• Temperature Distribution:

  • Fluid temperature rises with increasing Biot number and Brinkman number, as higher viscous dissipation and improved heat conduction generate more thermal energy.
  • The temperature profile exhibits a dual behavior regarding the Biot number: it decreases with increasing $Bi$ under injection conditions but increases under suction conditions.
  • Increasing the suction/injection parameter generally reduces the temperature by transporting cooler fluid closer to the surface, thinning the thermal boundary layer.
  • The impact of the mixed convection parameter on temperature is non-linear; temperature rises with $\lambda$ when viscous dissipation is negligible ($Br=0$) but declines as $\lambda$ increases when $Br$ is significant.

• Skin-Friction Coefficient ($C_f$):

  • The skin-friction coefficient (surface drag) increases with higher Brinkman number and mixed convection parameter.
  • It decreases with increasing suction/injection parameter.
  • For injection cases, an increase in the Biot number leads to a reduction in the skin-friction coefficient.

• Nusselt Number (Heat Transfer Rate):

  • The heat transfer rate generally increases with the mixed convection parameter and the suction/injection parameter.
  • The Brinkman number shows an inverse relationship with the heat transfer rate; higher viscous dissipation leads to a lower Nusselt number.
  • The effect of the Biot number on heat transfer is conditional: it enhances the rate under suction but reduces it under injection.
  • Overall, the rate of heat transfer is higher for suction flows compared to injection flows.

Significance and Claims
The authors claim that this study is the first to assess the interplay of convective boundary conditions, magnetic fields, and suction/injection on mixed convection hybrid nanofluid flow over stretching surfaces within Darcy-Forchheimer porous media. The research provides a comprehensive numerical framework using the DTM to solve complex non-linear flow problems.

The study suggests that these findings have potential implications for various industrial and engineering applications, including:

• Crude oil extraction
• Food processing
• Geothermal energy systems
• Heat exchangers
• Groundwater remediation
• MHD generator technologies

The paper concludes that optimizing the composition of hybrid nanofluids and controlling parameters like suction/injection and magnetic fields can significantly enhance thermal performance in these systems.