[Paper Review] A mathematical Study of Magnetohydrodynamic Casson Fluid via Special Functions with Heat and Mass Transfer embedded in Porous Plate

Kashif Ali Abro, Hina Saeed Shaikh+1 more|arXiv (Cornell University)|Jun 8, 2017

Nanofluid Flow and Heat Transfer6 citations

TL;DR

This study investigates heat and mass transfer in a magnetohydrodynamic Casson fluid over a porous plate using fractional calculus and special functions. It derives analytical solutions for velocity, temperature, and concentration profiles using the Caputo fractional derivative, with results expressed via Mittag-Leffler, Wright, and Robotnov-Hartley functions, demonstrating the influence of magnetic fields, porosity, and permeability on fluid dynamics and thermal behavior.

ABSTRACT

This article is proposed to investigate the impacts of heat and mass transfer in magnetohydrodynamic casson fluid embedded in porous medium. The generalized solutions have been traced out for the temperature distribution, mass concentration and velocity profiles under the existence and non-existence of transverse magnetic field, permeability and porosity. The corresponding solutions of temperature distribution and mass concentration, velocity profiles are expressed in terms of newly defined generalized Robotnov-Hartley function, wright function and Mittage-Leffler function respectively. All the corresponding solutions fulfill necessary conditions (initial, natural and boundary conditions) as well. Caputo Fractionalized solutions have been converted for ordinary solutions by substituting {\\zeta}=1. Some similar solutions for the temperature distribution, mass concentration and velocity profiles have been particularized form generalized solutions. Owing to the rheology of problem, graphical illustrations of distinct parameters are discussed in detail by depicting figures using Mathcad software (15).

Motivation & Objective

• To analyze the effects of a transverse magnetic field, porosity, and permeability on heat and mass transfer in a Casson fluid.
• To develop generalized analytical solutions for velocity, temperature, and concentration profiles using fractional calculus.
• To express solutions in terms of special functions such as Mittag-Leffler, Wright, and Robotnov-Hartley functions.
• To validate solutions against initial, boundary, and natural conditions.
• To recover classical (ordinary) solutions by setting the fractional order ζ = 1.

Proposed method

• Employment of the Caputo fractional derivative to model memory effects in the fluid flow system.
• Formulation of dimensionless governing equations for momentum, energy, and concentration transfer.
• Use of Laplace and Sumudu transforms to solve the fractional-order partial differential equations.
• Expression of solutions in terms of special functions: Mittag-Leffler for velocity, Wright for concentration, and Robotnov-Hartley for temperature.
• Application of initial and boundary conditions to ensure physical consistency of solutions.
• Numerical visualization of results using Mathcad 15 to analyze the impact of key parameters.

Experimental results

Research questions

• RQ1 How does the presence of a transverse magnetic field affect the velocity, temperature, and concentration profiles in a Casson fluid?
• RQ2 What is the influence of porosity and permeability on heat and mass transfer in the porous medium?
• RQ3 How do the fractional-order parameters affect the dynamics of the fluid system?
• RQ4 What are the analytical solutions for velocity, temperature, and concentration when using special functions?
• RQ5 How do the generalized solutions reduce to classical solutions when ζ = 1?

Key findings

• The velocity profile decreases with increasing magnetic field strength due to the Lorentz force inhibition.
• The temperature distribution increases with higher values of the fractional order ζ, indicating enhanced thermal memory effects.
• The concentration profile is significantly reduced under higher porosity, indicating slower mass diffusion.
• The generalized solutions in terms of Mittag-Leffler, Wright, and Robotnov-Hartley functions satisfy all initial, boundary, and natural conditions.
• Classical solutions are recovered when ζ = 1, confirming consistency with integer-order models.
• Graphical illustrations using Mathcad 15 confirm the physical consistency and sensitivity of the solutions to key parameters.