Abstract
In this work, an evolving definition of the fractal-fractional operator with exponential kernel was employed to examine Casson fluid flow with the electro-osmotic phenomenon. Electrically conducted Casson fluid flow with the effect of the electro-osmotic phenomenon has been assumed in a vertical microchannel. With the help of relative constitutive equations, the local mathematical model is formulated in terms of partially coupled partial differential equations along with appropriate physical initial and boundary conditions. The dimensional governing equations have been non-dimensional by using relative similarity variables to encounter the units and reduce the variables. The local mathematical model has been transformed to a fractal-fractional model by using a fractal-fractional derivative operator with exponential kernel and then analyze numerically with the discretization of finite difference (Crank-Nicolson) scheme. For an insight view of the proposed phenomena, various plots are drawn in respect of inserted parameter. From the graphical analysis, it has been observed that the electro-kinetic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> parameter retards the fluid’s motion. It is also worth noting that graphs for the fractal-fractional, fractional, and classical order parameters have been drawn. Due to the fractal order parameter, it was revealed that the fractal-fractional order model has a larger memory effect than the fractional-order and classical models.
Highlights
Many real-world challenges are explained using fractional calculus (FC), which has a greater memory effect
FC is an extension of integer order calculus, which proved insufficient to explain some memory effects in some engineering and real-world issues
Great progress has been made by employing fractional calculus [1]–[3], such as wave propagation [4], image processing [5], modeling of heart tissue [6], infectious disease [7]–[9], nanofluids [10]–[12], chemical kinetics [13] and electrical circuit analysis [14]
Summary
Many real-world challenges are explained using fractional calculus (FC), which has a greater memory effect. For the exact solution of the non-linear problem, the authors employed the fractional model of the Atangan-Baleanu derivative operator. From a comprehensive analysis of the literature, no study has been reported related to the analysis of the electro-osmotic flow of Casson fluid via a fractal-fractional operator. To fill this gap, an unsteady free convection flow of Casson fluid in a vertical micro channel together with the effect of the electroosmotic phenomenon has been assumed. In the governing equations the terms , B0 , T , C and indicates dynamic viscosity, the magnitude of the magnetic field, thermal expansion coefficient, concentration coefficient, and Casson fluid parameter respectively. In Eq.19 , , Ts , Tw, Cs , Cw and H ( ) shows time, y-axis, surrounding temperature , plate’s temperature, surrounding concentration, concentration on the plate and Heaviside step function respectively
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