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Non-Newtonian fluid flow is significant in engineering and biomedical applications such as thermal exchangers, electrical cooling mechanisms, nuclear reactor cooling, drug delivery, blood flow analysis, and tissue engineering. The Caputo operator has emerged as a prevalent tool in fractional calculus, garnering widespread recognition. This research aims to introduce a novel derivative by merging the proportional and Caputo operators, resulting in the fractional operator known as the constant proportional Caputo. In order to demonstrate this newly defined operator's dynamic qualities, it was employed in the analysis of the unsteady Casson flow model. In addition, the current work shows an analytical analysis to determine the Soret effect on the fractionalized MHD Casson fluid over an oscillating vertical plate. Fractional partial differential equations (PDEs) are used to formulate the problem along with IBCs. The introduction of appropriate nondimensional variables converts the PDEs into dimensionless form. The precise solutions to the fractional governing PDEs are then determined by the Laplace transform method. Velocity, concentration, and temperature profiles; the impacts of the Prandtl number; fractional parameter β and γ; and Soret and Schmidt numbers are graphically depicted. The profiles of temperature, concentration, and velocity rise with rising time and fractional parameters. Interestingly, as the Casson flow parameter is higher, fluid velocity decreases closest to the plate but increases away from the plate. Tables showing the findings for the skin-friction coefficient, Sherwood, and Nusselt numbers for a range of flow-controlling parameter values are provided. Furthermore, an investigation is undertaken to compare fractionalized and ordinary velocity fields. The results suggest that the fractional model employing a constant proportional derivative exhibits a quicker decay than the model incorporating conventional Caputo and Caputo-Fabrizio operators.
Abbas et al. (Fri,) studied this question.