Fractional Model of Couple Stress Fluid for Generalized Couette Flow: A Comparative Analysis of Atangana–Baleanu and Caputo–Fabrizio Fractional Derivatives

Different from a Newtonian fluid, couple stress fluid (CSF) includes a new material constant, which is responsible for couple stress and the lubricant viscosity. This material constant comes with the fourth-order spatial derivative term, and due to this higher-order derivative term in the momentum equation, this fluid (CSF) is comparatively less investigated even for the classical fluid problems. This paper aims to study the fractional model of CSF, based on the Atangana-Baleanu (AB) fractional derivatives definition. Since this AB definition is new, therefore, for the sake of comparison and correctness, this problem is also solved using the Caputo-Fabrizio (CF) fractional derivative definition. The CSF is considered to flow between two parallel plates, one of which is at rest and the other is moving with constant velocity. The external pressure gradient is also applied. This type of flow situations is usually called generalized Couette flow. The problem is first written in dimensionless form and then solved for the exact solution using the Laplace transform and the finite Fourier sine transform. The CSF velocity obtained via an AB fractional derivative is compared with the CSF velocity obtained via a CF fractional derivative approach, and the results obtained are shown graphically. The CSF results for interesting fluid parameters are displayed in various graphs for both the AB and CF fractional derivatives. It is observed that the CSF velocities obtained with the AB and CF fractional derivatives are the same for unit time. For time less than one and greater than one, variation in CSF velocities is observed. In limiting sense, the present CSF solutions are reduced to a similar Newtonian fluid problem solution in the absence of external pressure gradient.