Unsteady Couette Flow in a Composite Channel with Porous and Clear Regions Using Caputo–Fabrizio Fractional Derivative Approach

Abstract

This study examines fully developed unsteady Couette flow in a composite channel partially filled with a porous medium and partially with a clear fluid, within a fractional derivative framework. The Brinkman-extended Darcy model governs the porous region, while the Stokes equation describes the clear region, both formulated using Caputo–Fabrizio fractional derivatives to account for memory effects. The governing equations are transformed via Laplace techniques, solved analytically in the Laplace domain, and inverted to the time domain using the Riemann-sum approximation. Compared with classical integer-order models, the fractional-order formulation provides a more comprehensive and physically realistic description of the flow. Parametric analysis explores the effects of fractional order, time, and other physical parameters on velocity, interfacial velocity, skin friction, and volumetric flow rate. Results, presented as MATLAB-generated contour plots, reveal that transient and interfacial velocities and flow rate increase with time, whereas skin friction decreases. These findings demonstrate the effectiveness of fractional calculus in capturing the complex dynamics of fluid flow in composite domains.