The Effects of Shear Stress Memory and Variable Viscosity on Viscous Fluids Flowing Between Two Horizontal Parallel Plates

Dumitru Vieru (), Constantin Fetecau and Zulkhibri Ismail
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Dumitru Vieru: Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India
Constantin Fetecau: Academy of Romanian Scientists, 3 Ilfov, 050044 Bucharest, Romania
Zulkhibri Ismail: Centre for Mathematical Sciences, College of Computing and Applied Sciences, Universiti Malaysia Pahang Al-Sultan Abdullah, Lebuh Persiaran Tun Khalil Yaacob, Kuantan 26300, Pahang, Malaysia

Mathematics, 2025, vol. 13, issue 18, 1-16

Abstract: This article investigates a mathematical model with the Caputo derivative for the transient unidirectional flow of an incompressible viscous fluid with pressure-dependent viscosity. The fluid flows in the spatial domain bounded by two parallel plates extended to infinity. The plates translate in their planes with time-dependent velocities, and the fluid adheres to the solid boundaries. The generalization of the model consists of formulating a fractional constitutive equation to introduce the memory effect into the mathematical model. In addition, the fluid’s viscosity is assumed to be pressure-dependent. More precisely, in this article, the viscosity is considered a power function of the vertical coordinate of the channel. Analytic solutions of the dimensionless initial and boundary value problems have been determined using the Laplace transform and Bessel equations. The inversion of Laplace transforms is conducted using both the methods of complex analysis and the Stehfest numerical algorithm. In addition, we discuss the explicit solution in some meaningful particular cases. Using numerical simulations and graphical representations, the results of the ordinary model ( α = 1 ) are compared with those of the fractional model ( 0 < α < 1 ) , highlighting the influence of the memory parameter on fluid behavior.

Keywords: pressure-dependent viscosity; Caputo fractional derivative; analytical solution; Laplace transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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