Study of solution for a nonlinear Poiseuille flow with time-dependent boundary conditions and memory effects in viscoelastic fluids

Abstract

This paper presents an analysis of the nonlinear Poiseuille flow of viscoelastic fluids exhibiting temperature-dependent properties and memory effects. We formulate the problem as a system of nonlinear, time-dependent partial differential equations incorporating dynamic boundary conditions that vary with time. Utilizing the Galerkin approximation method alongside functional analysis techniques, we establish the existence, uniqueness, and regularity of weak and strong solutions within appropriate Sobolev spaces. Further, we investigate the qualitative properties of these solutions, including their asymptotic stability and long-term behavior, demonstrating that under suitable conditions on the memory kernel and external forcing terms, perturbations decay exponentially,ensuringthesystem’sreturntoequilibrium.Additionally,explicitsolutionsarederivedundersimplifyingassumptions, in particular, linear temperature dependence of viscosity and exponentially decaying memory kernels, so that we provide concrete descriptions of the fluid’s dynamic response