Analytical Solutions to the Unsteady Poiseuille Flow of a Second Grade Fluid with Slip Boundary Conditions

Abstract

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm–Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods.

An Extended Thermodynamics Study for Second-Grade Adiabatic Fluids

“…An interesting further study could be the investigation of solutions for different kinds of flow for second-order fluids described by the obtained field equations and the comparison with the analogue classical results like the ones obtained [30][31][32][33][34].…”

“…In the literature, there are also some controversial discussions about the coefficients and the stability criteria [25][26][27]. More recently, viscoelastic fluids have been described in different volumes in continuum mechanics like [28,29] and there have been a lot of different applications to different kind of flows with both analytical [30][31][32][33][34] and numerical solutions [35][36][37].…”

Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition

“…The traditional second-grade fluid constitutive relation is grounded in integer differentials [15,16]. By considering the historical memory during their flow process, meaning that their current state is not solely determined by the present stress state but is also influenced by the stress history of the past, the fractional second-grade fluid is developed.…”

Polymer Physics: From Theory to Experimental Applications

“…The enthalpy recovery of annealed PS was quantitatively described using retardation spectra of the primary Tg as well as the kinetic process of the sub-Tg relaxation process. Baranovskii [ 5 ] studied the unidirectional pressure-driven flow of a second-grade fluid within a planar channel bounded by impermeable solid walls. These authors examined the well-posed nature of this problem and derived its analytical solution while imposing weak regularity conditions on a function representing the initial velocity distribution.…”