Positive Solutions of a Fifth-Order Boundary Value Problem for Couple-Stress Porous-Channel Flow

Abstract

In this paper, we study the existence of positive solutions for a nonlinear fifth-order boundary value problem arising from fully developed couple-stress fluid flow through a porous channel. Under a natural restriction relating the couple-stress and permeability parameters, the associated linear operator factorizes into two positive second-order operators; we construct the corresponding Green function, show that it is strictly positive, and obtain it in an explicit closed form. By reformulating the problem as a completely continuous operator on a cone in C[0,1] and applying Krasnosel’skiĭ’s fixed point theorem of cone compression–expansion type, sufficient conditions for the existence of positive solutions are established in terms of the load parameter. The positivity of the velocity further yields a strictly positive cumulative-flow solution on (0,1]. Several examples and a numerically calibrated application are given to illustrate the main results.