Existence of Strong Solutions for Nonlinear Systems of PDEs Arising in Convective Flow

Khaled Bouazzaoui,Sameh Elsayed Ahmed,Mohammed Aiboudi,Sining Zheng

doi:10.1155/2022/7331913

Abstract

In this paper, we will study the existence of strong solutions for a nonlinear system of partial differential equations arising in convective flow, modeling a phenomenon of mixed convection created by a heated and diving plate in a porous medium saturated with a fluid. The main tools are Schäfer’s fixed-point theorem, the Fredholm alternative, and some theorems on second-order elliptic operators.

Highlights

In recent years, many authors have studied the case of the semi-infinite vertical plane plate immersed in a porous medium saturated with a fluid. e following problem is derived from this phenomenon: z2Ψ z2Ψ zx2 + zy2 kzyT,(1) z2T z2T λ􏼠zx2 + zy2􏼡 zxTzyΨ − zyTzxΨ, with mixed boundary conditions zxΨ(x, 0) − ωx(m− 1)/2, (2)In the framework of boundary layer approximations, by introducing similarity variables, we can transform the system of partial differential equations into a system of ordinary differential equations of the third order with appropriate boundary values. ese two-point boundary value problems can be studied by using a shooting method
Many authors have studied the case of the semi-infinite vertical plane plate immersed in a porous medium saturated with a fluid. e following problem is derived from this phenomenon: z2Ψ z2Ψ zx2 + zy2 kzyT, (1) z2T z2T λ􏼠zx2 + zy2􏼡 zxTzyΨ − zyTzxΨ, with mixed boundary conditions zxΨ(x, 0) − ωx(m− 1)/2, T(x, 0) Tω(x) T∞ + Axm, (2)
We study a problem given by two strongly coupled partial differential equations in a two-dimensional bounded domain, modeling a phenomenon of convective flow created by a heated and diving plate in a porous medium saturated with a fluid

Summary

Introduction

Many authors have studied the case of the semi-infinite vertical plane plate immersed in a porous medium saturated with a fluid. e following problem is derived from this phenomenon: z2Ψ z2Ψ zx2 + zy kzyT,. Ese two-point boundary value problems can be studied by using a shooting method (see for example [2, 3]). We refer the reader to [5]. We refer the reader to [7]. E second natural way of dealing with this problem, which is the framework of this paper, is straightly related to the coupled partial differential equations (see [1, 8, 9]). E aim of this paper is to generalize the existence of strong solutions of the problem introduced in [1, 8, 9]. Let us introduce the problem in which we are interested. Where λ is a strictly positive constant, ] is the unit outward normal vector on zΩ, and ∇Ψ⊥ (zyΨ, − zxΨ)

Preliminaries

Main Results

Conclusion