On the Convex and Convex-Concave Solutions of Opposing Mixed Convection Boundary Layer Flow in a Porous Medium

Abstract

In this paper, we are concerned with the solution of the third-order nonlinear differential equation f″′+ff″+βf′(f′-1)=0, satisfying the boundary conditions f(0)=a∈R, f′(0)=b<0, and f′(t)→λ, as t→+∞, where λ∈{0,1} and 0<β<1. The problem arises in the study of the opposing mixed convection approximation in a porous medium. We prove the existence, nonexistence, and the sign of convex and convex-concave solutions of the problem above according to the mixed convection parameter b<0 and the temperature parameter 0<β<1.

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

“…Proposition 2. Let us assume, in addition to assumptions ( 11)- (13), that zΩ is of class C 1,1 , Γ 1 � zΩ, and K ∈ (D(Ω)) 2 . en, problem ( 15)-( 18) have at least one solution (Ψ, H) in the space (W 2,4 (Ω)) 2 .…”

“…By the compactness of the embedding H 1 (Ω)↬L p (Ω), p ≥ 2, there exists a subsequence 2 and strongly in L p (Ω), p ≥ 2, to an element denoted (Ψ * , H * ), and we have H * ∈ L ∞ (Ω). Now, we shall complete the proof in three steps.…”

“…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 (see [1], p.3). ese two-point boundary value problems can be studied by using a shooting method (see for example [2,3]). For the auxiliary dynamical system, we refer the reader to [4].…”

“…More precisely, we will give new results about the existence of strong solutions, in particular, without assuming that ‖Ψ‖ H 2 is small (as in [8], eorem 4). We choose the data (the function K) in the larger space inspired by other works, and we will give conditions which make each solution in (C ∞ (Ω)) 2 .…”

An Extension Result on Similarity Solutions Arising During Mixed Convection

“…Therefore, by letting in Eqs. ( 5) and ( 6), we obtain the following boundary-layer approximation equations (see [5]): (7) and (8) where is the mixed convection parameter.…”