Similarity solutions of mixed convection boundary-layer flows in a porous medium

Abstract

Abstract. The similarity differential equation f + f f + β f ( f − 1) = 0 with β > 0 is considered. This differential equation appears in the study of mixed convection boundary-layer flows over a vertical surface embedded in a porous medium. In order to prove the existence of solutions satisfying the boundary conditions f (0) = a 0 , f (0) = b 0 and f (+∞) = 0 or 1, we use shooting and consider the initial value problem consisting of the differential equation and the initial conditions f (0) = a , f (0) = b and f (0) = c . For 0 < β 1 , we prove that there exists a unique solution such that f (+∞) = 0 , and infinitely many solutions such that f (+∞) = 1 . For β > 1 , we give only partial results and show some differences with the previous case.Mathematics subject classification (2010): 34B15, 34C11, 76D10.

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

“…The case where ≥ 0, ≥ 0, > 0 and ∈ {0,1} was treated by Aïboudi and al. in [3], and the results obtained generalize the ones of [6]. In [4], Brighi and Hoernel established some results about the existence and uniqueness of convex and concave solution of P 1( , ) where −2 < < 0 and > 0.…”

“…The interested reader can consult references [1,2] for more details on the physical derivation and the numerical treatments. Mathematical results about the problem P ( , ) with = 1 can be found in [3][4][5][6][7]. The case where ≥ 0, ≥ 0, > 0 and ∈ {0,1} was treated by Aïboudi and al.…”