Recent Advances on Similarity Solutions Arising During Free Convection

Abstract

This paper reviews results about free convection near a vertical flat plate embedded in some saturated porous medium. We focus on a third order autonomous differential equation that gives a special class of solutions called similarity solutions. Two cases are under consideration: in the first one we prescribe the temperature on the plate and in the second one we prescribe the heat flux on it. We will also see that the same equation appears in other industrial processes.

Parameter-dependent systems on the half-line and free convection boundary-layers in porous media

“…This situation has been almost completely solved by Brighi et al [2][3][4]. We do not obtain any new result in this case.…”

On a Nonlinear System of PDE’s Arising in Free Convection

“…This gives the second inequality of (9). To obtain the other one, we set l ′ = inf Γ Θ and H − = inf{H + Θ − l ′ ; 0} and proceed in the same way.…”

“…These boundary value problems have been widely studied, and mathematical results about them can be found, for example, in [1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [17], [18], [20] and [21].…”

Chaotic behavior in the flow along a wedge modeled by the Blasius equation

“…Then the coefficients become: a 1 = m+1 2 1, a 2 = 0, a 3 = m. In this case, the equation takes the form (Brighi and Hoernel, 2005): As all the forms of the Blasius equation are third order ordinary differential equations (ODE), it is convenient to split it in three first order ODE's before solving. With f = g and f = g = h, the Blasius equation becomes:…”

“…The nonlinear differential equation can be solved by using different approaches (Brighi and Hoernel, 2005;Zaturska and Banks, 2001). One common method is to convert the boundary value problem to an initial value problem, where the value of the velocity gradient f (0) = h(0) = s is estimated by using a shooting method for which f exists for [ 0,∞) and f (η L ) = g(η L ) = 1; η L is large value of η.…”