A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations

A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations

“…In what follows we consider the model introduced in [28] (see also [13,Section 2]), which is given by a steady double-diffusive convection system in a fluid saturated porous medium. More precisely, we focus on solving the coupling of the incompressible Brinkman-Forchheimer and double-diffusion equations, which reduces to finding a velocity field u, a pressure field p, a temperature field φ 1 and a concentration field φ 2 , the latter two defining a vector φ := (φ 1 , φ 2 ), such that…”

“…Next, in order to derive a new fully-mixed formulation for (2.1)-(2.5), and unlike [13], we do not employ any augmentation procedure and simply proceed as in [17] (see also [18]). More precisely, we now introduce as further unknowns the velocity gradient t, the pseudostress tensor σ, the temperature/concentration gradient t j , and suitable auxiliary variables ρ j depending on t j , u, and φ j , all of which are defined, respectively, by…”

“…This study allowed the evaluation of the control parameters effect on the flow structure, and heat and mass transfer. Meanwhile, an augmented fully-mixed formulation based on the introduction of the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors (besides the velocity, temperature and concentration fields) was analyzed in [13]. In there, the well-posedness of the problem is achieved by combining a fixed-point strategy, the Lax-Milgram and Banach-Nečas-Babuška theorems, and the well-known Schauder and Banach fixed-point theorems.…”

“…We point out that the augmented formulation introduced in [13], and the consequent use of classical Raviart-Thomas spaces and continuous piecewise polynomials to define the discrete scheme, are originated by the wish of performing the respective solvability analysis of the Brinkman-Forchheimer equations within a Hilbertian framework. However, it is well known that the introduction of additional terms into the formulation, while having some advantages, also leads to much more expensive schemes in terms of complexity and computational implementation.…”

“…According to the above bibliographic discussion, the goal of the present paper is to continue extending the applicability of the aforementioned Banach spaces framework by proposing now a new fully-mixed formulation, without any augmentation procedure, for the coupled problem studied in [13] and [28]. To this end, we proceed as in [17] and introduce the velocity gradient and pseudostress tensors as auxiliary unknowns, and subsequently eliminate the pressure unknown using the incompressibility condition.…”