Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations

Abstract

We introduce and analyse an augmented mixed variational formulation for the coupling of the Stokes and heat equations. More precisely, the underlying model consists of the Stokes equation suggested by the Oldroyd model for viscoelastic flow, coupled with the heat equation through a temperature-dependent viscosity of the fluid and a convective term. The original unknowns are the polymeric part of the extra-stress tensor, the velocity, the pressure, and the temperature of the fluid. In turn, for convenience of the analysis, the strain tensor, the vorticity, and an auxiliary symmetric tensor are introduced as further unknowns. This allows to join the polymeric and solvent viscosities in an adimensional viscosity, and to eliminate the polymeric part of the extra-stress tensor and the pressure from the system, which, together with the solvent part of the extra-stress tensor, are easily recovered later on through suitable postprocessing formulae. In this way, a fully mixed approach is applied, in which the heat flux vector is incorporated as an additional unknown as well. Furthermore, since the convective term in the heat equation forces both the velocity and the temperature to live in a smaller space than usual, we augment the variational formulation by using the constitutive and equilibrium equations, the relation defining the strain and vorticity tensors, and the Dirichlet boundary condition on the temperature. The resulting augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram theorem and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point and Lax–Milgram theorems, we employ Raviart–Thomas approximations of order k for the stress tensor and the heat flux vector, continuous piecewise polynomials of order ≤ k + 1 for velocity and temperature, and piecewise polynomials of order ≤ k for the strain tensor and the vorticity. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence.

A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

“…We end this section emphasizing from that we can recover the polymeric and solvent parts of the extra‐stress tensor as a simple postprocess of θ and t , whereas from the fourth equation of , we can compute the pressure in terms of σ conserving the same rate of convergence of the solution as we show theoretical and numerically in Ref. (Lemma 4.14 and Section 5), respectively. However, for the sake of simplicity and physical interest, in Section 5 we will focus only on the formulae suggested for the polymeric part of the extra‐stress tensor and the pressure.…”

“…Now, to derive our mixed approach (see , Section 2.1 for details), we begin by introducing the strain tensor as an additional unknown t : = e ( u ), whence the polymeric and solvent parts of the extra‐stress tensor can be written, respectively, as …”

“…In this section we recall from Ref. , Section 2.2, the weak formulation of the coupled problem given by . To this end, let us first group appropriately some of the unknowns and spaces as follows: where ℍ is endowed with the norm …”

“…Indeed, the following result taken from Ref. establishes the well‐posedness of .Theorem Assume that and κ 2 , κ 6 , κ 7 > 0, with , δ 2 , δ 3 ∈ (0, 2), and . Let and assume that the datum f satisfy …”

“…Furthermore, for any tensor fields τ = ( τ ij ) i , j = 1, n and ζ = ( ζ ij ) i , j = 1, n , we let div τ be the divergence operator div acting along the rows of τ , and define the transpose, the trace, the tensor inner product, and the deviatoric tensor, respectively, as where is the identity matrix in ℝ n × n . Additionally, we define the following tensorial and vectorial functional spaces (see , Section 2.2 for details): and …”