Mathematical analysis and numerical resolution of a heat transfer problem arising in water recirculation

Abstract

This work is devoted to the analysis and resolution of a well-posed mathematical model for several processes involved in the artificial circulation of water in a large waterbody. This novel formulation couples the convective heat transfer equation with the modified Navier-Stokes system following a Smagorinsky turbulence model, completed with a suitable set of mixed, nonhomogeneous boundary conditions of diffusive, convective and radiative type. We prove several theoretical results related to existence of solution, and propose a full algorithm for its computation, illustrated with some realistic numerical examples.

An optimization problem related to water artificial recirculation for controlling eutrophication

“…Then, {g n } n∈N is bounded in [H 1 (0, T )] N CT , which implies, thanks to the estimates (63), ( 59), ( 57)and ( 61), and to the Hypotheses of Theorem 2, that the sequence {u n } n∈N is bounded in W 3 . We also have, thanks to estimates obtained in [7] and [8] that the sequence {(v n , θ n )} n∈N is also bounded in W 1 ×W 2 . Thus, we can take a subsequence of…”

“…In particular, the sequence {g n } n∈N is bounded in [H 1 (0, T )] N CT and then, thanks to the estimates obtained in Lemma 4.1 of [7], in Theorem 9 of [8], and in Lemmas 1, 2 and 3, we have that the sequence…”

“…So, we are able to pass to the limit in the corresponding variational formulations using the same arguments that we have employed for proving the compactness of operator M u , and in the Galerkin approximations for systems ( 4) and ( 7) (cf. [7] and [8]). The only difficulty here is to prove that β = β( (ζ g + z)) (ζ g + z).…”

“…Under these hypotheses we will state now two lemmas (whose demonstrations can be found in [7] and [8], respectively), which will allow us to reformulate the state systems (4), (7) and (12) as homogeneous Dirichlet problems.…”

“…Therefore, we can separate the mathematical analysis of systems ( 4)-(7) from system (12). The coupled system (4)-( 7) has been fully analyzed by the authors in [7] and [8]. Thus, following the results there, we can assure that, for each control g ∈ [H 1 (0, T )] N CT , there exists a solution (v, θ) ∈ W 1 × W 2 of the thermohydrodynamic system (4)-( 7), and we can focus our attention in the solution u of the eutrophication system (12) or, equivalently, in the solution w of the modified system (38).…”