We provide in this article an investigation of the globally modied
Navier-Stokes problem coupled with the heat equation. After deriving the
variational formulation of this problem, we prove the existence and the
uniqueness of the solution using the method of Faedo-Galerkin and some
compactness results. Next, we propose a time discretization of these
equations based on Euler's implicit scheme. We prove the existence of
solution with the aid of Brouwer's xed point and study the stability of
discrete in time solution by using the energy approach.