Abstract. The following is a study of a shape optimal design problem for a stationnary Navier-Stokes flow with thermal transfers. The thermal transfers are convective, diffusive and radiative with multiple reflections (grey bodies). The state equation is a non-linear integrodifferential system and the problem is to minimize a cost function which depends on the solution, with respect to the domain of the equations. We prove the differentiability of the solution with respect to the domain, the differentiability of the cost function, we introduce a classical adjoint state equation and we give an expression of the differential of the cost function.
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