SHAPE DIFFERENTIABILITY OF DRAG FUNCTIONAL AND BOUNDARY VALUE PROBLEM SOLUTIONS FOR FLUID MIXTURE EQUATIONS  Abstract (English):
Problems of optimal design of various elements of technical structures stimulate mathematical statements of new problems of continuum mechanics and hydrodynamics in particular. This study refers to problems of shape optimization of profiles in a flow of fluid or gas. The paper deals with properties of solutions and their functional of inhomogeneous boundary value problem for nonlinear composite type partial differential equation system, simulating the a mixture of viscous compressible fluids (gases) flowing around an obstacle. Methods of the theory of partial differential equations, functional analysis and, in particular, the results on the solvability of boundary value problems for transport and Stokes equations established the well-posedness of a linear boundary value problem with singular coefficients (the problem of the original problem solution difference). This result allowed to obtain the uniqueness theorem to determine the nature of solutions dependence on the shape of the flow range and to prove domain differentiability of the solutions considered. Domain differentiability of the solution functional reflecting the force of the obstacle resistance to the incident flow is proved. A formula to equate this derivative as a sum of two summands, one of which clearly depends on mapping setting the domain shape, and the other can be expressed in terms of the so-called adjoint state, depending only on the solution of the original problem in a non-deformed domain. The functional derivative formulas may be used as the basis for building a numerical algorithm for finding the optimal shape of the body in a flow of mixture of viscous compressible fluids.

Keywords:
boundary value problem, mixture of viscous compressible fluids, conjugate problem, flowing around an obstacle, material derivative, shape derivative

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