We consider a model of a jet engine with liquid fuel
and high-boiling oxidizer. The mixture of fuel and oxidizer has the structure of a fluid with small bubbles. After ignition the pressure grows up and the movement of gas and fluid starts. In the combustion chamber the flow is subsonic but inside the variable cross-section nozzle the flow becomes supersonic and gas becomes homogeneous.
To describe this model we use the approach of G.Tricomi. He suggested using the equation of changing type, because the behavior of subsonic flow is similar to the behavior of the solution of elliptic equation and supersonic flow behaves as the solution of hyperbolic equation. For simplicity we consider a homogenization problem for the Lavrentiev--Bitsadze equation in a partially perforated domain with a Neumann boundary condition on the boundary of holes. The perforated part of the domain lies in the upper half--plane and has a locally periodic structure. The small parameter $\varepsilon$ characterizes the cell of periodicity. We study the asymptotic behavior of the solution of such aproblem as $\varepsilon\to 0$. We prove the existence and the uniqueness theorem for the posed problem. Then, we construct the homogenized problem and on the base of the a priori estimates of the solution we prove the weak convergence of the solutions of this problem to the solution of the homogenized problem.
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