Mathematical problems of dynamical interaction of fluids and multiferroic solids

Abstract

In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region ${\mathrm{\Omega }}^{+}$ is immersed in an inviscid fluid occupying an unbounded domain ${\mathrm{\Omega }}^{-}={\mathbb{R}}^3\setminus \overline{{\mathrm{\Omega }}^{+}}$. In the solid region, we consider the generalized Green–Lindsay's model of the thermo-electro-magneto-elasticity theory. In this case, in the domain ${\mathrm{\Omega }}^{+}$ we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain ${\mathrm{\Omega }}^{-}$. The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundary-transmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ. We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727–9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electro-magneto-elasticity theory are studied by the potential method and the theory of pseudodifferential equations.

Keywords:

• Dynamical problems
• thermo-electro-magneto-elasticity theory
• acoustic wave scattering

Maths:

• 74H20
• 74H25
• 35B65
• 74F15
• 34L25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSF) [grant numberFR-18-126].