Mathematical problems of interaction of different dimensional physical fields

Chkadua, George

doi:10.1088/1742-6596/451/1/012025

Cited by 2 publications

(1 citation statement)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many papers are devoted to the similar interaction problems, when in the solid region simpler mathematical models are considered. The Dirichlet type, Neumann type, and mixed type steady state oscillation interaction problems of acoustic waves and piezoelectric structures are studied in the papers Chkadua, 4 Chkadua and Natroshvili, 5 and Chkadua. 6 Similar interaction problems with the classical model of elasticity have been investigated by a number of authors.…”

Section: Introductionmentioning

confidence: 99%

Mathematical aspects of fluid–multiferroic solid interaction problems

Chkadua

Natroshvili

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

In the paper, we consider a three-dimensional model of fluid-solid interaction when a thermo-electro-magneto-elastic body occupying a bounded region Ω + is embedded in an inviscid fluid occupying an unbounded domain Ω − = R 3 ∖Ω + . In this case, we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function) in the domain Ω + , while we have a scalar acoustic pressure field in the unbounded domain Ω − . The physical kinematic and dynamic relations are described mathematically by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, we prove the uniqueness and existence theorems for the corresponding boundary-transmission problems in appropriate Sobolev-Slobodetskii and Hölder continuous function spaces.

show abstract

Section: Introductionmentioning

confidence: 99%

Mathematical aspects of fluid–multiferroic solid interaction problems

Chkadua

Natroshvili

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

Solvability, asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

Chkadua

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

In the paper, we investigate the mixed type transmission problem arising in the model of fluid-solid acoustic interaction when a piezoceramic elastic body ( C ) is embedded in an unbounded fluid domain ( ). The corresponding physical process is described by the boundary-transmission problem for second-order partial differential equations. In particular, in the bounded domain C , we have a 4 4 dimensional matrix strongly elliptic second-order partial differential equation, while in the unbounded complement domain , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener-Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev-Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions.The present paper is devoted to the investigation of a mixed type .M ! / interaction boundary-transmission problem associated with the previously described mathematical model with a piezoelectric solid component. The Dirichlet type .D ! / and Neumann type .N ! / problems are studied in [7].Similar interaction problems for the classical model of elasticity are studied in the references [8][9][10][11][12][13][14][15][16][17][18][19][20]. In this case, one has a threedimensional elastic field, the displacement vector with three components in the bounded domain C and a scalar pressure field in the unbounded domain . In our case, in the domain C , we have an additional electric field that essentially complicates investigation of the transmission problems in question. In particular, except transmission conditions, electric potential is given on one part of the boundary of C (the Dirichlet type condition), while on the other part, normal component of electric displacement is given (the Neumann type condition).In the paper [21], uniqueness and existence theorems for the mixed type interaction problem of acoustic waves and piezoelectric structures are stated without proof.We investigate the aforementioned problem with the use of the potential method and the theory of pseudodifferential equations on manifolds with boundary and prove existence and uniqueness theorems in Sobolev-Slobodetskii spaces.The paper is organized as follows. In Section 2, we formulate mixed type .M ! / boundary-transmission problem for steady state oscillation equation in appropriate function spaces. In Section 3, we define Jones modes and Jones eigenfrequencies and prove the uniqueness theorem for the problem .M ! /. In Section 4, we describe properties of potentials related to the Helmholtz equation and the steady-state oscillation equations of piezoelectricity. In Section 5, mixed interaction boundary-transmission problem .M / is formulated for the pseudo-oscillation equations and the unique...

show abstract

Mathematical problems of interaction of different dimensional physical fields

Cited by 2 publications

References 19 publications

Mathematical aspects of fluid–multiferroic solid interaction problems

Mathematical aspects of fluid–multiferroic solid interaction problems

Solvability, asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

Contact Info

Product

Resources

About