ABSTRACT
We consider small linearized vibrations of a viscous compressible fluid in a bounded or unbounded domain G with a smooth boundary Γ under different boundary conditions. The system of linear differential equations describing small motions of a viscous compressible fluid in the region G is https://www.w3.org/1998/Math/MathML"> ∂ v ∂ t = μ Δ v + μ 3 + ζ grad div v − grad ρ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn1a.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> ∂ ρ ∂ t = − div v , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn2a.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where v(x, t) = (υ1, υ2, υ3) is a fluid velocity vector, p(x, t) is a deviation of a fluid pressure from the equilibrium, x = (x1, x2, x3) are the Cartesian coordinates in G. Hereinafter we use the dimensionless system of units in which a characteristic curvature of Γ, a fluid density and a speed of sound are equal to 1. By μ and ς we denote the dimensionless viscosity coefficients which must satisfy the thermodynamic assumptions μ > 0 and ζ > 0. Everywhere for simplicity we assume ζ = β μ, β > 0. Eq. (1) is the system of three equations of motion, Eq. (2) is the equation of continuity.
