In a cylindrical coordinate system, a homogeneous and isotropic circular cylindrical elastic layer is considered, with internal and external radii. The thickness of the layer is assumed to take arbitrary values, depending on the radial and axial coordinates. Moreover, it is assumed that the cylindrical layer, as a three-dimensional body, strictly follows the mathematical theory of elasticity and is described by its three-dimensional equations. The inner cavity of the layer is filled with a viscous, incompressible, and stationary fluid, which is described by the linearized Navier–Stokes equations [1], [2].

Torsional oscillations in elastic cylindrical structures containing viscous fluids are a significant subject in engineering mechanics, with wide applications in structural dynamics, fluid-structure interaction, and mechanical engineering systems. These oscillations occur when cylindrical systems, such as pipelines, shafts, or layered shells, are subjected to rotational disturbances around their longitudinal axes. Understanding such phenomena is essential for optimizing the design and improving the performance and stability of modern mechanical and aerospace systems.

The dynamics of an elastic cylindrical shell interacting with a viscous incompressible fluid require rigorous treatment using elasticity theory and fluid dynamics. In this context, the behavior of both the solid and the fluid domains must be coupled, especially when the layer thickness varies along the radial or axial directions. Classical approaches to such problems have often employed the theory of elasticity and linearized Navier–Stokes equations for modeling the internal fluid [3], [4].

In practical engineering systems, such as heat exchangers, viscoelastic dampers, or rotating machinery with fluid-filled cavities, torsional oscillations can significantly influence the stress-strain distribution, energy dissipation, and operational stability. These systems frequently exhibit layered configurations with varying wall thicknesses, leading to additional complexity in mathematical modeling and necessitating advanced analytical or numerical methods.

The theoretical foundation of this study builds upon prior research by A. N. Guz, who investigated wave propagation in cylindrical shells filled with viscous compressible fluids [5], [6]. He demonstrated the critical role of initial stress conditions and boundary interactions in defining the wave characteristics within such media. The current research extends this framework to encompass incompressible fluids and focuses on the implications of layer thickness variation, a factor that has received limited attention in previous works.

The hydrodynamic behavior of the enclosed fluid is governed by the incompressibility condition and the linearized Navier–Stokes equations,forming the basis for modeling small-amplitude oscillatory motion[7]. In cylindrical coordinates, the coupling of fluid motion and elastic deformation introduces non-trivial boundary conditions and stress continuity requirements at the fluid-structure interface. This coupling is crucial in accurately predicting the dynamic response of the system under torsional excitation.

Furthermore, studies such as those by Khudoynazarov have contributed to the understanding of longitudinal-radial oscillations in similar viscoelastic shell configurations [8]. However, their application to torsional dynamics in a variable-thickness setting remains an open research area, motivating the present investigation. A key innovation in this paper is the representation of the dynamic variables through Bessel function-based expansions, which allow for effective approximation and solution of the resulting high-order differential equations[9].

The outcomes of this study are relevant for addressing vibration suppression, enhancing structural integrity, and developing vibration-insulated equipment in engineering applications. The modeling strategy employed herein—through scalar and vector potential functions, transformation techniques, and continuity enforcement—provides a general framework applicable to a variety of practical configurations involving layered cylindrical systems filled with viscous media.

Recent advancements in vibration analysis have shown that even slight variations in geometrical parameters—such as wall thickness—can dramatically alter the dynamic response of structural systems. This is particularly true in cylindrical systems with fluid-structure interaction, where oscillatory phenomena are not only affected by mechanical properties but also by hydrodynamic resistance and viscosity-induced damping. In such systems, torsional oscillations can lead to fatigue failure or excessive vibration if not properly understood and controlled. Moreover, as industries move toward lightweight, layered composite materials with non-uniform cross-sections, the need for refined models incorporating spatial variability in thickness becomes critical. This study addresses this gap by integrating spatially dependent parameters into the classical theory of elasticity and coupling it with modified fluid dynamics formulations. Furthermore, modern computational tools and analytical approximations enable the reduction of complex differential equations to manageable forms without losing essential physical behavior. The present research not only enhances theoretical understanding but also aims to offer design principles for vibration mitigation in engineering structures such as submarines, aerospace fuselages, and high-speed rotors filled with damping fluids. By establishing a framework that rigorously combines elastic shell mechanics with viscous fluid motion, the study contributes to the development of more accurate predictive models for advanced mechanical systems.

The methodology employed in this study is grounded in the mathematical modeling of torsional oscillations in a circular cylindrical elastic layer of varying thickness filled with a viscous incompressible fluid[10]. The problem is formulated within a cylindrical coordinate system, assuming the layer to be homogeneous, isotropic, and governed by the three-dimensional equations of classical elasticity theory. The internal cavity of the elastic shell is occupied by a stationary, viscous, and incompressible fluid, whose motion is described using the linearized Navier–Stokes equations[11]. The torsional oscillations are induced along the longitudinal axis of the cylindrical structure, and the corresponding motion equations are expressed through scalar and vector potentials to simplify the wave propagation analysis. The layer's elastic response is captured through wave equations derived from elasticity theory, while the fluid's behavior is governed by the incompressibility condition and modified Navier–Stokes formulations. Continuity and boundary conditions are imposed at the inner and outer surfaces of the layer and at the fluid-structure interface. General solutions are constructed using modified Bessel functions and power series expansions, accounting for the radial and axial dependence of thickness. To solve the resulting differential equations, operators are introduced to represent the dynamic interaction between the fluid and the elastic shell. Approximations are applied to reduce the problem to a tractable form by considering zero and first-order harmonics, which enables the derivation of engineering-relevant solutions under assumptions of low-frequency excitation. This approach facilitates understanding the stress-strain state and displacement fields within the system[12].

The study's results are derived from solving the system of coupled differential equations that govern the torsional oscillations of the elastic cylindrical layer and the viscous incompressible fluid. To examine the system's dynamic response, the solid and fluid domains are considered interdependent, and the governing equations are resolved under continuity and boundary conditions[13]. The oscillatory process is predominantly influenced by torsional excitations along the cylinder's axis, and fluid-structure interaction is essential in ascertaining the system's overall behaviour. The primary aim in the results phase is to ascertain the displacement fields, stress distribution, and pressure variations within both media. Due of the geometric complexity arising from the changeable thickness of the cylindrical layer, the analytical solutions are articulated using generalised functions that include modified Bessel functions. This mathematical formulation facilitates the description of radial and axial fluctuations in the system's response[14].The impact of augmenting the thickness of the cylindrical layer on the natural frequencies of torsional vibrations is of particular significance. The numerical and graphical results indicate that an increase in thickness results in a substantial reduction in vibration frequency. This result corresponds with theoretical predictions, as an increased thickness of the elastic layer enhances inertia and damping, hence modifying the resonant properties of the shell-fluid system. The generated displacement functions and resultant stress states illustrate the practicality of simplifying the governing system to low-order approximations—zero and first harmonic modes—adequate for encapsulating the fundamental physical processes pertinent to engineering applications[15].

The equations of motion for the layer

(1)

Are used in the form of wave equations

(2)

For the potentials of longitudinal f and transverse waves, introduced by the formula

where is the Laplace operator in the coordinate system

- components of the stress tensor and displacement vector;

- Lame coefficients; - density; - volume occupied by the layer.For a viscous incompressible fluid exhibiting minor oscillations, the following correlations are established: state of incompressibility

(3)

Navier-Stokes equation taking into account (3)

(4)

It is important to acknowledge that limitations are placed on frequency, current, and wave number, indicating that the truncated equations do not account for high-frequency and short-wavelength phenomena, thereby rendering them applicable just to low-frequency external impacts. Furthermore, the resultant approximate equations, regardless of the amount of approximation, are inadequate for scenarios with concentrated impacts on the system. Nonetheless, this does not suggest that these equations are limited to a narrow range of situations or entirely inapplicable, as the functions expressible in this manner encompass a sufficiently extensive class. Thus, the truncated equations exhibit a considerable scope of applicability. It is important to acknowledge that equations of infinite order render them inappropriate for addressing practical situations. Consequently, provided that the conditions established in prior studies are met, and by imposing restrictions on the oscillation frequency and wave number of propagating waves, while confining the analysis to the zero (n=0), first (n=1), and additional approximations, it is feasible to derive oscillation equations applicable to engineering problems.