This paper presents an in-depth analytical approach to understanding the free vibration dynamics of compressible fluids within rigid-walled rectangular tanks, emphasizing the significance of fluid-structure interaction (FSI). The study derives the governing partial differential equations and boundary conditions, providing exact solutions for two-dimensional and three-dimensional tank geometries. Through the calculation of natural frequencies using derived analytical expressions, the research examines the impact of varying tank dimensions on these frequencies via comprehensive sensitivity analyses.

The introduction underscores the importance of understanding FSI in engineering fields like storage tanks, pipelines, and offshore platforms. It traces the historical development of research in this domain, highlighting key contributions from pioneers such as Westergaard, Housner, Jacobsen, Lemm, and others. The paper acknowledges the significant advancements made through analytical and numerical modeling techniques, as well as the growing role of machine learning in enhancing FSI simulation accuracy.

The paper then presents the mathematical framework governing the behavior of compressible fluids within rectangular tanks. The core principles of mass and momentum conservation are expressed through the continuity and Navier-Stokes equations, respectively. The relationship between stress and strain rate is defined by the constitutive equation, while the Reynolds Transport Theorem relates the rate of change of a quantity within a control volume to the flux across the control surface and the rate of change within the volume itself.

For the three-dimensional tank analysis, the paper derives the governing Helmholtz equation in the frequency domain, representing the spatial variation of the fluid velocity field within the tank. The boundary conditions are specified, ensuring no flow normal to the tank boundaries, mass conservation within the tank, and no normal gradient of velocity at the boundaries. The method of separation of variables is employed to solve the partial differential equation, leading to a general solution for the velocity field.

The sensitivity analysis section explores the effects of varying tank dimensions on the first five natural frequencies. Each subsection focuses on a specific dimension (length, height, or width), systematically investigating how changes in that dimension influence the natural frequencies while keeping the other two dimensions constant. The results are presented in tabular and graphical forms, revealing a consistent decrease in natural frequencies across all modes as the length, height, or width of the tank increases.

The paper attributes this phenomenon to the fascinating interplay between the fluid's kinetic and potential energies. As the tank dimensions increase, the fluid has more space for deformation, reducing the restoring force acting on the fluid elements when compressed. This decrease in potential energy is identified as the primary contributor to the observed lowering of natural frequencies with increasing reservoir size.

Furthermore, the paper introduces the Rayleigh-Ritz method, a work and energy-based approach to analyzing vibrating systems. By utilizing approximate shape functions to represent the displacement of the vibrating body, the method calculates the potential and kinetic energies, leading to the derivation of fluid frequencies. The results obtained through the Rayleigh-Ritz method align well with the accurate method, validating its effectiveness in calculating fluid frequencies in reservoirs.

The discussion section delves deeper into the theoretical underpinnings of the observed behavior. Analogies are drawn to familiar concepts, such as the sound produced by flutes of varying lengths and the behavior of waves in the ocean, to provide intuitive explanations for the observed trends. The paper emphasizes the significance of understanding this relationship between tank dimensions and resonance frequencies, as it offers engineers the ability to optimize tank designs for safe and efficient performance under various loading conditions, including seismic events.

In conclusion, this study provides a comprehensive analytical framework for understanding the free vibration dynamics of compressible fluids in rigid-walled rectangular tanks, considering the crucial effects of fluid-structure interaction. The derived analytical expressions, combined with the Rayleigh-Ritz method, offer a powerful tool for calculating natural frequencies and analyzing the impact of tank dimensions on these frequencies. The results not only validate the analytical model but also provide valuable insights for engineers working on seismic-resistant storage facilities, acoustically tuned containers, and other applications involving fluid-structure interaction phenomena.