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Transient and periodic shear wave propagation in a solid-fluid coupled system

Aaron D'Cruz, Pierre Ricco

TL;DR

This work analyzes a Newtonian fluid lying atop a linear elastic solid, driven by sinusoidal shear at the solid base. It develops a two-pronged analytical approach: transient behavior via Laplace transforms reveals a cascade of wave reflections and fluid dissipation, while long-time forcing yields a periodic, Stokes-layer-like response through Fourier analysis. The fluid-only limit recovers classical Stokes problems, and numerical simulations verify the analytical results with good accuracy. The study quantifies transient durations via energy dissipation, linking them to practical timescales in viscometry, drag reduction, and biosensing, and demonstrates how resonance and coupling control interface velocity and penetration depth. This provides a rigorous, transferable framework for predicting and designing shear-driven solid-fluid devices across multiple engineering applications.

Abstract

A coupled system composed of a Newtonian fluid located on a sinusoidally-forced elastic solid is studied analytically and numerically. The focus is on the transient evolution from the beginning of the forced oscillations and on the periodic behaviour established once the transient has vanished. The analytical solution is expressed as series summations that elucidate the propagation and reflections of elastic transverse waves through the solid layer and the viscous dissipation of oscillations in the fluid layer. Short-term transients in both the fluid and the solid form at every interaction between an elastic wave and a solid boundary. The long-term transient, quantified by the power balance in the fluid layer, instead pertains to the formation of all the elastic waves in the solid layer. The system can be viewed as a generalised transient Stokes layer generated by the elastic waves or as a damped resonant oscillator when the velocity at the fluid-solid interface increases significantly with respect to the amplitude. A parametric study is carried out for three applications of technological interest, i.e. the indirect measurement of fluid viscosity, the turbulent drag reduction by travelling shear waves and the sensing and manipulation of biological flows.

Transient and periodic shear wave propagation in a solid-fluid coupled system

TL;DR

This work analyzes a Newtonian fluid lying atop a linear elastic solid, driven by sinusoidal shear at the solid base. It develops a two-pronged analytical approach: transient behavior via Laplace transforms reveals a cascade of wave reflections and fluid dissipation, while long-time forcing yields a periodic, Stokes-layer-like response through Fourier analysis. The fluid-only limit recovers classical Stokes problems, and numerical simulations verify the analytical results with good accuracy. The study quantifies transient durations via energy dissipation, linking them to practical timescales in viscometry, drag reduction, and biosensing, and demonstrates how resonance and coupling control interface velocity and penetration depth. This provides a rigorous, transferable framework for predicting and designing shear-driven solid-fluid devices across multiple engineering applications.

Abstract

A coupled system composed of a Newtonian fluid located on a sinusoidally-forced elastic solid is studied analytically and numerically. The focus is on the transient evolution from the beginning of the forced oscillations and on the periodic behaviour established once the transient has vanished. The analytical solution is expressed as series summations that elucidate the propagation and reflections of elastic transverse waves through the solid layer and the viscous dissipation of oscillations in the fluid layer. Short-term transients in both the fluid and the solid form at every interaction between an elastic wave and a solid boundary. The long-term transient, quantified by the power balance in the fluid layer, instead pertains to the formation of all the elastic waves in the solid layer. The system can be viewed as a generalised transient Stokes layer generated by the elastic waves or as a damped resonant oscillator when the velocity at the fluid-solid interface increases significantly with respect to the amplitude. A parametric study is carried out for three applications of technological interest, i.e. the indirect measurement of fluid viscosity, the turbulent drag reduction by travelling shear waves and the sensing and manipulation of biological flows.

Paper Structure

This paper contains 15 sections, 87 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Analytical results
  4. Transient solution by Laplace transforms
  5. Periodic solution by Fourier modes
  6. Fluid-only solution
  7. Numerical procedures
  8. Physical results
  9. Physical interpretation of the analytical solutions
  10. Visualisation of the system dynamics
  11. Parameter dependence
  12. Duration of long-term transients
  13. Conclusions
  14. Contour integration for transient solution
  15. Numerical procedures

Figures (16)

  • Figure 1: Schematic of the solid-fluid system.
  • Figure 2: Comparison between the numerical solutions (circles) and the transient analytical solutions (lines).
  • Figure 3: Plots of solid displacement (red) and fluid velocity (blue) for the first transmission and reflection.
  • Figure 4: Comparison of the periodic (solid blue) and transient (solid grey) contributions to the total fluid velocity (dashed blue), at the start of the first fluid transmission.
  • Figure 5: Periodic (red) and transient (grey) contributions to the total solid displacement for the first reflected wave (without the superposed incident wave). (d) and (e) show the consistent profile shape in the vicinity of the wavefront.
  • ...and 11 more figures