Abstract In the stability analysis of flows past deformable surfaces, the critical Reynolds number is computed assuming the solid and fluid densities are equal. In real applications, the solid-to-fluid density ratio (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>) varies from O(1) for liquid flows to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(10^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for gas flows past deformable surfaces. The effect of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> on the linear stability of a Couette flow past a viscoelastic continuum solid is studied. An increase in the density ratio is found to have a significant destabilizing effect. The most unstable mode is an inviscid mode, which is an elastic wave of the solid that is destabilised due to a coupling with fluid fluctuations when the wave speed is smaller than the maximum of the flow velocity. As the density ratio is increased, the wave speed decreases and the flow becomes unstable at a lower flow speed or Reynolds number. From numerical results and theoretical reasoning, the critical Reynolds number (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\text{ Re}_c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <msub> <mtext>Re</mtext> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>) is shown to scale as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\text{ Re}_c \sim \rho _r^{-1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <msub> <mtext>Re</mtext> <mi>c</mi> </msub> <mo>∼</mo> <msubsup> <mi>ρ</mi> <mi>r</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. At moderate to high Reynolds numbers, a small increase in density ratio can cause a relatively large decrease in the critical Reynolds number. Increasing solid-to-fluid viscosity ratio (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>) stabilises the system at all density ratios. For dissipative solids, the critical Reynolds numbers also decrease with an increase in the ratio of density to viscosity at all density ratios. This study shows that it is important to use the correct values of density ratio in computation of the stability boundaries. Graphic Abstract

Effect of density ratio on the stability of a Couette flow past viscoelastic compliant solid

期刊论文
发表日期：

2025年11月4日

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Mandeep Deka,
V. Kumaran

摘要

Abstract

In the stability analysis of flows past deformable surfaces, the critical Reynolds number is computed assuming the solid and fluid densities are equal. In real applications, the solid-to-fluid density ratio ( $\rho _r$ $ρ_{r}$ ) varies from O(1) for liquid flows to $O(10^3)$ $O (10^{3})$ for gas flows past deformable surfaces. The effect of $\rho _r$ $ρ_{r}$ on the linear stability of a Couette flow past a viscoelastic continuum solid is studied. An increase in the density ratio is found to have a significant destabilizing effect. The most unstable mode is an inviscid mode, which is an elastic wave of the solid that is destabilised due to a coupling with fluid fluctuations when the wave speed is smaller than the maximum of the flow velocity. As the density ratio is increased, the wave speed decreases and the flow becomes unstable at a lower flow speed or Reynolds number. From numerical results and theoretical reasoning, the critical Reynolds number ( $\text{ Re}_c$ ${Re}_{c}$ ) is shown to scale as $\text{ Re}_c \sim \rho _r^{-1/2}$ ${Re}_{c} \sim ρ_{r}^{- 1 / 2}$ . At moderate to high Reynolds numbers, a small increase in density ratio can cause a relatively large decrease in the critical Reynolds number. Increasing solid-to-fluid viscosity ratio ( $\mu _r$ $μ_{r}$ ) stabilises the system at all density ratios. For dissipative solids, the critical Reynolds numbers also decrease with an increase in the ratio of density to viscosity at all density ratios. This study shows that it is important to use the correct values of density ratio in computation of the stability boundaries.

Graphic Abstract