Linear motion of multiple superposed viscous fluids

In this paper the small-amplitude motion of multiple superposed viscous fluids is studied as a linearized initial-value problem. The analysis results in a closed set of equations for the Laplace transformed amplitudes of the interfaces that can be inverted numerically. The derived equations also contain the general normal mode equations, which can be used to determine the asymptotic growth rates of the systems directly. After derivation, the equations are used to study two different problems involving three fluid layers. The first problem is the effect of initial phase difference on the development of a Rayleigh-Taylor instability and the second is the damping effect of a thin, highly viscous, surface layer.

Figure 1

Schematic illustration of the multilayered initial-value problem. $a_i$ is the amplitude of the disturbance on interface $i$. $H_i,{\rho }_i$, and ${\mu }_i$, are the thickness, density, and dynamic viscosity of fluid $i$, respectively.

Figure 2

The amplitudes of the disturbances on the lower interface for the case $A=0.9,\epsilon ={10}^{-3}$, and $a_r=1$, normalized by the infinite layer thickness solution $a_p$. The legend denotes the nondimensional layer thickness $h$.

Figure 3

Critical amplitude ratio as a function of nondimensional layer thickness for $A=0.1$. Initial amplitude ratio (black) and normal mode ratio (gray). The legend denotes the value of the viscosity parameter $\epsilon$.

Figure 4

Critical amplitude ratio as a function of nondimensional layer thickness for $A=0.5$. Initial amplitude ratio (black) and normal mode ratio (gray). The legend denotes the value of the viscosity parameter $\epsilon$.

Figure 5

Critical amplitude ratio as a function of nondimensional layer thickness for $A=0.9$. Initial amplitude ratio (black) and normal mode ratio (gray). The legend denotes the value of the viscosity parameter $\epsilon$.

Figure 6

Normalized minimal growth rate as a function of nondimensional layer thickness ($h$) for $A=0.1$ (black), $A=0.5$ (gray), and $A=0.9$ (silver). The legend denotes the value of the viscosity parameter $\epsilon$.

Figure 7

Amplitudes of the disturbances of the oil-air interface for the highly viscous surface layer case with $kH=0.01$. The legend denotes the viscosity ratio of oil to water ($\overline{\nu }={\nu }_2/{\nu }_1$). The dashed lines are the damping rates corresponding to a pure air-water interface and an inextensible film over water.

Figure 8

Amplitudes of the disturbances of the oil-air interface for the highly viscous surface layer case with $kH=0.1$. The legend denotes the viscosity ratio of oil to water ($\overline{\nu }={\nu }_2/{\nu }_1$).

Figure 9

Amplitudes of the disturbances of the oil-air interface for the highly viscous surface layer case with $kH=0.6$. The legend denotes the viscosity ratio of oil to water ($\overline{\nu }={\nu }_2/{\nu }_1$).