Hydrodynamic stress correlations in fluid films driven by stochastic surface forcing

We study hydrodynamic fluctuations in a compressible and viscous fluid film confined between two rigid, no-slip, parallel plates, where one of the plates is kept fixed while the other one is driven in small-amplitude, translational displacements around its reference position. This jiggling motion is assumed to be driven by a stochastic, external, surface forcing of zero mean and finite variance. Thus, while the transverse (shear) and longitudinal (compressional) hydrodynamic stresses produced in the film vanish on average on either of the plates, these stresses exhibit fluctuations that can be quantified through their equal-time, two-point, correlation functions. For transverse stresses, we show that the correlation functions of the stresses acting on the same plate (self-correlators) as well as the correlation function of the stresses acting on different plates (cross correlators) exhibit universal, decaying, power-law behaviors as functions of the interplate separation. At small separations, the exponents are given by $-1$, while at large separations, the exponents are found as $-2$ (self-correlator on the fixed plate), $-4$ (excess self-correlator on the mobile plate), and $-3$ (cross correlator). For longitudinal stresses, we find much weaker power-law decays in the large separation regime, with exponents $-3/2$ (excess self-correlator on the mobile plate) and $-1$ (cross correlator). The self-correlator on the fixed plate increases and levels off upon increasing the interplate separation, reflecting the nondecaying nature of the longitudinal forces acting on the fixed plate.

Figure 1

Side view of a plane-parallel film of a compressible, viscous fluid, driven at its lower boundary ($z=0$) with an external forcing, giving the uniform surface velocity $\mathbf{u}\left(t\right)$.

Figure 2

The self- and cross correlators of the transverse stresses acting on the plates as functions of the redefined dimensionless frequency, ${\omega }_{{}_{\parallel }}{h}_{{}_{\parallel }}^2$, for fixed $h_{{}_{\parallel }}/{\gamma }_{{}_{\parallel }}=5$.

Figure 3

Profiles of the rescaled, equal-time, transverse stress correlators $h_{{}_{\parallel }}^2{\mathcal{C}}_{{}_{\parallel }}(z_{{}_{\parallel }},z_{{}_{\parallel }})$ and $h_{{}_{\parallel }}^2{\mathcal{C}}_{{}_{\parallel }}(0,z_{{}_{\parallel }})$ across the fluid film ($0\le z_{{}_{\parallel }}\le h_{{}_{\parallel }}$) for fixed $h_{{}_{\parallel }}/{\gamma }_{{}_{\parallel }}=5$.

Figure 4

Log-log plots of equal-time transverse stress (self- and cross) correlators ${\mathcal{C}}_{{}_{\parallel }}(h_{{}_{\parallel }},h_{{}_{\parallel }})$, $\mathrm{\Delta }{\mathcal{C}}_{{}_{\parallel }}(0,0)$, and ${\mathcal{C}}_{{}_{\parallel }}(0,h_{{}_{\parallel }})$, in panels (a)–(c), respectively, as functions of the dimensionless interplate separation, $h_{{}_{\parallel }}$, for ${\gamma }_{{}_{\parallel }}=2,10,30,$ and 100.

Figure 5

The rescaled universal forms of the three equal-time transverse stress correlators, as indicated on the graph [see also Figs. 4], as functions of $h_{{}_{\parallel }}/{\gamma }_{{}_{\parallel }}$, demonstrating the crossover between the two power-law regimes found at small and large values of $h_{{}_{\parallel }}/{\gamma }_{{}_{\parallel }}$ in each case.

Figure 6

Log-log plot of the longitudinal stress self-correlator ${\tilde{\mathcal{C}}}_{{}_{\perp }}(h_{{}_{\perp }},h_{{}_{\perp }};{\omega }_{{}_{\perp }})$ in panel (a) and linear plots of the excess longitudinal stress (self- and cross-) correlators $\mathrm{\Delta }{\tilde{\mathcal{C}}}_{{}_{\perp }}(0,0;{\omega }_{{}_{\perp }})$ and $Re{\tilde{\mathcal{C}}}_{{}_{\perp }}(0,h_{{}_{\perp }};{\omega }_{{}_{\perp }})$ in panels (b) and (c), respectively, as functions of the dimensionless frequency, ${\omega }_{{}_{\perp }}$, for fixed ${\gamma }_{{}_{\perp }}=2$ and $h_{{}_{\perp }}=10,30,$ and 50 as indicated on the graphs.

Figure 7

Profiles of the equal-time longitudinal stress correlators ${\mathcal{C}}_{{}_{\perp }}(z_{{}_{\perp }},z_{{}_{\perp }})$ and ${\mathcal{C}}_{{}_{\perp }}(0,z_{{}_{\perp }})$ across the fluid film ($0\le z_{{}_{\perp }}\le h_{{}_{\perp }}$) for fixed $h_{{}_{\perp }}=50$ and ${\gamma }_{{}_{\perp }}=2$.

Figure 8

Log-log plots of equal-time longitudinal stress (self- and cross) correlators ${\mathcal{C}}_{{}_{\perp }}(h_{{}_{\perp }},h_{{}_{\perp }})$, $\mathrm{\Delta }{\mathcal{C}}_{{}_{\perp }}(0,0),$ and ${\mathcal{C}}_{{}_{\perp }}(0,h_{{}_{\perp }})$, in panels (a)–(c), respectively, as functions of the dimensionless interplate separation, $h_{{}_{\perp }}$, for ${\gamma }_{{}_{\perp }}=2,10,30,$ and 100.

Figure 9

Poles of the longitudinal stress response functions in the complex frequency ($\varsigma$) plane for $h_{{}_{\perp }}=50$ and ${\gamma }_{{}_{\perp }}=2$. The poles are shown up to the order $\left|n\right|\le 36$, of which the last four poles fall over the imaginary axis (become purely diffusive). The rest of the poles (shown in two left and right branches in red and blue colors, respectively) represent partially attenuated acoustic modes propagating across the film.

Figure 10

Log-log plots of equal-time self-correlators on the fixed plate, ${\mathcal{C}}_{{}_{\parallel }}(h_{{}_{\parallel }},h_{{}_{\parallel }})$ and ${\mathcal{C}}_{{}_{\perp }}(h_{{}_{\perp }},h_{{}_{\perp }})$, in panels (a) and (b) respectively, as functions of the corresponding dimensionless frequency cutoffs for different values of the dimensionless system parameters as indicated on the graphs.