Generalized Knudsen Number for Unsteady Fluid Flow

We explore the scaling behavior of an unsteady flow that is generated by an oscillating body of finite size in a gas. If the gas is gradually rarefied, the Navier-Stokes equations begin to fail and a kinetic description of the flow becomes more appropriate. The failure of the Navier-Stokes equations can be thought to take place via two different physical mechanisms: either the continuum hypothesis breaks down as a result of a finite size effect or local equilibrium is violated due to the high rate of strain. By independently tuning the relevant linear dimension and the frequency of the oscillating body, we can experimentally observe these two different physical mechanisms. All the experimental data, however, can be collapsed using a single dimensionless scaling parameter that combines the relevant linear dimension and the frequency of the body. This proposed Knudsen number for an unsteady flow is rooted in a fundamental symmetry principle, namely, Galilean invariance.

Figure 1

(a) Dissipation in ${\mathrm{N}}_2$ as a function of pressure for a quartz crystal (inset) oscillating in shear mode at $f_0\approx 5\mathrm{MHz}$. Solid line is a fit to Eq. (1). Transition from the kinetic to viscous regime occurs at $p_c\approx 18\mathrm{Torr}$. (b) The inset shows $p_c$ vs $f_0$ for different quartz crystals in ${\mathrm{N}}_2$ (${\mathrm{Kn}}_l\approx 0$). The linear fit gives the empirical $\tau$ as a function of $p$. The main figure shows $\tau /{\mathcal{C}}_g$ for He, ${\mathrm{N}}_2$, and Ar as a function of $p$. Normalization by ${\mathcal{C}}_g$ accounts for the differences between gases [11]. Dashed line is $1/p$. Error bars are not shown when smaller than symbols.

Figure 2

(a) Dissipation vs pressure for a microcantilever (inset) with $l_x\times l_y\times l_z\approx 32\times 350\times 1\mu {\mathrm{m}}^3$ and $f_0\approx 18.8\mathrm{kHz}$ in ${\mathrm{N}}_2$. Solid line is a fit to Eq. (4); dotted (blue) line is a fit to the cylinder solution; $p_c\approx 1.2\mathrm{Torr}$. (b) $p_c$ vs $f_0$ in ${\mathrm{N}}_2$ for three sets of devices with different characteristic dimensions. Diamonds are nanocantilevers from Ref. [29]; circles are microcantilevers; squares are macroscopic resonators from Fig. 1. (c) $\widetilde{\mathrm{Wi}}$ and ${\widetilde{\mathrm{Kn}}}_l$ in He, ${\mathrm{N}}_2$, and Ar for all devices. Dashed line is $\widetilde{\mathrm{Wi}}+{\widetilde{\mathrm{Kn}}}_l=1$. The inset shows the same data using linear axes; the large data points correspond to binned average values.

Figure 3

(a) Dissipation vs $p$ for two cantilevers with different length scales but similar frequencies ($l_x\approx 800\mathrm{nm}$, $f_0\approx 894\mathrm{kHz}$; and $l_x\approx 32\mu \mathrm{m}$, $f_0\approx 924\mathrm{kHz}$) in ${\mathrm{N}}_2$. Transitions are determined by ${\mathrm{Kn}}_l\approx 1$ at $p_c\approx 56$ and 3.6 Torr, respectively. (b) Dissipation for a nanocantilever ($l_x\approx 1300\mathrm{nm}$, $f_0\approx 28.6\mathrm{MHz}$) and a macroscopic quartz crystal ($l_x\sim 5\mathrm{mm}$, $f_0\approx 32.7\mathrm{MHz}$ [4]); the transitions take place around 190 and 150 Torr, respectively. (c) Collapse plot for all the data in different gases. The thick solid line shows the scaling function $f$. The inset is a collapse of select cantilever data based on the viscous cylinder solution. Squares and diamonds correspond to microcantilevers ($l_x\approx 32\mu \mathrm{m}$) and nanocantilevers ($l_x\approx 800\mathrm{nm}$), respectively. Dashed line shows the imaginary part of the complex hydrodynamic function for a cylinder. The lower inset shows the parameters of the model.