Time-averaged transport in oscillatory squeeze flow of a viscoelastic fluid

Periodically driven flows are known to generate nonzero, time-averaged fluxes of heat or solute species, due to the interactions of out-of-phase velocity and temperature/concentration fields, respectively. Herein, we investigate such transport (a form of the well-known Taylor-Aris dispersion) in the gap between two parallel plates, one of which oscillates vertically, generating a time-periodic squeeze flow of either a Newtonian or a Maxwell fluid. Using the method of multiple time-scale homogenization, the mass/heat balance equation describing transport in this flow is reduced to a one-dimensional advection-diffusion-reaction equation. This result indicates three effective mechanisms in the mass/heat transfer in the system: an effective diffusion that spreads mass/heat along the concentration/temperature gradient, an effective advective flux, and an effective reaction that releases or absorbs mass/heat—in the time-averaged frame. Our results demonstrate that there exist resonant modes under which the velocity peaks when the dimensionless plate oscillation frequency (embodied by the Womersley number, the ratio of the transient inertia to viscous forces) approaches specific values. As a result, transport in this flow is significantly influenced by the dimensionless frequency. On the one hand, the effective, time-averaged dispersion coefficient is always larger than the molecular diffusivity and is sharply enhanced near resonance. The interaction between the fluid elasticity and the oscillatory forcing enhances the efficiency of transport in the system. On the other hand, the identified effective advection and reaction mechanisms may transport mass/heat from regions of high concentration/temperature to those of low concentration/temperature, or vice versa, depending on the value of the dimensionless frequency. Ultimately, it is shown that the oscillatory squeeze flow can either enhance or diminish transport, depending on the interplay of these three effective (homogenized) mechanisms.

Figure 1

Schematic of the oscillatory squeeze flow between two disks.

Figure 2

Dimensionless streamwise velocity $u$ variation across the gap height $z$ at $r=1$. Different values of $\text{De}$ and $\alpha$ highlight the effects of viscoelasticity and oscillatory forcing.

Figure 3

The dependence of the spatiotemporal average of the $r$-direction velocity, $u_{\mathrm{a}}$ from Eq. (28), and the absolute value of the denominator of $u, \left|\mathrm{\Theta }\right|$ from Eq. (30), on $\alpha$ at $r=1$.

Figure 4

The dependence of ${\lambda }_{\mathrm{D}}, {\lambda }_{\mathrm{U}}$ (or ${\lambda }_{\mathrm{S}}/2$), and $\mathrm{\Delta }r$ on $\alpha$ for different values of $\text{De}$, at $r=1$ with $\sigma =0.6$. $\mathrm{\Delta }r$ is obtained with $\delta =\epsilon$. The value of $\text{Pe}$ is determined by $\text{Pe}={\alpha }^2\sigma \delta /\epsilon$. Note that ${\lambda }_{\mathrm{U}}={\lambda }_{\mathrm{S}}/2$ for $r=1$.

Figure 5

The dependence of $D_{\mathrm{eff}}$ and $U_{\mathrm{eff}}$ (or $S_{\mathrm{eff}}/2$) on $\text{Pe}$, at $r=1$ when $\text{De}=30$ and $\sigma =3$. Two values of $\alpha$ are shown: $\alpha =1$ (no resonance) and ${\alpha }_1^{*}=1.64$ (first resonant mode). The value of ${\alpha }_1^{*}$ was obtained numerically. Note that $U_{\mathrm{eff}}=S_{\mathrm{eff}}/2$ for $r=1$.

Figure 6

The dependence of ${\lambda }_{\mathrm{D}}, {\lambda }_{\mathrm{U}}$ (or ${\lambda }_{\mathrm{S}}/2$), and $\mathrm{\Delta }r$ on $\alpha$ with different values of $\sigma$, at $r=1$ when $\text{De}=10$. The displacement of the upper plate satisfies $\delta =\epsilon$. The value of $\text{Pe}$ is determined by $\text{Pe}={\alpha }^2\sigma \delta /\epsilon$. Note that ${\lambda }_{\mathrm{U}}={\lambda }_{\mathrm{S}}/2$ for $r=1$.

Figure 7

An example of an OSF configuration for enhancing mass or heat transfer.

Figure 8

The dependence of ${\dot{m}}^{*}, \eta$, and ${\tilde{W}}_T$ on $\alpha$ at steady state for a Maxwell fluid with $\text{De}=30$ and $\sigma =0.6$. The amplitude of the oscillation of the upper plate is $\delta =10\epsilon$. The value of $\text{Pe}$ is determined by $\text{Pe}={\alpha }^2\sigma \delta /\epsilon$.