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

Periodically-driven flows are known to generate non-zero, 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 Maxwellian 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 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 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.