Possible Gigantic Variations on the Width of Viscoelastic Fingers
We analyze the effect of frequency on the width of a single finger displacing a viscoelastic fluid. We derive a generalized Darcy's law in the frequency domain for a linear viscoelastic fluid flowing in a Hele Shaw cell. This leads to an analytic expression for the dynamic permeability that has maxima which are several orders of magnitude larger than the static permeability. We then follow an argument of de Gennes to obtain the smallest possible finger width when viscoelasticity is important. Using this, and a conservation law, we obtain a lowest bound for the width of a single finger displacing a viscoelastic fluid. Our results indicate that when a small amplitude signal of the frequency that maximizes the permeability is overimposed to a constant pressure drop, gigantic variations are obtained for the finger width.