This article describes an unexplored transport phenomenon where a mildly viscoelastic medium encroaches a narrow capillary channel under the action of surface-tension force. The ultimate goal of the study is to provide the penetration length and the intrusion rate of the liquid as functions of time. The resulting analysis would be instrumental in building an inexpensive and convenient rheometric device which can measure the temporal scale for viscoelastic relaxation from the stored data of the aforementioned quantities. The key step in the formulation is a transient eigenfunction expansion of the instantaneous velocity profile. The time-dependent amplitude of the expansion as well as the intruded length are governed by a system of integro-differential relations which are derived by exploiting the mass and momentum conservation principles. The obtained integro-differential equations are simultaneously solved by using a fourth-order Runge-Kutta method assuming a start-up problem from rest. The resulting numerical solution properly represents the predominantly one-dimensional flow which gradually slows down after an initial acceleration and subsequent oscillation. The computational findings are independently verified by two separate perturbation theories. The first of these is based on a Weissenberg number expansion revealing the departure in the unsteady imbibition due to small but finite viscoelasticity. In contrast, the second one explains the long-time behaviour of the system by analytically predicting the decay features of the dynamics. These asymptotic results unequivocally corroborate the simulation inferring the accuracy of the numerics as well as the utility of the simplified mathematical models.
- capillary flows
- non-Newtonian flows