WAITING-TIME SOLUTIONS OF A NONLINEAR DIFFUSION EQUATION - EXPERIMENTAL-STUDY OF A CREEPING FLOW NEAR A WAITING FRONT

We investigate an unsteady plane viscous gravity current of silicone o il on a horizontal glass substrate. Within the lubrication approximati on with gravity as the dominant force, this current is described by th e nonlinear diffusion equation phi(t)=(phi(m) phi(x))(x) (phi is propo rtional to the liquid thickness h and m=3>0), which is of interest in many other physical processes. The solutions of this equation display a fine example of the competition between diffusive smoothening and no nlinear steepening. This work concerns the so-called waiting-time solu tions, whose distinctive character is the presence of an interface or front, separating regions with h not equal 0 and h=0, that remains mot ionless for a finite time interval t(w) meanwhile a redistribution of h takes place behind the interface. We start the experiments from an i nitial wedge-shape configuration [h(x)approximate to alpha'(x(0)-x)] w ith a small angle (alpha'less than or equal to 0.12 rad). In this situ ation, the tip of the wedge, situated at x(0) from the rear wall (15 c m less than or equal to x(0) less than or equal to 75 cm), waits at le ast several Seconds before moving. During this waiting stage, a region characterized by a strong variation of the free surface slope (corner layer) develops and propagates toward the front while it gradually na rrows and partial derivative(2)h/partial derivative x(2) peaks. The st age ends when the corner layer overtakes the front. At this point, the liquid begins to spread over the uncovered substrate. We measure the slope of the free surface in a range approximate to 10 cm around x(0), and, by integration, we determine the fluid thickness h(x) there. We find that the flow lends to a self-similar behavior when the corner la yer position tends to x(0); however, near the end of the waiting stage , it is perturbed by capillarity. Even if some significant effects are not included in the above equation, the main properties of its soluti ons are well displayed in the experiments