Waiting-Time Solutions of a Nonlinear Diffusion Equation - Experimental-Study of a Creeping Flow Near a Waiting Front

MARINO, BM; THOMAS, LP; GRATTON, R; DIEZ, JA; BETELU, S

PHYSICAL REVIEW E - STATISTICAL PHYSICS, PLASMAS, FLUIDS AND RELATED INTERDISCIPLINARY TOPICS

Americal Physical Society

Lugar: New York; Año: 1996 vol. 54 p. 2628 - 2636

1063-651X

We investigate an unsteady plane viscous gravity current of silicone oil on a horizontal glass substrate. Within the lubrication approximation with gravity as the dominant force, this current is described by the nonlinear diffusion equation φt=(φmφx)x (φ is proportional 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 nonlinear steepening. This work concerns the so-called waiting-time solutions, whose distinctive character is the presence of an interface or front, separating regions with h≠/0 and h=0, that remains motionless for a finite time interval tw meanwhile a redistribution of h takes place behind the interface. We start the experiments from an initial wedge-shape configuration [h(x) aprox α′(x0-x)] with a small angle (α′ < 0.12 rad). In this situation, the tip of the wedge, situated at x0 from the rear wall (15 cm<x0<75 cm), waits at least 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 narrows and ∂2h/∂x2 peaks. The stage 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 aprox10 cm around x0, and, by integration, we determine the fluid thickness h(x) there. We find that the flow tends to a self-similar behavior when the corner layer position tends to x0; 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 solutions are well displayed in the experiments