Modeling gravity-driven flows on an inclined plane

We develop an exact analytic solution for unconfined flows having an assumed rheology advancing on an inclined plane, We consider the time-dependent flow movement to be driven by gravitational transport and hydrostatic pressure. We examine how these two forces drive flow movement in the downstream and cross-stream directions by adopting a volume conservation approach. Simplifying assumptions reduce the governing equation to the dimensionless form partial derivative/partial derivative x(alpha h(m)) = partial derivative/partial derivative y(alpha h(m) partial derivative h/partial derivative y), where x and y are the downstream and cross-stream directions, respectively; h is the flow depth; and alpha = alpha(x) and m are prescribed by the rheology of the fluid, We solve this equation analytically for flows of arbitrary m and alpha using a similarity transformation. This method involves transforming variables and reducing the governing equation to a nonlinear ordinary differential equation. Our solution determines how flow depth and width change with distance from the source of the now for different alpha and m based on known or assumed initial parameters. Consequently, from the traditional geometric dimensions of the deposits, these rheological parameters can be inferred. We have applied the model to basaltic lava flows and found m values typically between 1 and 2. This contrasts with Newtonian fluids, for which m=3. The model of alpha(x) corresponding to constant viscosity approximates the field data of pahoehoe toes (<5 meters in length), whereas models of alpha(x) corresponding to linearly increasing and exponentially increasing viscosities better approximate the remote sensing data of longer flows (several kilometers in length).