# The Granular Blasius Problem

Last updated 18 November 2017.

The classical Blasius problem considers the formation of a boundary layer
through the change at *x* = 0 from a free-slip to a no-slip boundary
beneath an otherwise steady uniform flow. Discrete particle model (DPM)
simulations of granular gravity currents show that a similar phenomenon exists
for a steady flow over a uniformly sloped surface that is smooth upstream
(allowing slip) but rough downstream (imposing a no-slip condition). The
boundary layer is a region of high shear rate and therefore high inertial number
*I*; its dynamics are governed by the asymptotic behaviour of the
granular rheology as *I* → ∞. The
*μ*(*I*) rheology (Jop *et al.* 2006) asserts that
d*μ*/d*I* = *O*(1/*I*^{2}) as
*I* → ∞, but current experimental evidence is
insufficient to confirm this (Saingier *et al.* 2015). We show that
`generalised *μ*(*I*) rheologies', with different behaviours as
*I* → ∞, all permit the formation of a boundary
layer. We give approximate solutions for the velocity profile under each
rheology. The change in boundary condition considered here mimics more complex
topography in which shear stress increases in the streamwise direction (e.g. a
curved slope). Such a system would be of interest in avalanche modelling.

Here is my talk for APS DFD 2017.