689 research outputs found
A note on Stokes' problem in dense granular media using the --rheology
The classical Stokes' problem describing the fluid motion due to a steadily
moving infinite wall is revisited in the context of dense granular flows of
mono-dispersed beads using the recently proposed --rheology. In
Newtonian fluids, molecular diffusion brings about a self-similar velocity
profile and the boundary layer in which the fluid motion takes place increases
indefinitely with time as , where is the kinematic
viscosity. For a dense granular visco-plastic liquid, it is shown that the
local shear stress, when properly rescaled, exhibits self-similar behaviour at
short-time scales and it then rapidly evolves towards a steady-state solution.
The resulting shear layer increases in thickness as analogous
to a Newtonian fluid where is an equivalent granular kinematic
viscosity depending not only on the intrinsic properties of the granular media
such as grain diameter , density and friction coefficients but also
on the applied pressure at the moving wall and the solid fraction
(constant). In addition, the --rheology indicates that this growth
continues until reaching the steady-state boundary layer thickness , independent of the grain size, at about a finite
time proportional to , where is
the acceleration due to gravity and is the
relative surplus of the steady-state wall shear-stress over the
critical wall shear stress (yield stress) that is needed to bring the
granular media into motion... (see article for a complete abstract).Comment: in press (Journal of Fluid Mechanics
On the splash singularity for the free-surface of a Navier-Stokes fluid
In fluid dynamics, an interface splash singularity occurs when a locally
smooth interface self-intersects in finite time. We prove that for
-dimensional flows, or , the free-surface of a viscous water wave,
modeled by the incompressible Navier-Stokes equations with moving
free-boundary, has a finite-time splash singularity. In particular, we prove
that given a sufficiently smooth initial boundary and divergence-free velocity
field, the interface will self-intersect in finite time.Comment: 21 pages, 5 figure
On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
In this article, we study the axisymmetric surface diffusion flow (ASD), a
fourth-order geometric evolution law. In particular, we prove that ASD
generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older
regular surfaces of revolution embedded in R^3 and satisfying periodic boundary
conditions. We also give conditions for global existence of solutions and prove
that solutions are real analytic in time and space. Further, we investigate the
geometric properties of solutions to ASD. Utilizing a connection to
axisymmetric surfaces with constant mean curvature, we characterize the
equilibria of ASD. Then, focusing on the family of cylinders, we establish
results regarding stability, instability and bifurcation behavior, with the
radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana
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