689 research outputs found

    A note on Stokes' problem in dense granular media using the μ(I)\mu(I)--rheology

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    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 μ(I)\mu(I)--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 tt as νt\sqrt{\nu t}, where ν\nu 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 νgt\sqrt{\nu_g t} analogous to a Newtonian fluid where νg\nu_g is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular media such as grain diameter dd, density ρ\rho and friction coefficients but also on the applied pressure pwp_w at the moving wall and the solid fraction ϕ\phi (constant). In addition, the μ(I)\mu(I)--rheology indicates that this growth continues until reaching the steady-state boundary layer thickness δs=βw(pw/ϕρg)\delta_s = \beta_w (p_w/\phi \rho g ), independent of the grain size, at about a finite time proportional to βw2(pw/ρgd)3/2d/g\beta_w^2 (p_w/\rho g d)^{3/2} \sqrt{d/g}, where gg is the acceleration due to gravity and βw=(τwτs)/τs\beta_w = (\tau_w - \tau_s)/\tau_s is the relative surplus of the steady-state wall shear-stress τw\tau_w over the critical wall shear stress τs\tau_s (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

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    In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for dd-dimensional flows, d=2d=2 or 33, 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

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    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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