2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

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Abstract. Molecular dynamics (MD) simulations have been carried out to investigate the slip of fluid in the lid driven cavity flow where the no-slip boundary condition causes unphysical stress divergence. The MD results not only show the existence of fluid slip but also verify the validity of the Navier slip boundary condition. To better understand the fluid slip in this problem, a continuum hydrodynamic model has been formulated based upon the MD verification of the Navier boundary condition (NBC) and the Newtonian stress. Our model has no adjustable parameter because all the material parameters (density, viscosity, and slip length) are directly determined from MD simulations. Steady-state velocity fields from continuum calculations are in quantitative agreement with those from MD simulations, from the molecular-scale structure to the global flow. The main discovery is as follows. In the immediate vicinity of the corners where moving and fixed solid surfaces intersect, there is a core partial-slip region where the slippage is large at the moving solid surface and decays away from the intersection quickly. In particular, the structure of this core region is nearly independent of the system size. On the other hand, for a sufficiently large system, an additional partial-slip region appears where the slippage varies as 1/r, with r denoting the distance from the corner along the moving solid surface. The existence of this wide power-law region is in accordance with the asymptotic 1/r variation of stress and the NBC.

Computations of Marangoni convection are usually performed in two- or three-dimensional domains with rigid boundaries. In two dimensions, allowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure. We describe a finite-element technique for calculating bifurcations that arise due to thermal gradients in a two-dimensional domain with a deformable free surface. The fluid is assumed to be Newtonian, to conform to the Boussinesq approximation, and to have a surface tension that varies linearly with temperature. An orthogonal mapping from the physical domain to a reference domain is employed, which is determined as the solution to a pair of elliptic partial differential equations. The mapping equations and the equilibrium equations for the velocity, pressure, and temperature fields and their appropriate nonlinear boundary conditions are discretized using the finite-element method and solved simultaneously by Newton iteration. Contact angles other than 90 degrees are shown to disconnect the transcritical bifurcations to flows with an even number of cells in the expected manner. The loss of stability to single cell flows is associated with the breaking of a reflectional symmetry about the middle of the domain and therefore occurs at a pitchfork bifurcation point for contact angles both equal to, and less than, 90 degrees.

This paper presents numerical analysis of the hydrodynamic flow focusing in rectangular microchannels. The low Reynolds number pressure driven flow in symmetric system of crossed channels with three inlets and one outlet is investigated. The numerical model is used to elucidate the origin of broadening of the focused flow sheet observed experimentally close to the side walls of the outlet channel. It is found that the observed broadening is mainly due to the residual flow inertia and can be totally eliminated if flow Reynolds number is less than one. 1.

this paper, we present a spectral method for solving Dirichlet and Neumann gauge formulations. The test problem is formulated in Section 2. The spatial and temporal discretization schemes are described in Section 3. The numerical results are compared to benchmark results in Section 4. Finally, conclusions about the method, which we call the spectral gauge method, are made in Section 5

Abstract. Discontinuous velocity boundary data for the lid driven cavity flow has long been causing difficulties in both theoretical analysis and numerical simulations. In finite element methods, the variational form for the driven cavity flow is not valid since the velocity is not in H 1. Hence standard error estimates do not work. By using only W 1,r (1 <r<2) regularity and constructing a continuous approximation to the boundary data, here we present error estimates for both the velocity-pressure formulation and the pseudostressvelocity formulation of the two-dimensional Stokes driven cavity flow. 1.

The root structure of the equation sin(z) = cz is studied for |c| <= 1, and an iterative root finding method for the nonreal roots, based on an equation x = f(x) for the real part, is presented.

This paper presents numerical and experimental analysis of the hydrodynamic flow focusing in a rectangular microchannel. Aim of the study is to improve performance of the Particle Image Velocimetry (PIV) technique applied to micro-scale flow analysis. The symmetric flow focusing system of two channels crossed at right angle is investigated. The numerical model is used to analyse the effect of Reynolds number on the flow focusing mechanism. In the experiment, the flow focusing is applied to concentrate seeding tracers into a thin sheet at the channel axis. Such a modification removes the out of focus images of the seeding particles, effectively improving PIV evaluation of vector fields in microchannel. Based on the experimental and numerical results we have found that expected improvement is possible for the flow at Reynolds number less than 10 only. ∗) Key words: selective seeding, flow focusing. Copyright c ○ 2011 by IPPT PAN Notations c mass concentration, channel hydraulic diameter, dp particle diameter, D diffusion coefficient, dH

Poloidal-toroidal decomposition in a finite