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L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 12 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Driven cavity flow: From molecular dynamics to continuum hydrodynamics, Multiscale Model Sim
, 2005
"... Molecular dynamics (MD) simulations have been carried out to investigate the slip of fluid in the lid driven cavity flow where the noslip 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 ..."
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Cited by 8 (3 self)
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Molecular dynamics (MD) simulations have been carried out to investigate the slip of fluid in the lid driven cavity flow where the noslip 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 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. Steadystate velocity fields from continuum calculations are in quantitative agreement with those from MD simulations, from the molecularscale 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 partialslip 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 sufficiently large system, an additional partialslip region appears where the slippage varies as To whom correspondence should be addressed.
Marangoni–Bénard Convection with a Deformable Free Surface
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1998
"... Computations of Marangoni convection are usually performed in two or threedimensional 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 finiteelement technique for ..."
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Cited by 6 (4 self)
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Computations of Marangoni convection are usually performed in two or threedimensional 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 finiteelement technique for calculating bifurcations that arise due to thermal gradients in a twodimensional 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 finiteelement 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.
Tapping of magmas from ubiquitous mantle heterogeneities: an alternative to mantle plumes
, 1984
"... Longlived heterogeneities in the mantle are indicated by NdSrPb isotopic systematics. Explanations for these variations have invoked either distinct layers in the earth that are preferentially tapped by some source regions (such as plumes rising from the base of the mantle) or smallscale hetero ..."
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Longlived heterogeneities in the mantle are indicated by NdSrPb isotopic systematics. Explanations for these variations have invoked either distinct layers in the earth that are preferentially tapped by some source regions (such as plumes rising from the base of the mantle) or smallscale heterogeneities which are ubiquitously distributed through the mantle and tapped differentially in various source regions. Consideration of the thermal and mechanical aspects of the latter hypothesis provides an explanation for the differences between midoceanic ridge basalts and offaxis volcanism. The melting behavior in a region with heterogeneities is modified by lateral conduction of heat, so that the first melted regions are tapped preferentially at offaxis volcanoes where small degrees of melting occur. A small (less than a few kilometers), easily melted heterogeneity draws heat from its surroundings as it melts during ascent. An increase of melting by a factor of 2 over adiabatic ascent is probable in the most easily melted regions if the melted and unmelted regions have comparable volume. The increase is larger in the earliest stages if melting is confined to small isolated heterogeneities. The depleted isotopic ratios of midoceanic ridge basalts can be explained as follows. Heterogeneities with enriched components melt before those with depleted components as material ascends beneath the ridge. Then the enriched melts are removed from the source region before midoceanic ridge basalts are tapped.
FLOW FOCUSING IN MICROFLUIDIC DEVICES
"... 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 broaden ..."
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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.
Numerical solutions of 2d steady incompressible flow over a backwardfacing step, part i: High reynodls number solutions
 Computers & Fluids
"... ..."
Local Fluid and Heat Flow Near Contact Lines
, 1993
"... We consider steady, twodimensional fluid flow and heat transfer near contact lines in singlephase and twophase systems. Both single and double wedge geometries admit separable solutions in plane polar coordinates for both thermal and flow fields. We consider the class of functions which have bound ..."
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We consider steady, twodimensional fluid flow and heat transfer near contact lines in singlephase and twophase systems. Both single and double wedge geometries admit separable solutions in plane polar coordinates for both thermal and flow fields. We consider the class of functions which have bounded temperatures and velocities at the corner. When free surfaces are present, we seek LOCAL SOLUTIONS, those which satisfy all local boundary conditions, and PARTIAL LOCAL SOLUTIONS, those which satisfy all but the normalstress boundary condition. Our aim in this work is to describe local fluid and heat flow in problems where these fields are coupled by determining for which wedge angles solutions exist, identifying singularities in the heat flux and stress which are present at contact lines, and determining the dependence of these singularities on the wedge angles. For thermal fields in two phases we identify two modes of heat transfer that are analogous to the two modes identified by Proud...
Observations of viscoelastic instabilities in recirculation flows of
, 1996
"... A purely elastic flow instability in the recirculation flow of a driven haltcavity or blocked channel is discovered and examined both experimentally and numerically. Video image analysis is used to determine the critical Weissenberg number and characteristic wavenumber of the instability while lase ..."
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A purely elastic flow instability in the recirculation flow of a driven haltcavity or blocked channel is discovered and examined both experimentally and numerically. Video image analysis is used to determine the critical Weissenberg number and characteristic wavenumber of the instability while laser sheet imaging provides information about the structure and location of the secondary flow. To understand the basic flow and ultimately the flow instabilities, we numerically solve the combined momentum conservation and constitutive quations describing the flow of elastic fluids in the block flow geometry. Local rates of polymer stretch are estimated with a weakly elastic expansion using the OldroydB constitutive quation for the polymer stress. Numerical solutions for the kinematics and polymer stretch are also obtained using the ChilcottRallison FENE model to describe the polymeric stress. These results are discussed in the context of known purely elastic instability mechanisms in order to better understand the stability of general recirculation flows.
Theoretical Investigation of Electroosmotic Flows and Chaotic Stirring in Rectangular Cavities
"... Two dimensional, timeindependent and timedependent electroosmotic flows driven by a uniform electric field in a closed rectangular cavity with uniform and nonuniform zeta potential distributions along the cavity’s walls are investigated theoretically. First, we derive an expression for the onedi ..."
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Two dimensional, timeindependent and timedependent electroosmotic flows driven by a uniform electric field in a closed rectangular cavity with uniform and nonuniform zeta potential distributions along the cavity’s walls are investigated theoretically. First, we derive an expression for the onedimensional velocity and pressure profiles for a flow in a slender cavity with uniform (albeit possibly different) zeta potentials at its top and bottom walls. Subsequently, using the method of superposition, we compute the flow in a finite length cavity whose upper and lower walls are subjected to nonuniform zeta potentials. Although the solutions are in the form of infinite series, with appropriate modifications, the series converge rapidly, allowing one to compute the flow fields accurately while maintaining only a few terms in the series. Finally, we demonstrate that by timewise periodic modulation of the zeta potential, one can induce chaotic advection in the cavity. Such chaotic flows can be used to stir and mix fluids. Since devices operating on this principle do not require any moving parts, they may be particularly