stochastic analysis of transport in tubes with rough walls flow and transport in tubes with rough surfaces play an important role in a variety of applications often the topology of such surfaces cannot be accurately described in all of its relevant details due to either insufficient data or measurement errors or both in such cases this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we use a computational framework which is based on stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a generalized polynomial chaos and monte carlo simulations to solve the transformed stochastic problem we use our approach to describe transport of a passive scalar in stokes flow and to quantify the corresponding predictive uncertainty

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

homogenization of the boundary value for the neumann problem in this paper we study the convergence rates for homogenization problems for solutions of partial differential equations with rapidly oscillating neumann boundary data such a problem raised due to its importance for higher order approximation in homogenization theory high order approximation gives rise to the so called boundary layer phenomenon as a consequence we obtain the pointwise and w1 p convergence results our techniques are based on fourier analysis

asymptotic analysis of the laminar viscous flow over a porous bed we consider the laminar viscous channel flow over a porous surface the size of the pores is much smaller than the size of the channel and it is important to determine the effective boundary conditions at the porous surface we study the corresponding boundary layers and by a rigorous asymptotic expansion we obtain saffman s modification of the interface condition observed by beavers and joseph the effective coefficient in the law is determined through an auxiliary boundary layer type problem whose computational and modeling aspects are discussed in detail furthermore the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size ep finally we give the interface condition linking the effective pressure fields in the porous medium and in the channel and we determine the jump of the effective pressures explicitly

vanishing porosity limit of the coupled stokes brinkman system abstract we investigate the small porosity asymptotic behavior of the coupled stokes brinkman system in the presence of a curved interface between the stokes region and the brinkman region in particular we derive a set of approximate solutions validated via rigorous analysis to the coupled stokes brinkman system of particular interest is that the approximate solution satisfies a generalized beavers joseph saffman jones interface condition 1 9 with the constant of proportionality independent of the curvature of the interface

on the asymptotic limit of the navier stokes system on domains with rough boundaries abstract we study the asymptotic behavior of solutions to the incompressible navier stokes system considered on a sequence of spatial domains whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non degenerate meaning not oriented in a single direction

geometric regularity criteria for incompressible navier stokes equations with navier boundary conditions abstract we study the regularity criteria for weak solutions to the 3d incompressible navier stokes equations in terms of the direction of vorticity taking into account the boundary conditions a boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions provided that the directions of vorticity are coherently aligned up to the boundary as an application we establish the boundary regularity for weak solutions to navier stokes equations in round balls half spaces and right circular cylindrical ducts subject to the classical navier and kinematic boundary conditions

a one domain approach for modeling and simulation of free fluid over a porous medium we propose a one domain approach based on the brinkman model for the modeling and simulation of the transport phenomenon between free fluid and a porous medium a thin transition layer is introduced between the free fluid region and the porous media region across which the porosity and permeability undergo a rapid but continuous change we study the behavior of the solution to the one domain model analytically and numerically using the method of matched asymptotic expansion we recover the beavers joseph saffman bjs interface condition as the thickness of the transition layer goes to zero we also calculate the error estimates between the leading order solution of the one domain model and the standard darcy stokes model of two domain model with bjs condition numerical methods are developed for both the one domain model and the two domain model numerical results are presented to support the analytical results thereby justifying the one domain model as a good approximation to the two domain stokes darcy model

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

derivation of the wenzel and cassie equations from a phase field model for two phase flow on rough surface in this paper the equilibrium behavior of an immiscible two phase fluid on a rough surface is studied from a phase field equation derived from minimizing the total free energy of the system when the size of the roughness becomes small we derive the effective boundary condition for the equation by the multiple scale expansion homogenization technique the wenzel and cassie equations for the apparent contact angles on the rough surfaces are then derived from the effective boundary condition the homogenization results are proved rigorously by the convergence theory

the navier slip thin film equation for 3d fluid films existence and uniqueness abstract we consider the thin film equation t h h 2 h 0 in physical space dimensions i e one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension which corresponds to the lubrication approximation of the navier stokes equations of a three dimensional viscous thin fluid film with navier slip at the substrate this equation can have a free boundary the contact line moving with finite speed at which we assume a zero contact angle condition complete wetting regime previous results have focused on the 1 1 dimensional version where it has been found that solutions are not smooth as a function of the distance to the free boundary in particular a well posedness and regularity theory is more intricate than for the second order counterpart the porous medium equation or the thin film equation with linear mobility corresponding to darcy dynamics in the hele shaw cell here we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling wave profile this leads to control on the free boundary and in particular its velocity

free vibrations of completely free doubly curved laminated composite shallow shells abstract vibrational frequencies are determined for completely free laminated plates and shallow shells having rectangular planforms the ritz method with algebraic polynomial displacement functions is used convergence studies are made and accurate results are obtained by using 192 displacement terms for spherical circular cylindrical hyperbolic paraboloidal shallow shells and 64 terms for plates results are compared with those obtained experimentally and by finite element methods it is found that the present method needs considerably fewer degrees of freedom than the finite element method to obtain the same accuracy and that the results compare well with experimental ones the effect of various parameters material number of layers fiber orientation upon the frequencies is studied

deciding equivalences among conjunctive aggregate queries equivalence of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators count count distinct min max and sum essentially this class contains unnested sql queries with the above aggregate operators with a where clause consisting of a conjunction of comparisons and without a having clause the comparisons are either interpreted over a domain with a dense order like the rationals or with a discrete order like the integers characterizations of equivalence differ for the two cases for queries with either max or min equivalence is characterized in terms of dominance mappings which can be viewed as a generalization of containment mappings for queries with the count distinct operator a sufficient condition for equivalence is given in terms of equivalence of conjunctive queries under set semantics for some special cases it is shown that this condition is also necessary for conjunctive queries with comparisons but without aggregation equivalence under bag set semantics is characterized in terms of isomorphism this characterization essentially remains the same also for queries with the count operator moreover this characterization also applies to queries with the sum operator if the queries have either constants or comparisons but not both in the general case i e both comparisons and constants the characterization of the equivalence of queries with the sum operator is more elaborate all the characterizations given in the paper are decidable in polynomial space

average of the navier s law on the rapidly oscillating boundary we study the flow of newtonian fluid in a domain with periodically wrinkled boundary with slip navier s boundary condition the goal of this paper is to replace a microscopic boundary condition posed on the rough boundary by some macroscopic boundary condition posed on the middle surface of the oscillating boundary depending on the shape of wrinkles and the friction coefficient we get four different effective models

mhd flow in a rectangular duct with a perturbed boundary abstract the unsteady magnetohydrodynamic mhd flow of a viscous incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary is investigated a small boundary perturbation e is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries the mhd equations which are coupled in the velocity and the induced magnetic field are solved with no slip velocity conditions and by taking the side walls as insulated and the hartmann walls as perfectly conducting both the domain boundary element method dbem and the dual reciprocity boundary element method drbem are used in spatial discretization with a backward finite difference scheme for the time integration these mhd equations are decoupled first into two transient convection diffusion equations and then into two modified helmholtz equations by using suitable transformations then the dbem or drbem is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady state convection diffusion or modified helmholtz equations the dbem and drbem results are presented and compared by equi velocity and current lines at steady state for several values of hartmann number and the boundary perturbation parameter

gevrey regularity for navier stokes equations under lions boundary conditions abstract the navier stokes system is considered in a compact riemannian manifold gevrey class regularity is proven under lions boundary conditions in 2d for the rectangle cylinder and hemisphere and in 3d for the rectangle the cases of the 2d sphere and 2d and 3d torus are also revisited

effective behavior of a free fluid in contact with a flow in a curved porous medium the appropriate boundary condition between an unconfined incompressible viscous fluid and a porous medium is given by the law of beavers and joseph the latter has been justified both experimentally and mathematically using the method of periodic homogenization however all results so far deal only with the case of a planar boundary in this work we consider the case of a curved macroscopically periodic boundary by using a coordinate transformation we obtain a description of the flow in a domain with a planar boundary for which we derive the effective behavior the effective velocity is continuous in normal direction tangential to the interface a slip occurs additionally a pressure jump occurs the magnitude of the slip velocity as well as the jump in pressure can be determined with the help of a generalized boundary layer function the results indicate the validity of a generalized beavers and joseph type law where the geometry of the interface has an influence on the slip and jump constants

a multiscale finite element method for partial differential equations posed in domains with rough boundaries we propose and analyze a finite element scheme of multiscale type to deal with elliptic partial differential equations posed in domains with rough boundaries there is no need to assume that the boundary is periodic in any sense so the method is quite general on the other hand if the boundary is periodic we prove convergence of the scheme

flows of incompressible viscous liquids with anisotropic wall slip abstract this paper deals with a boundary value problem for the stokes equations with a general direction dependent navier type slip boundary condition this problem models the steady laminar flow of an incompressible linearly viscous liquid in a bounded domain with an impermeable rough boundary with variable and anisotropic roughness it is proved that the problem has a unique weak solution

droplet spreading intermediate scaling law by pde methods we consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity the standard lubrication approximation leads to an evolution equation for the film height h that is ill posed when the spreading is limited by the no slip boundary condition at the liquid solid interface due to a singularity at the moving contact line the most common relaxation of the no slip boundary condition removes this singularity but introduces a new physical length scale the slippage length b it is believed that this microscopic length scale only enters logarithmically in the effective that is macroscopic spreading behavior in this paper we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b more precisely we prove a scaling law for the diameter of the apparent that is macroscopic support of the droplet in time this is an intermediate scaling law it takes an initial layer to forget the initial droplet shape whereas after a long time the droplet is so thin that its spreading is governed by the physics on the scale b our proof works by deriving suitable estimates for physically relevant integral quantities the free energy the length of the apparent support and their respective rates of change as opposed to matched asymptotic methods this pde approach closely mimics a simple heuristic argument based on the gradient flow structure 2002 john wiley sons inc

a combined finite element method for elliptic problems posted in domains with rough boundaries abstract a combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes while good quality meshes often over refine unnecessarily the interior of the domain the basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size h h for other portions of the domain to reduce some unnecessary computational effort the transmission conditions across the fine coarse mesh interface are treated by the penalty technique the key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form which avoids the affection of the ratio h h in the error estimate we prove a quasi optimal convergence in terms of elements since there is no whole h 2 regularity in the domain with rough boundaries numerical results are provided for elliptic equations in domains with non oscillating or oscillating boundaries to illustrate the theoretical results

non linear vibrations of laminated cylindrical shallow shells under thermomechanical loading abstract the geometrically non linear vibrations of linear elastic composite laminated shallow shells under the simultaneous action of thermal fields and mechanical excitations are analysed for this purpose a model based on a very efficient p version first order shear deformation finite element with hierarchical basis functions is employed the equations of motion are solved in the time domain by a newmark implicit time integration method the model and code developed are partially validated by comparison with published data parametric studies are carried out in order to study the influence of temperature change initial curvature panel thickness and fibre orientation on the shells dynamics

vanishing viscosity limit for the 3d magnetohydrodynamic system with a slip boundary condition abstract this work investigates the solvability regularity and vanishing viscosity limit of the 3d viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition

decidable containment of recursive queries one of the most important reasoning tasks on queries is checking containment i e verifying whether one query yields necessarily a subset of the result of another one query containment is crucial in several contexts such as query optimization query reformulation knowledge base verification information integration integrity checking and cooperative answering containment is undecidable in general for datalog the fundamental language for expressing recursive queries on the other hand it is known that containment between monadic datalog queries and between datalog queries and unions of conjunctive queries are decidable it is also known that containment between unions of conjunctive two way regular path queries which are queries used in the context of semistructured data models containing a limited form of recursion in the form of transitive closure is decidable in this paper we combine the automata theoretic techniques at the base of these two decidability results to show that containment of datalog in union of conjunctive two way regular path queries is decidable in 2exptime by sharpening a known lower bound result for containment of datalog in union of conjunctive queries we show also a matching lower bound

on conjunctive queries containing inequalities conjunctive queries are generalized so that inequality comparisons can be made between elements of the query algorithms for containment and equivalence of such inequality queries are given under the assumption that the data domains are dense and totally ordered in general containment does not imply the existence of homomorphisms containment mappings but the homomorphism property does exist for subclasses of inequality queries a minimization algorithm is defined using the equivalence algorithm it is first shown that the constants appearing in a query can be divided into essential and nonessential subgroups the minimum query can be nondeterministically guessed using only the essential constants of the original query

thermal effects on geometrically nonlinear vibrations of rectangular plates with fixed edges nonlinear forced vibrations and postbuckling of isotropic rectangular plates subjected to thermal variations are studied geometric imperfections are taken into account since they play a fundamental role the plate is modelled by using the von karman hypothesis and the equations of motion are obtained by using an energy approach plates with fixed edges and additional rotational elastic constraint are considered a pseudo arclength continuation method is used in order to obtain numerical results laboratory experiments have been performed on two plates of different thickness

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

homogenization of parabolic equation in an evolving domain with an oscillating boundary abstract in this article we study the existence uniqueness and homogenization of the heat equation in a domain with highly oscillating and evolving time dependent boundary and an inhomogeneous time dependent data on the oscillating part of boundary motivated by fluid structure interaction problems we rewrite the parabolic equation in a reference configuration which transforms the boundary oscillations into rapidly oscillating coefficients finally we obtain the effective coefficients and corrector results in the reference configuration

couette flows over a rough boundary and drag reduction we consider the couette flow between two plates the lower plate is fixed and has periodically placed riblets of the characteristic size on it in the limit 0 we find the effective couette navier flow as an o 2 approximation for the effective mass flow and an o 2 l 1 approximation for the velocity in the effective solution the effect of roughness enters through the navier slip condition with the matrix coefficient in front of the effective shear stress calculated using a boundary layer problem furthermore an o 2 approximation for the tangential drag force is found in all estimates explicit dependence on the kinematic viscosity the velocity of the upper plate and the distance between the plates l 3 is kept also the uniqueness of the solution is expressed through a non linear algebraic condition linking and l 3 then the result is applied to the viscous sub layers around immersed bodies strictly containing the surface riblets it is found that for the riblets of the characteristic size being of the order smaller or equal to the approximation obtained for the tangential drag could be applied we compare and for realistic data and our results lead to the conclusion that the riblets reduce significantly tangential drag which may explain their presence on the skin of nektons

numerical approximation of laminar flows over rough walls with sharp asperities abstract we consider a viscous incompressible fluid filling an infinite horizontal domain bounded at the bottom by a plane wall and at the top by a rough wall the latter is assumed to consist of a plane wall covered with periodically distributed asperities whose size depends on a small parameter e 0 the assumption of sharp asperities is made that is the height of the asperities does not vanish as e 0 the rough wall is at rest and the plane wall is moving at a constant horizontal velocity we assume that the flow is governed by the stationary stokes equations we give an asymptotic approximation of the stokes flow under the rugose region by a couette flow depending on the size e of the asperities we then build a numerical approximation of the couette flow based on a partial domain decomposition method

boundary behavior of viscous fluids influence of wall roughness and friction driven boundary conditions we consider a family of solutions to the evolutionary navier stokes system supplemented with the complete slip boundary conditions on domains with rough boundaries we give a complete description of the asymptotic limit by means of convergence arguments and identify a general class of boundary conditions

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

viscous boundary layers for the navier stokes equations with the navier slip conditions we tackle the issue of the inviscid limit of the incompressible navier stokes equations when the navier slip with friction conditions are prescribed on impermeable boundaries we justify an asymptotic expansion which involves a weak amplitude boundary layer with the same thickness as in prandtl s theory and a linear behavior this analysis holds for general regular domains in both dimensions two and three

effective boundary condition for stokes flow over a very rough surface abstract the main purpose of this paper is to derive a wall law for a flow over a very rough surface we consider a viscous incompressible fluid filling a 3 dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a very rough wall the latter consists in a plane wall covered with periodically distributed asperities which size depends on a small parameter e 0 and with a fixed height we assume that the flow is governed by the stationary stokes equations using asymptotic expansions and boundary layer correctors we construct and analyze an asymptotic approximation of order o e 3 2 0 being arbitrary small in the h 1 norm for the velocity and in the l 2 norm for the pressure we derive an effective boundary condition of navier type then expressing the boundary layer terms in terms of the homogenized solution and the solution of a cell problem we obtain an effective approximation in the whole domain of the flow

first order system least squares for coupled stokes darcy flow the coupled problem with stokes flow in one subdomain and a darcy flow model in a second subdomain is studied in this paper both flow problems are treated as first order systems involving pseudostress and velocity in the stokes case and using a flux pressure formulation in the darcy subdomain as process variables respectively the beavers joseph saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems a combination of h mathrm div conforming raviart thomas and standard h 1 conforming elements is used for the stokes as well as for the darcy subsystem the homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates it also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies

eigenfunctions of the koch snowflake domain we obtain non tangential boundary estimates for the dirichlet eigenfunctions n and their gradients n for a class of planar domains with fractal boundaries this class includes the quasidiscs and in particular snowflake type domains when applied to the case when is the koch snowflake domain one of our main results states that 1 tends to or 0 as approaches certain types of boundary points where 1 0 denotes the ground state eigenfunction of the dirichlet laplacian on more precisely let ob resp ac denote the set of boundary points which are vertices of obtuse resp acute angles in an inner polygonal approximation of the snowflake curve then given eob resp e ac we show that 1 resp 0 as tends to in within a cone based at moreover we show that blowup of 1 also occurs at all boundary points in a cantor type set these results have physical relevance to the damping of waves by fractal coastlines as pointed out by sapovalet al in their experiments on the koch drum

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

a shape hessian based boundary roughness analysis of navier stokes flow the influence of boundary roughness characteristics on the rate of dissipation in a viscous fluid is analyzed using shape calculus from the theory of optimal control of systems governed by partial differential equations to study the mapping mathscr d from surface roughness topography to the dissipation rate of a navier stokes flow expressions for the shape gradient and hessian are determined using the velocity method in the case of couette and poiseuille flows a flat boundary is a local minimum of the dissipation rate functional thus for small roughness heights the behavior of mathscr d is governed by the flat wall shape hessian operator whose eigenfunctions are shown to be the fourier modes for stokes flow the shape hessian is determined analytically and its eigenvalues are shown to grow linearly with the wavenumber of the shape perturbation for navier stokes flow the shape hessian is computed numerically and the ratio of its eigenvalues to those of a stokes flow depend only on the re

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

generation of optimal periodic oscillations for the control of boundary layers this paper is devoted to the optimal design problem of periodic surfaces solutions to elliptic partial differential equations occurring in oscillating domains exhibit boundary layer behavior and we intend to control the first order correctors using a mathematical framework derived from the homogenization techniques the existence of an optimal boundary layer control is proved

on the vanishing viscosity limit of 3d navier stokes equations under slip boundary conditions in general domains we consider the vanishing viscosity limit for the navier stokes equations with certain slip without friction boundary conditions in a bounded domain with non flat boundary in particular we are able to show convergence in strong norms for a solution starting with initial data belonging to the special subclass of data with vanishing vorticity on the boundary the proof is obtained by smoothing the initial data and by a perturbation argument with quite precise estimates for the equations of the vorticity and for that of the curl of the vorticity

effective boundary conditions for laminar flows over periodic rough boundaries effective boundary conditions or wall laws are proposed for a laminar flow over a rough wall with periodic roughness elements these effective conditions are posed on a regularized boundary which allows the details of the wall to be avoided and dramatically reduces the computational cost the effective conditions stem from an asymptotic expansion of the solution which is presented here both first and second order conditions are discussed and tested numerically on bidimensional cases

vanishing viscosity limit for incompressible navier stokes equations with navier boundary conditions for small slip length in this paper we will analyze the vanishing viscosity limit of the incompressible navier stokes equations with the navier slip boundary conditions in a bounded domain of r2 when the slip length is smaller than or equal to the order of viscosity by using an energy method and developing kato s approach given in the work of kato math sci res inst publ 2 85 98 1984 we obtain several conditions to guarantee that the solution of the navier stokes equations with the navier slip boundary conditions goes to the solution of the associated problem of the euler equations in the energy space l2 uniformly in time as the viscosity goes to zero

transparent boundary conditions for the helmholtz equation in some ramified domains with a fractal boundary the paper addresses a class of boundary value problems in some self similar ramified domains with the laplace or helmholtz equations much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains a self similar finite element method is proposed and tested it can be used for numerically computing the spectrum of the laplace operator with neumann boundary conditions as well as the eigenmodes the eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out finally a numerical method for the wave equation is addressed

artificial conditions for the linear elasticity equations in this paper we consider the equations of linear elasticity in an exterior domain we exhibit artificial boundary conditions on a circle which lead to a non coercive second order boundary value problem in the particular case of an axisymmetric geometry explicit computations can be performed in fourier series proving the well posedness except for a countable set of parameters a perturbation argument allows to consider near circular domains we complete the analysis by some numerical simulations

vibrations of strongly irregular or fractal resonators it is shown on a specific example that fractal boundary conditions drastically alter the properties of wave excitations in space the low frequency part of the vibration spectrum of a finite range fractal drum is computed using an analogy between the helmoltz equation and the diffusion equation the irregularity of the frontier is found to influence strongly the density of states at low frequency the fractal perimeter generates a specific screening effect very near the frontier the decrease of the wave form is related directly to the behavior of the harmonic measure the possibility of localization of the vibrations is qualitatively discussed and we show that localized modes may exist at low frequencies if the geometrical structures possess narrow paths possible application of these results to the interpretation of thermal properties of binary glasses is briefly discussed

upper bounds on waiting times for the thin film equation the case of weak slippage we derive upper bounds on the waiting time of solutions to the thin film equation in the regime of weak slippage n in 2 frac 32 11 in particular we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary for n in 2 frac 32 11 our estimates are sharp for n 2 they are sharp up to a logarithmic correction term note that the case n 2 corresponds with a grain of salt to the assumption of the navier slip condition at the fluid solid interface we also obtain results in the regime of strong slippage n in 1 2 however in this regime we expect them not to be optimal our method is based on weighted backward entropy estimates hardy s inequality and singular weight functions we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long

relevance of the slip condition for fluid flows near an irregular boundary we consider the navier stokes equation in a domain with a rough boundary the roughness is modeled by a small amplitude and small wavelength oscillation with typical scale varepsilon ll 1 for periodic oscillation it is well known that the best homogenized that is regular in varepsilon boundary condition is of navier type such result still holds for random stationary irregularities as shown recently by the first author 5 15 we study here arbitrary irregularity patterns

finite element modal formulation for hypersonic panel flutter analysis with thermal effects a finite element time domain modal formulation for analyzing nonlinear flutter of panels subjected to hypersonic airflow has been developed von karman large deflection plate theory is used for description of the structural nonlinearity and third order piston theory is employed to consider the aerodynamic nonlinearity the thermal loadings of uniformly distributed surface temperatures and temperature gradients through the panel thickness are considered by the application of the modal truncation technique the number of governing equations of motion is reduced dramatically so that the computational costs are reduced significantly all possible types of panel behavior including flat and stable buckled but dynamically stable limit cycle oscillation lco periodic motion and chaotic motion were observed examples of the applications of the proposed methodology were flutter responses of isotropic and composite panels special emphasis was placed on the boundary between lco and chaos and on the route to chaos time history phase plane plot poincare map bifurcation diagram and lyapunov exponent are employed in the chaos study it is found that at low or moderately high dynamic pressures the fluttering panel typically takes a period doubling route to evolve into chaos whereas at high dynamic pressures the route generally involves bursts of chaos and rejuvenations of periodic motions

roughness induced effects on the quasi geostrophic model we study in this paper the effect of small scale irregularities on the quasi geostrophic model this study is motivated by some problems related to oceanography as the gulf stream separation or the impact of the topography on the global circulation we first consider the role of coastal roughness in the phenomenon of western intensification of boundary currents we show that the roughness is responsible for a nonlinear dynamics of the boundary layers governed by a quasilinear elliptic equation we thus extend substantially the classical derivation of munk layers 15 and the results of convergence obtained in 10 we then discuss the effect of a rough topography by generalizing and justifying some formal computations of 17 in particular we derive rigorously a simplified model of oceanic circulation with a nonlinear and nonlocal dissipative term due to the roughness

a posteriori error estimate for the h d i v conforming mixed finite element for the coupled darcy stokes system an h d i v conforming mixed finite element method has been proposed for the coupled darcy stokes flow in kanschat and riviere 2010 12 which imposes normal continuity on the velocity field strongly across the darcy stokes interface here we develop an a posteriori error estimator for this h d i v conforming mixed method and prove its global reliability and efficiency due to the strong coupling on the interface special techniques need to be employed in the proof this is the main difference between this paper and babuska and gatica s 2010 work 38 in which they analyzed an a posteriori error estimator for the mixed formulation using weakly coupled interface conditions

effective boundary condition at a rough surface starting from a slip condition abstract we consider the homogenization of the navier stokes equation set in a channel with a rough boundary of small amplitude and wavelength it was shown recently that for any non degenerate roughness pattern and for any reasonable condition imposed at the rough boundary the homogenized boundary condition in the limit e 0 is always no slip we give in this paper error estimates for this homogenized no slip condition and provide a more accurate effective boundary condition of navier type our result extends those obtained in basson and gerard varet 2008 6 and gerard varet and masmoudi 2010 13 in which the special case of a dirichlet condition at the rough boundary was examined

a lagrange multiplier method for a stokes biot fluid poroelastic structure interaction model we study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium the free fluid is governed by the stokes equations while the flow in the poroelastic medium is modeled using the biot poroelasticity system equilibrium and kinematic conditions are imposed on the interface a mixed darcy formulation is employed resulting in continuity of flux condition of essential type a lagrange multiplier method is employed to impose weakly this condition a stability and error analysis is performed for the semi discrete continuous in time and the fully discrete formulations a series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters

navier stokes equations in 3d thin domains with navier friction boundary condition abstract in this article we study the 3d navier stokes equations with navier friction boundary condition in thin domains we prove the global existence of strong solutions to the 3d navier stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small we generalize the techniques developed to study the 3d navier stokes equations in thin domains see g raugel g sell navier stokes equations on thin 3d domains i global attractors and global regularity of solutions j amer math soc 6 1993 503 568 g raugel g sell navier stokes equations on thin 3d domains ii global regularity of spatially periodic conditions in nonlinear partial differential equations and their application college de france seminar vol xi longman harlow 1994 pp 205 247 r temam m ziane navier stokes equations in three dimensional thin domains with various boundary conditions adv differential equations 1 1996 499 546 r temam m ziane navier stokes equations in thin spherical shells in optimization methods in partial differential equations in contemp math vol 209 amer math soc providence ri 1996 pp 281 314 to the navier friction boundary condition by introducing a new average operator m e in the thin direction according to the spectral decomposition of the stokes operator a e our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the stokes operator a e with navier friction boundary condition

constructing wall laws with domain decomposition or asymptotic expansion techniques abstract this paper is concerned with the numerical simulation of flows of viscous fluids over rough surfaces the proposed methodology introduces consistant equivalent wall laws using domain decomposition techniques with nonmatching grids and local periodicity assumptions the paper describes this approach with its theoretical background studies its limitations and its potential applicability and compares its results to the boundary conditions that are obtained by alternative asymptotic strategies

analysis and improvement of brinkman lattice boltzmann schemes bulk boundary interface similarity and distinctness with finite elements in heterogeneous porous media this work focuses on the numerical solution of the stokes brinkman equation for a voxel type porous media grid resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude it is first analytically demonstrated that the lattice boltzmann method lbm and the linear finite element method fem both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity this numerical artefact may lead to an apparent negative viscosity in low permeable blocks inducing spurious velocity oscillations the two relaxation times trt lbm may control this effect thanks to free tunable two rates combination moreover the brinkman force based bf trt schemes may maintain the nondimensional darcy group and produce viscosity independent permeability provided that the spatial distribution ofis fixed independently of the kinematic viscosity such a property is lost not only in the bf bgk scheme but also by partial bounce back trt gray models as shown in this work further we propose a consistent and improved ibf trt model which vanishes viscosity correction via simple specific adjusting of the viscous mode relaxation rate to local permeability value this prevents the model from velocity fluctuations and in parallel improves for effective permeability measurements from porous channel to multidimensions the framework of our exact analysis employs a symbolic approach developed for both lbm and fem in single and stratified unconfined and bounded channels it shows that even with similar bulk discretization bf ibf and fem may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no slip conditions while fem enforces them on the grid vertexes the lbm prescribes them implicitly we derive effective lbm continuity conditions and show that the heterogeneous viscosity correction impacts them a property also shared by fem for shear stress but in contrast with fem effective velocity conditions in lbm give rise to slip velocity jumps which depend on i neighbor permeability values ii resolution and iii control parameter ranging its reliable values from poiseuille bounce back solution in open flow to zero in darcy s limit we suggest an upscaling algorithm for from multilayers to multidimensions in random extremely dispersive samples finally on the positive side for lbm besides its overall versatility the implicit boundary layers allow for smooth accommodation of the flat discontinuous darcy profiles quite deficient in fem

decoupled schemes for a non stationary mixed stokes darcy model we study numerical methods for solving a non stationary mixed stokes darcy problem that models coupled fluid flow and porous media flow a decoupling approach based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model error estimates are derived and numerical experiments are conducted to demonstrate the computational effectiveness of the decoupling approach

existence of a weak solution to a fluid elastic structure interaction problem with the navier slip boundary condition abstract we study a nonlinear moving boundary fluid structure interaction fsi problem between an incompressible viscous newtonian fluid modeled by the 2d navier stokes equations and an elastic structure modeled by the shell or plate equations the fluid and structure are coupled via the navier slip boundary condition and balance of contact forces at the fluid structure interface the slip boundary condition might be more realistic than the classical no slip boundary condition in situations e g when the structure is rough and in modeling fsi dynamics near or at a contact cardiovascular tissue and cell seeded tissue constructs which consist of grooves in tissue scaffolds that are lined with cells are examples of rough elastic interfaces interacting with an incompressible viscous fluid the problem of heart valve closure is an example of a fsi problem with a contact involving elastic interfaces we prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting this is the first existence result for fluid structure interaction problems involving elastic structures satisfying the navier slip boundary condition

asymptotic behavior of a non newtonian flow in a thin domain with navier law on a rough boundary abstract we consider a non newtonian flow in a thin domain of thickness e the flow is described by the 3d incompressible navier stokes stokes system with a nonlinear viscosity being a power of the shear rate power law of flow index p the bottom of the domain is irregular by the presence of slight roughness of amplitude e and period e satisfying the relation 1 assuming pure slip or partial slip with a friction coefficient e with 0 on the rough boundary we consider the limit when domain thickness tends to zero and we obtain different models depending on the magnitude with respect to 2 p 1 p p 1 p and the magnitude with respect to 1 p 1

a diffuse interface method for the navier stokes darcy equations perfusion profile for a patient specific human liver based on mri scans abstract we present a diffuse interface method for coupling free and porous medium type flows modeled by the navier stokes and darcy equations its essential component is a diffuse geometry model generated from the phase field solution of a separate initial boundary value problem that is based on the allen cahn equation phase field approximations of the interface and its gradient are then employed to transfer all interface terms in the coupled variational flow formulation into volumetric terms this eliminates the need for an explicit interface parametrization between the two flow regimes we illustrate accuracy and convergence for a series of benchmark examples using standard low order stabilized finite element discretizations our diffuse interface method is particularly attractive for coupled flow analysis on imaging data with complex implicit interfaces where procedures for deriving explicit surface parametrizations constitute a significant bottleneck we demonstrate the potential of our method to establish seamless imaging through analysis workflows by computing a perfusion profile for a full scale 3d human liver based on mri scans

weighted hilbert spaces for the stationary exterior stokes problem with navier slip boundary conditions abstract in this paper we analyze the three dimensional exterior stokes problem with the navier slip boundary conditions describing the flow of a viscous and incompressible fluid past an obstacle where it is assumed that the fluid may slip at the boundary because the flow domain is unbounded we set the problem in weighted spaces in order to control the behavior at infinity of the solutions this functional framework also allows to prescribe various behaviors at infinity of the solutions growth or decay existence and uniqueness of solutions are shown in a hilbert setting which gives the tools for a possible numerical analysis of the problem weighted korn s inequalities are the key point in order to study the variational problem

stability analysis of maccormack rapid solver method for evolutionary stokes darcy problem abstract a numerical method based on the explicit implicit scheme of maccormack finite difference scheme was applied to the solution of parabolized navier stokes equations partitioned methods which solve the coupled problem by successively solving the sub physics problems have recently been studied for the full evolutionary groundwater surface water flow with convergence established over both bounded and long time intervals a maccormack rapid solver method based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model under a time step limitation of the form c 1 h t c 2 h where h t are mesh size and time step respectively and c j j 1 2 are two physical parameters we prove both uniform and asymptotic stabilities over long time intervals some numerical experiments are presented and discussed

effective surface and boundary conditions for heterogeneous surfaces with mixed boundary conditions to deal with multi scale problems involving transport from a heterogeneous and rough surface characterized by a mixed boundary condition an effective surface theory is developed which replaces the original surface by a homogeneous and smooth surface with specific boundary conditions a typical example corresponds to a laminar flow over a soluble salt medium which contains insoluble material to develop the concept of effective surface a multi domain decomposition approach is applied in this framework velocity and concentration at micro scale are estimated with an asymptotic expansion of deviation terms with respect to macro scale velocity and concentration fields closure problems for the deviations are obtained and used to define the effective surface position and the related boundary conditions the evolution of some effective properties and the impact of surface geometry peclet schmidt and damkohler numbers are investigated finally comparisons are made between the numerical results obtained with the effective models and those from direct numerical simulations with the original rough surface for two kinds of configurations

wall laws for fluid flows at a boundary with random roughness the general concern of this paper is the effect of rough boundaries on fluids we consider a stationary flow governed by incompressible navier stokes equations in an infinite domain bounded by two horizontal rough plates the roughness is modeled by a spatially homogeneous random field with characteristic size e a mathematical analysis of the flow for small e is performed the navier s wall law is rigorously deduced from this analysis this substantially extends former results obtained in the case of periodic roughness notably in 16 17 2007 wiley periodicals inc

vanishing viscous limits for the 2d lake equations with navier boundary conditions abstract the vanishing viscosity limit is considered for the viscous lake equations with navier friction boundary conditions we prove that the inviscid limit satisfies the inviscid lake equations and the results include flows generated by l p initial vorticity with 1 p

homogenization of the brush problem with a source term in l 1 we consider a domain which has the form of a brush in 3d or the form of a comb in 2d i e an open set which is composed of cylindrical vertical teeth distributed over a fixed basis all the teeth have a similar fixed height their cross sections can vary from one teeth to another one and are not supposed to be smooth moreover the teeth can be adjacent i e they can share parts of their boundaries the diameter of every tooth is supposed to be less than or equal to e and the asymptotic volume fraction of the teeth as e tends to zero is supposed to be bounded from below away from zero but no periodicity is assumed on the distribution of the teeth in this domain we study the asymptotic behavior as e tends to zero of the solution of a second order uniformly elliptic equation with a zeroth order term which is bounded from below away from zero when the homogeneous neumann boundary condition is satisfied on the whole of the boundary first we revisit the problem where the source term belongs to l 2 this is a classical problem but our homogenization result takes place in a geometry which is more general that the ones which have been considered before moreover we prove a corrector result which is new then we study the case where the source term belongs to l 1 working in the framework of renormalized solutions and introducing a definition of renormalized solutions for degenerate elliptic equations where only the vertical derivative is involved such a definition is new we identify the limit problem and prove a corrector result

navier wall law for nonstationary viscous incompressible flows abstract we study the navier wall law for the two dimensional initial boundary value problem of the navier stokes equations in a domain with a rough boundary the navier wall law is verified for the initial data in c 1 class under the natural compatibility condition our proof relies on the boundary layer analysis and the l theory of the navier stokes equations in the half space

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions