alien attractors and memory annihilation of structured sets in hopfield networks this paper considers the encoding of structured sets into hopfield associative memories a structured set is a set of vectors with equal hamming distance h from one another and its centroid is an external vector that has distance h 2 from every vector of the set structured sets having centroids are not infrequent when such a set is encoded into a noiseless hopfield associative memory using a bipolar outer product connection matrix and the network operates with synchronous neuronal update the memory of all encoded vectors is annihilated even for sets with as few as three vectors in dimension n 5 four for n 5 in such self annihilating structured sets the centroid emerges as a stable attractor we call it an alien attractor for canonical structured sets self annihilation takes place only if h n 2 self annihilation does not occur and alien attractors do not emerge in dimensions less than five

truncation errors and the rotational invariance of three dimensional lattice models in the lattice boltzmann method abstract the application of the lattice boltzmann method lbm in three dimensional isothermal hydrodynamic problems often adopts one of the following models d3q15 d3q19 or d3q27 although all of them retrieve consistent navier stokes dynamics in the continuum limit they are expected to behave differently at discrete level the present work addresses this issue by performing a lbm truncation error analysis as a conclusion it is theoretically demonstrated that differences among the aforementioned cubic lattices lie in the structure of their non linear truncation errors while reduced lattice schemes such as d3q15 and d3q19 introduce spurious angular dependencies through non linear truncation errors the complete three dimensional cubic lattice d3q27 is absent from such features this result justifies the superiority of the d3q27 lattice scheme to cope with the rotational invariance principle in three dimensional isothermal hydrodynamic problems particularly when convection is not negligible such a theoretical conclusion also finds support in numerical tests presented in this work a poiseuille duct flow and a weakly rotating duct flow

a theoretical investigation into the performance of the hopfield model an analysis is made of the behavior of the hopfield model as a content addressable memory cam and as a method of solving the traveling salesman problem tsp the analysis is based on the geometry of the subspace set up by the degenerate eigenvalues of the connection matrix the dynamic equation is shown to be equivalent to a projection of the input vector onto this subspace in the case of content addressable memory it is shown that spurious fixed points can occur at any corner of the hypercube that is on or near the subspace spanned by the memory vectors analysed is why the network can frequently converge to an invalid solution when applied to the traveling salesman problem energy function with these expressions the network can be made robust and can reliably solve the traveling salesman problem with tour sizes of 50 cities or more

analysis of the absorbing layers for the weakly compressible lattice boltzmann methods abstract it has been demonstrated that lattice boltzmann methods lbms are very efficient for computational aeroacoustics caa in order to address the issue of absorbing acoustic boundary conditions for lbm three kinds of damping terms are proposed and added to the right hand side of the lbm governing equations according to the classical theory these terms play an important role to damp and minimize the acoustic wave reflections from computational boundaries the corresponding macroscopic equations with the damping terms are recovered for analyzing the macroscopic behaviors of the these damping terms and determining the critical absorbing strength the dissipative and dispersive properties of the proposed absorbing layer terms are then further analyzed considering the linearized lbm equations they are explored in the wave number spaces via the von neumann analysis the related damping strength critical values and the optimal absorbing term are discussed finally some benchmark problems are implemented to assess the theoretical results

introducing a modified exact difference method for incorporating unsteady and non uniform force terms in lattice kinetic models abstract the exact difference method edm of kupershtokh 2004 in this work is modified to accurately incorporate unsteady non uniform force terms into lattice kinetic models we use the characteristic scheme for integrating the boltzmann equation along with the trapezoidal rule to estimate the integration results the method is based on discretizing the total derivative of the equilibrium distribution function edf in the edm using a central second order scheme instead of a first order method it is shown that the assumption of constant density in calculating the total derivative of the edf is not necessary in the limit of incompressible fluid flows using the chapman enskog analysis it is shown that the present method can be applied to both the standard lattice boltzmann method and the recently introduced constant speed kinetic model cskm of zadehgol and ashrafizaadeh 2014 the following benchmark flows are simulated i womersley flow ii hartmann flow iii unsteady taylor green vortex flow and iv the circular interface using the cskm and the results are in excellent agreement with the analytical solutions

the immersed boundary lattice boltzmann method for solving fluid particles interaction problems a new computational method the immersed boundary lattice boltzmann method is presented this method is a combination and utilizes the most desirable features of the lattice boltzmann and the immersed boundary methods the method uses a regular eulerian grid for the flow domain and a lagrangian grid to follow particles that are contained in the flow field the rigid body conditions for the fluid and the particles are enforced by a penalty method which assumes that the particle boundary is deformable with a high stiffness constant the velocity field of the fluid and particles is solved by adding a force density term into the lattice boltzmann equation this novel method preserves the advantages of lbm in tracking a group of particles and at the same time provides an alternative and better approach to treating the solid fluid boundary conditions the method also solves the problems of fluctuation of the forces and velocities on the particles when the bounce back boundary conditions are applied this method enables one to simulate problems with particle deformation and fluid structure deformation its results are validated by comparison with results from other methods

mesoscopic modeling of flow and dispersion phenomena in fractured solids the problem of hydrodynamic dispersion in porous media is considered and numerical predictions of the mixing degree in a single intersection are provided the flow field in the intersection and adjacent pores or fractures is calculated using a lattice boltzmann model for single phase flow a particle tracking scheme is used subsequently that monitors the migration of solute particles in the area of the intersection taking into account the local flow field and a brownian field mixing is quantified in terms of the probability of solute transfer across the junction into the opposite fracture to circumvent the problem of large computational times for cases of fast flow compared to diffusion a lattice boltzmann advection diffusion model is used that offers significant savings on computational time without sacrificing accuracy it is shown that the solute dispersion in a fracture network is a strong function of the reynolds number even if the peclet number remains constant due to the extensive recirculation areas that may develop in regions close to the junction

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

an inverse design analysis of mesoscopic implementation of non uniform forcing in mrt lattice boltzmann models abstract in this paper the mesoscopic representation of non uniform forcing is investigated by an inverse design approach for several mrt multiple relaxation time lattice boltzmann models first the mesoscopic forcing representation of a standard lbm mrt model on a square lattice is re visited by the multiscale chapman enskog expansion we derive the most general form of the representation by taking advantage of the mrt formulation in particular we show that there are three free components in the mesoscopic representation of forcing second by the same methodology the forcing scheme of two new rectangular mrt lattice boltzmann models are derived based on the requirements of the navier stokes equations these theoretical results are then validated by numerical simulations of a forced taylor green vortex flow with several different forms of non uniform forcing to alter the kinetic energy evolution of the system the numerical results are in excellent agreement with the corresponding time dependent analytical solution of flow

singular forces and pointlike colloids in lattice boltzmann hydrodynamics we present an accurate method to include arbitrary singular distributions of forces in the lattice boltzmann formulation of hydrodynamics we validate our method with several examples involving stokeslet stresslet and rotlet singularities finding excellent agreement with analytical results a minimal model for sedimenting particles is presented using the method in the dilute limit this model has accuracy comparable to but computational efficiency much greater than algorithms that explicitly resolve the size of the particles

an efficient framework for fluid structure interaction using the lattice boltzmann method and immersed moving boundaries this paper presents a serial computational framework that hydrodynamically couples the lattice boltzmann method lbm and the discrete element method dem for the solution of particle suspension problems in two and three dimensions the single relaxation time bhatnagar gross krook lbgk form of the lattice boltzmann equation is employed with an immersed moving boundary method for the fluid structure interaction similar algorithms have been previously reported in the literature however this work deliberately utilizes solution options that minimize the computational overheads of the framework to facilitate simulations of multibody structural fields in large fluid domains in particular mixed boundary conditions are employed which combine the simple bounce back technique with the immersed moving boundary method and the relatively inexpensive d3q15 lattice is employed for 3d solutions the fundamentals of the lbm are briefly discussed followed by a review of the coupling techniques available for fsi using the lbm options for mapping solid obstacles to the lbm grid are presented and an algorithm for automatic dynamic subcycling of the two explicit solution schemes is outlined the lbm dem framework is then validated and benchmarked against previously published lbm results with comments made where appropriate on the comparative accuracy and convergence characteristics finally a multi particle suspension simulation is presented to qualitatively assess the performance of the framework when a large number of dynamic contacts exist

mixed problems for hyperbolic functional differential equations with unbounded delay the phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed the set of axioms for generalized solutions of initial boundary value problems is presented a theorem on the existence and continuous dependence upon initial boundary data is given the mixed problem is transformed into a system of integral functional equations the existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities examples of phase spaces are given

adjoint based fluid flow control and optimisation with lattice boltzmann methods a lattice boltzmann lb framework to solve fluid flow control and optimisation problems numerically is presented problems are formulated on a mesoscopic basis in a side condition the dynamics of a newtonian fluid is described by a family of simplified boltzmann like equations namely bgk boltzmann equations which are linked to an incompressible navier stokes equation it is proposed to solve the non linear optimisation problem by a line search algorithm the needed derivatives are obtained by deriving the adjoint equations referred to as adjoint bgk boltzmann equations the primal equations are discretised by standard lattice boltzmann methods lbm while for the adjoint equations a novel discretisation strategy is introduced the approach follows the main ideas behind lbm and is therefore referred to as adjoint lattice boltzmann methods albm the corresponding algorithm retains most of the basic features of lb algorithms in particular it enables a highly efficient parallel implementation and thus solving large scale fluid flow control and optimisation problems the overall solution strategy the derivation of a prototype adjoint bgk boltzmann equation the novel albm and its parallel realisation as well as its validation are discussed in detail in this article numerical and performance results are presented for a series of steady state distributed control problems with up to approximately 1 6 million unknown control parameters obtained on a high performance computer with up to 256 processing units

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

a new method for the numerical solution of vorticity streamfunction formulations abstract a lattice boltzmann model for vorticity streamfunction formulations is proposed in this paper the present model was validated by several benchmark problems excellent agreement between the present results and other numerical data shows that this model is an efficient numerical method for the numerical solution of vorticity streamfunction formulations

low and high order accurate boundary conditions the present contribution focuses on the accuracy of reflection type boundary conditions in the stokesbrinkmandarcy modeling of porous flows solved with the lattice boltzmann method lbm which we operate with the two relaxation time trt collision and the brinkman force based scheme bf called bf trt scheme in parallel we compare it with the stokesbrinkmandarcy linear finite element method fem where the dirichlet boundary conditions are enforced on grid vertices in bulk both bf trt and fem share the same defect in their discretization a correction to the modeled brinkman equation appears given by the discrete laplacian of the velocity proportional resistance force this correction modifies the effective brinkman viscosity playing a crucial role in the triggering of spurious oscillations in the bulk solution while the exact form of this defect is available in lattice aligned straight or diagonal flows in arbitrary flow lattice orientations its approximation is constructed at boundaries we verify that such a brinkman viscosity correction has an even more harmful impact already at the first order it shifts the location of the no slip wall condition supported by traditional lbm boundary schemes such as the bounce back rule for that reason this work develops a new class of boundary schemes to prescribe the dirichlet velocity condition at an arbitrary wall boundary node distance and that supports a higher order accuracy in the accommodation of the trt brinkman solutions for their modeling we consider the standard bf scheme and its improved version called ibf this latter is generalized in this work to suppress or to reduce the viscosity correction in arbitrarily oriented flows our framework extends the one and two point families of linear and parabolic link wise boundary schemes respectively called b li and b mli which avoid the interference of the brinkman viscosity correction in their closure relations the performance of lbm and fem is thoroughly evaluated in three benchmark tests which are run throughout three distinctive permeability regimes the first configuration is a horizontal porous channel studied with a symbolic approach where we construct the exact solutions of fem and bf ibf with different boundary schemes the second problem refers to an inclined porous channel flow which brings in as new challenge the formation of spurious boundary layers in lbm that is numerical artefacts that arise due to a deficient accommodation of the bulk solution by the low accurate boundary scheme the third problem considers a porous flow past a periodic square array of solid cylinders which intensifies the previous two tests with the simulation of a more complex flow pattern the ensemble of numerical tests provides guidelines on the effect of grid resolution and the trt free collision parameter over the accuracy and the quality of the velocity field spanning from stokes to darcy permeability regimes it is shown that with the use of the high order accurate boundary schemes the simple uniform mesh based trt lbm formulation can even surpass the accuracy of fem employing hardworking body fitted meshes

computation of gas solid flows by finite difference boltzmann equation abstract in this paper we will discuss in detail how to use a finite difference lattice boltzmann equation model in which an external force term is involved to simulate two way coupling gas solid flows the numerical results are found to be in good agreement with analytical data and some other numerical results

generalized lattice boltzmann method with multirange pseudopotential the physical behavior of a class of mesoscopic models for multiphase flows is analyzed in details near interfaces in particular an extended pseudopotential method is developed which permits to tune the equation of state and surface tension independently of each other the spurious velocity contributions of this extended model are shown to vanish in the limit of high grid refinement and or high order isotropy higher order schemes to implement self consistent forcings are rigorously computed for 2d and 3d models the extended scenario developed in this work clarifies the theoretical foundations of the shan chen methodology for the lattice boltzmann method and enhances its applicability and flexibility to the simulation of multiphase flows to density ratios up to o 100

consistent lattice boltzmann modeling of low speed isothermal flows at finite knudsen numbers in slip flow regime ii application to curved boundaries gaseous flows inside microfluidic devices often fall in the slip flow regime according to this theoretical description the navier stokes model remains applicable in bulk while at solid walls a slip velocity boundary model shall be considered physically it is well established that to properly account for the wall curvature the wall slip velocity must be determined by the shear stress rather than the normal component of the velocity derivative alone as commonly applied to planar surfaces it follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics cfd techniques this paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice boltzmann method lbm to substantiate this conclusion this work undertakes the following three studies first we examine the conditions under which the generic reflection type boundary rules used by lbm become consistent models for the slip velocity boundary condition this effort makes use of the second order chapman enskog expansion method where we address both planar and curved boundaries the analysis also clarifies the capabilities and limitations behind the considered reflection type slip schemes second we revisit the family of parabolic accurate lbm slip boundary schemes originally formulated in phys rev e 96 013311 2017 2470 004510 1103 physreve 96 013311 on the basis of the multireflection framework and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate lbm slip boundary formulations such as the linearly accurate slip schemes and the widely popular kinetic based boundary schemes in addition we also discuss the numerical stability of the parabolic slip schemes previously developed providing an heuristic strategy to improve their stable range of operation third we evaluate the performance of the several slip boundary schemes debated in this paper the numerical tests correspond to two classical 2d benchmark flow problems of slip over non planar solid surfaces namely i the velocity profile of the cylindrical couette flow and ii the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles the obtained numerical results confirm the competitiveness of the lbm when equipped with slip boundary schemes of parabolic accuracy as cfd tool to simulate slippage phenomena over arbitrarily non planar surfaces indeed although operating on a simple uniform mesh discretization the lbm yields a similar or even superior level of accuracy compared to state of the art fem simulations conducted on hardworking body fitted meshes this conclusion establishes the lbm as a very appealing cfd technique for simulating microfluidic flows in the slip flow regime a result that deserves further exploration in future studies

lattice boltzmann modeling with discontinuous collision components hydrodynamic and advection diffusion equations irrespective of the nature of the modeled conservation laws we establish first the microscopic interface continuity conditions for lattice boltzmann lb multiple relaxation time link wise collision operators with discontinuous components equilibrium functions and or relaxation parameters effective macroscopic continuity conditions are derived for a planar implicit interface between two immiscible fluids described by the simple two phase hydrodynamic model and for an implicit interface boundary between two heterogeneous and anisotropic variably saturated soils described by richard s equation comparing the effective macroscopic conditions to the physical ones we show that the range of the accessible parameters is restricted e g a variation of fluid densities or a heterogeneity of the anisotropic soil properties when the interface is explicitly tracked the interface collision components are derived from the leading order continuity conditions among particular interface solutions a harmonic mean value is found to be an exact lb solution both for the interface kinematic viscosity and for the interface vertical hydraulic conductivity function we construct simple problems with the explicit and implicit interfaces matched exactly by the lb hydrodynamic and or advection diffusion schemes with the aid of special solutions for free collision parameters

a mass conserving boundary condition for the lattice boltzmann equation method in lattice boltzmann lb simulations the widely used wall boundary conditions bcs proposed by filipova and hanel fh and mei luo and shyy mls result in constant mass leakage in certain circumstances in this paper we have analyzed the source of the leakage based on this analysis we propose a second order accurate mass conserving wall bc in our bc the distribution function at a wall node is decomposed into its equilibrium and non equilibrium parts the mass conservation is guaranteed by enforcing a mass conserving rule in the construction of the fictitious equilibrium distribution part we have shown through several benchmark test problems involving steady and unsteady flows that our new bc not only eliminates the constant mass leakage but also has many other advantages over the fh and mls bcs

designing asymmetric hopfield type associative memory with higher order hamming stability the problem of optimal asymmetric hopfield type associative memory ham design based on perceptron type learning algorithms is considered it is found that most of the existing methods considered the design problem as either 1 finding optimal hyperplanes according to normal distance from the prototype vectors to the hyperplane surface or 2 obtaining weight matrix w w sub ij by solving a constraint optimization problem in this paper we show that since the state space of the ham consists of only bipolar patterns i e v v sub 1 v sub 2 v sub n sup t spl isin 1 1 sup n the basins of attraction around each prototype training vector should be expanded by using hamming distance measure for this reason in this paper the design problem is considered from a different point of view our idea is to systematically increase the size of the training set according to the desired basin of attraction around each prototype vector we name this concept the higher order hamming stability and show that conventional minimum overlap algorithm can be modified to incorporate this concept experimental results show that the recall capability as well as the number of spurious memories are all improved by using the proposed method moreover it is well known that setting all self connections w sub ii spl forall i to zero has the effect of reducing the number of spurious memories in state space from the experimental results we find that the basin width around each prototype vector can be enlarged by allowing nonzero diagonal elements on learning of the weight matrix w if the magnitude of w sub ii is small for all i then the condition w sub ii 0 spl forall i can be relaxed without seriously affecting the number of spurious memories in the state space therefore the method proposed in this paper can be used to increase the basin width around each prototype vector with the cost of slightly increasing the number of spurious memories in the state space

boundary conditions for free interfaces with the lattice boltzmann method abstract in this paper we analyze the boundary treatment of the lattice boltzmann method lbm for simulating 3d flows with free surfaces the widely used free surface boundary condition of korner et al 27 is shown to be first order accurate the article presents a new free surface boundary scheme that is suitable for second order accurate simulations based on the lbm the new method takes into account the free surface position and its orientation with respect to the computational lattice numerical experiments confirm the theoretical findings and illustrate the different behavior of the original method and the new method

gpu accelerated edge region based level set evolution constrained by 2d gray scale histogram due to its intrinsic nature which allows to easily handle complex shapes and topological changes the level set method lsm has been widely used in image segmentation nevertheless lsm is computationally expensive which limits its applications in real time systems for this purpose we propose a new level set algorithm which uses simultaneously edge region and 2d histogram information in order to efficiently segment objects of interest in a given scene the computational complexity of the proposed lsm is greatly reduced by using the highly parallelizable lattice boltzmann method lbm with a body force to solve the level set equation lse the body force is the link with image data and is defined from the proposed lse the proposed lsm is then implemented using an nvidia graphics processing units to fully take advantage of the lbm local nature the new algorithm is effective robust against noise independent to the initial contour fast and highly parallelizable the edge and region information enable to detect objects with and without edges and the 2d histogram information enable the effectiveness of the method in a noisy environment experimental results on synthetic and real images demonstrate subjectively and objectively the performance of the proposed method

gravity in a lattice boltzmann model in this paper we consider the introduction of a body force in the incompressible limit into the lattice boltzmann model a number of methods are considered and their suitability to our objectives determined when there is no density variation across the fluid gravity can be introduced in the form of an altered pressure gradient this method correctly satisfies the navier stokes equation however if there is a non negligible density variation present produced by the body force or otherwise this method becomes less accurate as the density variation increases and the constant density approximation becomes less valid three other methods are also considered for application when there is a non negligible density variation the equations of motion satisfied by these models are found up to second order in the knudsen number and it is seen that only one of these methods satisfies the true navier stokes equation numerical simulations are performed to compare the different models and to assess the range of application of each

linear and logarithmic capacities in associative neural networks a model of associate memory incorporating global linearity and pointwise nonlinearities in a state space of n dimensional binary vectors is considered attention is focused on the ability to store a prescribed set of state vectors as attractors within the model within the framework of such associative nets a specific strategy for information storage that utilizes the spectrum of a linear operator is considered in some detail comparisons are made between this spectral strategy and a prior scheme that utilizes the sum of kronecker outer products of the prescribed set of state vectors which are to function nominally as memories the storage capacity of the spectral strategy is linear in n the dimension of the state space under consideration whereas an asymptotic result of n 4 log n holds for the storage capacity of the outer product scheme computer simulated results show that the spectral strategy stores information more efficiently the preprocessing costs incurred in the two algorithms are estimated and recursive strategies are developed for their computation

a general weight matrix formulation using optimal control classical methods from optimal control theory are used in deriving general forms for neural network weights the network learning or application task is encoded in a performance index of a general structure consequently different instances of this performance index lead to special cases of weight rules including some well known forms comparisons are made with the outer product rule spectral methods and recurrent back propagation simulation results and comparisons are presented

application of the lattice boltzmann method to transition in oscillatory channel flow in this study the applicability of the lattice boltzmann method to oscillatory channel flow with a zero mean velocity has been evaluated the model has been compared to exact analytical solutions in the laminar case re 100 where re is the reynolds number based on the stokes layer for the womersley parameter 1 31 in this regime there was good agreement between numerical and exact analytical solutions the model was then applied to study the primary instability of oscillatory channel flow with a zero mean velocity for these transitionary flows the parameters were varied in the range 400 re 1000 and 4 16 disturbances superimposed on the numerical solution triggered the two dimensional primary instability this phenomenon has not been numerically evaluated over the range of or re currently investigated the results are consistent with quasi steady linear stability theories and previous numerical investigations

random interactions in higher order neural networks recurrent networks of polynomial threshold elements with random symmetric interactions are studied precise asymptotic estimates are derived for the expected number of fixed points as a function of the margin of stability in particular it is shown that there is a critical range of margins of stability depending on the degree of polynomial interaction such that the expected number of fixed points with margins below the critical range grows exponentially with the number of nodes in the network while the expected number of fixed points with margins above the critical range decreases exponentially with the number of nodes in the network the random energy model is also briefly examined and links with higher order neural networks and higher order spin glass models are made explicit

the vallee poussin problem for higher order nonlinear hyperbolic equations for higher order nonlinear hyperbolic equations nonclassical boundary value problems of vallee poussin type are studied and unimprovable in a sense sufficient conditions of solvability unique solvability and well posedness are established

axisymmetric multiphase lattice boltzmann method a lattice boltzmann method for axisymmetric multiphase flows is presented and validated the method is capable of accurately modeling flows with variable density we develop the classic shan chen multiphase model phys rev e 47 1815 1993 for axisymmetric flows the model can be used to efficiently simulate single and multiphase flows the convergence to the axisymmetric navier stokes equations is demonstrated analytically by means of a chapmann enskog expansion and numerically through several test cases in particular the model is benchmarked for its accuracy in reproducing the dynamics of the oscillations of an axially symmetric droplet and on the capillary breakup of a viscous liquid thread very good quantitative agreement between the numerical solutions and the analytical results is observed