lattice boltzmann model for the incompressible navier stokes equation in this paper a lattice boltzmann lb model to simulate incompressible flow is developed the main idea is to explicitly eliminate the terms of o m2 where m is the mach number due to the density fluctuation in the existing lb models in the proposed incompressible lb model the pressure p instead of the mass density is the independent dynamic variable the incompressible navier stokes equations are derived from the incompressible lb model via chapman enskog procedure numerical results of simulations of the plane poiseuille flow driven either by pressure gradient or a fixed velocity profile at entrance as well as of the 2d womersley flow are presented the numerical results are found to be in excellent agreement with theory

painleve test and exact similarity solutions of a class of nonlinear diffusion equations for 1 1 dimensional diffusion equations where the diffusion coefficient is an arbitrary power of the dependent variable a painleve test is performed on both the partial differential equation and the similarity reductions the invariance of the painleve property under a large class of similarity reductions is shown for the reductions with the painleve property the explicit similarity solutions are given it is demonstrated how results of a painleve analysis can be used to obtain further exact similarity solutions even in cases without the painleve property

self similar solutions of certain coupled integrable systems similarity reductions of the coupled nonlinear schrodinger equation and an integrable version of the coupled maxwell bloch system are obtained by applying non translational symmetries the reduced system of coupled ordinary differential equations are solved in terms of painleve transcendents leading to new exact self similar solutions for these integrable equations

equivalence groupoids and group classification of multidimensional nonlinear schrodinger equations abstract we study admissible and equivalence point transformations between generalized multidimensional nonlinear schrodinger equations and classify lie symmetries of such equations we begin with a wide superclass of schrodinger type equations which includes all the other classes considered in the paper showing that this superclass is not normalized we partition it into two disjoint normalized subclasses which are not related by point transformations further constraining the arbitrary elements of the superclass we construct a hierarchy of normalized classes of schrodinger type equations this gives us an appropriate normalized superclass for the non normalized class of multidimensional nonlinear schrodinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses after a preliminary study of lie symmetries of nonlinear schrodinger equations with potentials and modular nonlinearities for an arbitrary space dimension we exhaustively solve the group classification problem for such equations in space dimension two

a lattice boltzmann scheme for incompressible multiphase flow and its application in simulation of rayleigh taylor instability in this paper we propose a new lattice boltzmann scheme for simulation of multiphase flow in the nearly incompressible limit the new scheme simulates fluid flows based on distribution functions the interfacial dynamics such as phase segregation and surface tension are modeled by incorporating molecular interactions the lattice boltzmann equations are derived from the continuous boltzmann equation with appropriate approximations suitable for incompressible flow the numerical stability is improved by reducing the effect of numerical errors in calculation of molecular interactions an index function is used to track interfaces between different phases simulations of the two dimensional rayleigh taylor instability yield satisfactory results the interface thickness is maintained at 3 4 grid spacings throughout simulations without artificial reconstruction steps

an improved multiphase lattice boltzmann flux solver for three dimensional flows with large density ratio and high reynolds number an improved multiphase lattice boltzmann flux solver mlbfs is proposed in this work for effective simulation of three dimensional 3d multiphase flows with large density ratio and high reynolds number as a finite volume scheme the mlbfs originally proposed in 27 applies the finite volume method to solve for macroscopic flow variables directly the fluxes are reconstructed locally at each cell interface by using the standard lbm solutions due to the modeling error of the standard lbm the reconstructed fluxes deviate from those in the navier stokes equations and to compensate this error a complex tensor is introduced in the original mlbfs however the computation of the tensor introduces additional complexity and usually needs a relatively thicker interface thickness to maintain numerical stability which makes the solver be complex and inefficient in the 3d case to remove this drawback in this work a theoretical analysis to the formulations obtained from the chapman enskog expansion is conducted it is shown that the modeling error can be effectively removed by modifying the computation of the equilibrium density distribution function with this improvement the proposed 3d mlbfs not only avoids the calculation of the compensation tensor but also is able to maintain numerical stability with very thin interface thickness several benchmark cases including the challenging droplet impacting on a dry surface head on collisions of binary droplets and droplet splashing on a thin film with density ratio 1000 and reynolds number up to 3000 are studied to validate the proposed solver the obtained results agree well with the published data

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

lie transformations similarity reduction and solutions for the nonlinear madelung fluid equations with external potential the application of lie group methods to a system of coupled nonlinear partial differential equations representing what is usually called a madelung fluid is shown the generating operators of the transformation group that depends on five arbitrary group constants will be constructed and all subclasses of systems of ordinary differential equations derived by similarity reduction will be presented in tabular form two subclasses of physical interest are investigated in detail and the similarity solutions are compared with solutions found earlier by the application of inverse scattering transform techniques to the cubic nonlinear schrodinger equation similarity solutions for the madelung equations with linear external potential x f0x are presented

an efficient lattice boltzmann multiphase model for 3d flows with large density ratios at high reynolds numbers we report on the development implementation and validation of a new lattice boltzmann method lbm for the numerical simulation of three dimensional multiphase flows here with only two components with both high density ratio and high reynolds number this method is based in part on but aims at achieving a higher computational efficiency than inamuro et al s model inamuro et al 2004 here we use a lbm to solve both a pressureless navier stokes equation in which the implementation of viscous terms is improved and a pressure poisson equation using different distribution functions and a d3q19 lattice scheme additionally we propose a new diffusive interface capturing method based on the cahn hilliard equation which is also solved with a lbm to achieve maximum efficiency the entire model is implemented and solved on a heavily parallel gpgpu co processor the proposed algorithm is applied to several test cases such as a splashing droplet a rising bubble and a braking ocean wave in all cases numerical results are found to agree very well with reference data and or to converge with the discretization

on groups in which engel sinks are cyclic abstract for an element g of a group g an engel sink is a subset e g such that for every x g all sufficiently long commutators x g g g belong to e g we conjecture that if g is a profinite group in which every element admits a sink that is a procyclic subgroup then g is procyclic by locally nilpotent we prove the conjecture in two cases when g is a finite group or a soluble pro p group

a stable discretization of the lattice boltzmann equation for simulation of incompressible two phase flows at high density ratio a stable discretization of the lattice boltzmann equation lbe for non ideal gases is presented for simulation of incompressible two phase flows having high density and viscosity ratios the stiffness of the discretized forcing terms in lbe for non ideal gases is known to trigger severe numerical instability and restrict practical application of the lbe method use of a proper pressure updating scheme is also crucial to the stability of the lbe method because of non negligible pressure variation across the phase interface to deal with these issues we propose a stable discretization scheme which assumes the low mach number approximation and utilizes the stress and potential forms of the surface tension force the incompressible transformation and the consistent discretization of the intermolecular forcing terms the proposed stable discretization scheme is applied to simulate 1 d advection equation with a source term a stationary droplet droplet oscillation and droplet splashing and deposition on a thin film at a density ratio of 1000 with varying reynolds numbers the numerical solutions of stationary and oscillatory droplets agree well with analytic solutions including the laplace s law the time history of the spread factor of the liquid sheet emitted after the droplet impact also follows the known spreading power law

moment equations for the diffusion of the particles of a mixture via the scattering kernel formulation of the nonlinear boltzmann equation abstract there are two goals in this paper essentially devoted to the application of the scattering kernel formulation of the nonlinear boltzmann equation to the case of a mixture the first goal is to show the equivalence between the scattering kernel representation of the collision term and the two other representations usually adopted for it in the literature namely the kinetic and the transition probability representation respectively the second goal is to derive instead the general scalar nonlinear boltzmann equation that is the equation governing an isotropic distribution function for the mixture considered and to establish then the moment equations associated with it

singular point analysis and integrals of motion for coupled nonlinear schrodinger equations for a system of two coupled nonlinear schrodinger equations and the corresponding madelung fluid equations we calculate the similarity transformation and reductions to ordinary differential equations the symmetries can be used to classify several types of solutions besides a system of coupled painleve ii equations we obtain a two dimensional quartic hamiltonian system using noether s theorem and performing a painleve test we identify those parameters which lead to regular motion on tori using the results of the p test we obtain heretofore undiscovered integrals of motion for the madelung fluid by direct calculations investigating the dynamics of the system for parameters that are different from those obtained by the painleve test we calculate numerically the surface of section kam tori and study the transition into chaotic behaviour

applications of nonclassical symmetries in this paper we discuss new applications of the nonclassical method of symmetry reduction the equations treated are well known in mathematical physics and are used to describe multisoliton solutions of the burger equation they are also useful in the study of kac moody algebras and in the examination of nonlinear waves in the theory of elasticity with the help of the nonclassical symmetry method the partial differential equations are reduced to ordinary differential equations which are solvable explicitly

the nonlinear schrodinger equation as a galilean invariant dynamical system the invariance of the nonlinear schrodinger equation under the galilei group is analyzed from the point of view of the inverse scattering transform it is shown that this group induces an infinite dimensional nonlinear canonical realization which is locally equivalent to a direct product of the two well known galilean actions describing classical particles and the free schrodinger equation

new classes of similarity solutions for the nonlinear carleman boltzmann equation abstract we apply the lie group based similarity method to the carleman model and calculate the generators of the group depending on four group constants heretofore undiscovered classes of similarity solutions will be presented and already discovered solutions by ernst and wick will be rediscovered and classified by specifying the group constants

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

a lattice boltzmann model for multiphase flows with large density ratio a lattice boltzmann model for simulating multiphase flows with large density ratios is described in this paper the method is easily implemented it does not require solving the poisson equation and does not involve the complex treatments of derivative terms the interface capturing equation is recovered without any additional terms as compared to other methods m r swift w r osborn j m yeomans lattice boltzmann simulation of liquid gas and binary fluid systems phys rev e 54 1996 5041 5052 t inamuro t ogata s tajima n konishi a lattice boltzmann method for incompressible two phase flows with large density differences j comput phys 198 2004 628 644 t lee c l lin a stable discretization of the lattice boltzmann equation for simulation of incompressible two phase flows at high density ratio j comput phys 206 2005 16 47 besides it requires less discrete velocities as a result its efficiency could be greatly improved especially in 3d applications it is validated by several cases a bubble in a stationary flow and the capillary wave the numerical surface tension obtained from the laplace law and the interface profile agrees very well with the respective analytical solution the method is further verified by its application to capillary wave and the bubble rising under buoyancy with comparison to other methods all the numerical experiments show that the present approach can be used to model multiphase flows with large density ratios

lie transformations nonlinear evolution equations and painleve forms we present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one parameter continuous lie group of transformations it is shown that in general the corresponding invariant variables the subclass of which is the usual similarity variables lead to ordinary differential equations of painleve type in the case of inverse scattering transform solvable equations as conjectured by ablowitz ramani and segur this is found to be also true for certain higher spatial dimensional versions such as the kadomtsev petviashivilli two dimensional sine gordon and ernst equations for the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points

gradient theorem for completely integrable hamiltonian systems for evolution equations which can be written in hamiltonian form two ways there exists a relation between two functions q 1 and q 2 both of which are gradients of conserved functionals the relation can be extended to define recursively functions q n it is shown that the q n corresponding to the general evolution equation associated with the zakharov shabat eigenvalue problem are all gradients of conserved functionals this in turn implies all these functionals are in involution

numerical simulation of 3d bubbles rising in viscous liquids using a front tracking method the rise of bubbles in viscous liquids is not only a very common process in many industrial applications but also an important fundamental problem in fluid physics an improved numerical algorithm based on the front tracking method originally proposed by tryggvason and his co workers has been validated against experiments over a wide range of intermediate reynolds and bond numbers using an axisymmetric model j hua j lou numerical simulation of bubble rising in viscous liquid j comput phys 22 2007 769 795 in the current paper this numerical algorithm is further extended to simulate 3d bubbles rising in viscous liquids with high reynolds and bond numbers and with large density and viscosity ratios representative of the common air water two phase flow system to facilitate the 3d front tracking simulation mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh on the latter mesh the governing navier stokes equations for incompressible newtonian flow are solved in a moving reference frame attached to the rising bubble specifically the equations are solved using a finite volume scheme based on the semi implicit method for pressure linked equations simple algorithm and it appears to be robust even for high reynolds numbers and high density and viscosity ratios the 3d bubble surface is tracked explicitly using an adaptive unstructured triangular mesh the numerical model is integrated with the software package paramesh a block based adaptive mesh refinement amr tool developed for parallel computing paramesh allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer further peskin distribution function is applied to interpolate the variable values between the front and the background meshes detailed sensitivity analysis about the numerical modeling algorithm has been performed the current model has also been applied to simulate a number of cases of 3d gas bubbles rising in viscous liquids e g air bubbles rising in water simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities in addition we applied this model to simulate the interaction between two bubbles rising in a liquid which illustrated the model s capability in predicting the interaction dynamics of rising bubbles