Similarity solutions for creeping flow and heat transfer in second grade fluids

The two-dimensional equations of motions for the slowly flowing and heat transfer in second grade fluid are written in cartesian coordinates neglecting the inertial terms. When the inertia terms are simply omitted from the equations of motions the resulting solutions are valid approximately for Re?1. This fact can also be deduced from the dimensionless form of the momentum and energy equations. By employing Lie group analysis, the symmetries of the equations are calculated. The Lie algebra consist of four finite parameter and one infinite parameter Lie group transformations, one being the scaling symmetry and the others being translations. Two different types of solutions are found using the symmetries. Using translations in x and y coordinates, an exponential type of exact solution is presented. For the scaling symmetry, the outcoming ordinary differential equations are more involved and only a series type of approximate solution is presented. Finally, some boundary value problems are discussed.     

非Четаев型非完整系统的Lie对称性与守恒量

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

On the Noether symmetry and Lie symmetry of mechanical systems

The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates. The Lie symmetry is an invariance of the differential equations of motion under the transformations. In this paper, the relation between these two symmetries is proved definitely and firstly for mechanical systems. The results indicate that all the Noether symmetries are Lie symmetries for Lagrangian systems meanwhile a Noether symmetry is a Lie symmetry for the general holonomic or nonholonomic systems provided that some conditions hold. The project supported by the National Natural Science Foundation of China (19972010)     

Integration of third order ordinary differential equations possessing two-parameter symmetry group by Lie’s method

The solution of a class of third order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that reduction to quadratures is possible according to two scenarios: (1) if upon first reduction of order the obtained second order ordinary differential equation besides the inherited point symmetry acquires at least one more new point symmetry (possibly a hidden symmetry of Type II). (2) First, reduction paths of the fourth order differential equations with four parameter symmetry group leading to the first order equation possessing one known (inherited) symmetry are constructed. Then, reduction paths along which a third order equation possessing two-parameter symmetry group appears are singled out and followed until a first order equation possessing one known (inherited) symmetry are obtained. The method uses conditions for preservation, disappearance and reappearance of point symmetries.     

Lie Point Symmetries,Conservation and Balance Laws in Linear Gradient Elastodynamics

The aim of this work is the derivation of Lie point symmetries, conservation and balance laws in linear gradient elastodynamics of grade-2 (up to second gradients of the displacement vector and the first gradient of the velocity). The conservation and balance laws of translational, rotational, scaling variational symmetries and addition of solutions are derived using Noether’s theorem. It turns out that the scaling symmetry is not a strict variational symmetry in gradient elasticity.      

Exact solutions to drift-flux multiphase flow models through Lie group symmetry analysis

In the present paper, Lie group symmetry method is used to obtain some exact solutions for a hyperbolic system of partial differential equations (PDEs), which governs an isothermal no-slip drift-flux model for multiphase flow problem. Those symmetries are used for the governing system of equations to obtain infinitesimal transformations, which consequently reduces the governing system of PDEs to a system of ODEs. Further, the solutions of the system of ODEs which in turn produces some exact solutions for the PDEs are presented. Finally, the evolutionary behavior of weak discontinuity is discussed.     

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

Group theoretical analysis and similarity solutions for stress boundary layers in viscoelastic flows

Lie group theory is used to obtain point symmetries of the boundary layer equations derived in the literature for the high Weissenberg number flow of upper convected Maxwell (UCM) and Phan-Tien-Tanner (PTT) type of viscoelastic fluids. The equations are reduced to ordinary differential equation systems with the use of scaling and spiral transformation groups. Similarity solutions are obtained and discussed for different cases such as flow around corners, flow over moving and stretching walls, and exponential shear flows.     

Optimal system,group invariant solutions and conservation laws of the CGKP equation

In this paper, the (2 + 1)-dimensional cubic generalized Kadomtsev–Petviashvili (CGKP) equation that is derived from the Maxwell–Bloch equations is investigated. By means of Lie symmetry analysis method, we obtain the Lie point symmetries for the equation and the optimal system of the symmetry algebra. Based on the optimal system, a lot of group invariant solutions are obtained. In addition, explicit conservation laws of the equation are studied.     

Painlevé analysis,Lie symmetries,and exact solutions for the time-dependent coefficients Gardner equations

In this paper, the three variable-coefficient Gardner (vc-Gardner) equations are considered. By using the Painlevé analysis and Lie group analysis method, the Painlevé properties and symmetries for the equations are obtained. Then the exact solutions generated from the symmetries and Painlevé analysis are presented.     

变质量非完整系统的Lie对称性逆问题

本文研究质量非完整系统的Lie对称性逆问题：根据已知积分求相应的Lie对称性,具体研究了受Chetaev型和非Chetaev型非完整约束的变质量系统的Lie对称性逆问题。首先,根据Lie对称所满足的确定方程和限制方程,给出Lie对称的结构方程和相应的守恒量及其表达式;其次,由已知守恒量求出相应的Noether对称性;最后,根据Noether对称性求出相应的Lie对称性。     

Analysis of the von Karman equations by group methods

One of the systems of equations approximating the large deflection of plates consists of two coupled non-linear fourth order partial differential equations, known as the von Karman equations. The full symmetry group for the steady equations is a finitely generated Lie group with ten parameters. For the time-dependent system the full symmetry group is an infinite parameter Lie group. Several subgroups of the full group are used to generate exact solutions of the time-independent and the time-dependent systems. These include the dilatation group (similar solutions), rotation group, screw group and others. Physical implications and applications are discussed.     

Symmetry analysis of modified 2D Burgers vortex equation for unsteady case

In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations(PDEs).     

Nonlocal symmetries of evolution equations

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution equations in one spatial variable invariant under Lie algebras of the dimension up to three. As a result, we construct the broad families of new nonlinear evolution equations possessing nonlocal symmetries which in principle cannot be obtained within the classical Lie approach.     

Application of Lie transformation group methods to classical linear theories of rods and plates

In the present paper, a class of partial differential equations governing various rod and plate theories of Bernoulli–Euler and Poisson–Kirchhoff type is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived and the general statement of the associated group-classification problem is given. A simple relation is deduced allowing to recognize easily the variational symmetries among the “ordinary” symmetries of a self-adjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Bessel-Hagen’s extension of Noether’s theorem) conservation laws are suggested. Solutions of group-classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods, fluid conveying pipes and plates resting on variable elastic foundations. The obtained group-classification results are used to derive conservation laws and group-invariant solutions readily applicable in rod dynamics and plate statics and dynamics. New generalized symmetries and conservation laws for the theories of Timoshenko beams, Reissner–Mindlin plates and three-dimensional elastostatics are presented.     

SIMILARITY SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A SPECIAL NON-NEWTONIAN FLUID IN A SPECIAL COORDINATE SYSTME

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By ~sing Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.     

Group theoretic methods for approximate invariants and Lagrangians for some classes of y″+εF(t)y′+y=f(y,y′)

Some recent results on the Lie symmetry generators of equations with a small parameter and the relationship between symmetries and conservation laws for such equations are used to construct first integrals and Lagrangians for autonomous weakly non-linear systems, y″+εF(t)y′+y=f(y,y′). An adaptation of a theorem that provides the point symmetry generators that leave the invariant functional involving a Lagrangian for such equations is presented. A detailed example to illustrate the method is given (and other examples are discussed). The (approximate) symmetry generators, invariants and Lagrangians maintain the perturbation order of the ‘small parameter’ stipulated in the equation — first order in this case.     

Symmetry-breaking at non-positive solutions of semilinear elliptic equations

We consider symmetry-breaking bifurcations at non-positive, radially symmetric solutions of semilinear elliptic equations on a ball with Dirichlet boundary conditions. For nonlinearities which are asymptotically affine linear, we find solutions at which the symmetry breaks. The kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group O(n). For this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough. Thus we obtain, by the equivariant branching lemma, a large variety of isotropy subgroups of O(n) which occur as symmetries of the bifurcating solution branches.     

Addendum to: Symmetry classification of quasi-linear PDE’s. II: an exceptional case

This short note completes the symmetry analysis of a class of quasi-linear partial differential equations considered in the previous paper (Nonlinear Dynamics 51: 309–316, 2008): it deals with an “exceptional” Lie point symmetry which is admitted only if the involved parameters are fixed by precise relationships. The peculiarity of this symmetry is enhanced by the fact that, combined with the presence of a conditional symmetry of “weak” type, it leads to a family of solutions which include, as a particular case, a relevant solution of the Grad–Schlüter–Shafranov equation, well known in plasma physics.