Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.     

Invariant solution for an axisymmetric turbulent free jet using a conserved vector

An axisymmetric turbulent free jet described by an effective viscosity, which is the sum of the kinematic viscosity and the kinematic eddy viscosity, is investigated. The conservation laws of the jet are derived using the multiplier method. A second conserved vector, in addition to the elementary conserved vector, exists provided the effective viscosity has a special form. The Lie point symmetry associated with the elementary conserved vector is obtained and used to generate the invariant solution. The analytical solution is derived when the effective viscosity depends only on the distance along the jet. The numerical solution is obtained when the effective viscosity depends also on the distance across the jet. The eddy viscosity causes an apparent increase in the viscosity of the mean flow which produces an increase in the width of the jet due to an increase in diffusion and also a decrease in the maximum mean velocity along the axis of the jet.     

Mutation invariant functions on cluster ensembles

We define the notion of a mutation invariant function on a cluster ensemble with respect to a group action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We show that these invariants have geometric and number theoretic interpretations, and classify them for ensembles associated to affine Dynkin diagrams. The primary tool used in this classification is the relationship between cluster algebras and the Teichmüller theory of surfaces.     

Mathematical modeling of solidification process near the inner core boundary of the Earth

Radially symmetric analytic solutions of the heat and mass transfer equations governing convection in the Earth’s fluid core are found in terms of deviations from the adiabatic reference state. We demonstrate that an increase of the convective velocity leads to a decrease of the light constituent mass fraction and specific entropy. Where fluid is rising/descending, convective motions decrease/increase the mass fraction and entropy at the inner core boundary (ICB). The influence of convective motions on the thermal fluxes at the core mantle boundary is studied. On the basis of exact solutions we demonstrate that the liquid is supercooled near the ICB. An important point is that an increase in the convective velocity directed to the ICB increases the constitutional supercooling. We show that the anelastic model (AM) can be used only at small supercoolings near the ICB. The most probable solidification scenario “constitutional supercooling and morphological instability” should be described by a mushy layer theory near the ICB and by the AM in the rest region of the fluid outer core. On the basis of dendritic theory and selection mechanisms of crystal growth the dendrite tip radius and interdendritic spacing in the mushy layer at the ICB are determined in the presence of convection.     

Group invariant solution for a free jet on a hemi-spherical shell

When two circular jets impinge upon each other along the axis of a hemi-spherical shell then a free jet on a hemi-spherical shell is formed. The governing equations are Prandtl’s momentum boundary layer equation and the continuity equation. The conserved quantity is required for the free jet on a hemi-spherical shell. The conserved quantity for the free jet on a hemi-spherical shell is established with the help of a conserved vector. The group invariant solution for the third-order partial differential equation for the stream function is constructed.     

圆形垂直浮力射流的稳定性与混合特性研究

建立浅水静止环境中圆形轴对称垂直浮力射流的k-ε模型,采用混合有限分析方法进行了数值计算．针对两种不同的流动形态:近区的混合流体以浮力表面层的形式沿径向扩散的稳定排放;近区产生旋涡,浮力热水对混合热水形成二次挟带的非稳定排放,并对稳定性判据进行了验证,最后对两种不同流动结构下的远区的混合特性进行了数值模拟,结果同Lee和Jirka的试验和理论资料均十分吻合．     

Symmetries in equations of incompressible viscoelastic Maxwell medium<Superscript>*</Superscript>

We consider unsteady flows of incompressible viscoelastic Maxwell medium with upper, low, and Jaumann convective derivatives in the rheological constitutive law. We give characteristics of a system of equations that describe a three-dimensional motion of such a medium for all three types of convective derivative. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. We study group properties of this system in the two-dimensional case. On this basis, we choose submodels of the Maxwell model that can be reduced to hyperbolic ones. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. We also present concrete examples of invariant solutions.     

The Coupling of Gravity Waves and Convection: Amplitude Equations and Planform Selection

Convective motion in a layer of fluid heated from below is considered where the boundaries are stress free and the upper surface supports interfacial gravity waves. Inviscid, immiscible, constant density, ambient fluid is separated from the convecting layer below by a stable density jump. An important parameter in the problem is δ representing the ratio of the interfacial density jump to the density change across the convecting layer. Amplitude equations are derived describing convective motion in the plane and a planform selection analysis performed. It is demonstrated that the breaking of the translational and Galilean invariance of the problem allows a strong coupling between a large-scale interfacial mode and convection. The resulting phase dynamics is third order in time.     

Geometric Invariant Theory in a Model-Independent Analysis of Spontaneous Symmetry and Supersymmetry Breaking

Functions which are covariant or invariant under the transformations of a reductive linear algebraic group can be advantageously expressed in terms of functions defined in the orbit space of the group, i.e. as functions of a finite set of basic invariant polynomials. This fact and the tools of geometric invariant theory can be exploited in many physical contexts where the study of covariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry and/or supersymmetry breaking in possibly supersymmetric quantum field theories of elementary particles, in the analysis of phase spaces and structural phase transitions in solid state physics (Landau's theory), in covariant bifurcation theory, in crystal field theory and in most areas of solid state theory where use is made of symmetry adapted functions. We shall present some elements of geometric invariant theory and illustrate some of the possible applications in the study of spontaneous symmetry and supersymmetry breaking.     

A criterion of the linear long wave stability of steady magnetohydrodynamic jet flows of an ideal fluid

The problem of the linear stability of steady axisymmetric shear magnetohydrodynamic jet flows of an inviscid ideally conducting incompressible fluid with a free boundary is investigated. It is assumed that the jet is of unlimited length, there is a longitudinal constant electric current along its surface, and it is directed along the axis of a cylindrical shell with infinite conductivity, such that there is a vacuum layer between its free boundary and the inner surfaces of the shell. The necessary and sufficient condition for the stability of such flows with respect to small axisymmetric long-wave perturbations of special form is obtained by Lyapunov's direct method. Bilateral exponential estimates of the growth of small perturbations are constructed in the case when this stability condition breaks down, where the indices in their exponents are calculated from the parameters of the steady flows and the initial data for the perturbations. An example of a steady axisymmetric shear magnetohydrodynamic jet flow and of the initial small axisymmetric long-wave perturbations imposed on it is given, which, at the linear stage, will evolve in time and space in accordance with the estimates constructed.     

Threshold model of a neural ensemble

Networks of threshold elements whose inputs are assigned positive and negative (inhibitory) weights and outputs take the values 0 and 1 are considered. A stationary ensemble is defined as a connected subnetwork of a threshold network for which the unit state (1, 1, …, 1) = 1 is stable. The transfer of an ensemble into the state 1 is called switching on. Necessary and sufficient conditions for a network to be an ensemble are given. It is shown that, in the proposed model, the switching on of one of two ensembles having common elements does not necessarily lead to the switching on of the other.     

Magnetic catalysis in a P-even, chiral-invariant three-dimensional model with four-fermion interaction

We study how an external magnetic field H affects the phase structure of a P-even, chiral-invariant three-dimensional model of field theory with a four-fermion interaction. An arbitrarily small, but nonzero, magnetic field induces the spontaneous breaking of the initial symmetry (the magnetic catalysis phenomenon). Depending on the values of the coupling constants, the model vacuum is either P-even or chiral invariant for H≠0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 323–338, August, 2000.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

A convection model for fire spread simulation

We present a convection model which can be coupled with fire propagation models in order to take into account the wind and the slope which are two of the most relevant factors affecting surface fire spread. An asymptotic analysis gives a three-dimensional convective model governed by a two-dimensional equation.     

The invariant solution of thermal diffusion equations for a non-linear buoyancy force

A model of the thermal-diffusion convection of a binary mixture when there is a non-linear dependence of the buoyancy force on the temperature and concentration is considered. An invariant solution, which describes the steady flow of the mixture in a plane vertical layer, is constructed and investigated. The effect of non-linearity of the buoyancy force on the type of flow is examined.     

Singularity formation in thin jets with surface tension

We derive and study asymptotic models for the dynamics of a thin jet of fluid that is separated from an outer immiscible fluid by fluid interfaces with surface tension. Both fluids are assumed to be incompressible, inviscid, irrotational, and density-matched. One such thin jet model is a coupled system of PDEs with nonlocal terms—Hilbert transforms—that result from expansion of a Biot-Savart integral. In order to make the asymptotic model well-posed, the Hilbert transforms act upon time derivatives of the jet thickness, making the system implicit. Within this thin jet model, we demonstrate numerically the formation of finite-time pinching singularities, where the width of the jet collapses to zero at a point. These singularities are driven by the surface tension and are very similar to those observed previously by Hou, Lowengrub, and Shelley in large-scale simulations of the Kelvin-Helmholtz instability with surface tension and in other related studies. Dropping the nonlocal terms, we also study a much simpler local model. For this local model we can preclude analytically the formation of certain types of singularities, though not those of pinching type. Surprisingly, we find that this local model forms pinching singularities of a very similar type to those of the nonlocal thin jet model. © 1998 John Wiley & Sons, Inc.     

Poincaré bifurcation of a three-dimensional system

Consider a three-dimensional system having an invariant surface. By using bifurcation techniques and analyzing the solutions of bifurcation equations, we study the spatial bifurcation phenomena near a family of periodic orbits and a center in the invariant surface respectively. New formula of Melnikov function is derived and sufficient conditions for the existence of periodic orbits are obtained. An application of our results to a modified van der Pol–Duffing electronic circuit is given.     

Ein Tropfenakkreszenzmodell in Atmosphäre von homogen isotroper Turbulenz

Summary In the present work a turbulence model is set up which shows accrescence qualities for small droplets during its time of diffusion dissociation. At the same time the life time of the single element, and by that the quantitative accrescence ratio which originates in an turbulence element, is defined by the genral turbulence statistic.By means of a calculated example it is shown how the accrescence density in one real situation is being computed for a whole statistical ensemble. The results show good conformity with available experimental data.     

Likelihood ratio tests for equality of shape under varying degrees of orientation invariance

We consider a problem from image cytometry where the objective is to describe possible changes in the shape and orientation of cellular nuclei after treatment with a toxin. The shapes of nuclei are represented by individual ellipses. It is argued that the shape comparison problem can be formulated as a generalization of a hypothesis test for the equality of covariance matrices. For many cell types, the test statistic should be invariant with respect to orientations of the cells. For other cell types, the test statistic should be equivariant with respect to orientations of the cells, but invariant with respect to orientations of the images. Likelihood ratio tests (LRTs) are derived under a Wishart model. The likelihood maximization uses a new result about the minimization of the determinant of a sum of matrices under individual rotations. The applicability and limitations of these LRTs are demonstrated by means of simulation experiments. The reference distributions of the test statistics under the null hypothesis are obtained using unrestricted and restricted randomization procedures. Justification for the Wishart model is provided using a residual diagnostic method. The scientific implications of the results are considered.     

Study of free convective boundary layer of isothermal lateral surface of axisymmetrical horizontal body

Approximate analytical solution of simplified Navier–Stokes and Fourier–Kirchhoff equations describing free convective heat transfer from isothermal surface has been presented. It is supposed that the surface has the horizontal axis of symmetry and its axial cross-section lateral boundary is a concave function. The equation for the boundary layer thickness is derived for typical for natural convection assumptions. The most important are that the convective fluid flow is stationary and the normal to the surface component of velocity is negligibly small in comparison with the tangential one. The theoretical results are verified by two characteristic cases of the revolution surfaces namely for horizontal conic and vertical round plate. Both limits of presented solution coincide with known formulas.