The generation of beams of three-dimensional periodic internal waves in an exponentially stratified fluid

The spectral method is used to construct an exact solution of the linearized problem of the generation of disturbances by localized sources that execute arbitrary periodic motions in a viscous exponentially stratified fluid. The expressions obtained do not contain any adjusting parameters and describe conical beams of three-dimensional periodic internal waves and two types of boundary layers, the spatial scale of which is given by the kinematic viscosity and the buoyancy frequency of the medium. The thickness of one of them, which is analogous to Stokes periodic flow in a homogeneous viscous fluid, is specified by the kinematic viscosity and the wave frequency, that is, it additionally depends on a ratio of the wave and buoyancy frequencies. The thickness of the specific internal boundary layer also depends on the geometry of the problem. In the approximation of weak stratification and low viscosity, asymptotic estimates of the expressions obtained are presented for two types of generators, namely, in the form of a plane inclined rectangle that vibrates along its surface (a frictional source) and along the normal to it (a piston source) in the non-degenerate case when the wave cone does not touch the radiating plane. In limiting cases the analytical expressions obtained agree with known exact solutions of the problem of generating axially symmetric and two-dimensional periodic internal waves.     

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.     

Non-Boussinesq turbulent buoyant jet of a low-density gas leaks into high-density ambient

In this article, we study the problem of low-density gas jet injected into high-density ambient numerically which is important in applications such as fuel injection and leaks. It is assumed that the local rate of entrainment is consisted of two components; one is the component of entrainment due to jet momentum while the other is the component of entrainment due to buoyancy. The integral models of the mass, momentum and concentration fluxes are obtained and transformed to a set of ordinary differential equations using some similarity transformations. The resulting system is solved to determine the centerline quantities which are used to get the mean axial velocity, mean concentration and mean density of the jet. Therefore, the centerline and mean quantities are used together with the governing equation to determine some important turbulent quantities such as, cross-stream velocity, Reynolds stress, velocity-concentration correlation, turbulent eddy viscosity and turbulent eddy diffusivity. Throughout this paper the developed model is verified by comparing the present results with experimental results and jet/plume theory from the literature.     

Conservation laws and conserved quantities for laminar two-dimensional and radial jets

A systematic way to derive the conserved quantities for the liquid jet, free jet and wall jet using conservation laws is presented. Both two-dimensional and radial jets are considered. The jet flows are described by Prandtl’s momentum boundary layer equation and the continuity equation. The multiplier approach (also know as variational derivative approach) is first applied to construct a basis of conserved vectors for the system. The basis consists of two conserved vectors. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived for the liquid jet and the free jet. The multiplier approach is then applied to construct a basis of conserved vectors for the third-order partial differential equation for the stream function. The basis consists of two local conserved vectors one of which is a non-local conserved vector for the system. The conserved quantities for the free jet and the wall jet are derived from the corresponding conservation laws and boundary conditions. The approach gives a unified treatment to the derivation of conserved quantities for jet flows and may lead to a new classification of jets through conserved vectors and their multipliers.     

Numerical analysis of a corrected Smagorinsky model

The classical Smagorinsky model's solution is an approximation to a (resolved) mean velocity. Since it is an eddy viscosity model, it cannot represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. This model has recently been corrected to incorporate this flow and still be well-posed. Herein we first develop some basic properties of the corrected model. Next, we perform a complete numerical analysis of two algorithms for its approximation. They are tested and proven to be effective.     

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.     

平面射流中纳米粒子积聚的矩方法

应用大涡模拟方法求解平面湍射流场,矩方法求解纳米粒子的一般动力学方程．通过对每种情况3 000个时间步的平均,得到了Schmidt数和Damkohler数对纳米粒子动力学特性的影响．结果发现, 当气体参数不变时,Schmidt数的变化只对直径小于1 nm的颗粒数密度的分布产生影响．直径小的颗粒其颗粒数密度沿流动方向下降迅速,而具有大Schmidt数的颗粒,沿横向的分布较窄．较小的颗粒容易发生积聚和扩散,并且体积增长较快,因而颗粒多分散性较为明显．小的颗粒积聚时间尺度能增强颗粒的碰撞和积聚频率,导致颗粒尺寸迅速增大．Damkohler数越大,颗粒的多分散也越明显．     

Fluid flow in an inhomogeneous granular medium

The seepage of a compressible fluid in an inhomogeneous undeformable granular medium is investigated. It is assumed that the fluid flow in a porous space is described by the Navier–Stokes equations. It is shown that, in the case of an inhomogeneous velocity field, a tensor of additional effective stresses occurs in connection with the transfer of fluid particles in a transverse direction when flow occurs around the granules of the medium in a longitudinal direction. Using the fundamental propositions of Reynolds’ averaging theory and Prandtl's mixing path, the structure of the effective viscosity coefficient is determined and hypotheses are formulated which enable it to be assumed to be independent of the flow velocity. It is established by comparison with experimental data that the effective viscosity coefficient can exceed the viscosity coefficient of the flowing fluid by an order of magnitude. The equations of average motion are obtained, which in the case of an incompressible fluid have the form of the Navier–Stokes equations with body forces proportional to the velocity. It is established that, in addition to the well-known dimensionless flow numbers, there is a new number which characterizes the ratio of the Darcy porous drag forces to the effective viscosity forces. The proposed equations are extended to the case of the flow of an aerated fluid. The components of the angular momentum vector are used as the required functions instead of the components of the velocity vector. This enables a solving system of equations to be obtained, which, apart from the notation, is identical with the similar equations for the case of an incompressible fluid. The solution of a new problem of the fluid flow in a plane channel with permeable walls is presented using three models: Darcy's law for an incompressible and aerated fluid, and also of an aerated fluid taking the effective viscosity into account. It is established that, for the same pressure drop, the maximum flow rate corresponds to Darcy's law. Compressibility leads to its reduction, but by simultaneously taking into account the compressibility and the effective viscosity one obtains minimum values of the flow rate. The effective viscosity and aeration of the fluid has a considerable effect on the flow parameters.     

Turbulent diffusion of mass in circular pipe flow

An implicit finite difference scheme was used to solve the convective-diffusion equation to predict the steady-state transport of a conservative, neutrally bouyant tracer injected along the centreline into a fully developed turbulent pipe flow. Three different distributions for the radial mass diffusivity have been compared with two independent sets of experimental data. The results indicate that the distribution based on the turbulent kinematic eddy viscosity predicted by a k?l model produces the closest agreement between the numerical model predictions and the experimentally observed tracer distribution.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.     

Invariance for stochastic differential systems with time-dependent constraining sets

The first part of this article presents invariance criteria for a stochastic differential equation whose state evolution is constrained by time-dependent security tubes. The key results of this section are derived by considering an equivalent problem where the square of distance function represents a viscosity solution to an adequately defined partial differential equation. The second part of the paper analyzes the broader context when solutions are constrained by more general time-dependent convex domains. The approach relies on forward stochastic variational inequalities with oblique reflection, the generalized subgradients acting as a reacting process that operates only when the solution reaches the boundary of the domain.     

Numerical solution of turbulent jet flows

The work deals with numerical modelling of several turbulent 3D jet flows: steady impinging jet, steady free jet in cross–flow, synthetic free jet (unsteady) and synthetic impinging jet (unsteady). The numerical method is based on artificial compressibility method with dual time extension for unsteady cases. Space discretization uses cell–centered finite volume method with third order accurate upwind approximation for convection, the time discretisations are implicit. Turbulence is modelled using two–equation eddy viscosity models and by explicit algebraic Reynolds stress model (EARSM by Wallin and Hellsten). The results of first three cases are compared with measurements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Stabilizing the Strongly Nonlinear Internal Wave Model

A strongly nonlinear asymptotic model describing the evolution of large amplitude internal waves in a two-layer system is studied numerically. While the steady model has been demonstrated to capture correctly the characteristics of large amplitude internal solitary waves, a local stability analysis shows that the time-dependent inviscid model suffers from the Kelvin–Helmholtz instability due to a tangential velocity discontinuity across the interface accompanied by the interfacial deformation. An attempt to represent the viscous effect that is missing in the model is made with eddy viscosity, but this simple ad hoc model is shown to fail to suppress unstable short waves. Alternatively, when a smooth low-pass Fourier filter is applied, it is found that a large amplitude internal solitary wave propagates stably without change of form, and mass and energy are conserved well. The head-on collision of two counter-propagating solitary waves is studied using the filtered strongly nonlinear model and its numerical solution is compared with the weakly nonlinear asymptotic solution.     

Lie group analysis for the effect of temperature-dependent fluid viscosity with thermophoresis and chemical reaction on MHD free convective heat and mass transfer over a porous stretching surface in the presence of heat source/sink

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and two second-order ordinary differential equation corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. It is found that with the increase of magnetic field intensity the fluid velocity decreases but the temperature increases at a particular point of the heated stretching surface. Impact of thermophoresis particle deposition with chemical reaction in the presence of heat source/sink plays an important role on the concentration boundary layer. The results thus obtained are presented graphically and discussed.     

Multi-dimensional BSDE with oblique reflection and optimal switching

In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimates. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.     

Diffraction of a non-stationary linearly inhomogeneous acoustic wave which slides off a semi-infinite soft screen

An exact analytical solution of a new non-stationary scalar diffraction problem is obtained and analysed. A plane acoustic wave with a profile in the form of a delta function propagates along a semi-infinite soft screen. The wave amplitude varies linearly along the wave front. After reaching the end of the screen it “slides” off the screen, generating a diffraction field. A special modification of the Smirnov–Sobolev method is used to find this field. The solution is obtained in the form of an elementary function. It is shown that the sliding wave excites a travelling perturbation that is unlimited along the length of the screen. A similar phenomenon obviously also occurs when elastic waves slide from a cut (crack), which must be taken into account, in particular, in fracture theory.     

A simple solution of the bridged crack problem

A simple, yet effective solution is derived to the problem of a penny-shaped crack, whose faces are bridged by fibers, and subjected to normal loading. The mathematical apparatus developed by the author earlier (Fabrikant, 1989, 1991) is used to reduce the governing integral equation to a sequence of integrals of elementary functions, which can be easily solved by the method of collocations. The case of uniform loading is considered as an example. The accuracy of the obtained solution is rigorously verified by different criteria.     

Continual mechanics of monodisperse suspensions. rheological equations of state for suspensions of moderate concentration : PMM vol. 37, n6, 1973, pp. 1059–1077

Rheological relationships linking mean and moment stresses and, also, the force and moment of interphase reaction in a macroscopic flow of small solid sphere suspension with the kinematic characteristics of the flow are derived. This makes it possible to close the system of equations of suspension hydrodynamics. Coefficients of viscosity and of moment viscosity of a suspension are obtained and calculated.The equations of conservation of mass, momentum and moment of momentum of suspension and of its phases, considered (from the macroscopic point of view) to be coexistent continuous media, were formulated in a general form in [1]. These equations contain unknown vectors and tensors which define the interaction between the considered continuous media and, also, stresses and moment stresses appearing when these are in motion. To close the equations of conservation it is necessary to express all these quantities in terms of unknown variables of these equations (mean concentration of suspension, pressure in the fluid phases, and phase velocities). This problem is the second of the fundamental problems of hydromechanics of suspensions indicated in [1].Here this problem is solved with the use of a kind of self-consistent field theory, which is essentially an extension and generalization of methods developed in [2 – 7]. Expressions for all of the quantities mentioned above are derived. They can be considered to be rheological equations of state for suspensions. Expressions for the various coefficients of these equations and their dependence on parameters of phases and on the flow frequency spectrum are also considered.