Numerical analysis of oscillatory Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for a hard sphere molecular gas

The time-dependent motion of a rarefied gas between two parallel planes caused by an oscillatory motion of the plane is studied based on the linearized Boltzmann equation for a hard sphere molecular gas. With the aid of a deterministic numerical method, an accurate numerical analysis is carried out for a wide range of gas rarfaction and oscillatory frequency. The detailed data of the shear stress acting on the planes is provided in a complete form for a wide range of the parameters. The transition of the solution from low to high frequencies under various degrees of gas rarfaction is discussed.     

Implicit finite element schemes for stationary compressible particle-laden gas flows

The derivation of macroscopic models for particle-laden gas flows is reviewed. Semi-implicit and Newton-like finite element methods are developed for the stationary two-fluid model governing compressible particle-laden gas flows. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers. This is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters and the additional strong nonlinearity caused by interfacial coupling terms. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. The original Jacobian is replaced by a low-order approximation. Special emphasis is laid on the numerical treatment of weakly imposed boundary conditions. It is shown that the proposed approach offers unconditional stability and faster convergence rates for increasing CFL numbers. The strongly coupled solver is compared to operator splitting techniques, which are shown to be less robust.     

Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom

A system of model kinetic equations is proposed to describe flows of a diatomic rarefied gas (nitrogen). A conservative numerical method is developed for its solution. A shock wave structure in nitrogen is computed, and the results are compared with experimental data in a wide range of Mach numbers. The system of model kinetic equations is intended to compute complex-geometry three-dimensional flows of a diatomic gas with rotational degrees of freedom.     

Heat transfer in a gas mixture between parallel plates

The one-dimensional steady-state heat flux and the temperature distribution in a rarefied gas mixture between two parallel plates with different temperatures are studied using the kinetic theory. The Boltzmann equation is solved by the projection method assuming that the gas consists of elastic hard spheres and the reflection from the surfaces is diffuse. The flow features are analyzed for a wide range of the Knudsen number. The molecular numerical densities of the components, the total temperature of the mixture, and the mixture heat flux are obtained. The behavior of the distribution functions for the components is discussed. A comparison with other authors’ results shows that the accuracy of the given method is good.     

Geometrical series and phase space in a finite oscillatory motion

This article discusses some interesting physical propertiesof oscillatory motion of a particle on two joined inclined planes.The geometrical series demonstrates that the particle will oscillateduring a finite time. Another detail is the converging pathto the origin of the phase space. Due to its simplicity, thismotion may be used as a didactic example for the non-lineardynamic study.     

On the shock wave spectrum for isentropic gas dynamics with capillarity

We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities.     

Propagation of a shock wave in an isentropic gas with small viscosity

A new method is considered for constructing the front and the dynamics of motion of a shock wave propagating in an isentropic gas with small viscosity with a speed close to the speed of sound. The proposed algorithm of solution makes it possible to considerably raise the accuracy and effectiveness of numerical computations on the computer as compared with known universal numerical methods based on the use of difference schemes.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 273–308, 1977.     

Implicit numerical method for computing three-dimensional rarefied gas flows on unstructured meshes

The method based on the numerical solution of a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The basic idea behind the method is the use of a second-order accurate TVD scheme on hybrid unstructured meshes in physical space and a fast implicit time discretization method without iterations at the upper level. The performance of the method is illustrated by computing test examples of three-dimensional rarefied gas flows in variously shaped channels in a wide range of Knudsen numbers.     

An analytical and numerical study of liquid dynamics in a one‐dimensional capillary under entrapped gas action

Capillary dynamics has been and is yet an important field of research, because of its very relevant role played as the core mechanism at the base of many applications. In this context, we are particularly interested in the liquid penetration inspection technique. Because of the obviously needed level of reliability involved with such a non‐destructive test, this paper is devoted to study how the presence of an entrapped gas in a close‐end capillary may affect the inspection outcome. This study is carried out through a one‐dimensional ordinary differential model that despite its simplicity is able to point out quite well the capillary dynamics under the effect of an entrapped gas. The paper is divided into two main parts; the first starts from an introductory historical review of capillary flows modelling, goes on presenting the one‐dimensional second order ordinary differential model, taking into account the presence of an entrapped gas and therefore ends by showing some numerical simulation results. The second part is devoted to the analytical study of the model by separating the initial transitory behaviour from the stationary one. Besides, these solutions are compared with the numerical ones, and finally, an expression is deduced for the threshold radius switching from a fully damped transitory to an oscillatory one. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiscale approach to computation of three-dimensional gas mixture flows in engineering microchannels

A multiscale approach to computing real gas flows in engineering microchannels on high-performance computer systems in a wide range of Knudsen numbers is described. The numerical implementation of the approach combines the solution of quasigasdynamic equations and the molecular dynamics method. Following the approach, the parameters of the real gas equation of state are found at the molecular level, the kinetic gas properties are calculated, and the form of boundary conditions on the microchannel walls are determined. The technique is verified by computing several test problems. The results agree well with available theoretical and experimental data.     

Synchronous motion of two vertically excited planar elastic pendula

The dynamics of two planar elastic pendula mounted on the horizontally excited platform have been studied. We give evidence that the pendula can exhibit synchronous oscillatory and rotation motion and show that stable in-phase and anti-phase synchronous states always co-exist. The complete bifurcational scenario leading from synchronous to asynchronous motion is shown. We argue that our results are robust as they exist in the wide range of the system parameters.     

Numerical Challenges for Resolving Spike Dynamics for Two One-Dimensional Reaction-Diffusion Systems

Asymptotic and numerical methods are used to highlight different types of dynamical behaviors that occur for the motion of a localized spike-type solution to the singularly perturbed Gierer–Meinhardt and Schnakenberg reaction-diffusion models in a one-dimensional spatial domain. Depending on the parameter range in these models, there can either be a slow evolution of a spike toward the midpoint of the domain, a sudden oscillatory instability triggered by a Hopf bifurcation leading to an intricate temporal oscillation in the height of the spike, or a pulse-splitting instability leading to the creation of new spikes in the domain. Criteria for the onset of these oscillatory and pulse-splitting instabilities are obtained through asymptotic and numerical techniques. A moving-mesh numerical method is introduced to compute these different behaviors numerically, and results are compared with corresponding results computed using a method of lines based software package.     

Some features of viscous gas flow injected into a supersonic stream from the lateral surface of a cylinder

The article presents a procedure and results of numerical analysis of the structure of flow near an elongated cylinder with a spherically blunted nose that moves with a supersonic velocity with local injection of gas from its lateral surface. The analysis is based on the solution of the full Navier-Stokes equations for a compressible gas. Nonstationary fluctuations in the region of the condensation discontinuity are detected in the given range of Mach and Reynolds numbers. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996. pp. 119–124.     

Visualization in mechanics: the dynamics of an unbalanced roller

It is well known that mechanical engineering students often find mechanics a difficult area to grasp. This article describes a system of equations describing the motion of a balanced and an unbalanced roller constrained by a pivot arm. A wide range of dynamics can be simulated with the model. The equations of motion are embedded in a graphical user interface for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the system dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration the simulation tool has been applied to a range of roller–pivot arm configurations. In addition, approximations to the equations of motion are explored and a second-order model is shown to be accurate for a limited range of parameters.     

High-order accurate conservative method for computing the poiseuille rarefied gas flow in a channel of arbitrary cross section

A high-order accurate method is proposed for analyzing the isothermal rarefied gas flow in an infinitely long channel with an arbitrarily shaped cross section (Poiseuille flow). The basic idea behind the method is the use of hybrid unstructured meshes in physical space and the application of a conservative technique for computing the gas velocity. Examples of calculations are provided for channels of various cross sections in a wide range of Knudsen numbers. Schemes of the first-, second-, and third orders of accuracy in space are compared.     

Application of moment equations to the mathematical simulation of gas microflows

An approach to the simulation of moderately rarefied gas flows in a transition zone is developed. The applicability of the regularized Grad 13-moment (R13) equations to the numerical simulation of a transition flow between the continual and free-molecular gas flow regimes is explored. For the R13 equations, a numerical method is proposed that is a higher order accurate version of Godunov’s explicit method. A numerical procedure for implementing solid-wall boundary conditions is developed. One- and two-dimensional test problems are solved, including the shock wave structure and the Poiseuille flow in a plane channel. The possibility of applying the R13 equations to the simulation of plane channel and jet flows in a transition regime is explored. To this end, the flow in a square cavity generated by the motion of one of the walls is studied and the operation of the Knudsen pump is analyzed.     

On the effective viscosity of a dilute emulsion of gas bubbles

The problem of the motion of a rigid spherical body in a homogeneous emulsion of gas bubbles is considered in the Stokes approximation, using the self-consistent field method. An expression is obtained for the correction factor in the Stokes formula for the drag of the body in the first approximation with respect to the volume concentration of the dispersed phase. An analytical relation between the correction factor and the ratio of the sizes of the bubbles and the body is found. It is shown that, in the limit when this ratio tends to zero, the correction factor obtained is identical to Taylor's result for the effective viscosity of an emulsion of gas bubbles. In the case of non-point bubbles, the coefficient on the volume concentration in the expression for the effective viscosity of the emulsion can be considerably different from Taylor's result. A similar conclusion was also obtained in the case of the problem of the motion of a spherical bubble of arbitrary size in an emulsion of gas bubbles.     

Nonlinear nonequilibrium kinetic model of the boltzmann equation for monatomic gases

A model kinetic equation approximating the Boltzmann equation in a wide range of nonequilibrium gas states was constructed to describe rarefied gas flows. The kinetic model was based on a distribution function depending on the absolute velocity of the gas particles. Highly efficient in numerical computations, the model kinetic equation was used to compute a shock wave structure. The numerical results were compared with experimental data for argon.     

Simulation of rarefied gas flows on the basis of the Boltzmann kinetic equation solved by applying a conservative projection method

Flows of a simple rarefied gas and gas mixtures are computed on the basis of the Boltzmann kinetic equation, which is solved by applying various versions of the conservative projection method, namely, a two-point method for a simple gas and gas mixtures with a small difference between the molecular masses and a multipoint method in the case of a large mass difference. Examples of steady and unsteady flows are computed in a wide range of Mach and Knudsen numbers.     

Weak oscillations of a gas bubble in a spherical volume of compressible liquid

The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.