Analytical solutions for thermal forcing vortices in boundary layer and its applications

Using the Boussinesq approximation, the vortex in the boundary layer is assumed to be axisymmetrical and thermal-wind balanced system forced by diabatic heating and friction, and is solved as an initial-value problem of linearized vortex equation set in cylindrical coordinates. The impacts of thermal forcing on the flow field structure of vortex are analyzed. It is found that thermal forcing has significant impacts on the flow field structure, and the material representative forms of these impacts are closely related to the radial distribution of heating. The discussion for the analytical solutions for the vortex in the boundary layer can explain some main structures of the vortex over the Tibetan Plateau.     

Analytical solutions for dynamics of dual-spin spacecraft and gyrostat-satellites under magnetic attitude control in omega-regimes

The attitude dynamics of a dual-spin spacecraft (a gyrostat with one rotor) with magnetic actuators attitude control is considered in the constant external magnetic field at the presence of the spacecraft’s own magnetic dipole moment, which is created proportionally to the angular velocity components (this motion regime can be called as “the omega-regime” or “the omega-maneuver”). The research of the dual-spin spacecraft angular motion under the action of the magnetic restoring torque is fulfilled in the generalized formulation close to the classical mechanics’ task of the heavy body/gyrostat motion in the Lagrange top. Analytical exact solutions of differential equations of the motion are obtained for all parameters in terms of elliptic integrals and the Jacobi functions. New obtained analytical solutions can be classified as results developing the classical fundamental problem of the rigid body and gyrostat motion around the fixed point. The technical application of the omega-regime to the angular reorientation of the spacecraft longitudinal axis along the angular momentum vector is considered.     

The method of fundamental solutions for Stokes flow in a rectangular cavity with cylinders

Numerical solutions based on the method of fundamental solutions are discussed for Stokes flow inside a rectangular cavity in the presence of circular cylinders. The Stokeslets are used as the fundamental solutions to obtain the solution for the flow field by a linear combination of fundamental solutions. Flow results on the cellular structure of flow field resulting from the dynamics of cylinders and the horizontal walls of the cavity are reported for (i) one rotating cylinder in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions, (ii) two rotating cylinders kept apart in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions. The effect of aspect ratio of the rectangular cavity, direction of movement of the two parallel horizontal sides of the cavity and the diameter of the rotating cylinder on the flow structure are studied. The flow results obtained for the single cylinder case are in accordance with the results available in the literature. From the computational point of view, the present numerical procedure based on the method of fundamental solutions is efficient and simple to implement as compared to the mesh-dependent schemes, which needs complex mesh generation procedure for the multiply connected geometrical domains considered in this article.     

Competitive modes for the Baier–Sahle hyperchaotic flow in arbitrary dimensions

In this article, the stretch-twist-fold (STF) flow is numerically studied using phase portraits, sensitive dependence on initial conditions, Lyapunov exponents, power spectrum, and the Poincaré map. The stretch-twist-fold flow is a two-parameter family of Stokes flows defined in a unit sphere that is associated with the fluid particle motion that naturally arises in the dynamo theory, which proposes a mechanism by which celestial bodies, such as earth and sun can maintain and amplify the magnetic field continuously. For this continuous growth of magnetic field, scientists are interested to invent new tools for the nonfuel consumption magnetism propulsion for the low earth orbit of spacecrafts or satellites. General properties of a chaotic dynamical system reference to the stretch-twist-fold flow model are addressed and numerical solutions are generated to explain some of these properties. Analytically, we studied the local behavior at equilibrium points. The predictability of chaos in the STF flow with the numerical calculation of Lyapunov exponents and Poincaré map is presented in this paper.     

Unsteady shear flow of a viscoplastic material

A viscoplastic, or yield-stress, liquid occupies the space between two infinite parallel plates. Initially the whole system is at rest. The lower plate is suddenly jerked into motion with given speed or shear stress, while the upper plate is kept fixed. The flow consists of two regions; (1) a lower sheared region bounded above by the yield surface, (2) an upper unyielded region bounded below by the yield surface. The yield surface propagates to the upper plate as time proceeds. We first consider the equivalent one plate problem of flow over a jerked plate, and find similarity solutions and small time asymptotic solutions for prescribed shear and speed cases respectively. These solutions are used as initial solutions for the two plate case. The motion of the yield surface and the time taken for the entire material to yield are investigated. The problems considered here are two dimensional representations of some control devices, for example the light duty clutch, which consists of two corotating, coaxial discs separated by a layer of electrorheological material. In this application it is useful to know the time taken for all the material to yield.     

Stability of the motion of a rigid body in an unsteady flow

The problem of rigid-body motion in an unsteady gas flow is considered using a flow model [1] in which the motion of the body is described by a system of integrodifferential equations. The case in which among the characteristic exponents of the fundamental system of solutions of the linearized equations there are not only negative but also one zero exponent is analyzed. The instability conditions established with respect to the second-order terms on the right sides of the equations are noted. The problem may be regarded as a generalization of the problem of the lateral instability of an airplane in the critical case solved by Chetaev [2], pp. 407–408.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 18–22, May–June, 1989.     

An ‘assumed deviatoric stress–pressure–velocity’ mixed finite element method for unsteady,convective, incompressible viscous flow: Part I: Theoretical development

A formulation of a mixed finite element method for the analysis of unsteady, convective, incompressible viscous flow is presented in which: (i) the deviatoric-stress, pressure, and velocity are discretized in each element, (ii) the deviatoric stress and pressure are subject to the constraint of the homogeneous momentum balance condition in each element, a priori, (iii) the convective acceleration is treated by the conventional Galerkin approach, (iv) the finite element system of equations involves only the constant term of the pressure field (which can otherwise be an arbitrary polynomial) in each element, in addition to the nodal velocities, and (v) all integrations are performed by the necessary order quadrature rules. A fundamental analysis of the stability of the numerical scheme is presented. The method is easily applicable to 3-dimensional problems. However, solutions to several problems of 2-dimensional Navier-Stokes' flow, and their comparisons with available solutions in terms of accuracy and efficiency, are discussed in detail in Part II of this paper.     

An invariant solution of the turbulent boundary-layer equations

The problem of the group stratification of the system of equations describing motion in the laminar sublayer and the turbulent core is considered. The fundamental group admissible by the initial system is constructed; invariant solutions constructed on one of the subgroups lead to a system of ordinary differential equations. Joining of the solutions and interchange of the equations occur at the boundary of the laminar sublayer. A class of power-law flows of a turbulent boundary layer is investigated. In the region of decelerated motion a double-valued solution is found corresponding to attached or separated flow. The commonly used integral characteristics are calculated and presented in the form of an interpolation polynomial.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 126–132, July–August, 1975.     

Rotationally symmetric flow over a rotating disk

The rotationally symmetric flow over a rotating disk in an incompressible viscous fluid is analyzed by a new method when the fluid at infinity is in a state of rigid rotation (in the same or in the opposite sense) about the same axis as that of the disk. Asymptotic expansions for the velocity field over the entire flow field are obtained for the general class of one-parameter rotationally symmetric flows. This method is further extended to the case when a uniform suction or injection is assumed at the rotating disk. Fluid motion induced by oscillatory suction of small amplitude at the rotating disk is also discussed.An initial-value analysis reveals that resonance is possible only when the angular velocity of the rotating fluid is greater than that of the rotating disk.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

Axial Couette flow of an Oldroyd-B fluid in an annulus

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated.     

Limiting self-similar motion of a solid particle in a free-molecular gas flow escaping from an orifice

A problem of motion of a single spherical solid particle in the far wake of a free-molecular gas flow escaping from an orifice is considered. It is shown that the spatial distributions of macroscopic parameters of the gas are completely determined by functions of one variable: coordinate along an arbitrary straight line normal to the axis of symmetry. Based on this property, a dimensionless equation of particle motion is derived, which has self-similar solutions: trajectories of motion and particle coordinates (traces) on the target for different initial conditions. Conditions of determining the gas flow behavior on the basis of particle traces on the target are considered.     

Nonlinear effects in absolute and convective instabilities of a near-wake

Linear and nonlinear initial-value problems are discussed for planar inviscid disturbances in streamlined near-wakes. This is mostly for those areas of near-wake flow where the basic motion comprises nearly uniform shear with or without normal influx into the accompanying viscous interfacial layer, although agreement is found with linear properties for full velocity profiles of double-Blasius, double-Jobe–Burggraf, Hakkinen–Rott and Goldstein form. With nonlinear disturbances, wavelike initial conditions yield a known critical-layer development, whereas more general, non-wave, initial conditions lead to a new integro-partial-differential amplitude equation which is studied analytically and numerically. The solutions show decay, finite-time blowup or nonlinear upstream-travelling disturbances. The normal influx proves crucial. Absolute and upstream- or downstream-convective instability is encountered (depending on the profiles, and flow reversal, for example); and in generic cases (for any thin airfoil) nonlinearity is shown analytically to provoke upstream convection. Increased nonlinearity drives the typical transition point extremely close to the trailing edge. Comparisons are made with three-dimensional behaviour in the linear case and with a direct simulation in the nonlinear regime.     

Boundary-layer non-Newtonian flow over vertical plate in porous medium saturated with nanofluid

The free convective heat transfer to the power-law non-Newtonian flow from a vertical plate in a porous medium saturated with nanofluid under laminar conditions is investigated. It is considered that the non-Newtonian nanofluid obeys the mathematical model of power-law. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The partial differential system governing the problem is transformed into an ordinary system via a usual similarity transformation. The numerical solutions of the resulting ordinary system are obtained. These solutions depend on the power-law index n, Lewis number Le, buoyancy-ratio number N _r, Brownian motion number N _b, and thermophoresis number N _t. For various values of n and Le, the effects of the influence parameters on the fluid behavior as well as the reduced Nusselt number are presented and discussed.     

A Method Based on the Radon Transform for Three-Dimensional Elastodynamic Problems of Moving Loads

An integral transform procedure is developed to obtain fundamental elastodynamic three-dimensional (3D) solutions for loads moving steadily over the surface of a half-space. These solutions are exact, and results are presented over the entire speed range (i.e., for subsonic, transonic and supersonic source speeds). Especially, the results obtained here for the tangential loads (one of these loads is along the direction of motion and the other is orthogonal to the direction of motion) are quite new in the literature. The solution technique is based on the use of the Radon transform and elements of distribution theory. It also fully exploits as auxiliary solutions the ones for the corresponding plane-strain and anti-plane shear problems (the latter problems are 2D and uncoupled from each other). In particular, it should be noticed that the plane-strain problem here is completely analogous to the original 3D problem, not only with respect to the field equations but also with respect to the boundary conditions. This makes the present technique more advantageous than other techniques, which require first the determination of a fictitious auxiliary plane-strain problem through the solution of an integral equation. Our approach becomes particularly simple when there is no angular dependence in the boundary conditions (i.e., when axially symmetric problems regarding their boundary conditions are considered). On the contrary, no such constraint needs to be fulfilled as regards the material response (and, therefore, the governing equations of the problem) and/or also possible loss of axisymmetry due to the generation of shock (Mach-type) waves in the medium. Therefore, the solution technique can easily deal with general 3D problems having a largely arbitrary radial dependence in the boundary conditions and involving: (i) material anisotropy in static and dynamic situations, and (ii) asymmetry caused by changes in the nature of governing PDEs due to the existence of different velocity regimes (super-Rayleigh, transonic, supersonic) in dynamic situations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Uniqueness theorems in the theory of thermoelasticity for solids with double porosity

In this paper the coupled linear theory of thermoelasticity for solids with double porosity is considered. The governing system of field equations of this theory is based on motion equations, conservation of fluid mass, constitutive equations, extended Darcy’s law for materials with double porosity and Fourier’s law for heat conduction. A wide class of the basic internal and external boundary value problems (BVPs) of steady vibrations is formulated and uniqueness theorems for regular (classical) solutions of these BVPs are proved.     

Water waves in a suspended container

Analysis and experiments are carried out on a horizontal rectangular wave tank which swings at the lower end of a pendulum. The walls of the tank generate waves which affect the motion of the pendulum. For small displacements of the tank, linearised shallow water equations are used to model the motion, and there exist time-periodic solutions for the system whose periods are governed by a transcendental relation. Numerical and analytic solutions of this relation show that the fundamental period is greater than both the period of the empty tank (moving like a simple pendulum) and the fundamental period of the standing wave which occurs when the tank is removed from its supports and held fixed. For a rectangular tank the theory compares well with some experimental measurements. Qualitative observations are also made of the effect of breaking waves on the tank motion: for a tank which has a mass small compared with its load the energy dissipated by breaking waves can rapidly reduce the amplitude of swing of the tank. Potential flow theory is used with linearised free-surface boundary conditions to find time periodic motions for a tank with a hyperbolic cross section.     

The effect of long-range forces on the dynamics of a bar

The one-dimensional dynamic response of an infinite bar composed of a linear “microelastic material” is examined. The principal physical characteristic of this constitutive model is that it accounts for the effects of long-range forces. The general theory that describes our setting, including the accompanying equation of motion, was developed independently by Kunin (Elastic Media with Microstructure I, 1982), Rogula (Nonlocal Theory of Material Media, 1982) and Silling (J. Mech. Phys. Solids 48 (2000) 175), and is called the peridynamic theory. The general initial-value problem is solved and the motion is found to be dispersive as a consequence of the long-range forces. The result converges, in the limit of short-range forces, to the classical result for a linearly elastic medium. Explicit solutions in elementary form are given in a broad class of special cases. The most striking observations arise in the Riemann-like problem corresponding to a constant initial displacement field and a piecewise constant initial velocity field. Even though, initially, the displacement field is continuous, it involves a jump discontinuity for all later times, the Lagrangian location of which remains stationary. For some materials the magnitude of the discontinuity-jump oscillates about an average value, while for others it grows monotonically, presumably fracturing the material when it exceeds some critical level.