On contact instabilities of viscoplastic fluids in two-dimensional setting

The Richtmyer–Meshkov and Rayleigh–Taylor instabilities in viscoplastic (Bingham) fluids are studied in two-dimensional setting. The evolution of the Richtmyer–Meshkov instability in a Bingham fluid is analyzed as compared with its evolution in a Newtonian fluid. The critical amplitude of the initial perturbation in the velocity field is estimated. Numerical results obtained for Richtmyer–Meshkov and Rayleigh–Taylor instabilities in a Bingham fluid are presented and compared with those obtained for a Newtonian fluid.     

Oscillatory convection and the Cattaneo law of heat conduction

The problem of thermal convection is investigated for a layer of fluid when the heat flux law of Cattaneo is adopted. The boundary conditions are those appropriate to two fixed surfaces. It is shown that for small Cattaneo number the critical Rayleigh number initially increases from its classical value of 1707.765 until a critical value of the Cattaneo number is reached. For Cattaneo numbers greater than this critical value a notable Hopf bifurcation is observed with convection occurring at lower Rayleigh numbers and by oscillatory rather than stationary convection. The aspect ratio of the convection cells likewise changes.     

水平来流对扰动成长和对流周期性的影响

对Pr=0.0272的纯流体在矩形腔体外加水平来流时,进行二维流体力学基本方程组的数值模拟.研究了该纯流体Rayleigh-Benard对流的一维行波斑图的成长及时空的演化.发现对流成长过程可以划分为3个阶段,即对流发展、对流指数成长和周期变化。在对流指数成长阶段对不同相对Rayleigh(瑞利)数Ra_r的最大垂直流速场随时间变化的情况进行分析,获得了最大垂直流速场指数成长阶段的线性成长率γ_m和相对Rayleigh数Ra_r的关系公式.研究了行波周期受水平来流Reynolds(雷诺)数的影响,揭示了行波对流周期性及其对水平来流Reynolds数的依赖性.     

平均温度分布随时间变化时的Bénard对流

本文假定上、下平板之间温差随时间按指数规律变化,研究当界于两平板之间流体层的平均温度分布随时间变化时的Bénard对流,文中将临界Rayleigh数当作时间的函数,并将其按小参数展开成级数。在时间远离零点时,得到临界Rayleigh数的一个近似到二阶的非常简单的表达式。     

Well-posedness of the free boundary problem in compressible elastodynamics

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.     

Enhanced and reduced natural convection using magnetic fluid in a square cavity

Natural convection using a magnetic fluid was studied in a square cavity under the influence of a permanent magnet. The aim was to explore the degree by which heat transfer may be controlled, enhanced or reduced, by investigating a set of different distances of a permanent magnet to the cavity. These distances of the magnet were set such that the cavity was in some cases fully dominated by buoyancy or by the magnetic body force and in other cases partly dominated by either of both body forces in different parts of the fluid. The effect on heat transfer was characterised by an averaged Nusselt number, Rayleigh and magnetic Rayleigh number. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The long‐time behaviour of the thermoconvective flow in a porous medium

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd.     

On the Rayleigh wave on a curvilinear boundary between elastic and fluid media

Along the boundary between elastic and fluid media, the surface Rayleigh wave propagates. The velocity of this wave v _R0 in the case of a plane boundary is less than the velocity of the Rayleigh wave v _R on a free plane boundary of an elastic medium and less than the velocity v _P0 in a fluid medium. To investigate the velocity v _R0 in the case of curvilinear boundaries, the propagation of Rayleigh waves under consideration along cylindrical and spherical surfaces is studied. The velocity of the Rayleigh wave depends on the curvature of the wave trajectory and the curvature in the direction perpendicular to the trajectory. Furthermore this velocity depends on the presence or absence of a fluid medium. Bibliography: 5 titles.     

Thermal convection problem of micropolar fluid subjected to hall current

Thermal instability of a micropolar fluid layer heated from below in the presence of hall currents is investigated. Using the appropriate boundary conditions on the boundary surfaces of the fluid layer, the frequency equation is derived and then critical Rayleigh number is determined. It is found that hall current parameter has destabilizing effect on the system. For specific values of parameters, oscillatory convection in observed in the system. The behavior of Rayleigh number with wavenumber is also computed for different values of various parameters. The results of some earlier workers have been reduced as a special case from the present problem.     

On the linear stability analysis of a Maxwell fluid with double-diffusive convection

The problem of double-diffusive convection and cross-diffusion in a Maxwell fluid in a horizontal layer in porous media is re-examined using the modified Darcy–Brinkman model. The effect of Dufour and Soret parameters on the critical Darcy–Rayleigh numbers is investigated. Analytical expressions of the critical Darcy–Rayleigh numbers for the onset of stationary and oscillatory convection are derived. Numerical simulations show that the presence of Dufour and Soret parameters has a significant effect on the critical Darcy–Rayleigh number for over-stability. In the limiting case some previously published results are recovered.     

Solution of the boussinesq equations by the finite element method

A finite element procedure is presented for the calculation of two-dimensional transient convective/conductive heat transfer in a fluid region. The governing equations are expressed in terms of the primitive variables; the flow is assumed to be laminar, and the fluid incompressible within the Boussinesq approximation. Three typical problems are examined: flow through a sudden enlargement, natural convection in rectangular enclosures, and natural convection between horizontal concentric cylinders. An assessment of the characteristics of the flow regime is made in association with varying dimensionless Prandtl and Rayleigh numbers, as well as cavity aspects ratios. The upper limit for the Rayleigh number in the present paper is 10⁷. Wherever possible, the results are compared with existing solutions obtained by other numerical methods.     

Non-torsional oscillations of a disc in rotating second order fluid

This paper deals with the non-torsional oscillations of a disc in rotating second-order fluid. The disc and the fluid are initially in a state of rigid rotation and the non-torsional oscillations in its own plane are then imposed on the disc. The depth of penetration of the oscillations is increased due to the presence of the coefficient of visco-elasticity. It tends to infinity when the frequency of the oscillations is twice the angular velocity of rotation, meaning thereby that no equilibrium boundary layer exist. An initial value problem for two cases—(i) one disc bounding a semi-infinite mass of the fluid, (ii) two discs containing the fluid in between them is discussed. The classical Rayleigh layer for second-order fluid is derived as a particualr case and it is also found that steady Ekman layer is reached for large time.     

A boundary value problem of thermal convection

The two-dimensional thermal convection of a viscous fluid in a rectangular region is analyzed using the Boussinesq theory, and the Rayleigh boundary conditions. The existence of a unique convention state bifurcating from each eigenvalue of the linearized theory is established by using the Morse lemma. This establishes the validity of the formal perturbation method for determining the convection states. The effects of imperfections on the transition from the conduction to the convection states are studied. A theorem of Thom is used to justify a previous asymptotic expansion method near the critical Rayleigh number.     

Effect of dust particles on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

The problem of the effect of dust particles on the thermal convection in micropolar ferromagnetic fluid saturating a porous medium subject to a transverse uniform magnetic field has been investigated theoretically. Linear stability analysis and normal mode analysis methods are used to find an exact solution for a flat micropolar ferromagnetic fluid layer contained between two free boundaries. In case of stationary convection, the effect of various parameters like medium permeability, dust particles, non-buoyancy magnetization, coupling parameter, spin-diffusion parameter and micropolar heat conduction parameter are analyzed. For sufficiently large values of magnetic parameter M₁, the critical magnetic thermal Rayleigh number for the onset of instability is determined numerically and results are depicted graphically. It is also observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and dust particles.     

Stability analysis of Rayleigh�CB��nard convection in a porous medium

In this paper we study the problem of Rayleigh?CBénard convection in a porous medium. Assuming that the viscosity depends on both the temperature and pressure and that it is analytic in these variables we show that the Rayleigh?CBénard equations for flow in a porous media satisfy the idea of exchange of stabilities. We also show that the static conduction solution is linearly stable if and only if the Rayleigh number is less than or equal to a critical Rayleigh number. Finally, we show that a measure of the thermal energy of the fluid decays exponentially which in turn implies that the L² norm of the perturbed temperature and velocity also decay exponentially.     

悬浮固粒对二维混合层流动失稳特性的影响*

本文在不可压缩二维混合层流动方程的基础之上,通过添加固粒的作用项,推导得到了修正的瑞利方程;然后用数值计算方法解其特征方程,得到了悬浮固粒的质量密度、固粒和气流的速度比值以及Stokes数不同时二维混合层流动中扰动频率与空间增长率的关系曲线,给出了关于悬浮固粒对流场失稳特性影响的几个重要结论。     

Thermal convection in a simple fluid with fading memory

The problem of thermal convection in a viscoelastic fluid with fading memory is studied. After describing what is meant by a fluid with fading memory, we establish existence and uniqueness for the linearized thermal convection written in an arbitrary bounded domain of the three-dimensional space. Finally, we derive conditions on the Rayleigh number, which guarantee the exponential decay in the linearized stability problem.     

膜与不可压缩有界流体接触时的自由振动

圆形膜与流体接触时的振动被广泛应用于工业中．推导了圆形膜在与不可压缩有界流体接触时,非对称自由振动的自振频率．鉴于膜在不可压缩、非粘性流体中振动引起的小振幅,采用速度势函数来描述流体场．使用两种方法来推导系统的自由振动频率．它们包括变分法及一种近似解法——Rayleigh商法．在用两种方法求得的自由振动频率值之间具有良好的相关性．最后,研究了流体的深度、质量密度以及径向张力对耦合系统自由振动频率的影响．     

Energy stability in the Bénard problem for a fluid of second grade

The critical Rayleigh number for stability of convection in a fluid of second grade is obtained by means of the energy method. If the thermodynamic restrictions derived by Dunn and Fosdick [1] are adopted the nonlinear energy boundary is shown to be the same as that of linear theory, and the nonexistence of subcritical instabilities is deduced.     

On the Influence of Thermal Convection in Classical Taylor‐Couette Flows

The presented work deals with the instabilities that occur in the flow of a viscous fluid between axisymmetric cylinders with a rotating inner and stationary outer cylinder. The results of a numerical study of convective flows are presented. The inner cylinder is rotating and heated from within, while the outer cylinder is stationary and cooled outside. Stationary horizontal endplates are used to seal the annulus, forming an enclosure. The working fluid is silikon oil M3. The flow of oil was rendered visible by injecting aluminium powder. By increasing the Reynolds number with angular velocity of the driving inner cylinder, the flow bifurcates into different types of instabilities. Investigation was aimed to find the values of critical Reynolds and Rayleigh numbers corresponding to the critical speeds and temperature differences at which these instabilities set in. The three‐dimensional problem was modelled numerically using software package FLUENT in which discretization is performed by means of finite volume techniques. Computational grid was created in preprocessor Gambit. Numerical experiments are conducted to determine the interdependence between the heat transfer mechanism and the structure of secondary flows