Oscillatory convection in a dilute3He-superfluid4He solution

Convective instabilities in a rectangular, unity-aspect-ratio Rayleigh-Bénard cell with a solution of 1.46%³He in superfluid⁴He have been studied in the temperature range 0.70–1.05 K, with a corresponding Prandtl number range of 0.045<<0.15. The onset of stationary convection is much like that in a classical, one-component fluid. The oscillatory instability is studied by using an extremely sensitive local temperature probe. It is found that the total heat transport efficiency is suppressed by the oscillations in the entire range of Prandtl number we have studied. The local temperature probe indicates a striking difference in the oscillatory amplitude when the sense of rotation of the convective rolls is reversed. The magnitude of the convective velocity is deduced from both the initial slope of the Nusselt number near the onset of the stationary convection and the frequency of the oscillations. Determinations of the temperature dependence of the convective velocity using these two methods agree very well. The observed behavior of the oscillatory frequency and onset condition supports the theory of oscillatory convection for a classical, low-Prandtl-number, one-component fluid.     

Marangoni Convection Instabilities Induced by Evaporation of Liquid Layer in an Open Rectangular Pool

In order to investigate the Marangoni convection instability of 0.65cSt silicone oil induced by evaporation in liquid layer, a series of experiments are carried out in an open rectangular pool. The effects of side wall temperature as well as ambient temperature on competitions between BM convection and thermocapillary convection are analyzed thoroughly. Increasing of the side wall temperature would inevitably enhance thermocapillary convection and suppress the formation of BM cells by transferring hot fluid from border to surface. As long as the side wall temperature is high enough, BM cells would disappear completely and multicellular rolls as well as hydrothermal waves would occur in the whole layer. Increasing ambient temperature would enhance both BM convection and thermocapillary convection, but the later one benefits more from it because hydrothermal waves can occur at a lower Ma number. Critical Marangoni numbers for the incipience of hydrothermal waves and that disappearance of BM convection cells are obtained under different ambient temperatures.     

Thermosolutal convection between poorly conducting plates

Thermosolutal convection is considered in a fluid layer between poorly conducting horizontal boudaries. The horizontal scale of the motions is much greater than the depth of the fluid layer, provided the motion is not too vigorous, and this disparity between horizontal and vertical scales provides the basis foe an asymptotic expansion of the solution. Under the assumption of near-constant solute-flux at the horizontal boundaries, a pair of evolution equations is derived for the depth-averaged temperature and solute concentration fields.These long-wave equations are investigated for two-dimensional convection by numerical integrations, and the results are compared with linear and weakly non-linear theory. The asymptotic expansion is shown to break down for large values of R, when the form assumed for the convection becomes inappropriate.The onset of three-dimensional convection is analysed. Steady square convection cells are stable. Oscillatory convection in the form of two-dimensional travelling-wave rolls is stable to three-dimensional disturbances near onset.     

Mixed convection of micropolar fluids along a vertical wavy surface

Summary This paper presents numerical results for the steady-state mixed convection in micropolar fluids along a vertical wavy surface. The problem has been formulated by a simple trnasposition theorem, and the spline alternating-direction implicit method has been applied to solve the governing momentum, angular momentum and energy equations. The influence of the micropolar parameters (R and ), the amplitude-wave length ratio and the Gr/Re² number on the skin-friction coefficient and Nusselt number have been studied. Results demonstrate that the skin friction coefficient and local Nusselt number consist of a mixture of two harmonics in micropolar fluids and in Newtonian fluids. As the vortex viscosity parameter (R) increases, the heat transfer rate decreases but the skin friction increases. In addition, when the spin gradient viscosity parameter () increases, the heat transfer rate and the skin friction decreases. However, the heat transfer rate of a micropolar fluid is smaller than a Newtonian fluid, but the skin friction of a micropolar fluid is larger than a Newtonian fluid under all circumstances.     

Thermal convection and nonlinear effects of a superfluid3He-4He mixture in a porous medium

The convective instability of superfluid³He-⁴He mixtures in porous media is investigated. The general hydrodynamic equations are derived and reduced to a single nonlinear equation for a scalar field. The superfluid mixtures in a porous medium have a constant⁴He chemical potential and behave essentially like a classical fluid in a porous medium. Two-fluid effects are calculated both at the onset of steady convection and the subsequent boundary of instability. The shift of critical Rayleigh number is about 1% or less at the onset of convection, but can be as large as 20% or more at the instability boundary for some regions aroundT 1 K. This two-fluid shift is quite large compared to the corresponding 0.001% shift at the onset of convection for bulk superfluid³He-⁴He mixtures.     

Oscillatory Marangoni convection in a conducting fluid layer with a deformable free surface in the presence of a vertical magnetic field

Summary. Linear stability theory is applied to the problem of the onset of oscillatory Marangoni convection in a horizontal layer of electrically conducting fluid heated from below in the presence of a vertical magnetic field. The fluid layer is bounded from below by a rigid boundary and from above by a deformable free surface. The critical Marangoni number M _c, the critical wavenumber a _c and the critical frequency _c are obtained for wide ranges of the Prandtl number #E5/E5#₁, the magnetic Prandtl number #E5/E5#₂, the crispation number C _r and the Chandrasekhar number Q. We present numerically a necessary and sufficient condition for oscillatory Marangoni convection to occur.     

Convective Turbulence in Superfluid Solutions 3He–4He

In present work, we continue our experimental investigations of heat instability in superfluid ³He–⁴He solutions heated from below. We research two solutions with ³He concentrations 5.0% and 9.5% for temperature of 270 mK. It is found that for 5% solution the dependence is linear in temperature range studied whereas for the solution of 9.5% we observed the deviation from linear dependence above some critical value . This effect manifests the thermal instability which appears under start of phase separation in 9.5% solution if heat flow is switched on. For 5.0% solution where one does not observe the phase separation at the values of applied, the instability was not observed. To identify the possible mechanism of a thermal instability in stratified solution, we estimated the dependence of the Nusselt number on relative Raileigh number Ra/Ra _c . One observes that the dependence can be fitted as Nu=(Ra/Ra _c ) ^b where b=0.31±0.04. Note that the dependence obtained agrees rather good with the empiric expression of (Busse in Rep. Prog. Phys. 41:1929, 1978) and connecting the numbers Nu and Ra for turbulent convection. This gives grounds to conclude the heat transfer in a stratified solution is realized by transition to the regime of turbulent convection.      

Buoyancy driven convection in vertically shaken granular matter: experiment, numerics, and theory

Buoyancy driven granular convection is studied for a shallow, vertically shaken granular bed in a quasi 2D container. Starting from the granular Leidenfrost state, in which a dense particle cluster floats on top of a dilute gaseous layer of fast particles (Meerson et al. in Phys Rev Lett 91:024301, 2003; Eshuis et al. in Phys Rev Lett 95:258001, 2005), we witness the emergence of counter-rotating convection rolls when the shaking strength is increased above a critical level. This resembles the classical onset of convection—at a critical value of the Rayleigh number—in a fluid heated from below. The same transition, even quantitatively, is seen in molecular dynamics simulations, and explained by a hydrodynamic-like model in which the granular material is treated as a continuum. The critical shaking strength for the onset of granular convection is accurately reproduced by a linear stability analysis of the model. The results from experiment, simulation, and theory are in good agreement. The present paper extends and completes our earlier analysis (Eshuis et al. in Phys Rev Lett 104:038001, 2010).     

Effect of Brinkman boundary layer on the onset of Marangoni convection in a fluid-saturated porous layer

Summary The effect of Brinkmann boundary layer on the onset of convection driven by surface tension gradients, called Marangoni convection, in a thin horizontal fluid-saturated sparsely packed porous layer bounded by adiabatic free boundaries is studied analytically by means of linear stability analysis. The single-term Galerkin expansion technique is shown to be convenient and instructive to establish eigenvalues. The convergence of the results obtained by this approximate technique is checked by comparing the results with those of exact solutions obtained using a regular perturbation technique. By comparing the results of the two techniques, we found that the single-term Galerkin expansion is accurate only for small values of the porous parameter (<10). The effect of large values of on the onset of Marangoni convection is determined using a method of matched asymptotic expansions. The effect of a boundary layer that exists for large values of is shown to increase the critical Marangoni number by an amount of 2 compared to that for small values of . This uniformly valid solution permits a unified treatment of Marangoni convection and provides the means for a deeper explanation of the physical phenomena. The results obtained are valid for highly porous materials of current practical importance.     

Thermocapillary instability of an infinite Prandtl number fluid with negligible gravitational effects

Summary Finite amplitude fluid motion is investigated in a horizontal layer of an infinite Prandtl number fluid with an upper free surface for the case where thermocapillary effects are significant and gravitational effects are negligible. It is found that subcritical instability exists and that two-dimensional rolls and down-hexagons (where motion is downward at the cells' centers) are always unstable. But up-hexagons (where motion is upward at the cells' centers) are stable for sufficiently small amplitude , while both uphexagons and squares are stable in a range of larger where hysteresis effects exist.With 1 Figure     

Fatigue crack growth of PMMA in organic agents

Crack growth tests under cyclic loading were executed at 295 K in various organic agents using compact tension and pure bending specimens of polymethylmethacrylate (PMMA). The cyclic frequencies f for the two kinds of test were 0.4 to 1 and 33 Hz, respectively. Two interesting features are pointed out: (i) transitional behaviour is observed on a crack growth rate against stress intensity factor range K diagram, and (ii) the fatigue fracture surfaces tested in highly viscous agents are covered with a new type of striation named wavy striation, as reported previously. The crack growth rate at the transition was analysed based on fluid flow through the pores within the craze forming at the crack tip. The wavy striation was also investigated by use of the theory of meniscus instability. It is found that both the phenomena may be well described by a parameter P = T(K) ²/f where T and are the surface tension and viscosity of the organic agents, respectively.     

Mixed convection flow of micropolar fluid over an isothermal plate with variable spin gradient viscosity

Summary A steady two-dimensional mixed convection flow of viscous incompressible micropolar fluid past an isothermal horizotal heated plate with uniform free stream and variable spin-gradient viscosity is considered. With appropriate transformations the boundary layer equations are transformed into nonsimilar equations appropriate for three distinct regimes, namely, the forced convection regime, the free convection regime and the mixed convection regime. Solutions of the governing equations for these regimes are obtained by an implicit finite difference scheme developed for the present problem. Results are obtained for the pertinent parameters, such as the buoyancy parameter, in the range of 0 to 10 and the vortex viscosity parameters, =0.0, 1.0, 3.0, 5.0 and 10.0 for fluid with Prandtl number Pr=0.7 and are presented in terms of local shear-stress and the local rate of heat transfer. Effects of these parameters are also shown graphically on the velocity, temperature and the couple stress distributions. From the present analysis, it is observed that both the momentum boundary layer and the thermal boundary layer increase due to an increase in the vortex viscosity of the fluid.List of symbols f, F, dimensionless stream function for forced convection free convection and mixed convection, respectively - g acceleration due to gravity - Gr_x local Grashof number - j micro-inertia density - m ₂₃ distribution of couple stress - N microrotation component normal to (x, y)-plane - p pressure of the fluid - q dimensionless rate of heat transfer - Re_x local Reynolds number - T temperature of the fluid in the boundary layer - T temperature of the ambient fluid - T temperature at the surface - u, v thex andy-components of the velocity field - U free stream velocity - x, y axis in direction along and normal to the plate Greek thermal diffusivity - coefficient of volume expansion - vortex viscosity parameter - stream function - , , nondimensional similarity variables - buoyancy parameter (=Gr _x Re _x ^/5/2 ) - vortex viscosity - density of the fluid - v kinematic coefficient of viscosity - spin-gradient viscosity - stream function - dimensionless skin-friction - fluid viscosity     

Oscillatory convection induced by gravity and centrifugal forces in a rotating porous layer distant from the axis of rotation

The linear stability theory is used to investigate analytically the effects of gravity on centrifugally driven convection in a rotating porous layer offset from the axis of rotation. The stability of a basic solution is analysed with respect to the onset of stationary and oscillatory convection. It is also demonstrated that the stationary mode is the critical mode of convection thereby resulting in the convection rolls being aligned parallel to the axis of rotation. Besides providing a non-motionless basic solution and dictating the direction of the wave number, gravity plays a passive role and does not affect the stability results.     

Bénard-Marangoni convection in two thin immiscible liquid layers with a uniform magnetic field

Summary The effect of a uniform magnetic field on Bénard-Marangoni convection in a shallow cavity, with differentially heated side walls, filld with two viscous, immiscible, incompressible an electrically conducting fluids is studied in the presence of a buoyancy force. The fluid-fluid interface and the free surface are assumed to be flat, and the driving forces for the flow are the thermocapillary and the buoyancy forces. Closed-form solutions, under thin layer approximation neglecting the side wall effects, are obtained for the stream function and the temperature. The solutions are obtained in various limiting cases, namely: absence of buoyancy force, absence of thermocapillary force and absence of magnetic field, and they coincide with the results existing in the literature. The velocity is calculated, and the resulting cell patterns are discussed for different values of , the ratio of the temperature gradients of surface tension at the interface and the free surface and the Hartmann number. There exist four different flow regimes depending on the values of but with reduced convection compared to the non-magnetic case for Marangoni convection. It is observed that it is possible to control the convection in the lower layer by a suitable choice of the magnetic field.     

The stability of the narrow-gap Taylor-Couette system with an axial flow

Summary The stability of viscous flow between concentric rotating cylinders with an axial flow due to an axial pressure gradient is considered. The governing equations with respect to three-dimensional disturbances are derived and solved by a direct numerical procedure. Results are given for the case of small-gap approximation. Three typical cases =–1,0 and 0.5 are studied, where represents the ratio of angular velocity of the outer cylinder to that of the inner cylinder. The value of the axial Reynolds numberR is up to 100. It is found that the critical disturbance is a non-axisymmetric mode when the value ofR is sufficiently large, and the transition of the onset mode withR is demonstrated in detail. Results for the critical Taylor number, wave number, vortex incline angle, and relative wave velocity are also determined. The present stability analysis is found to be in agreement with previous experimental studies and particularly reveals the stability characteristics with the variation of .     

Three-dimensional evaluation of the T-stress in centre cracked plates

Thick three-dimensional (3-D) finite element models of centre cracked plates are used to study the variation in the biaxiality factor with the crack aspect ratio a/w. The use of two widely accepted methods to evaluate the T-stress in two-dimensions, namely the boundary layer method and the displacement field method, to calculate the T-stress in three-dimensions is studied. It is shown that the boundary layer method gives results that compare rather well with the two-dimensional plane strain values (maximum difference of 6 percent), while the displacement field method results are about 15 percent lower. Two parameters are shown to affect the three-dimensional evaluation of the biaxiality factor, namely the material's Poisson's ratio and the specimen's thickness t. The biaxiality factor is directly proportional to and inversely proportional to t. Three-dimensional analysis is required to assess correctly the effect of and t on .     

Numerical Simulation of Thermocapillary Convection in a Shallow Rectangular Cavity Under the Action of Combining Horizontal Temperature Gradient with Vertical Heat Flux

In order to understand the influence of the vertical heat flux on thermocapillary convection, we conducted a series of unsteady two-dimensional numerical simulations of thermocapillary convection in a differently heated shallow rectangular cavity with vertical heat flux on the bottom by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and aspect ratio is 30. It is found that a small vertical heat flux has slightly influence on the flow pattern of stable or unstable thermocapillary convection. However, the critical Marangoni number increases first, and then decreases with the increase of the heat flux. And the flow pattern of the oscillatory thermocapillary convection transits from a series of the rolls rotating clockwise and moving from the cold wall to the hot wall to the single roll near the hot wall and a series of rolls near the cold wall, further, two series of rolls moving from the hot wall and cold wall towards the hot spot with the maximum temperature. With the increase of the Marangoni number, the period and the wavelength of the oscillatory thermocapillary convection increase, but the wave speed decreases.     

Boundary-layer analysis of forced convection with a plate and porous substrate

Summary. The classical boundary-layer analysis of Blasius and Pohlhausen for forced convection from a flat plate in the presence of a uniform external stream is extended to the case where a thin layer of a porous medium is attached to the plate. It is found that the effect of the porous layer is to increase the heat flux as represented by a Nusselt number, and to within the approximations made this increase is independent of the Darcy number. The deviation increases with the Prandtl number, and also increases with the new parameter defined by Eq. (9).     

Non-Darcy regime mixed convection on vertical plates in saturated porous media with lateral mass flux

Summary A unified treatment is presented of mixed convection on vertical plates embedded in fluid saturated porous media with prescribed variable plate temperature or surface heat flux for the case of non-Darcy limiting regime. The plates are permeable with lateral mass flux. By suitable similarity transformations, it is shown that the two problems of prescribed temperature and prescribed heat flux lead to identical differential equations with two common boundary conditions and third boundary condition differing in the two cases. The effect of lateral mass flux and the free stream on the Nusselt number and the energy transport by the boundary layer is investigated. The unified approach to the mixed convection problem includes free convection problem as a special case. Exact analytical solutions are obtained for two cases of free convection problem.Notation b inertial coefficient - c specific heat - D _p pore diameter - E rate of upward energy transport - Ê dimensionless rate of energy transport - f dimensionless stream function - f _w mass flux parameter - g acceleration due to gravity - k thermal conductivity - K permeability of the porous medium - m exponent in the variation of heat flux - M mixed convection parameter - Nu _x Nusselt number - Pe _x Peclet number - q _w surface heat flux - Ra _x local Rayleigh number - Ra _x ^* modified local Rayleigh number - T temperature - T _e ambient temperature - T _w plate temperature - T _w temperature difference=T _w-T_e - u velocity in thex-direction - u _e free stream velocity - v velocity in they-direction - v _w lateral velocity at the plate - x coordinate along the plate in the upward direction - y coordinate normal to the plate - equivalent thermal diffusivity =k/c _e - coefficient of thermal expansion - porosity - dimensionless variable - dimensionless temperature - viscosity - fluid density - _e ambient density - exponent in the variation of plate temperature - stream function