Experimental study of single bubble motion in a liquid metal column exposed to a DC magnetic field

The motion of single Argon bubbles rising in the eutectic alloy GaInSn under the influence of a DC longitudinal magnetic field (parallel to the direction of bubble motion) was examined. The magnetic field strength was varied up to 0.3 T corresponding to a magnetic interaction parameter N (which measures the ratio of electromagnetic forces to inertial forces) slightly greater than 1. The liquid metal was at rest in a cylindrical container. Bubble and liquid velocities were measured using ultrasound Doppler velocimetry (UDV). The measured bubble terminal velocity showed oscillations indicating a zigzag movement of ellipsoidal bubbles. For small bubbles (d_e ⩽ 4.6 mm) an increase of the drag coefficient with increasing magnetic interaction parameter N was observed, whereas for larger bubbles (d_e ⩾ 5.4 mm) the application of the magnetic field reduces the drag coefficient. The measurements revealed a distinct electromagnetic damping of the bubble induced liquid velocity leading to more rectilinear bubble trajectories when the magnetic field is applied. Moreover, significant modifications of the bubble wake structure were observed. Raising of the magnetic field strength caused an enlargement of the eddies in the wake. The Strouhal number decreases with increasing magnetic interaction parameter N.     

Heat transfer in nucleate pool boiling of aqueous SDS and triton X-100 solutions

Variation in degree of surface wettability is presented through the application of Cooper’s correlative approach (h ∝ M ^−0.5 q _w ^″0.67) for computing enhancement (ϕ) in nucleate pool boiling of aqueous solutions of SDS and Triton X-100 and its presentation with Marangoni parameter (χ) that represents the dynamic convection effects due to surface tension gradients. Dynamic spreading coefficient defined as σ _dyn N _a , which relates spreading and wetting characteristics with the active nucleation site density on the heated surface and bubble evolution process, represents cavity filling and activation process and eliminates the concentration dependence of nucleate pool boiling heat transfer in boiling of aqueous surfactant solutions. Using the dynamic spreading coefficient (σ _dyn N _a  = 0.09q _w ^″0.71), correlation predictions within ±15% for both SDS and Triton X-100 solutions for low heat flux boiling condition (q _w^″ ≤ 100 kW/m²) characterised primarily by isolated bubble regime are presented.     

Marangoni instabilities in small circular containers under microgravity

Circular containers of various aspect ratios a with flat free upper liquid surfaces were heated from below under microgravity to generate the Marangoni instability (MI). We realized “liquid lateral sidewalls” for the containers to come near to the “slippery sidewalls” introduced by Rosenblat et al. (J Fluid Mech 120:91–122, 1982a) and Echebarría et al. (Physica D 99:487–502, 1997), henceforth referred to as RHD and EKP, respectively. The flow structure was visualized by aluminium flakes and recorded on videotape. The MI was clearly observed in all containers above a critical Marangoni number Ma ^c which depends on a. In the first microgravity experiment in a container with a=7.5, we found significant convective heat transport and reported a Nusselt number Nu=1.8 for Ma=4×Ma ^c . In a second microgravity experiment with containers with a=0.5, 0.75, 1.0, 1.5, 2.0, 4.0 and 5.0, various flow structures (azimuthal and radial wave numbers) were observed, depending on a and Ma. The observed scenario compares qualitatively well with the stability curves calculated by RHD and EKP. Frequent switching between modes (2,1) and (1,1) was observed in the container with a=2 at supercritical Ma that is exactly the case for which this behaviour was predicted by EKP for reduced gravity.     

Numerical simulation of a bubble rising in shear-thinning fluids

The motion of a single bubble rising freely in quiescent non-Newtonian viscous fluids was investigated experimentally and computationally. The non-Newtonian effects in the flow of viscous inelastic fluids are modeled by the Carreau rheological model. An improved level set approach for computing the incompressible two-phase flow with deformable free interface is used. The control volume formulation with the SIMPLEC algorithm incorporated is used to solve the governing equations on a staggered Eulerian grid. The simulation results demonstrate that the algorithm is robust for shear-thinning liquids with large density (ρ₁/ρ_g up to 10³) and high viscosity (η₁/η_g up to 10⁴). The comparison of the experimental measurements of terminal bubble shape and velocity with the computational results is satisfactory. It is shown that the local change in viscosity around a bubble greatly depends on the bubble shape and the zero-shear viscosity of non-Newtonian shear-thinning liquids. The shear-rate distribution and velocity fields are used to elucidate the formation of a region of large viscosity at the rear of a bubble as a result of the rather stagnant flow behind the bubble. The numerical results provide the basis for further investigations, such as the numerical simulation of viscoelastic fluids.     

Topological structure bifurcation of temperature field of deformable drop in Marangoni migration

The unsteady processes of the Marangoni migration of deformable liquid drops are simulated numerically in a wider range of Marangoni number (up to Ma = 500) in the present work. A steady terminal state can always be reached, and the scaled terminal velocity is a monotonic function decreasing with increasing Marangoni number, which is generally in agreement with corresponding experimental data. The topological structure of flow field in the steady terminal state does not change as the Marangoni number increases, while bifurcation of the topological structure of temperature field occurs twice at two corresponding critical Marangoni numbers. A third critical value of Marangoni number also exists, beyond which the coldest point jumps from the rear stagnation to inside the drop though the topological structure of the temperature field does not change. It is found that the inner and outer thermal boundary layers may exist along the interface both inside and outside the drop if Ma > 70. But the thickness decreases with increasing Marangoni number more slowly than the prediction of potential flow at large Marangoni and Reynolds numbers.     

Marangoni效应下气泡上升过程的FTM模拟

采用界面跟踪法FTM(Front-Tracking Method),并结合CSF(continuum surface force)模型,研究了在垂直方向上温度分布不均匀的对称流场中由Marangoni效应引起的气泡上升运动问题。模拟了在不同的M a数和Pr数下单气泡及同轴双气泡的运动。研究结果表明,在不同的M a数下气泡的运动差异较大,M a数越大,气泡运动至稳态时的速度越大,且气泡运动的最大速度值与M a数呈正相关关系;增大Pr数所造成的粘度增大或热扩散率减小将削弱气泡的迁移运动;Marangoni对流中双气泡的局部运动证实了温度梯度和气泡运动速度紧密相关。     

Theoretical and experimental studies on the contact line motion of second-order fluid

In this study, we studied the contact line motion of second-order fluids theoretically and experimentally. The theoretical study showed that the positive first normal stress difference (N ₁) increases the contact line velocity while the second normal stress difference (N ₂) does not affect the contact line motion. The increased contact line velocity is caused by the hoop stress acting on the curved stream lines near the contact line. The hoop stress increases the liquid pressure near the contact line, and the increased pressure changes the surface profile to have the smaller curvature and smaller dynamic contact angle. The contribution of N ₁is 1 order of magnitude smaller than the contribution from the viscous component when the Deborah number remains O(1). For experiments, silicone oils of different kinematic viscosities (1,000–200,000 mm ²/s) were used while eliminating the drying problem and shear-thinning effect near the contact line. The silicone oils were well fitted to the second-order fluid model with the positive first normal stress difference. The spreading rate of a silicone oil drop on a solid surface was faster than the spreading rate predicted by the theory for Newtonian fluids. As the theory predicts that N ₁increases the contact line velocity and the experimental result confirms the theoretical prediction, the effect of N ₁is established.     

Numerical study of the laminar thermocapillary flow in a square cavity

The hydrodynamic and heat-transfer processes in the problem of a laminar thermocapillary flow of a viscous incompressible fluid in a square cavity with isothermal vertical and isentropic horizontal surfaces are investigated numerically under the assumption that the gravity is absent, the free surface is flat, and the surface tension depends linearly on the temperature. Calculations were performed by a compact-difference method on irregular grids with a fifth-order accuracy for four Prandtl numbers (Pr=1, 16, 200, and 3000) as the Marangoni (Ma) number varies from 10² to 10⁴. The maximum local heat transfer versus theMa number is obtained. It is shown that, for thePr values considered, the maxima of the distribution of the horizontal velocity component on the surface is displaced to the cold boundary according to a law inversely proportional to theMa number. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 81–89, July–August, 1999.     

Some remarks on “Drag coefficients of a slowly moving sphere in Non-Newtonian fluids”

Viscoelastic solutions were ejected vertically downwards into air and various Newtonian fluids. The measured swell increased significantly when ejected into a liquid rather than air. The observed increase is considered a result of both bouyancy and drag forces on the solution. The following dimensions expression relating the ratio of the swell diameter in liquid and air D_L/D_A to the elastic shear compliance of the ejected solution J_e was experimentally observed.(D_L/D_A)⁶-1=30(Δ?/?_s)^{?$\text{1}\text{2}$}([g²η²_N?_s]$\text{1}\text{3}$J_e)^{$\text{3}\text{5}$}, where Δ? is the density difference between the extruded and Newtonian fluid, ?_s is the solution density, g is the gravitational constant, and η_N is the Newtonian fluid viscosity. Thus with this expression a simple extrudate swell technique exists to estimate the elastic shear compliance of a viscoelastic solution.     

Soret effect and slow mass diffusion as a catalyst for overstability in Marangoni-Bénard flows

We study the onset of time dependent Marangoni-Bénard convection in binary mixtures subject to Soret effect by numerical computation of linear instability thresholds in infinite fluid layers and two-dimensional boxes. The calculations are done for positive Marangoni numbers (Ma > 0) and negative Marangoni Soret parameters S _M = –(D _S γ _c )/(Dγ _T ) where D _S and D are the Soret and mass diffusion coefficients, respectively, and γ _T , γ _c are the first derivatives of the surface tension with respect to temperature and concentration. Our purpose is to understand why for particular choices of Prandtl and Schmidt numbers, the increase of the stabilizing solutal contribution leads to a decrease of the critical temperature difference, a phenomenon already reported by Chen & Chen [5] and Skarda et al. [12] For various choices of Prandtl and Schmidt numbers we analyze the evolution of the critical Marangoni number Ma _c , critical wavenumber k _c and angular frequency ω _c with S _M and compute the corresponding eigenvectors. We next propose a physical mechanism which explains how the stabilizing solutal contribution acts as a catalyst for overstability. Finally, we extend our results to two dimensional boxes of small aspect ratio.     

Oscillatory Marangoni convection around the air bubble in a vertical surfactant stratificationConvection de Marangoni oscillante autour d'une bulle d'air dans une stratification verticale surfactante

The solutocapillary Marangoni convection around a gas bubble in the inhomogeneous binary mixture of miscible fluids with a vertical surfactant concentration gradient was studied experimentally. A new phenomenon, the oscillatory instability of the surfactant mass transfer, near the bubble boundary, was detected and investigated. The interpretation of this effect as an interaction between the surfactant adsorption at the bubble free surface and solutocapillary and buoyancy convective mechanisms is proposed. The experimental data on oscillation period in relation to bubble dimensions, time, liquid layer thickness, physico-chemical fluid parameters and concentration gradients are presented and discussed. To cite this article: K. Kostarev et al., C. R. Mecanique 332 (2004).     

Asymptotic behavior of Toeplitz matrices and determinants

We consider the inverse X _N and determinant D_N(c) of an N×N Toeplitz matrix C_N=[c_i?j] ₀ ^N?1 as N ar∞. Under the condition that there exists a monotonic decreasing summable bound b _n ≧|c _n |+|c _?n |, and that the generating function \(c(\theta ) = \sum\limits_{n = - \infty }^\infty {c_n e^{i{\text{ }}n{\text{ }}\theta } }\) does not vanish, we construct a matrix iterative process which yields (i) explicit asymptotic formulae for the elements of X_N when v(c) = (2π)^?1 [arg{c(2π)}?arg{c(0)}] is zero. Thence we obtain (ii) expressions for the constants, and bounds on the remainder, in the asymptotic formula $$\ln D_N (c) = N{\text{ }}k_0 (c) + E_0 (c) + E_{1,N} (c) + \mathcal{R}_N (c),$$ and (iii) the extension of this formula to the case of general integral v(c). Under certain further conditions the monotonicity of E_1,N+?_N is proved. We discuss various identities for D_N which apply when c(θ) is a rational function of e^iθ and mention a conjecture for D _N when c(θ) has zeros, and is discontinuous with arbitrary v(c).     

Inverse Estimation of the Effective Diffusion of the Filter in the In Situ Diffusion and Retention (DR) Experiment

Porous filters are often used in laboratory and in situ diffusion and retention experiments. The proper interpretation of these experiments requires knowing the effective diffusion, D _e, of the filter which is commonly determined from laboratory diffusion experiments or estimated from the filter porosity. The D _e of the filter in the in situ experiment may differ from the D _e of the filter measured in the laboratory due to pore clogging. Here, we present an inverse method to estimate the D _e of the filter of in situ diffusion experiments. The method has been tested for several sampling schemes, numbers of synthetic data, N, and standard deviations of the noise, ??. It has been applied to the following tracers used in the in situ diffusion and retention (DR) experiment performed in the Opalinus clay at Mont Terri underground research laboratory: HTO/HDO, Br^?,I^?, ²² Na⁺,¹³³ Ba²⁺,⁸⁵ Sr²⁺, Cs⁺/¹³⁷Cs⁺, and ⁶⁰Co²⁺. The estimation error increases with the standard deviation of the noise of the data and decreases with the number of data. It is smallest for sorbing tracers. The D _e of the filter can be properly estimated from 12 data collected within the first 3?days for conservative tracers as long as ????? 0.02 and for sorbing tracers as long as ??????0.05. The estimate of D _e for conservative tracers is poor when data are collected from a 10-day experiment with daily sampling. The convergence of the estimation algorithm for conservative tracers improves by starting with a value of the D _e smaller than the true value. The choice of the initial value of D _e does not affect the convergence of the estimation algorithm for sorbing tracers. Filter clogging and vertical flow though the filter can influence the tracer transport through the filter. The use of the D _e of the filter obtained from a laboratory test for the in situ experiment may result in large errors for strongly sorbing tracers. Such errors can be overcome by estimating the equivalent D _e of the filter with the proposed inverse method which will be useful for the design of in situ diffusion experiments.     

Oscillating convection modes in the surroundings of an air bubble under a horizontal heated wall

The development of different oscillatory modes and their transition into a non-periodic state of convection, initiated by the thermal Marangoni-effect in the vicinity of an air bubble under a horizontal, heated wall, was investigated. In the further surroundings of the air bubble a stably stratified thermal field was maintained. The flow phenomena in the vicinity of the bubble were studied using light sheet and shearing interferometer flow visualization techniques. The observed modes are described with regard to their kinematics. The influence of the Marangoni number and of the bubble geometry on the mode selection is discussed. The boundaries of the different modes and of the non-periodic state are indicated.List of symbols a thermal diffusivity - Bo Bond number, Eq. (4) - c phase velocity, Eq. (6) - g acceleration due to gravity - l characteristic length - Mg Marangoni number, Eq. (1) - n wavenumber - Pr Prandtl number ( = v/a) - r radial coordinate - r _B bubble radius - Ra Rayleigh number ( = ga¦T/r¦l ⁴/va) - Re Reynolds number ( = u _mg l/) - t _p oscillation period - T temperature - T _w wall temperature - u _mg characteristical Marangoni velocity, Eq. (2) - z axial coordinate normal to the heated wall - z _B bubble height Greek letters surface tension - kinematic viscosity - dynamic viscosity Dedicated to Professor Dr.-Ing. Julius Siekmann on the occasion of his 65th birthday     

Non-similar Solution for Natural Convective Boundary Layer Flow Over a Sphere Embedded in a Porous Medium Saturated with a Nanofluid

A boundary layer analysis is presented for the natural convection past an isothermal sphere in a Darcy porous medium saturated with a nanofluid. Numerical results for friction factor, surface heat transfer rate, and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter N _r, Brownian motion parameter N _b, thermophoresis parameter N _t, and Lewis number L _e. The dependency of the friction factor, surface heat transfer rate (Nusselt number), and mass transfer rate (Sherwood number) on these parameters has been discussed.     

Steady rise of a small spherical gas bubble along the axis of a cylindrical pipe at high Reynolds number

Steady irrotational flow of inviscid liquid of density ρ_l around a spherical gas bubble which lies on the axis of a cylindrical pipe is investigated using the analysis of Smythe (Phys. Fluids 4 (1961) 756). The bubble radius b=qa is assumed small compared to the pipe radius a, and the interfacial tension between gas and liquid is γ. Far from the bubble, in the frame in which the bubble is at rest, the liquid velocity along the pipe is v₀, whereas the liquid velocity at points on the wall closest to the bubble is U_zw=v₀(1+1.776q³+⋯). The decrease in wall pressure as the bubble passes is therefore Δp=1.776ρ_lv₀²q³. When the Weber number W=2bv₀²ρ_l/γ is small, the bubble deforms into an oblate spheroid with aspect ratio χ=1+9W(1+1.59q³)/64. If the fluid viscosity μ is non-zero, and the Reynolds number Re=2v₀ρ_lb/μ is large, a viscous boundary layer develops on the walls of the pipe. This decays algebraically with distance downstream of the bubble, and an exponentially decaying similarity solution is found upstream. The drag D on the bubble is D=12πμv₀b(1−2.21Re^−1/2)(1+1.59q³)+7.66μv₀bRe^1/2q^9/2, larger than that given by Moore (J. Fluid Mech. 16 (1963) 161) for motion in unbounded fluid. At high Reynolds numbers the dissipation within the viscous boundary layers might dominate dissipation in the potential flow away from the pipe walls, but such high Reynolds numbers would not be achieved by a spherical air bubble rising in clean water under terrestrial gravity.     

Co-current flow effects on a rising Taylor bubble

The effects of co-current flows on a rising Taylor bubble are systematically investigated by a front tracking method coupled with a finite difference scheme based on a projection approach. Both the upward (the co-current flows the same direction as the buoyancy force) and the downward (the co-current moves in the opposite direction of the buoyancy force) co-currents are examined. It is found that the upward co-current tends to elongate the bubble, while the downward co-current makes the bubble fatter and shorter. For large N_f (the inverse viscosity number), the upward co-current also elongates the skirted tail and makes the tail oscillate, while the downward co-current shortens the tail and even changes a dimpled bottom to a round shape. The upward co-current promotes the separation at the tail, while the downward co-current suppresses the separation. The terminal velocity of the Taylor bubble rising in a moving flow is a linear combination of the mean velocity (U_C) of the co-current and the terminal velocity (U₀) of the bubble rising in the stagnant liquid, and the constant is around 2 which agrees with the literature. The wake length is linearly proportional to the velocity ratio (U_C/U₀). The co-currents affect the distribution of the wall shear stresses near the bubble, but not the maximum.     

Universal Attractors¶for the Navier-Stokes Equations¶of Compressible and Heat-Conductive Fluid¶in Bounded Annular Domains in ℝn

This paper is concerned with the dynamics for the Navier-Stokes equations for a polytropic viscous heat-conductive ideal gas in bounded annular domains Ω _n in ? ⁿ (n= 2, 3). One of the important features of this problem is that the metric spaces H ⁽¹⁾ and H ⁽²⁾ we work with are two incomplete metric spaces, as can be seen from the constraints θ >0 and u> 0, withθ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, δ₃, δ₄, δ₅ satisfying certain conditions, two sequences of closed subspaces H ^{( i )} _δ?H ^{( i )} (i= 1,2) are found, and the existence of two (maximal) universal attractors in H ⁽¹⁾ _δ and H ⁽²⁾ _δ is proved.     

Equilibrium states in homogeneous turbulence with baroclinic instability

The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ?U _i /?x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise ${(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)}The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ∂U _i /∂x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise (?B/?x₂ = N_h² = -2WS){(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)} and vertical (?B/?x₃ = N_v²){(\partial B{/}\partial x_3\,{=}\,N_v^2)} directions. Computations based on the rapid distortion theory (RDT) are performed for several values of the rotation number R = 2Ω/S and the Richardson number R_i = N_v²/S² < 1{R_i\,{=}\,N_v^2/S^2 <1 }. It is shown that, during an initial phase, the energies and the buoyancy fluxes are sensitive to the effects of pressure and viscosity. At large time, the ratios of energies, as well as the normalized fluxes, evolve to an asymptotically constant value, while the pressure–strain correlation scaled with the product of the turbulent kinetic energy by the shear rate approaches zero. Accordingly, an analytical parametric study based on the “pressure-less” approach (PLA) is also presented. The analytical study indicates that, when R _i  < 1, there is an exponential instability and equilibrium states of turbulence, in agreement with RDT. The energies and the buoyancy fluxes grow exponentially for large times with the same rate (γ in St units). The asymptotic value of the ratios of energies yielded by RDT is well described by its PLA counterpart derived analytically. At R _i  = 0, the asymptotic value of γ increases with increasing R approaching 2 for high rotation rates. At low rotation rates, an important contribution to the kinetic energy comes from the streamwise kinetic energy, whereas, at high rotation rates, the contribution of the vertical kinetic energy is dominant. When 0 < R _i  < 1 and R 1 0{R\ne 0}, the asymptotic value of γ decreases as R _i increases so as it becomes zero at R _i  = 1.     

Axisymmetric simulation of the interaction of a rising bubble with a rigid surface in viscous flow

The interaction between a rising deformable gas bubble and a solid wall in viscous liquids is investigated by direct numerical simulation via an arbitrary-Lagrangian–Eulerian (ALE) approach. The flow field is assumed to be axisymmetric. The bubble is driven by gravity only and the motion of the gas inside the bubble is neglected. Deformation of the bubble is tracked by a moving triangular mesh and the liquid motion is obtained by solving the Navier–Stokes equations in a finite element framework. To understand the mechanisms of bubble deformation as it interacts with the wall, the interaction process is studied as a function of two dimensionless parameters, namely, the Morton number (Mo) and Bond number (Bo). We study the range of Bo and Mo from (2, 6.5 × 10⁻⁶) to (16, 0.1). The film drainage process is also considered in this study. It is shown that the deformation of a bubble interacting with a solid wall can be classified into three modes depending on the values of Mo and Bo.