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Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure
- Nikolaos A. Pelekasis, John A. Tsamopoulos
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- Journal:
- Journal of Fluid Mechanics / Volume 254 / September 1993
- Published online by Cambridge University Press:
- 26 April 2006, pp. 467-499
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It is well known from experiments in acoustic cavitation that two bubbles pulsating in a liquid may attract or repel each other depending on whether they oscillate in or out of phase, respectively. The forces responsible for this phenomenon are called ‘Bjerknes’ forces. When attractive forces are present the two bubbles are seen to accelerate towards each other and coalesce (Kornfeld & Suvorov 1944) and occasionally even breakup in the process. In the present study the response of two initially equal and spherical bubbles is examined under a step change in the hydrostatic pressure at infinity. A hybrid boundary–finite element method is used in order to follow the shape deformation and change in the potential of the two interfaces. Under the conditions mentioned above the two bubbles are found to attract each other always, with a force inversely proportional to the square of the distance between them when this distance is large, a result known to Bjerknes. As time increases the two bubbles continue accelerating towards each other and often resemble either the spherical-cap shapes observed by Davies & Taylor (1950), or the globally deformed shapes observed by Kornfeld & Suvorov (1944). Such shapes occur for sufficiently large or small values of the Bond number respectively (based on the average acceleration). It is also shown here that spherical-cap shapes arise through a Rayleigh–Taylor instability, whereas globally deformed shapes occur as a result of subharmonic resonance between the volume oscillations of the two bubbles and certain non-spherical harmonics (Hall & Seminara 1980). Eventually, in both cases the two bubbles break up due to severe surface deformation.
Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field
- Nikolaos A. Pelekasis, John A. Tsamopoulos
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- Journal:
- Journal of Fluid Mechanics / Volume 254 / September 1993
- Published online by Cambridge University Press:
- 26 April 2006, pp. 501-527
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The motion of two gas bubbles in response to an oscillatory disturbance in the ambient pressure is studied. It is shown that the relative motion of bubbles of unequal size depends on the frequency of the disturbance. If this frequency is between the two natural frequencies for volume oscillations of the individual bubbles, the two bubbles are seen to move away from each other; otherwise attractive forces prevail. Bubbles of equal size can only attract each other, irrespective of the oscillation frequency. When the Bond number, Bo (based on the average acceleration) lies above a critical region, spherical-cap shapes appear with deformation confined on the side of the bubbles facing away from the direction of acceleration. For Bo below the critical region shape oscillations spanning the entire bubble surface take place, as a result of subharmonic resonance. The presence of the oscillatory acoustic field adds one more frequency to the system and increases the possibilities for resonance. However, only subharmonic resonance is observed because it occurs on a faster timescale, O(1/ε), where ε is the disturbance amplitude. Furthermore, among the different possible periodic variations of the volume of each bubble, the one with the smaller period determines which Legendre mode will be excited through subharmonic resonance. Spherical-cap shapes also occur on a timescale O(1/ε). When the bubbles are driven below resonance and for quite large amplitudes of the acoustic pressure, ε ≈ 0.8, a subharmonic signal at half the natural frequency of volume oscillations is obtained. This signal is primarily associated with the zeroth mode and corresponds to volume expansion followed by rapid collapse of the bubbles, a behaviour well documented in acoustic cavitation experiments.
Forced convection and sedimentation past a flat plate
- Nikolaos A. Pelekasis, Andreas Acrivos
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- Journal:
- Journal of Fluid Mechanics / Volume 294 / 10 July 1995
- Published online by Cambridge University Press:
- 26 April 2006, pp. 301-321
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The steady laminar flow of a well-mixed suspension of monodisperse solid spheres, convected steadily past a horizontal flat plate and sedimenting under the action of gravity, is examined. It is shown that, in the limit as Re → ∞ and ∈ → 0, where Re is the bulk Reynolds number and ∈ is the ratio of the particle radius a to the characteristic length scale L, the analysis for determining the particle concentration profile has several aspects in common with that of obtaining the temperature profile in forced-convection heat transfer from a wall to a fluid stream moving at high Reynolds and Prandtl numbers. Specifically, it is found that the particle concentration remains uniform throughout the O(Re−1/2) thick Blasius boundary layer except for two O(∈2/3) thin regions on either side of the plate, where the concentration profile becomes non-uniform owing to the presence of shear-induced particle diffusion which balances the particle flux due to convection and sedimentation. The system of equations within this concentration boundary layer admits a similarity solution near the leading edge of the plate, according to which the particle concentration along the top surface of the plate increases from its value in the free stream by an amount proportional to X5/6, with X measuring the distance along the plate, and decreases in a similar fashion along the underside. But, unlike the case of gravity settling on an inclined plate in the absence of a bulk flow at infinity considered earlier (Nir & Acrivos 1990), here the concentration profile remains continuous everywhere. For values of X beyond the region near the leading edge, the particle concentration profile is obtained through the numerical solution of the relevant equations. It is found that, as predicted from the similarity solution, there exists a value of X at which the particle concentration along the top side of the plate attains its maximum value ϕm and that, beyond this point, a stagnant sediment layer will form that grows steadily in time. This critical value of X is computed as a function of ϕs, the particle volume fraction in the free stream. In contrast, but again in conformity with the similarity solution, for values of X sufficiently far removed from the leading edge along the underside of the plate, a particle-free region is predicted to form adjacent to the plate. This model, with minor modifications, can be used to describe particle migration in other shear flows, as, for example, in the case of crossflow microfiltration.
Secondary Bjerknes forces between two bubbles and the phenomenon of acoustic streamers
- NIKOLAOS A. PELEKASIS, ALEXANDRA GAKI, ALEXANDER DOINIKOV, JOHN A. TSAMOPOULOS
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- Journal:
- Journal of Fluid Mechanics / Volume 500 / April 2004
- Published online by Cambridge University Press:
- 03 February 2004, pp. 313-347
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The translational velocities of two spherical gas bubbles oscillating in water, which is irradiated by a high-intensity acoustic wave field, are calculated. The two bubbles are assumed to be located far enough apart so that shape oscillations can be neglected. Viscous effects are included owing to the small size of the bubbles. An asymptotic solution is obtained that accounts for the viscous drag on each bubble, for large ${\it Re}$ based on the radial part of the motion, in a form similar to the leading-order prediction by Levich (1962), $C_{D} = 48/{\it Re}_{T}$; ${\it Re}_{T} \to \infty$ based on the translational velocity. In this context the translational velocity of each bubble, which is a direct measure of the secondary Bjerknes force between the two bubbles, is evaluated asymptotically and calculated numerically for sound intensities as large as the Blake threshold. Two cases are examined. First, two bubbles of unequal size with radii on the order of $100\,\umu$m are subjected to a sound wave with amplitude $P_{A} < 1.0$ bar and forcing frequency $\omega_{f} = 0.51\omega_{10}$, so that the second harmonic falls within the range defined by the eigenfrequencies of the two bubbles, $\omega_{10} < 2\omega_{f} < \omega_{20}$. It is shown that their translational velocity changes sign, becoming repulsive as $P_{A}$ increases from 0.05 to 0.1 bar due to the growing second harmonic, $2\omega_{f}$, of the forcing frequency. However, as the amplitude of sound further increases, $P_{A} \approx 0.5$ bar, the two bubbles attract each other due to the growth of even higher harmonics that fall outside the range defined by the eigenfrequencies of the two bubbles. Second, the case of much smaller bubbles is examined, radii on the order of $10\,\umu$m, driven well below resonance, $\omega_{f}/2\pi = 20$ kHz, at very large sound intensities, $P_{A} \approx 1$ bar. Numerical simulations show that the forces between the two bubbles tend to be attractive, except for a narrow region of bubble size corresponding to a nonlinear resonance related to the Blake threshold. As the distance between them decreases, the region of repulsion is shifted, indicating sign inversion of their mutual force. Extensive numerical simulations indicate the formation of bubble pairs with constant average inter-bubble distance, consisting of bubbles with equilibrium radii determined by the primary and secondary resonance frequencies for small and moderate sound amplitudes or by the Blake threshold for large sound amplitudes. It is conjectured that in experiments where ‘acoustic streamers’ are observed, which are filamentary structures consisting of bubbles that are aligned and move rapidly in a cavitating fluid at nearly constant distances from each other, bubbles with size determined by the Blake threshold are predominant because those with size determined by linear resonance are larger and therefore become unstable due to shape oscillations.
Linear stability of a gas boundary layer flowing past a thin liquid film over a flat plate
- NIKOLAOS A. PELEKASIS, JOHN A. TSAMOPOULOS
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- Journal:
- Journal of Fluid Mechanics / Volume 436 / 10 June 2001
- Published online by Cambridge University Press:
- 22 June 2001, pp. 321-352
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The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water film that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, r˙, compared to the free-stream velocity, U∞, and (c) small film thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid film like x3/4; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the film. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the triple-deck regime. Two distinct families of eigenvalues are identified, one corresponding to the well-known Tollmien–Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air–water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We−1) increases, indicating the increasing effect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We−1 ≈ We−1c, beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr−1), indicating the increasing effect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr−1 ≈ Fr−1c, beyond which the family of interfacial waves becomes convectively unstable. As We−1 and Fr−1 are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.
Boundary layer flow of air past solid surfaces in the presence of rainfall
- DIMITRIS N. SMYRNAIOS, NIKOLAOS A. PELEKASIS, JOHN A. TSAMOPOULOS
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- Journal:
- Journal of Fluid Mechanics / Volume 425 / 25 December 2000
- Published online by Cambridge University Press:
- 01 December 2000, pp. 79-110
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The steady two-dimensional laminar flow of an air stream, flowing past a solid surface at high Reynolds number, is examined in the presence of rainfall. As raindrops sediment on the surface they coalesce and form a continuous water film that flows due to shear, pressure drop and gravity, in general. In the limit as the boundary layer and film thickness remain smaller than the radius of curvature of the surface a simplified lubrication-type formulation describes the flow field in the film, whereas the usual boundary layer formulation is applied in the gas phase. In the case of a flat plate and close to the leading edge, x → 0, a piecewise-self-similar solution is obtained, according to which creeping flow conditions prevail in the film and its thickness grows like x3/4, whereas the Blasius solution is recovered in the air stream. Numerical solution of the governing equations in the two phases and for the entire range of distances from the leading edge, x = O(1), shows that the film thickness increases as the rainfall rate, r˙, increases or as the free-stream velocity, U∞, decreases and that the region of validity of the asymptotic result covers a wide range of the relevant problem parameters. In the case of flow past a NACA-0008 airfoil at zero angle of attack a Goldstein singularity may appear far downstream on the airfoil surface due to adverse pressure gradients, indicating flow reversal and eddy formation inside the liquid film, and, possibly, flow separation. However, when the effect of gravity becomes evident in the film flow, as the Froude number decreases, and provided gravity acts in such a way as to negate the effect of the adverse pressure gradient, the location of the singularity is displaced towards the trailing edge of the airfoil and the flow pattern resembles that for flow past a flat plate. The opposite happens when gravity is aligned with the adverse pressure gradient. In addition it was found that there exists a critical water film thickness beyond which the film has a lubricating effect delaying the appearance of the singularity. Below this threshold the presence of the liquid film actually enhances the formation of the singularity.