Bubble-wall friction in a circular tube (original) (raw)

Effective rheology of bubbles moving in a capillary tube

Physical Review E, 2013

We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The calculation is based on a derivation of the equation of motion of a bubble train, considering the capillary forces and the entropy production associated with the viscous flow. We also calculate the configurational probability of the positions of the bubbles.

A long gas bubble moving in a tube with flowing liquid

International Journal of Multiphase Flow - INT J MULTIPHASE FLOW, 2009

The steady axisymmetric behavior of a long gas bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted mesh for wide ranges of capillary number Ca and Reynolds number Re. As illustrated with a series of computed results, the hydrodynamic stresses due to liquid flow around the bubble tend to shape the middle section of long bubbles into a cylinder of constant radius, with a uniform annular liquid flow adjacent to the tube wall. But the surface tension effect tends to cause nonuniformities in the annular liquid film thickness. The annular liquid film thickness generally increases with increasing Ca, but decreases with increasing Re. In units of the bubble velocity relative to the tube wall, the average liquid flow velocity U¯ is always less than unity, indicating that the bubble always moves faster than the average liquid flow. For convenient p...

Steady axisymmetric motion of a small bubble in a tube with flowing liquid

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010

The steady axisymmetric behaviour of a relatively small bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number C a and Reynolds number R e . Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of R e , the velocity of a bubble increases with bubble size for large values of C a but decreases with bubble size for small values of C a . At large values of R e , however, a bubble of large size appears to move at a slower velocity for any given value of C a . When R e is large (e.g. R e = 100) and C a > 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large R e makes a small bubble of radius greater than a quarter of the tube ...

Effect of buoyancy on the motion of long bubbles in horizontal tubes

As a confined long bubble translates along a horizontal liquid-filled tube, a thin film of liquid is formed on the tube wall. For negligible inertial and buoyancy effects, respectively, small Reynolds (Re) and Bond (Bo) numbers, the thickness of the liquid film depends only on the flow capillary number (Ca). However, buoyancy effects are no longer negligible as the diameter of the tube reaches millimeter length scales, which corresponds to finite values of Bo. We perform experiments and theoretical analysis for a long bubble in a horizontal tube to investigate the effect of Bond number (0.05 < Bo < 0.5) on the thickness of the liquid film and the bubble orientation at different capillary numbers 10 −3 < Ca < 10 −1. We investigate several features of the lubricating film around the bubble. (i) Due to the gravitational effects, the film deposited on the upper wall of the channel is thinner than the film at the bottom wall. We extend the available theory for the film thickness at the front of the bubble in a two-dimensional geometry at low capillary numbers Ca < 10 −3 and finite Bo to account for the effect of larger Ca. The resulting model shows very good agreement with the present experimental measurements. (ii) Due to the asymmetry in the liquid film thickness and the consequent drainage of the liquid from the top to the bottom of the tube, the bubble is inclined relative to the channel centerline and our side-view visualizations allow direct quantification of the inclination angle, which increases with both Bo and Ca. While the inclination angle at the top is smaller than that at the bottom of the tube, the average of these two values follows the predictions of a mass balance analysis in the central region of the bubble. (iii) The inclination of the bubble causes the thickness of the thin film at the back of the bubble to depend on the length of the bubble, whereas the thickness at the front of the bubble does not depend on the bubble length.

The motion of a large gas bubble rising through liquid flowing in a tube

Journal of Fluid Mechanics, 1978

The theory presented here describes the motion of a large gas bubble rising through upward-flowing liquid in a tube. The basis of the theory is that the liquid motion round the bubble is inviscid, with an initial distribution of vorticity which depends on the velocity profile in the liquid above the bubble. Approximate solutions are given for both laminar and turbulent velocity profiles and have the form \begin{equation} U_s = U_c+(gD)^{\frac{1}{2}}\phi(U_c/(gD)^{\frac{1}{2}}), \end{equation}Us being the bubble velocity, Uc the liquid velocity at the tube axis, g the acceleration due to gravity, and D the tube diameter; ϕ indicates a functional relationship the form of which depends upon the shape of the velocity profile. With a turbulent velocity profile, a good approximation to (1) which is suitable for many practical purposes is \begin{equation} U_s = U_s + U_{s0}, \end{equation}Us0 being the bubble velocity in stagnant liquid. Published data for turbulent flow are known to agree...

Consideration of the dynamic forces during bubble growth in a capillary tube

Chemical Engineering Science, 2010

Single bubbles were generated from a capillary tube in quiescent water and the bubble formation process was studied in detail using high-speed video at two pressures, 1.38 and 0.93 kPa. The bubble equivalent spherical radius, r, was derived from the data sets of 100 bubbles: (1) the a and b semi-axes values for an ellipsoidal model and (2) a (more accurate) cylindrical integration of the bubble image in 1-pixel layers. The two methods were compared and showed a significant improvement with the more accurate integration approach. Based on the time trends in the derivatives of a and b, three growth phases were identified and the different forces acting on the bubble were calculated. The analysis showed significant differences between the two cases, despite similar times from appearance to detachment. For the 0.93 kPa case, the bubble shape detachment is described by Cassini oval while for 1.38 kPa it is a lenmiscate. For the study conditions, the momentum force was negligible for both cases; however, the viscous drag force, added mass, and surface tension forces were not. The bubble eccentricity exhibited an oscillatory behavior, which we propose arose from highly non-linear wavephenomena. Finally, only the low-pressure case was in agreement with the predictions of Oguz and Prosperetti (1993); for flows less than the critical flow, the high pressure was not in agreement, indicating a smaller value for the transition to super-critical flow for the study conditions.

Influence of the velocity distribution on the motion of long bubbles in tube

In slug flow, the mechanism that controls the evolution of slug lengths is the hydrodynamic interaction between the long bubbles: the velocity difference between each long bubble and the previous one depends on the velocity distribution of the developing flow in the slug in between. The velocity of a long bubble should depend not only of the mean velocity of the carrying liquid but also of the velocity distribution. To demonstrate and quantify the influence of the velocity distribution, physical and numerical experiments have been undertaken. The physical experiment consists in studying the equilibrium of a stationary bubble in a developing flow created by two consecutive elbows. The numerical experiment is carried out for the inertial regime in the frame of the inviscid theory: the rotational flow is solved with the boundary element method. The sensitivity of the bubble velocity is highlighted.

The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow

Journal of Fluid Mechanics, 1995

Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure-velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The 'piston' is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments.