Interdisciplinary Applied Mathematics

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Maximum


/bubble speed


Electric Odd atrragbt volt/cm


/ °


°f


° 150


Coulombic


* 110


Breakdown


о 98


i


Theory



lower *


bound у


. / Vanishing



*


Cc-Cb


Mr

10″ 10″ Ct, nnl/titcr


FIGURE 8.15. Nondimensional bubble velocity expressed as the capillary number Ca versus ion concentration for different values of voltage drop. (Courtesy of H.-C. Chang.)


that the anionic surfactant produces. In particular, SDS introduces negative charges in the liquid-air interface, and thus a negative zeta potential Zb is created, in contrast to the positive capillary potential Zb at the wall. The corresponding electric double layers contain charge q of opposite sign, and thus    they    drag    the    flow along    opposite    directions    in    the    gap region


due to    the    corresponding    force qE.    If Zc    =    |Zb|,    then    the    net    flow in the


gap is zero and    the    bubble    will    move    as in the    case    of    the    pressure-driven


flow, thus at rather low speeds. However, if |Zb| > Zc, then net reverse flow occurs at the gap, which can cause liquid accumulation in the back of the bubble and eventually an ejection of the bubble. In contrast, the addition of a cationic surfactant will produce a positive Zb thus producing the opposite effect from before, i.e., no bubble motion.


The above arguments are in general valid, but they do not explain the observed window of bubble motion and specifically the upper and lower limits. The EDL scales inversely proportional to the square root of the ion concentration (see equation (7.1)), and thus at high ionic concentrations the EDL almost vanishes. Therefore, the aforementioned mechanism for flow reduction in the film gap is not there, and this explains the upper limit in    the    window    of    Figure    8.15. On    the    other hand, at    very low    ionic

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