tailieunhanh - Velocity of long bubbles transported by a liquid in horizontal channel

The exact solutions for the bubble velocity and the liquid film thickness are given under an integral form. An approximate analytical solution can be found when the mean liquid velocity is weak enough versus the drift velocity. Two cases are investigated: the Couette flow and the Poiseuille flow. At first order, the two cases lead to identical solutions. The influence of the velocity distribution appears only at the second order. | Vietnam Journal of Mechanics, VAST, Vol. 32, No. 3 (2010), pp. 182 – 190 VELOCITY OF LONG BUBBLES TRANSPORTED BY A LIQUID IN HORIZONTAL CHANNEL Ha Ngoc Hien1 , Jean Fabre2 1 Institute of Mechanics, VAST 2 Institut de Mécanique des Fluides de Toulouse, France Abstract. The influence of flowing liquid on the motion of long bubbles in horizontal channel is studied theoretically. The method of Benjamin (1968) is extended to the case of moving liquid: the liquid is inviscid, the surface tension is ignored and the liquid motion at infinity is characterized its velocity profile. The exact solutions for the bubble velocity and the liquid film thickness are given under an integral form. An approximate analytical solution can be found when the mean liquid velocity is weak enough versus the drift velocity. Two cases are investigated: the Couette flow and the Poiseuille flow. At first order, the two cases lead to identical solutions. The influence of the velocity distribution appears only at the second order. 1. INTRODUCTION The motion of bubbles in a channel or a tube is influenced by two mechanisms: gravity and transport by the mean liquid velocity. Nicklin et al. [1] have showed by experiments that the bubble velocity in vertical tube in a flowing liquid with the mean velocity u0 , can be expressed in the form: V = C0 u0 + V∞ , (1) where V∞ is the bubble drift velocity in stagnant liquid due to the force of Archimedes and C0 is a factor whose value depends on the liquid velocity profile imposed upstream the bubble. The influence of the liquid motion on the bubble velocity was analyzed theoretically by Collins et al. [2] and Bendiksen [3] for axis-symmetrical bubble in vertical pipes. The flow is supposed to be inviscid and rotational, the upstream vorticity distribution being determined by a given velocity profile. Collins et al. [2] have obtained expressions for the bubble velocity, which agree well with the experimental results and the results of Bendiksen taken into