Growth of Bubbles in Reservoirs and its Consequences on the Foam Formation

Abstract

The effect of partial confinement on the shape and volume of bubbles generated by injection of gas at a constant flow rate, into a highly viscous liquid is studied numerically and experimentally. By using the Boundary Element Method, numerical solutions of the Stokes equations for the viscous liquid yield the evolution of the surface of a bubble. These solutions and experiments show that cylindrical, conical, and pipe walls with periodic corrugations, concentric with the gas injection orifice in the horizontal bottom of the liquid, may strongly affect the shape and volume of the bubbles. Thus, the presence of walls could be used to control the size of the generated bubbles without changing the gas flow rate. A well-known scaling law for the volume of the bubbles generated by injection of gas at a high flow rate in a highly viscous, unconfined liquid is extended to take into account the presence of cylindrical or conical walls around the injection orifice. In addition, we study numerically the thickness film that is formed between the free surface of a bubble and the cylindrical walls in both cases.

Appendix A. The Green’s functions for Axi-symmetric flow.

This Appendix lists Green functions for axi-symmetric flow generated by a unit ring force located at $({x}_{0}, {r}_{0})$ and pointing in the direction $\mathbf{e}_{\alpha }$ with $\alpha = { r}, { x}$. Define

$$\begin{aligned} Z&= x - x_{0} ,\\ L&= \sqrt{Z^{2} + (r + r_{0} )} ,\\ D&= \sqrt{Z^{2} + \sqrt{Z^{2} + (r + r_{0} )} } , \\ S&= \sqrt{Z^{2} + r^{2} + r_{0}^{2} } , \\ m&= \frac{{2(rr_{0} )^{{\frac{1}{2}}} }}{L}, \\ \end{aligned}$$

and the elliptic integrals

$$\begin{aligned} K\left( m \right)&= \mathop \smallint \limits _{0}^{{\frac{\pi }{2}}} \frac{{d\theta }}{{\sqrt{1 - m^{2} sen^{2} \theta } }}~,\\ E\left( m \right)&= \mathop \smallint \limits _{0}^{{\frac{\pi }{2}}} \sqrt{1 - m^{2} sen^{2} \theta d\theta } \end{aligned}$$

Then

$$\begin{aligned} G_{x}^{x}&= 4\frac{r}{L}\left( {K + E\frac{{Z^{2} }}{{D^{2} }}} \right) \! , \\ G_{r}^{x}&= 2\frac{Z}{L}\left( {K - E\frac{{S^{2} - r^{2} }}{{D^{2} }}} \right) \! ,\\ G_{r}^{x}&= 2\frac{{rZ}}{{r_{0} L}}\left( { - K + E\frac{{S^{2} - 2r_{0}^{2} }}{{D^{2} }}} \right) \! ,\\ G_{r}^{r}&= 2\frac{1}{{r_{0} L}}\left[ {k\left( {S^{2} + Z^{2} } \right) - E\left( {L^{2} + \frac{{Z^{2} S^{2} }}{{D^{2} }}} \right) } \right] \! ,\\ {{T_{x}^{x}}}_{x}&= 8\frac{{rZ^{3} }}{{D^{2} }}\left( {K - E\frac{{4S^{2} }}{{D^{2} }}} \right) \! ,\\ {{T_{x}^{x}}}_{r}&= T^{{x}_{x}}_{r} = - 4\frac{{Z^{2} }}{{D^{2} L~}}\left[ {K\frac{{S^{2} - 2r^{2} }}{{L^{2} }} - E\left( {1 + \frac{{8r_{0}^{2} \left( {2r_{0}^{2} - S^{2} } \right) }}{{D^{2} L^{2} }}} \right) } \right] \! ,\\ {{T_{r}^{x}}}_{r}&= - 4\frac{{rZ}}{L}\left[ {K\left( {\frac{1}{{r^{2} }} + \frac{{2Z^{2} }}{{D^{2} L^{2} }}} \right) - \frac{E}{{D^{2} }}\left( {6 - S^{2} \left( {\frac{1}{{r^{2} }} + \frac{{8Z^{2} }}{{D^{2} L^{2} }}} \right) } \right) } \right] \! ,\\ {{T_{x}^{r}}}_{x}&= - 4\frac{{rZ^{2} }}{{r_{0} D^{2} L}}\left[ {K\frac{{2r_{0}^{2} - S^{2} }}{{L^{2} }} + E\left( {1 + \frac{{8r_{0}^{2} \left( {2r^{2} - S^{2} } \right) }}{{D^{2} L^{2} }}} \right) } \right] \! ,\\ {{T_{x}^{r}}}_{r}&= T^{r_r}_{x} = - 4\frac{Z}{{r_{0} }}\left[ {K\left( {\frac{{Z^{2} S^{2} }}{{D^{2} L^{2} }} - 2} \right) + \frac{E}{{D^{2} }}\left( {2S^{2} - Z^{2} - \frac{{16r^{2} r_{0}^{2} Z^{2} }}{{D^{2} L^{2} }}} \right) } \right] \! ,\\ {{T_{r}^{r}}}_{r}&= - 4\frac{r}{L}\left[ \frac{K}{{r_{0} }}\left( {\frac{{Z^{2} \left( {S^{2} - 2r_{0}^{2} } \right) }}{{D^{2} L^{2} }} - \frac{{r^{2} - r_{0}^{2} - 2Z^{2} }}{{r^{2} }}} \right) \right. \\&\qquad \left. + \frac{E}{{D^{2} }}\left( {\frac{{8r_{0} Z^{2} \left( {S^{2} - 2r^{2} } \right) }}{{D^{2} L^{2} }} + \frac{{r^{2} \left( {r^{2} + r_{0}^{2} } \right) - S^{2} \left( {r_{0}^{2} + 2Z^{2} } \right) }}{{r^{2} r_{0} }}} \right) \right] \! . \end{aligned}$$