How to move a bubble which is trapped by the capillary pressure? I have a question about how to move a trapped bubble in a tube.
If we assume to have a horizontal tube, with water on each side of the bubble. The point to the left of the bubble is point 1, while the point to the right is point 2. 
The capillary pressure equation is: $\Delta P_{cap}=\frac{2\cdot \sigma \cdot cos(\theta)}{R}$
Where $\theta$ is the contact angle, $\sigma$ is the interfacial tension between the gas buble and the water, while $R$ is the radius of the tube. 
Since the bubble is trapped, I assume that pressure in point 2 ($P_2$) is larger than the pressure in point 1 ($P_1$). In order to accomplish this, the contact angle between the bubble and point 2 must be smaller than the contact angle betweenthe bubble and point 1, or the radius of point 2 must be smaller than the radius of point 1:
$$P_2-P_1=\frac{2\sigma cos(\theta_1)}{R}-\frac{2\sigma cos(\theta_1)}{R}=?\tag1$$
My question is therefore how we can get this bubble moving by applying more force on the left side (I want to move it towards the right)? If we apply more force from the left side, won't that increase the pressure $P_2$ the same as $P_1$, because of the capillary pressure, so that $P_2$ will always be larger than $P_1$ (if not the contact angle changes),and the bubble cannot be moved with more pressure? 
 A: The stuck bubble problem happens when there is static equilibrium between the circumferential contact forces of surface tension and the forces due to the pressure difference upstream and downstream of the bubble. The gas pressure inside the bubble is uniform and between the upstream and downstream liquid pressures. In that equilibrium state there is a liquid flow from upstream to downstream of the bubble.
There are three things you can do to unbalance the equilibrium and get the bubble to move: 1. increase upstream liquid pressure. This will initially cause the bubble size to decrease but eventually the contact surface tension will become less than the pressure force. 2. Reduce the downstream pressure. This will initially cause the size of the bubble to expand, but eventually the contact forces will become smaller than the pressure force. 3. Finally you could add a surfactant to the upstream liquid flow. This would reduce surface tension when it reaches the bubble and unbalance the equilibrium.
There may also be another way. The stuck bubble problem is one of laminar flow. If you somehow create pulsatile flow and achieve the Womersly frequency the flow becomes turbulent. This may also unbalance the equilibrium.
A: From an engineering POV you could use sound, either in the audible or ultrasonic range to disrupt the adhesion. A transducer could be placed in the liquid at one end of the tube and arranged so some sound travels along the tube. If that is not possible them maybe attach a transducer to the outside. All experimental though. One of those "try it and see" answers with no theory to back it up.
