It is an interesting subject. Could you be a bit more specific in your question? Do you consider surface tension? What kind of decrease are you referring to (z=a vs. z=b or Newtonian liquid vs. non-Newtonian)?

Newtonian vs non-Newtonian. Bubbles rising in the non-Newtonian liquid are larger than bubbles rising in the Newtonian liquid. Both liquids have the same viscosity and surface tension.

If the non-Newtonian fluid is shear-thinning (I read this as less viscosity when shear is applied), I would expect an increased slip velocity of the bubbles. This would cause a reduction of the void fraction even if bubble size remains the same. Mind you, I don't know details such as Reynolds or Weber numbers. Bubbles merging or breaking may be another reason for a change variation of the slip velocity. Again, it is an interesting subject. Good luck with your work.

Dear Petor Gkotsis,

Can you tell me "Both liquids have the same viscosity" means???  Is your Newtonian and non-Newtonian  liquids have same value 

Yes. Both the Newtonian and the non-Newtonian liquids have the same viscosity value.

Is it real fluid ?? or you consider non-Newtonian having a viscosity of the Newtonian fluid. I think it is not practically possible in real cases.  However, you calculate the effective viscosity non-Newtonian fluids using theoretical equation. The effective viscosity is a function of  K,n, length scale, and velocity. Then compare both fluids bulk viscosity of liquid. If bulk liquid viscosity increases, higher viscous forces acts on the bubble, which leads increase in gas phase velocity in the middle of the bubble. This can be understood by analyzing the velocity field inside the bubble.

Dear Somasekhar Goud,

Thanks for the interest. Here is the case. We have two liquids:

1. A water/glycerol solution (70/30 %).

2. A water/glycerol solution (70/30 %) in which a small amount of Xanthan gum is added.

Viscosity of both liquids was measured by a rheometer. Liquid 1 exhibits Newtonian behavior having a constant viscosity of 4 mPas. Liquid 2 exhibits non-Newtonian behavior (Carrea-Yasuda model employed) having a viscosity of 4 mPas (at infinite shear rate).

In upward bubbly flow with liquid 2 bubbles are larger in comparison with bubbles rising in upward bubbly flow with liquid 1. Since both liquids have almost the same viscosity, I think that the increase in size cannot be attributed to viscous forces exerted on the bubble.

Dear Ingo Riess,

Thank you very much.

Dear Petros Gkotsis,

I think we need to look into velocity profiles in the liquid phase, shear rate variation and   viscosity distribution  for non-Newtonian liquids.

As surface tension is same for both, I am expecting viscous  forces  plays important role. For Newtonian liquids  viscosity is  constant  but in the case of non-Newtonian liquids it changes with shear rate or velocity gradient. I hope you are clear in this. I'm working on Power-law liquids. I am  also interested to know the solution for your problem.

Best regards 

sekhar