A variety of matter capable of flowing exhibits yield stress, amongst others foams. The generation of foams occurs by creating new gas-solid surfaces. This is often done by injecting and dispersing gas. Thus, a large body of publications related gas injection is available. 

However,  your question appears to be ambigous in the following sense: In general foams exhibit yield stress phenomena when they are considered as continous matter. On the other hand, the structure of the continous phase (in general liquid matter capable of flowing) can be per se a yield stress fluid. In both cases scarce systematic investigations have been published as yet. 

Best regards

Thank you for your reply.

You are right. My question is ambiguous.

The problem that interests me is the injection of a gas in a tank filled with a liquid showing a yield stress.

The gas jet is immersed in the yield stress fluid.

One of the applications is the injection of biogas in waste to stimulate the activity of bacteria.

Enhanced oil recovery (EOR) is used to describe various processes applied to increase hydrocarbon recovery from petroleum reservoir fields to above the level of natural depletion.Injection of gas is the most commonly applied EOR technique. The gas may either originate from the production from an oil or a gas condensate field or be produced specifically with the purpose of being used as injection gas.(Besides,the microorganisms could  also be injected to release oil form porous media)

the motion of fluid for the Injection of gas in reservoir is described by Darcy's Law( linear relation), and many commercial reservoir simulators haved been developed;In your case,the leading description may be Navier–Stokes equations.

We have done some work on gas injection in a viscoplastic fluid which is contained in a tube of constant or variable cross section. The refs are:

Dimakopoulos, Y. & Tsamopoulos, J., "Transient displacement of viscoplastic fluids by air in straight or suddenly constricted tubes", J. NonNewt. Fluid Mech., 112/1, 43-75 (2003).

Dimakopoulos, Y. & Tsamopoulos, J., "Transient displacement of Newtonian and Viscoplastic liquids by air from complex conduits", J. Non Newt. Fluid Mech., 142(1-3) 162-182 (2007),

or the opposite problem:

Papaioannou, J., Karapetsas, G., Dimakopoulos Y. and Tsamopoulos, J. "Injection of a viscoplastic material inside a tube or between parallel disks: conditions for wall detachment of the advancing front J. Rheol. 53(5), 1155-1191 (2009),

Maybe you will find them helpful for your problem.