Conditions for static bubbles in viscoplastic fluids

Abstract

We consider the slow motion of a gas bubble in a cylindrical column filled with a viscoplastic fluid, modeled here as a Herschel–Bulkley fluid. Because of the yield stress of the fluid, it is possible that a bubble will remain trapped in the fluid indefinitely. We adapt Prager’s two variational principles to our problem. From these variational principles we develop two general stopping conditions, i.e., for a given bubble we can calculate a critical Bingham number above which the bubble will not move. The first condition is derived by bounding the velocity field and the second condition by bounding the stress field. We illustrate these conditions by considering specific bubble shapes, e.g., axisymmetric bubbles. We also develop a condition for bubble motion.

Motion and shape of bubbles rising through a yield-stress fluid

“…They found for a Bingham fluid that motion occurred only for values of Y less than a yield criterion Y c . While recent experimental results [8] for spheres falling through simple (non-aging [9]) yieldstress fluids agree well with theoretical predictions [7,10,11], there are rather few previous papers concerned with the complementary problem of rising bubbles in yield-stress materials [4][5][6][12][13][14][15][16][17] or with the displacement of a yield-stress fluid by gas [18][19][20]. Theoretical treatment of this problem is complicated by the possibility of coupling among the fluid's rheological properties, the shape of the bubble, and its motion, as well as by the nonlinearity of model constitutive relations for viscoplastic fluids.…”

“…Theoretical treatment of this problem is complicated by the possibility of coupling among the fluid's rheological properties, the shape of the bubble, and its motion, as well as by the nonlinearity of model constitutive relations for viscoplastic fluids. Estimates of the yield criterion Y c at which the buoyant force on the bubble is balanced by the yield stress have been derived by Dubash and Frigaard [5], and the same group has performed experiments on bubbles rising through a cylinder containing Carbopol, a yield-stress polymer gel [6]. Tsamopoulos et al have carried out a numerical study of the rise of bubbles through a viscoplastic fluid [16] using a regularized form of the Bingham model [21] and obtained detailed results for the bubble shape and rise velocity as functions of the dimensionless parameters which characterize the problem.…”

“…These materials flow only when the applied stress is greater than a yield stress y ; for < y they behave as soft solids. As a result, while a gas bubble will always rise due to buoyancy through a fluid with zero yield stress, it can remain stationary in a yield-stress fluid if the buoyant force is insufficient to overcome the opposing force due to the yield stress [4][5][6]. Similarly, a solid sphere will not fall through a yield-stress fluid unless the net gravitational force is large enough to overcome the yield stress [7].…”

Isogeometric Analysis of the Navier–Stokes–Cahn–Hilliard equations with application to incompressible two-phase flows

“…This setup holds e.g. for the slow motion of a gas bubble in a viscoplastic fluid as described in [11]. With the bubble being at rest, the velocity field in the computational domain is expected to be zero.…”

The start of ebullition in quiescent, yield-stress fluids

“…The shear strength allows for trapping of bubbles of exsolved gas (Winterwerp et al 2004;Gauglitz et al 1996;Dubash and Frigaard 2004). Exsolution occurs by a first order phase transformation with the new gas phase separating from the liquid in stages: first by nucleation and growth, and second by ageing (Boudreau et al 2001).…”