Dynamic simulations of an encapsulated microbubble translating in a tube at low capillary and Reynolds numbers

The dynamic translation of a micron-sized encapsulated bubble is investigated numerically inside a horizontal tube where liquid flows under constant pressure drop, when the effect of gravity is neglected. The coating of the bubble is treated as an infinitesimally thin viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed to solve the axisymmetric flow configuration combined with the spine or elliptic mesh generation techniques for updating the mesh. The microbubble is initially elongated and the Reynolds number of the flow is relatively small, i.e. $$\hbox {Re}< 5$$ . Benchmark simulations for long free bubbles robustly recover the scaling of the film thickness with the 2/3 power of the capillary number based on surface tension. In the case of encapsulated bubbles, for a sufficiently small capillary number and after a short transient period, the bubble acquires a Bretherton type shape that slowly expands in order to accommodate changes in the liquid pressure. The speed of translation is nearly constant, close to the mean velocity of the flow, and does not depend on surface tension, shell elasticity or bending resistance. Fluid motion in the thin film “contact” region that forms in the gap between the tube and the shell is seen to be a stable flow arrangement that entails a mixture of pressure driven and shear driven flow, with the latter greatly affected by the area dilatation modulus via the tangential stress balance. By introducing a modified capillary number based on the area dilatational modulus, rather than surface tension, it is seen that the dimensionless film thickness that occupies the region between the bubble and the tube wall increases with the 1/3 power of the modified capillary number with increasing area dilatation. Simulations when surface tension is absent indicate that tangential shear generated due to variation of the membrane stress in the transition region that joins the bulk of the flow configuration with the contact region, leads to film thinning via the 5/7 power of the modified capillary number. Variations in the transverse shear of the viscoelastic shell generate large lubrication overpressures in the thin film region between the tube and the shell that are exerted radially on the shell and are conjectured to be responsible for the onset of 3d buckled shapes. The latter are often observed experimentally in similar flow configurations of capsules and are characterized by wrinkles that develop in the azimuthal direction around the shell equator.