Speeding up thermocapillary migration of a confined bubble by wall slip

It is usually believed that wall slip contributes small effects to macroscopic flow characteristics. Here we demonstrate that this is not the case for the thermocapillary migration of a long bubble in a slippery tube. We show that a fraction of the wall slip, with the slip length λ much smaller than the tube radius R, can make the bubble migrate much faster than without wall slip. This speedup effect occurs in the strongslip regime where the film thickness b is smaller than λ when the Marangoni number S=τ_TR=σ₀ .(1) is below the critical value S* ∼ .( λ=R)^1/2, where τ_T is the driving thermal stress and σ0 is the surface tension. The resulting bubble migration speed is found to be Ub ∼(σ0/μ)S³ (λ/R) which can be more than a hundred times faster than the no-slip result Ub ∼ (σ/Mu;)S⁵ (Wilson, J. Eng. Math., vol. 29, 1995, pp. 205-217; Mazouchi & Homsy, Phys. Fluids, vol. 12, 2000, pp. 542-549), with μ being the fluid viscosity. The change from the fifth power law to the cubic one also indicates a transition from the no-slip state to the strong-slip state, albeit the film thickness always scales as b ∼ RS ². The formal lubrication analysis and numerical results confirm the above findings. Our results in different slip regimes are shown to be equivalent to those for the Bretherton problem (Liao, Li & Wei, Phys. Rev. Lett., vol. 111, 2013, 136001). Extension to polygonal tubes and connection to experiments are also made. It is found that the slight discrepancy between experiment (Lajeunesse & Homsy, Phys. Fluids, vol. 15, 2003, pp. 308-314) and theory (Mazouchi & Homsy, Phys. Fluids, vol. 13, 2001, pp. 1594-1600) can be interpreted by including wall slip effects.