FIG. S6: Bifurcation diagram for x as a function of the period of p ( t ) for a thermal gradient of 7 K/nm. The data correspond to the (10,10)@(15,15) with inner tube and outer tube lengths of 5 nm and 1000 nm, respectively, for the following parameters: F$_{vdw}$ = 1.92 nN, α = 8.08 pN nm/K, and γ = 1.48 pN ps $^{2}$/ ˚ A $^{2}$. changing the double-walled carbon nanotube considered. Keeping the same α value of the 7 K/nm case and using γ = 0.5 pN ps $^{2}$/ ˚ A $^{2}$, we can see in Figure S7 that the steady velocity state will be achieved for even longer outer tubes for a thermal gradient of 0.1 K/nm. Thus, the value of 1 µ m for the outer tube length can be considered as a lower limit where stead velocity state is expected, for thermal gradients below ∼ 10 K/nm. 17