Traveling Waves and Capillarity Driven Spreading of Shear-thinning Fluids (Journal of Partial Differential Equations, 2018)
We study capillary spreadings of thin films of liquids of power-law rheology. These satisfy u_t+(u^{λ+2}|u_{xxx}^{λ-1}u_{xxx})_x=0, where u(x,t) represents the thickness of the one-dimensional liquid and λ > 1. We look for traveling wave solutions so that u(x,t)=g(x+ct) and thus g satisfies g'''=\frac{|g-ε|^\frac1λ}{g^{1+\frac2λ}}sgn(g-ε). We show that for each ε > 0 there is an infinitely oscillating solution, g_ε, such that lim_{t→∞}g_ε=ε and that g_ε→g_0 as ε→ 0, where g_0 ≡ 0 for t ≥ 0 and g_0=c_λ|t|^{\frac{3λ}{2λ+1}} for t < 0  for some constant c_λ.
Keywords: Shear-thinning fluid; traveling waves