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Volume 23, Issue 1
Traveling Waves and Capillarity Driven Spreading of Shear-thinning Fluids

Joseph A. Iaia

DOI: 10.4208/jpde.v23.n1.3

J. Part. Diff. Eq., 23 (2010), pp. 33-67.

Published online: 2010-02

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  • Abstract

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

  • AMS Subject Headings

35J35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-23-33, author = {}, title = {Traveling Waves and Capillarity Driven Spreading of Shear-thinning Fluids}, journal = {Journal of Partial Differential Equations}, year = {2010}, volume = {23}, number = {1}, pages = {33--67}, abstract = {

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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v23.n1.3}, url = {http://global-sci.org/intro/article_detail/jpde/5221.html} }
TY - JOUR T1 - Traveling Waves and Capillarity Driven Spreading of Shear-thinning Fluids JO - Journal of Partial Differential Equations VL - 1 SP - 33 EP - 67 PY - 2010 DA - 2010/02 SN - 23 DO - http://doi.org/10.4208/jpde.v23.n1.3 UR - https://global-sci.org/intro/article_detail/jpde/5221.html KW - Shear-thinning fluid KW - traveling waves AB -

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.

Joseph A. Iaia . (2019). Traveling Waves and Capillarity Driven Spreading of Shear-thinning Fluids. Journal of Partial Differential Equations. 23 (1). 33-67. doi:10.4208/jpde.v23.n1.3
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