Asymptotic behavior of traveling wave solutions of the equations for the flow of a fluid with small viscosity and capillarity
Author:
J. L. Boldrini
Journal:
Quart. Appl. Math. 44 (1987), 697-708
MSC:
Primary 35B40; Secondary 35Q20, 76D99
DOI:
https://doi.org/10.1090/qam/872822
MathSciNet review:
872822
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We study the oscillations of the traveling wave solutions of \[ \left \{ {_{{v_t} = - p{{\left ( u \right )}_x} + \epsilon {v_{xx}} - \delta {u_{xxx}}}^{{u_t} = {v_x},}} \right .\] for small $\epsilon$ and $\delta$. These solutions give information about the structure of the shock layers in fluids with small viscosity and capillarity. We conclude that the traveling wave has oscillations with increasing amplitude when $\epsilon$ and $\delta$ approach zero such that $\delta \ne O\left ( {{\epsilon ^2}} \right )$. When $\delta = o\left ( {{\epsilon ^2}} \right )$, if there are oscillations, their amplitude decreases to zero as $\epsilon$ and $\delta$ approach zero. When $\delta = {\epsilon ^2}$ the shape of the traveling wave is independent of the magnitude of $\epsilon$ and $\delta$.
J. L. Boldrini, Is elasticity the proper asymptotic theory for materials with viscosity and capillarity?, Proceedings of the Royal Society of Edinburgh.
- R. Hagan and M. Slemrod, The viscosity-capillarity criterion for shocks and phase transitions, Arch. Rational Mech. Anal. 83 (1983), no. 4, 333–361. MR 714979, DOI https://doi.org/10.1007/BF00963839
- Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
D. J. Korteweg, Sur la forme que prenent les équations du mouvement des fluids si l’on tient compte des forces capillaires par des variations de densité, Arch. Neerl. Sci. Exactes Nat., Ser. II 6, 1–24 (1901)
- M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301–315. MR 683192, DOI https://doi.org/10.1007/BF00250857
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
J. D. van der Waals, Théorie thermodynamique de la capillarité dans l’hypoth `ese d’une variation continue de densité. Arch. Neerl. Sci. Exactes Nat. 28, 121–209 (1895)
J. L. Boldrini, Is elasticity the proper asymptotic theory for materials with viscosity and capillarity?, Proceedings of the Royal Society of Edinburgh.
R. Hagan and M. Slemrod, The viscosity-capillarity criterion for shocks and phase transitions, Arch. Rat. Mech. Anal. 83, 333–361 (1983)
J. K. Hale, Ordinary differential equations, Krieger, New York (1980)
D. J. Korteweg, Sur la forme que prenent les équations du mouvement des fluids si l’on tient compte des forces capillaires par des variations de densité, Arch. Neerl. Sci. Exactes Nat., Ser. II 6, 1–24 (1901)
M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Wauls fluid. Arch. Rat. Mech. Anal. 81, Number 4, 301–315 (1983)
J. A. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York (1982)
J. D. van der Waals, Théorie thermodynamique de la capillarité dans l’hypoth `ese d’une variation continue de densité. Arch. Neerl. Sci. Exactes Nat. 28, 121–209 (1895)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35B40,
35Q20,
76D99
Retrieve articles in all journals
with MSC:
35B40,
35Q20,
76D99
Additional Information
Article copyright:
© Copyright 1987
American Mathematical Society