Exact Real Trinomial Solutions to the Inner and Outer Hele–Shaw Problems

• Published in: Journal of mathematical fluid mechanics Vol. 17; no. 1; pp. 9 - 22
• Main Author: Runge, Vincent
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.03.2015
• Subjects: Classical and Continuum Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Physics; Physics and Astronomy; domain of univalence; Hele–Shaw cell; exact real trinomial solution; Secondary 30C50; Primary 76B07
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-014-0194-1

We find an explicit representation of the evolution of t ↦ γ t = { z ( ζ , t ) , ζ ∈ C , | ζ | = 1 } of the contour γ t = ∂ ω t of fluid spots ω t = { z ( ζ , t ) , | ζ | < 1 } for t > 0 or t < 0 in the Hele–Shaw problem with a sink ( t > 0 ) or a source ( t < 0 ) localized at point z ( 0 , t ) described by trinomials z ( ζ , t ) = a 1 ( t ) ζ + a N ( t ) ζ N + a M ( t ) ζ M , where M = 2 N - 1 , and integer N ≥ 2 , for the classical formulation of the problem when ω t is within γ t (inner Hele–Shaw problem), or by z ( ζ , t ) = a - 1 ( t ) ζ - 1 + a N ( t ) ζ N + a M ( t ) ζ M , where M = 2 N + 1 , and integer N ≥ 1 , for the outer Hele–Shaw problem when ω t is outside of γ t . We obtained a sufficient condition for univalence of real trinomials, improving a result found by Ruscheweyh and Wirths (Ann Pol Math. 28:341–355, 1973 ). A sufficient condition is also found for functions used in the outer problem.