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

We find an explicit representation of the evolution of t bar right arrow gamma(t) = {z(zeta,t), zeta is an element of C, vertical bar zeta vertical bar = 1} of the contour gamma(t) = partial derivative w(t) of fluid spots w(t) = {z(zeta,t), vertical bar zeta vertical bar < 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(zeta,t) = a(1)(t)zeta + a(N()t)zeta(N) + a(M)(t)zeta(M) , where M = 2N - 1, and integer N >= 2, for the classical formulation of the problem when w(t) is within gamma(t) (inner Hele-Shaw problem), or by z(zeta,t) = a(-1)(t)zeta(-1) + a(N()t)zeta(N) + a(M)(t)zeta(M) , where M = 2N + 1, and integer N >= 1, for the outer Hele-Shaw problem when w(t) is outside of gamma(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.