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An open problem concerning breakup of fluid jets

An open problem concerning breakup of fluid jets. Michael Renardy Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123, USA. Supported by National Science Foundation. Governing equation Describes breakup of a jet of Newtonian fluid in Stokes flow.

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An open problem concerning breakup of fluid jets

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  1. An open problem concerningbreakup of fluid jets Michael Renardy Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123, USA Supported by National Science Foundation

  2. Governing equation Describes breakup of a jet of Newtonian fluid in Stokes flow. s(X,t) is the stretch factor, and  is the force in the jet. This force is not given, but is determined by the constraint Problem: Link the asymptotic behavior at jet breakup, i.e. near the point where s to the behavior of the initial condition near the point where s(X,0) has its maximum.

  3. Similarity solutions (Papageorgiou 1995, Eggers 1997) Here we have put the breakup time at t=0 and the breakup point at X=0. If we want all terms in the equations to have the same order of magnitude, we need =2, =1. The similarity ansatz leads to the following set of ODEs:

  4. Regular behavior at =0: The simplest case n=1 leads to Next, we substitute This leads to the implicit solution

  5. There is a one-parameter family of solutions which differ by rescaling. W.l.o.g. set C=1. We must satisfy the equation for k: This can be rewritten as The smallest positive root is at =2.17487.

  6. Finite time blowup Suppose s(X,0) has a unique maximum at X=0, and moreover, for X close to zero where C is positive and n is some integer. Then s(0,t) blows up in finite time. Reference: ZAMP 52 (2001), 881-887; 58 (2007), 904-905. Expected result: If s(X,0)~A-Bx2n, then the asymptotics of blowup is described by the similarity solution corresponding to that value of n. If we know a priori that (t)~k(tc-t), where tc is the blowup time, then this is true.

  7. Consider Substitute Leads to If we set tc-t=e-, we obtain Self-similar behavior follows from the closed form solution of this ODE. A perturbation argument can be used if we know that Reference: IMA J. Appl. Math. 70 (2005), 353-358.

  8. Questions? Answers?

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