Nonlinear equation for curved stationary flames

A nonlinear equation describing curved stationary flames with arbitrary gas expansion $\theta = \rho_{{\rm fuel}}/\rho_{{\rm burnt}}$, subject to the Landau-Darrieus instability, is obtained in a closed form without an assumption of weak nonlinearity. It is proved that in the scope of the asymptotic expansion for $\theta \to 1,$ the new equation gives the true solution to the problem of stationary flame propagation with the accuracy of the sixth order in $\theta - 1.$ In particular, it reproduces the stationary version of the well-known Sivashinsky equation at the second order corresponding to the approximation of zero vorticity production. At higher orders, the new equation describes influence of the vorticity drift behind the flame front on the front structure. Its asymptotic expansion is carried out explicitly, and the resulting equation is solved analytically at the third order. For arbitrary values of $\theta,$ the highly nonlinear regime of fast flow burning is investigated, for which case a large flame velocity expansion of the nonlinear equation is proposed.Comment: 29 pages 4 figures LaTe

Balancing torques in membrane-mediated interactions: Exact results and numerical illustrations

Torques on interfaces can be described by a divergence-free tensor which is fully encoded in the geometry. This tensor consists of two terms, one originating in the couple of the stress, the other capturing an intrinsic contribution due to curvature. In analogy to the description of forces in terms of a stress tensor, the torque on a particle can be expressed as a line integral along any contour surrounding the particle. Interactions between particles mediated by a fluid membrane are studied within this framework. In particular, torque balance places a strong constraint on the shape of the membrane. Symmetric two-particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime; in particular, the problem may be solved exactly in the case of two membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.Comment: 16 pages, 10 figures, REVTeX4 style. The Gaussian curvature term was included in the membrane Hamiltonian; section II.B was rephrased to smoothen the flow of presentatio