Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in R-2 or R-3, the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by partial derivative(t)v + (v . del(x))v = -del(xP), div(x)v = 0, where v(t, x) is the velocity of the particle located at x is an element of D at time t and p(t, x) is an element of R is the pressure. Solutions v and p to the Euler equation can be obtained by solving {del x {partial derivative t phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2)} = 0 at a = kappa(t, x), v(t, x) = del(x)phi(t, x, a) at a = kappa(t, x), partial derivative(t)kappa(t, x) | (v. del(x))kappa(t, x) = 0 div(x)v(t, x) = 0 where phi : R x D x R-l -> R and kappa : R x D -> R-l are additional unknown mappings (l >= 1 is prescribed). The third equation in the system says that kappa is an element of R-l is convected by the flow and the second one that phi can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition kappa(0, x) = x on D (and thus l = 2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52: 411-452, 1999) in his Eulerian-Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross. D and that carry each "particle" at time t = 0 at a prescribed location at time t = T > 0, that is, kappa(T, x) is prescribed in D for all x is an element of D. We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary partial derivative D of the bounded region D (a two-or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through partial derivative D of particles labelled by each value of. at each point of partial derivative D. One of the main novelties is the introduction of a prescribed "generalized" Bernoulli's function H : R-l -> R, namely, we add to (0.1) the requirement that partial derivative(t)phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2) = H(a) at a = kappa(t, x) with phi, p, kappa periodic in time of prescribed period T > 0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of "Lamb's surfaces" and "isotropic manifolds" in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier's formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional (kappa, v) -> integral(T)(0)integral(D) {1/2|v(t, x)|(2) + H(kappa(t, x))} dtdx defined for kappa and v that are T-periodic in t, such that partial derivative(t)kappa(t, x) + (v . del(x))kappa(t, x) = 0, div(x)v(t, x) = 0, and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize integral(]0, L[x]0,1[) {(1/2)|del psi|(2) + H(psi)}dx for psi is an element of W-1,W-2 (]0, L[x]0,1[) under appropriate boundary conditions, where. is the stream function. For a minimizer, corresponding functions phi and kappa are given in terms of the stream function psi.

Steady Three-Dimensional Rotational Flows: An Approach Via Two Stream Functions And Nash-Moser Iteration

Boris Buffoni

We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D = (0, L) x R-2. We are concerned with flows that are periodic in the second and third variables and that have prescribed flux through each point of the boundary aD. The Bernoulli equation states that the "Bernoulli function" H :=-1/2 vertical bar v vertical bar(2) + p (where v is the velocity field and p the pressure) is constant along stream lines, that is, each particle is associated with a particular value of H. We also prescribe the value of H on partial derivative D. The aim of this work is to develop an existence theory near a given constant solution. It relies on writing the velocity field in the form v = del f x del g and deriving a degenerate nonlinear elliptic system for f and g. This system is solved using the Nash Moser method, as developed for the problem of isometric embeddings of Riemannian manifolds; see, e.g., the book by Q. Han and J.-X. Hong (2006). Since we can allow H to be nonconstant on partial derivative D, our theory includes three-dimensional flows with nonvanishing vorticity.

MATHEMATICAL SCIENCE PUBL2019

On the coupling between an ideal fluid and immersed particles

Mathieu Desbrun, Tudor Ratiu

In this paper, we present finite-dimensional particle-based models for fluids which respect a number of geometric properties of the Euler equations of motion. Specifically, we use Lagrange-Poincare reduction to understand the coupling between a fluid and a set of Lagrangian particles that are supposed to simulate it. We substitute the use of principal connections in Cendra et al. (2001) [13] with vector field valued interpolations from particle velocity data. The consequence of writing evolution equations in terms of interpolation is two-fold. First, it provides estimates on the error incurred when interpolation is used to derive the evolution of the system. Second, this form of the equations of motion can inspire a family of particle and hybrid particle spectral methods, where the error analysis is "built in". We also discuss the influence of other parameters attached to the particles, such as shape, orientation, or higher-order deformations, and how they can help us achieve a particle-centric version of Kelvin's circulation theorem. (C) 2013 Elsevier B.V. All rights reserved.

Elsevier2013

A path integral formalism of DNA looping probability

Ludovica Cotta-Ramusino

In this thesis we describe a path integral formalism to evaluate approximations to the probability density function for the location and orientation of one end of a continuum polymer chain at thermodynamic equilibrium with a heat bath. We concentrate on those systems for which the associated energy density is at most quadratic in its variables. Our main motivation is to exploit continuum elastic rod models for the approximate computation of DNA looping probabilities. We first re-derive, for a polymer chain system, an expression for the second order correction term due to quadratic fluctuations about a unique minimal energy configuration. The result, originally stated for a quantum mechanical system by G. Papadopoulos (1975), relies on an elegant algebraic argument that carries over to the real-valued path integrals of interest here. The conclusion is that the appropriate expression can be evaluated in terms of the energy of the minimizer and the inverse square root of the determinant of a matrix satisfying a certain non-linear system of differential equations. We then construct a change of variables, which establishes a mapping between the solutions of the aforementioned non-linear Papadopoulos equations and a matrix satisfying an initial value problem for the classic linear system of Jacobi equations associated with the second variation of the energy functional. This conclusion is trivial if no cross-term is present in the second variation, but ceases to be so otherwise. Cross-terms are always present in the application of rod models to DNA. We therefore can conclude that the second order fluctuation correction term to the probability density function for a chain is always given by the inverse square root of the determinant of a matrix of solutions to the Jacobi equations. We believe this conclusion to be original for the real-valued case when the second-variation involves cross-terms. Similar results are known for quantum mechanical systems, and, in this context, a connection between the so called Van-Hove-Morette determinant, which involves partial derivatives of the classical action with respect to the boundary values of the configuration variable, and the Jacobi determinant have also been established. We next apply the formula described above to the specific context of rods, for which the configuration space is that of framed curves, or curves in R3 × SO(3). An immediate application of our theory is possible if the rod model encompasses bend, twist, stretch and shear. However the constrained case, where the rod is considered to be inextensible and unshearable, is more standard in polymer physics. In this last case, our results are more delicate as the Lagrangian description breaks down, and the Hamiltonian formulation must be invoked. It is known that the unconstrained local minimizers approach constrained minimizers as the coefficients in the shear and extension terms of the energy are sent to infinity. Here we observe that the Hamiltonian form of the unconstrained Jacobi system similarly has a limit, so that the fluctuation correction in the path integral can still be expressed as the square root of the determinant of a matrix solution of a set of Jacobi equations appropriate to the constrained problem. As in reality DNA or biological macromolecules are certainly at least slightly shearable and extensible, the limit of the fluctuation correction is undoubtedly physically appropriate. The above theory provides a computationally highly tractable approach to the estimation of the appropriate probability density functions. For application to sequence-dependent models of DNA the associated systems of equations has non-constant coefficients, which is of little consequence for a numerical treatment, but precludes the possibility of finding closed form expressions. On the other hand the theory also applies to simplified homogeneous models. Accordingly, we conclude by applying our approach in a completely analytic and closed-form way to the computation of the approximate probability density function for a uniform, non-isotropic, intrinsically straight and untwisted rod to form a circular loop.

EPFL2008

A path integral formalism of DNA looping probability

Ludovica Cotta-Ramusino

EPFL2008