In continuum mechanics, the equations of motion for mixtures were derived through the use of a variational principle by Bedford and Drumheller in [1978]. Only immiscible mixtures were investigated. We have chosen a different approach. In this paper, we first write the equations of motion for each constituent of an inviscid miscible mixture of fluids without chemical reactions or diffusion. The theory is based on Hamilton’s extended principle and regards the mixture as a collection of distinct continua. The internal energy is assumed to be a function of densities, entropies and successive spatial gradients of each constituent. Our work leads to the equations of motion in an universal thermodynamic form in which interaction terms subject to constitutive laws, difficult to interpret physically, do not occur. For an internal energy function of densities, entropies and spatial gradients, an equation describing the barycentric motion of the constituents is obtained. The result is extended for dissipative mixtures and an equation of energy is obtained. A form of Clausius-Duhem’s inequality which represents the second law of thermodynamics is deduced. In the particular case of compressible mixtures, the equations reproduce the classical results. Far from critical conditions, the interfaces between different phases in a mixture of fluids are layers with strong gradients of density and entropy. The surface tension of such interfaces is interpreted.

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.

Contents 1 Introduction 1 1.1 Travelling waves : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Invariant manifolds : : : : : : : : : : : : : : : : : : : : : : : : 2 1.3 Fast travelling waves : : : : : : : : : : : : : : : : : : : : : : : 4 2 The Problem and the Results 9 2.1 Two travelling wave problems : : : : : : : : : : : : : : : : : : 9 2.1.1 A convective reaction-diffusion equation : : : : : : : : 9 2.1.2 The Cahn-Hillard equation : : : : : : : : : : : : : : : : 10 2.2 The functional-analytic setting : : : : : : : : : : : : : : : : : : 10 2.2.1 Fractional power spaces : : : : : : : : : : : : : : : : : 11 2.2.2 Reformulation of the equations : : : : : : : : : : : : : 12 2.2.3 Conditions for the nonlinearities : : : : : : : : : : : : 13 2.2.4 Solutions of the abstract equations : : : : : : : : : : : 14 2.2.5

We develop a conservative, second order accurate fully implicit discretization in two dimensions of the Navier-Stokes NS and Cahn-Hilliard CH system that has an associated discrete energy functional. This system provides a diffuse-interface description of binary fluid flows with compressible or incompressible flow components [44,4]. In this work, we focus on the case of flows containing two immiscible, incompressible and density-matched components. The scheme, however, has a straightforward extension to multi-component systems. To efficiently solve the discrete system at the implicit time-level, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We analyze and prove convergence of the scheme in the absence of flow. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an external shear flow, the evolution

In this paper a partial Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn{Hilliard equation is established by explicit energy calculations. Strong nondegeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the ow on the global attractor is shown to be semi-conjugate to the ow on the global attractor of the Cha ee-Infante equation, and in the metastable case close to the nonlocal reaction{di usion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle. 1

In a diblock copolymer system the free energy field depends nonlocally on the monomer density field. In addition there are two positive parameters in the constitutive relation. One of them is small with respect to which we do singular perturbation analysis. The second one is of order 1 with respect to which we do bifurcation analysis. Combining the two techniques we find wriggled lamellar solutions of the Euler-Lagrange equation of the total free energy. They bifurcate from the perfect lamellar solutions. The stability of the wriggled lamellar solutions is reduced to a relatively simple finite dimensional problem, which may be solved accurately by a numerical method. Our tests show that most of them are stable. The existence of such stable wriggled lamellar solutions explains why in reality the lamellar phase is fragile and it often exists in distorted forms.

Abstract. We study the phase space of the evolution equation ht = −(f(h)hxxx)x − (g(h)hx)x by means of a dissipated energy (a Liapunov function). Here h(x, t) ≥ 0, and at h = 0 the coefficient functions f> 0 and g can either degenerate to 0, or blow up to ∞, or tend to a nonzero constant. We first show all positive periodic steady states are ‘energy unstable ’ fixed points for the evolution (meaning the energy decreases under some zero–mean perturbation) if (g/f) ′ ′ ≥ 0 or if the perturbations are allowed to have period longer than that of the steady state. For power law coefficients (f(y) = y n and g(y) = By m for some B> 0) we analytically determine the relative energy levels of distinct steady states. For example, with m−n ∈ [1, 2) and for suitable choices of the period and mean value, we find three fundamentally different steady states. The first is a constant steady state that is nonlinearly stable and is a local minimum of the energy. The second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state; it is a saddle point. The third is a periodic collection of ‘droplet ’ (compactly supported) steady states having lower energy than either the positive steady state or the constant one. Since the energy must decrease along every orbit, these results significantly constrain the dynamics of the evolution equation. Our results suggest that heteroclinic connections could exist between certain of the steady states, for example from the periodic steady state to the droplet one. In a companion article we perform numerical simulations to confirm their existence. We study the evolution equation 1.

The ABC lamellar phase of a triblock copolymer in the strong segregation region is studied with the periodic and Neumann boundary conditions. In the periodic case we prove the existence of local minimizers of the free energy functional with fine lamellar structure. Among these local minimizers we identify the ones most favored by the free energy, and hence find the thickness of lamellar microdomains. Under the Neumann condition we show that perfect lamellar structure does not exist due to the boundary effect. We view the strong segregation limit as a \Gamma-limit of the free energy by a proper choice of the material sample size. The key step is the spectral analysis of a large matrix resulted from the second derivative of a reduced free energy. 1

Abstract. The phase field approach has become a popular tool in modeling interface motion, microstructure evolution, and more recently the shape transformation of vesicle membranes under elastic bending energy. While it is advantageous to employ phase field models in numerical simulations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes, detecting topological events may also become important for many applications such as those in the simulation of blood cells. Motivated by such considerations, a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events. It can also be used as a control method to avoid unphysical changes of topology due to the numerical methods, should it become necessary for particular practical applications. Through numerical experiments, we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes.

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: