We study diffusive mixing in the presence of thermal fluctuations under the assumption of large Schmidt number. In this regime we obtain a limiting equation that contains a diffusive stochastic drift term with diffusion coefficient obeying a Stokes-Einstein relation, in addition to the expected advection by a random velocity. The overdamped limit correctly reproduces both the enhanced diffusion in the ensemble-averaged mean and the long-range correlated giant fluctuations in individual realizations of the mixing process, and is amenable to efficient numerical solution. Through a combination of Eulerian and Lagrangian numerical methods we demonstrate that diffusion in liquids is not most fundamentally described by Fick's irreversible law; rather, diffusion is better modeled as reversible random advection by thermal velocity fluctuations. We find that the diffusion coefficient is effectively renormalized to a value that depends on the scale of observation. Our work reveals somewhat unexpected connections between flows at small scales, dominated by thermal fluctuations, and flows at large scales, dominated by turbulent fluctuations.

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations represent a coarse-graining of the microscopic dynamics of the fluid molecules in both space and time, and eliminate the fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatiotemporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions and construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation of state constraint. The resulting spatio-temporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients, and investigate the validity of common simplifications such as neglecting the spatial non-homogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing.

Abstract. We present a general variable viscosity and variable density immersed boundary method that is first-order accurate in the variable density case and, for problems possessing sufficient regularity, second-order accurate in the constant density case. The viscosity and density are considered material properties and are defined by a dynamically updated tesselation. Empirical convergence rates are reported for a test problem of a two-dimensional viscoelastic shell with spatially varying material properties. The reduction to first-order accuracy in the variable density case can be avoided by using an iterative scheme, although this approach may not be efficient enough for practical use. In our timestepping scheme, both the inertial and viscous terms are split into two parts: a constant-coefficient part that is treated implicitly, and a variable-coefficient part that is treated explicitly. This splitting allows the resulting equations to be solved efficiently using fast constant-coefficient linear solvers, and in this work, we use solvers based on the Fast Fourier transform (FFT). As an application of this method, we perform fully three-dimensional, two-phase simulations of red blood cells accounting for variable viscosity and variable density. We study the behavior of red cells during shear flow and during capillary flow.Key words. immersed boundary method, variable viscosity, red blood cells AMS subject classifications. 65M06, 76D05, 76Z051. Introduction. The immersed boundary method is a general approach for simulating fluid-structure interaction. It has been used to study diverse phenomena including animal locomotion, DNA coiling, blood clotting, and heart valve dynamics [1,2,3,4,5,6,7]. It has also been applied to various problems in cellular dynamics, such as biofilm dynamics [8], cellular growth [9, 10, 11], cellular blebbing [12], and platelet margination [13]. Although the original immersed boundary method was formulated for fluids with uniform density and viscosity, several variable density versions of the method have been developed for applications in which the mass density of the structure is different from that of the background fluid, such as the dynamics of parachutes and flapping flags [14,15,16]. Another approach, the Front-Tracking method of Unverdi and Tryggvason [17], has been used together with the immersed boundary method to simulate multiphase fluids with piecewise constant density and viscosity. Situations more general than the piecewise constant case are also of significant interest. For instance, continuously varying viscosity appears in models of stratified fluids [18] and flow through pipes with viscous heating [19] (the viscosity of water is highly temperature dependent, varying between 1.79 cP to 0.28 cP in the range 0• C to 100 • C). To our knowledge, the immersed boundary method presented here is the first to handle the general variable viscosity case, in which the viscosity may be an arbitrary Eulerian or Lagrangian function.To test our methodology, we calculate empirical conv...

The size of the nucleus scales robustly with cell size so that the nuclear-to-cell volume ratio (N/C ratio) is maintained during cell growth in many cell types. The mechanism responsible for this scaling remains mysterious. Previous studies have established that the N/C ratio is not determined by DNA amount but is instead influenced by factors such as nuclear envelope mechanics and nuclear transport. Here, we developed a quantitative model for nuclear size control based upon colloid osmotic pressure and tested key predictions in the fission yeast Schizosaccharomyces pombe. This model posits that the N/C ratio is determined by the numbers of macromolecules in the nucleoplasm and cytoplasm. Osmotic shift experiments showed that the fission yeast nucleus behaves as an ideal osmometer whose volume is primarily dictated by osmotic forces. Inhibition of nuclear export caused accumulation of macromolecules and an increase in crowding in the nucleoplasm, leading to nuclear swelling. We further demonstrated that the N/C ratio is maintained by a homeostasis mechanism based upon synthesis of macromolecules during growth. These studies demonstrate the functions of colloid osmotic pressure in intracellular organization and size control.