Ivan C. Christov

Stokes' first problem for some non-Newtonian fluids: Results and mistakes

The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.

On the Propagation of Second-Sound in Nonlinear Media: Shock, Acceleration and Traveling Wave Results

An exhaustive study of thermal shock, acceleration (or temperature-rate) and traveling waves in media with temperature-dependent thermal conductivity, and within which the flux is described by the Maxwell–Catteneo law, is presented. The resulting quasilinear, hyperbolic system of equations predicts a variety of interesting phenomena in such media including, but not limited to, dynamic thermal shock waves with jump-dependent wave speed, finite-time temperature-rate wave blow-up, and temperature-rate wave formation as traveling waves reach a critical speed. Both analytical (integral transforms, singular surface theory, solution of transcendental equations using special functions) and numerical approaches are used; the former are benchmarked against the latter where appropriate. Parallels are drawn between the nonlinear heat waves discussed in the present work and nonlinear wave phenomena in related “classical” continuum theories such as acoustics.

From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

Through a combined computational–experimental study of flow in a slowly rotating quasi-two-dimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed “streamline jumping.” This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., “strange” eigenmodes of the advection-diffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincare section and finite-time Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, ...

A mapping method for distributive mixing with diffusion: Interplay between chaos and diffusion in time-periodic sine flow

We present an accurate and efficient computational method for solving the advection-diffusion equation in time-periodic chaotic flows. The method uses operator splitting, which allows the advection and diffusion steps to be treated independently. Taking advantage of flow periodicity, the advection step is solved using a mapping method, and diffusion is “added” discretely after each iteration of the advection map. This approach results in the construction of a composite mapping matrix over an entire period of the chaotic advection-diffusion process and provides a natural framework for the analysis of mixing. To test the approach, we consider two-dimensional time-periodic sine flow. By comparing the numerical solutions obtained by our method to reference solutions, we find qualitative agreement for large time steps (structure of concentration profile) and quantitative agreement for small time steps (low error). Further, we study the interplay between mixing through chaotic advection and mixing through diffusion leading to an analytical model for the evolution of the intensity of segregation with time. Additionally, we demonstrate that our operator splitting mapping approach can be readily extended to three dimensions.

Resolving a paradox of anomalous scalings in the diffusion of granular materials

Granular materials do not perform Brownian motion, yet diffusion can be observed in such systems when agitation causes inelastic collisions between particles. It has been suggested that axial diffusion of granular matter in a rotating drum might be “anomalous” in the sense that the mean squared displacement of particles follows a power law in time with exponent less than unity. Further numerical and experimental studies have been unable to definitively confirm or disprove this observation. We show two possible resolutions to this apparent paradox without the need to appeal to anomalous diffusion. First, we consider the evolution of arbitrary (non-point-source) initial data towards the self-similar intermediate asymptotics of diffusion by deriving an analytical expression for the instantaneous collapse exponent of the macroscopic concentration profiles. Second, we account for the concentration-dependent diffusivity in bidisperse mixtures, and we give an asymptotic argument for the self-similar behavior of such a diffusion process, for which an exact self-similar analytical solution does not exist. The theoretical arguments are verified through numerical simulations of the governing partial differential equations, showing that concentration-dependent diffusivity leads to two intermediate asymptotic regimes: one with an anomalous scaling that matches the experimental observations for naturally polydisperse granular materials, and another with a “normal” diffusive scaling (consistent with a “normal” random walk) at even longer times.

Acoustic traveling waves in thermoviscous perfect gases

We study one-dimensional traveling wave phenomena in thermoviscous perfect gases with constant material properties. First, we summarize the known forms of the solution of a classic acoustic signaling problem based on the linearized theory. Next, we review several weakly-nonlinear models, all of which admit traveling wave solutions (TWSs) in the form of classical Taylor shocks, and note some of their features. We then consider traveling waves under the fully-nonlinear theory of gas dynamics, and derive a new third-order equation of motion valid for arbitrary Mach number values. Focusing on the special cases of (a) inviscid, thermally-conducting, and (b) viscous, non-thermally-conducting (i.e., strictly adiabatic) perfect gases, exact and/or numerical solutions of this nonlinear ordinary differential equation (ODE) are determined, asymptotic expressions presented, and critical values of the physical parameters identified. It is shown that, in addition to kinks, the fully-nonlinear theory allows for discontinuous solutions (i.e., shocks) and solutions that exhibit acceleration waves.

Successive phase transitions and kink solutions in ϕ(8), ϕ(10), and ϕ(12) field theories.

We obtain exact solutions for kinks in ϕ(8), ϕ(10), and ϕ(12) field theories with degenerate minima, which can describe a second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higher-order field theories have kink solutions with algebraically decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order ϕ(4) and ϕ(6) theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the coexistence of up to three distinct kinks (for a ϕ(12) potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higher-order field theories have specific cases in which only nonlinear phonons are allowed. For the ϕ(10) field theory, which is a quasiexactly solvable model akin to ϕ(6), we are also able to obtain three analytical solutions for the classical free energy as well as the probability distribution function in the thermodynamic limit.

Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua

We compare and contrast two types of deformations inspired by mixing applications—one from the mixing of fluids (stretching and folding) and the other from the mixing of granular matter (cutting and shuffling). The connection between mechanics and dynamical systems is discussed in the context of the kinematics of deformation, emphasizing the equivalence between stretches and Lyapunov exponents. The stretching and folding motion exemplified by the baker’s map is shown to give rise to a dynamical system with a positive Lyapunov exponent, the hallmark of chaotic mixing. In contrast, cutting and shuffling does not stretch. When an interval exchange transformation is used as the basis for cutting and shuffling, we establish that all of the map’s Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per unit volume, is shown to be exponential when there is stretching and folding but linear when there is only cutting and shuffling. We also discuss how a simple computational approach can disc...

Successive phase transitions and kink solutions in

We obtain exact solutions for kinks in ϕ(8), ϕ(10), and ϕ(12) field theories with degenerate minima, which can describe a second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higher-order field theories have kink solutions with algebraically decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order ϕ(4) and ϕ(6) theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the coexistence of up to three distinct kinks (for a ϕ(12) potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higher-order field theories have specific cases in which only nonlinear phonons are allowed. For the ϕ(10) field theory, which is a quasiexactly solvable model akin to ϕ(6), we are also able to obtain three analytical solutions for the classical free energy as well as the probability distribution function in the thermodynamic limit.

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We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of intermaterial interfaces) that can be computed over given (finite) numbers of iterations. We find that good mixing can be achieved after a finite number of iterations of a one-dimensional cutting and shuffling map, even though such a map cannot be considered chaotic in the usual sense and/or it may not fulfill the conditions of the ergodic theorems for interval exchange transformations. Specifically, good shuffling can occur with only six or seven intervals of roughly the same length, as long as the rearrangement order is an irreducible permutation. This study has implications for a number of mixing processes in which discontinuities arise either by construction or due to the underlying physics.