arXiv subjects

Jiajun Tong

Publications and source records attributed to Jiajun Tong.

25 records · Page 2Linked to original sources

arXiv2019

On the Viscous Camassa-Holm Equations with Fractional Diffusion

We study Cauchy problem of a class of viscous Camassa-Holm equations (or Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and in the whole space in two and three dimensions. Order of the fractional diffusion is assumed to be $2s$ with $s\in [n/4,1)$, which seems to be sharp for the validity of the main results of the paper; here $n=2,3$ is the dimension of space. We prove global well-posedness in $C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2}))$ whenever the initial data $u_0\in D(A)$, where $A$ is the Stokes operator. We also prove that such global solutions gain regularity instantaneously after the initial time. A bound on a higher-order spatial norm is also obtained.

math.AP↗

arXiv2019

Regularized Stokes Immersed Boundary Problems in Two Dimensions: Well-posedness, Singular Limit, and Error Estimates

Inspired by the numerical immersed boundary method, we introduce regularized Stokes immersed boundary problems in two dimensions to describe regularized motion of a 1-D closed elastic string in a 2-D Stokes flow, in which a regularized $δ$-function is used to mollify the flow field and singular forcing. We establish global well-posedness of the regularized problems, and prove that as the regularization parameter diminishes, string dynamics in the regularized problems converge to that in the Stokes immersed boundary problem with no regularization. Viewing the un-regularized problem as a benchmark, we derive error estimates under various norms for the string dynamics. Our rigorous analysis shows that the regularized problems achieve improved accuracy if the regularized $δ$-function is suitably chosen. This may imply potential improvement in the numerical method, which is worth further investigation.

math.AP↗

arXiv2017

Directed migration of microscale swimmers by an array of shaped obstacles: modeling and shape optimization

Achieving macroscopic directed migration of microscale swimmers in a fluid is an important step towards utilizing their autonomous motion. It has been experimentally shown that directed motion can be induced, without any external fields, by certain geometrically asymmetric obstacles due to interaction between their boundaries and the swimmers. In this paper, we propose a kinetic-type model to study swimming and directional migration of microscale bimetallic rods in a periodic array of posts with non-circular cross-sections. Both rod position and orientation are taken into account; rod trapping and release on the post boundaries are modeled by empirically characterizing curvature and orientational dependence of the boundary absorption and desorption. Intensity of the directed rod migration, which we call the normalized net flux, is then defined and computed given the geometry of the post array. We numerically study the effect of post spacings on the flux; we also apply shape optimization to find better post shapes that can induce stronger flux. Inspired by preliminary numerical results on two candidate posts, we perform an approximate analysis on a simplified model to show the key geometric features a good post should have. Based on that, three new candidate shapes are proposed which give rise to large fluxes. This approach provides an effective tool and guidance for experimentally designing new devices that induce strong directed migration of microscale swimmers.

cond-mat.soft↗

arXiv2017

Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear non-local equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an $H^{5/2}$-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parameterized circular configuration, then global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as $t\rightarrow +\infty$. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions.

math.AP↗

arXiv2017

Guiding microscale swimmers using teardrop-shaped posts

The swimming direction of biological or artificial microscale swimmers tends to be randomised over long time-scales by thermal fluctuations. Bacteria use various strategies to bias swimming behaviour and achieve directed motion against a flow, maintain alignment with gravity or travel up a chemical gradient. Herein, we explore a purely geometric means of biasing the motion of artificial nanorod swimmers. These artificial swimmers are bimetallic rods, powered by a chemical fuel, which swim on a substrate printed with teardrop-shaped posts. The artificial swimmers are hydrodynamically attracted to the posts, swimming alongside the post perimeter for long times before leaving. The rods experience a higher rate of departure from the higher curvature end of the teardrop shape, thereby introducing a bias into their motion. This bias increases with swimming speed and can be translated into a macroscopic directional motion over long times by using arrays of teardrop-shaped posts aligned along a single direction. This method provides a protocol for concentrating swimmers, sorting swimmers according to different speeds, and could enable artificial swimmers to transport cargo to desired locations.

cond-mat.soft↗

arXiv2015

Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems

In this article, we discuss the stability of soft quasicrystalline phases in a coupled-mode Swift-Hohenberg model for three-component systems, where the characteristic length scales are governed by the positive-definite gradient terms. Classic two-mode approximation method and direct numerical minimization are applied to the model. In the latter approach, we apply the projection method to deal with the potentially quasiperiodic ground states. A variable cell method of optimizing the shape and size of higher-dimensional periodic cell is developed to minimize the free energy with respect to the order parameters. Based on the developed numerical methods, we rediscover decagonal and dodecagonal quasicrystalline phases, and find diverse periodic phases and complex modulated phases. Furthermore, phase diagrams are obtained in various phase spaces by comparing the free energies of different candidate structures. It does show not only the important roles of system parameters, but also the effect of optimizing computational domain. In particular, the optimization of computational cell allows us to capture the ground states and phase behavior with higher fidelity. We also make some discussions on our results and show the potential of applying our numerical methods to a larger class of mean-field free energy functionals.

cond-mat.soft↗

arXiv2015

Stability of Two-Dimensional Soft Quasicrystals

The relative stability of two-dimensional soft quasicrystals is examined using a recently developed projection method which provides a unified numerical framework to compute the free energy of periodic crystal and quasicrystals. Accurate free energies of numerous ordered phases, including dodecagonal, decagonal and octagonal quasicrystals, are obtained for a simple model, i.e. the Lifshitz-Petrich free energy functional, of soft quasicrystals with two length-scales. The availability of the free energy allows us to construct phase diagrams of the system, demonstrating that, for the Lifshitz-Petrich model, the dodecagonal and decagonal quasicrystals can become stable phases, whereas the octagonal quasicrystal stays as a metastable phase.

cond-mat.soft↗