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Abstract Given a family$$\mathcal{F}$$of bipartite graphs, theZarankiewicz number$$z(m,n,\mathcal{F})$$is the maximum number of edges in an$$m$$by$$n$$bipartite graph$$G$$that does not contain any member of$$\mathcal{F}$$as a subgraph (such$$G$$is called$$\mathcal{F}$$-free). For$$1\leq \beta \lt \alpha \lt 2$$, a family$$\mathcal{F}$$of bipartite graphs is$$(\alpha,\beta )$$-smoothif for some$$\rho \gt 0$$and every$$m\leq n$$,$$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$$(\alpha,\beta )$$-smooth family$$\mathcal{F}$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$$is bipartite. In this paper, we strengthen their result by showing that for every real$$\delta \gt 0$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\delta n^{\alpha -1}$$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$$\mathcal{F}$$consisting of the single graph$$K_{s,t}$$when$$t\gg s$$. We also prove an analogous result for$$C_{2\ell }$$-free graphs for every$$\ell \geq 2$$, which complements a result of Keevash, Sudakov and Verstraëte.
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Nonlinear electrophoretic velocity of a spherical colloidal particle
Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength$$\beta =a^*e^*E_\infty ^*/k_B^*T^*$$, defined as the ratio of the product of the applied electric field magnitude$$E_\infty ^*$$and particle radius$$a^*$$, to the thermal voltage$$k_B^*T^*/e^*$$, where$$k_B^*$$is Boltzmann's constant,$$T^*$$is the absolute temperature, and$$e^*$$is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density$$\sigma$$over a wide range of$$\beta$$. Here,$$\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$$, where$$\sigma ^*$$is the dimensional surface charge density, and$$\epsilon ^*$$is the permittivity of the electrolyte. For moderately charged particles ($$\sigma ={O}(1)$$), the electrophoretic velocity is linear in$$\beta$$when$$\beta \ll 1$$, and its dependence on the ratio of the Debye length ($$1/\kappa ^*$$) to particle radius (denoted by$$\delta =1/(\kappa ^*a^*)$$) agrees with Henry's formula. As$$\beta$$increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is$$\delta$$-dependent. For$$\beta \gg 1$$, the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all$$\delta$$. For highly charged particles ($$\sigma \gg 1$$) in the thin-Debye-layer limit ($$\delta \ll 1$$), our computations are in good agreement with recent experimental and asymptotic results.
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- Award ID(s):
- 2002120
- PAR ID:
- 10502186
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 968
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2023.537
