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2 results

  • Steady states of thin-film equations with van der Waals force with mass constraint

  • Part of
  • XINFU CHEN, HUIQIANG JIANG, GUOQING LIU
  • Journal:
    European Journal of Applied Mathematics / Volume 34 / Issue 2 / April 2023
    Published online by Cambridge University Press:
    30 May 2022, pp. 280-302
    • Article
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  • We consider steady states with mass constraint of the fourth-order thin-film equation with van der Waals force in a bounded domain which leads to a singular elliptic equation for the thickness with an unknown pressure term. By studying second-order nonlinear ordinary differential equation,

    \begin{equation*}h_{rr}+\frac{1}{r}h_{r}=\frac{1}{\alpha}h^{-\alpha}-p\end{equation*}
    we prove the existence of infinitely many radially symmetric solutions. Also, we perform rigorous asymptotic analysis to identify the blow-up limit when the steady state is close to a constant solution and the blow-down limit when the maximum of the steady state goes to the infinity.

  • 4 - Interactions of Particles, Droplets, and Bubbles

  • from Part I - Principles
  • Chao Zhu, New Jersey Institute of Technology, Liang-Shih Fan, Ohio State University, Zhao Yu
  • Book:
    Dynamics of Multiphase Flows
    Published online:
    10 September 2021
    Print publication:
    17 June 2021, pp 126-171

  • Summary

    Chapter 4 discusses the mechanisms and formulation of various basic particle–particle interactions. The essential modes of these interactions include a pair of spheres interacting by head-on approaching or by wake attraction, flow through a uniformly suspended sphere, electrostatic field induced by the suspended charged particles, normal collision dynamics involving forces, deformation, contact area and duration for a pair of elastic spheres, van der Waals force, and capillary force due to liquid bridge between two particles. The chapter further discusses the nonidealized particle–particle interactions and associated formulation, including the radiation transport equation for thermal radiation within a particle cloud, collision dynamics with tangential friction and torsional traction of elastic spheres, inelastic collisions, and the concept of restitution coefficient, heat and charge transfer by particle collisions, and deformation, breakup, and coalescence of fluid particles. These particle–particle interactions are critical to the model formation of dense-phase multiphase flows.