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Analytical solutions of one-way coupled magnetohydrodynamic free surface flow

机译：单向耦合磁流体动力自由表面流的解析解

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摘要

We study the flow in a layer of conductive liquid under the influence of surface tension, gravity, and Lorentz forces due to imposed potential differences and transverse magnetic fields, as a function of the Hartmann number, the Bond number, the Reynolds number, the capillary number and the height-to-width ratio A. For aspect ratios A ＜＜ 1 and Reynolds numbers Re ≤ A, lubrication theory is applied to determine the steady state shape of the liquid surface to lowest order. Assuming low Hartmann (Ha ≤ O(1)), capillary (Ca ≤ O(A~4)), Bond (Bo ≤ O(A~2)) numbers and contact angles close to 90°, the flow details below the surface and the free surface elevation for the complete domain are determined analytically using the method of matched asymptotic expansions. The amplitude of the free surface deformation scales linearly with the capillary number and decreases with increasing Bond number, while the shape of the free surface depends on the Bond number and the contact angle condition. The strength of the flow scales linearly with the magnetic field gradient and applied potential difference and vanishes for high aspect ratio layers (A → 0). The analytical model results are confirmed by numerical simulations using a finite volume moving mesh interface tracking method, where the Lorentz force is calculated from the equation for the electric potential. It is shown that the analytical result for the free surface elevation is accurate within 0.4% from the numerical results for Ha~2 ≤ 1, Ca ≤A~4, Bo ≤ A~2, Re ≤ A and A ≤ 0.1 and within 2% for A = 0.5. For A = 0.1, the solution remains accurate within 1% of the numerical solution when either Ha2 is increased to 400, Ca to 200A~4 or Bo to 100A~2.
机译：我们研究了由于施加的电势差和横向磁场而在表面张力，重力和洛伦兹力的影响下在导电液体层中的流动，它是哈特曼数，键数，雷诺数，毛细管的函数对于深宽比A ＜＜ 1且雷诺数Re≤A，采用润滑理论将液态表面的稳态形状确定为最低阶。假设哈特曼（Ha≤O（1））低，毛细管（Ca≤O（A〜4）），键（Bo≤O（A〜2））数和接触角接近90°，则流量在表面以下使用匹配渐近展开法解析地确定整个域的自由表面高程。自由表面变形的幅度与毛细管数成线性比例，并随着邦德数的增加而减小，而自由表面的形状取决于邦德数和接触角条件。流动强度与磁场梯度和施加的电势差成线性比例，对于高深宽比层（A→0）消失。分析模型的结果通过使用有限体积移动网格界面跟踪方法的数值模拟得到了证实，其中洛伦兹力是根据电势方程计算的。结果表明，自由表面高程的分析结果与Ha〜2≤1，Ca≤A〜4，Bo≤A〜2，Re≤A和A≤0.1的数值结果相比，准确度在0.4％以内对于A ＝ 0.5％。对于A = 0.1，当Ha2增加到400，Ca增加到200A〜4或Bo增加到100A〜2时，溶液的精确度保持在数值溶液的1％以内。

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  • 来源
    《Applied Mathematical Modelling》 |2016年第4期|2577-2592|共16页
  • 作者

    B.W. Righolt; S. Kenjeres; R. Kalter; M.J. Tummers; C.R. Kleijn;

  • 作者单位

    Department of Chemical Engineering, Faculty of Applied Sciences, Delft University of Technology, Delft, 2628BL, Netherlands,J. M. Burgerscentre for Fluid Mechanics, Netherlands;

    Department of Chemical Engineering, Faculty of Applied Sciences, Delft University of Technology, Delft, 2628BL, Netherlands,J. M. Burgerscentre for Fluid Mechanics, Netherlands;

    Department of Chemical Engineering, Faculty of Applied Sciences, Delft University of Technology, Delft, 2628BL, Netherlands,J. M. Burgerscentre for Fluid Mechanics, Netherlands;

    Laboratory for Aero and Hydrodynamics, Delft University of Technology, Netherlands,J. M. Burgerscentre for Fluid Mechanics, Netherlands;

    Department of Chemical Engineering, Faculty of Applied Sciences, Delft University of Technology, Delft, 2628BL, Netherlands,J. M. Burgerscentre for Fluid Mechanics, Netherlands;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    Magnetohydrodynamics; Free surface; Analytical solutions; Numerical solutions; Lubrication theory; Matched asymptotic expansions;

    机译：磁流体动力学;自由表面分析解决方案;数值解;润滑原理;匹配渐近展开;

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