Effects of magnetic field applied perpendicular to a shear plane in shear flow on the deformation of a ferrofluid droplet are numerically investigated. The boundary integral method is employed to solve the two-phase Stokes flow under a uniform magnetic field. When the magnetic field is applied perpendicular to the shear plane, the deformation of the droplet in the shear plane decreases. The magnetic field causes the droplet to elongate in the y-direction, and its cross-sectional radius in shear plane decreases. Consequently, the apparent capillary number in the shear plane decreases, thereby suppressing the droplet deformation. Droplet breakup is also suppressed by imposing a magnetic field perpendicular to the shear plane, thereby increasing the critical capillary numbers. The critical capillary numbers for the magnetic Bond numbers Bo = 2.0 and 4.0 increase to approximately 110% and 130%, respectively, than those without magnetic field. Furthermore, an equation for the theoretical prediction of the droplet deformation under a magnetic field in shear flow is presented, which is based on the small deformation theory, the decrease in the cross-sectional radius, and the boundary conditions at the droplet interface. The theoretical prediction agrees well with the numerical results for the variation in the magnetic susceptibility of the droplet as well as the viscosity ratio between the external fluid and the ferrofluid droplet under a small deformation. The critical capillary numbers under a magnetic field can also be predicted by using the numerical results without a magnetic field.

中文翻译：

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磁场下剪切流中铁磁流体液滴的变形和破碎

对剪切流中垂直于剪切面施加的磁场对铁磁流体液滴变形的影响进行了数值研究。采用边界积分法求解均匀磁场下的两相斯托克斯流。当垂直于剪切面施加磁场时，剪切面内液滴的变形减小。磁场导致液滴在 y 方向上拉长，并且其在剪切平面中的横截面半径减小。结果，剪切面中的表观毛细管数减少，从而抑制了液滴变形。还可以通过施加垂直于剪切面的磁场来抑制液滴破裂，从而增加临界毛细管数。与没有磁场的情况相比，磁性键合数 Bo = 2.0 和 4.0 的临界毛细管数分别增加到约 110% 和 130%。此外，基于小变形理论、横截面半径的减小和液滴界面的边界条件，提出了剪切流磁场下液滴变形的理论预测方程。对于小变形下液滴磁化率的变化以及外部流体与铁磁流体液滴之间的粘度比，理论预测与数值结果非常吻合。磁场下的临界毛细管数也可以通过使用没有磁场的数值结果来预测。