标题： 非定常杆爬升在Oldroyd-B流体中 显示英文标题

标题： Transient rod-climbing in an Oldroyd-B fluid

摘要： 韦森贝格效应，或称为杆爬现象，发生在非牛顿流体中，其中流体界面沿着旋转的杆上升。 尽管这一现象很显著，但对其的理论见解仍然有限。 在早期的工作中，Joseph & Fosdick（\emph{档案馆 比率 机械 分析}，第49卷，1973年，第321--380页）采用域摄动方法对二阶流体进行研究，通过基于旋转速度展开的解来确定平衡界面高度。 在本工作中，我们通过使用无量纲变量和方程的渐近分析来研究时间相关的界面高度，采用Oldroyd-B模型。 我们首先忽略表面张力和惯性，专注于重力与粘弹性之间的相互作用。 在小变形情况下，控制方程表明存在一个时间边界层，在这个时间边界层中，界面在短时间内迅速上升，然后逐渐趋于稳态。 通过采用拉伸的时间变量，我们在短时间内尺度上推导出瞬态速度场和相应的界面轮廓，并在更长时间尺度上恢复稳态轮廓。 随后，我们重新引入小但有限的惯性效应，以研究其与粘弹性的相互作用，并提出一个判断杆爬现象发生条件的标准。 显示英文摘要

摘要： The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph \& Fosdick (\emph{Arch. Rat. Mech. Anal.}, vol. 49, 1973, pp. 321--380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Oldroyd-B model. We begin by neglecting surface tension and inertia to focus on the interaction between gravity and viscoelasticity. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface profile on this short time scale and recover the steady-state profile on a longer time scale. Subsequently, we reintroduce small but finite inertial effects to investigate their interplay with viscoelasticity and propose a criterion for determining the conditions under which rod-climbing occurs.