黏弹性液滴热毛细迁移的对流不稳定

液滴在温度分布不均的固壁面上产生的热毛细迁移广泛存在于微流控、喷墨印刷等应用中, 对其流动进行稳定性分析对液滴迁移的精准控制具有重要意义. 本文采用线性稳定性理论研究了附壁黏弹性液滴在热毛细迁移中的对流不稳定性, 得到了不同Prandtl数($Pr$)下的临界Marangoni数($Ma_{\rm c})$与弹性数的函数关系, 并分析了临界模态的流场和能量机制. 研究发现: 流体弹性激发了更多不稳定模态, 小$Pr$的临界模态为斜波和流向波, 而中高$Pr$的临界模态为斜波和展向稳态模态. 强弹性使得$Ma_{\rm c}$显著下降, 而弱弹性略微增强了流动稳定性. 在中$Pr$下, $Ma_{\rm c}$随$Pr$的增大而增大. 对于斜波模态, 扰动温度的振幅可存在于流场中间区域, 而其他两种模态的温度振幅只存在于自由表面上, 并且在高$Pr$下的流线分布几乎是对称的. 能量分析表明: 随着弹性数增大, 基本流做功由正变负; 在小$Pr$中, 扰动应力做功既可能耗散能量又可能提供能量; 在高$Pr$中, 基本流做功可忽略不计. 对于同向流向波, 扰动速度和扰动应力做功在垂直方向上均存在多次振荡. 将液滴迁移与热毛细液层进行对比发现, 由于基本流和边界条件的不同, 两者在临界模态和扰动流场中均存在较大差异. 

CONVECTIVE INSTABILITY IN THERMOCAPILLARY MIGRATION OF A VISCOELASTIC DROPLET

Thermocapillary migration of a droplet placed on a non-uniformly heated solid surface appears in a variety of practical applications, such as microfluidics, inkjet printing, et al. The flow stability analysis is crucial for the precise control of droplet migration. In the present work, the convective instability in thermocapillary migration of a wall-attached viscoelastic droplet is examined by linear stability analysis. The relation between the critical Marangoni number ($Ma_{\rm c}$) and the elastic number is obtained at different Prandtl numbers ($Pr$). The flow fields and energy mechanisms of preferred modes are analyzed. The results show that more kinds of preferred modes are excited by the elasticity. The preferred modes at small $Pr$ are the oblique and streamwise waves, while those at moderate and high $Pr$ are oblique waves and spanwise stationary modes. The strong elasticity significantly reduces the $Ma_{\rm c}$, while the weak elasticity slightly enhances the flow stability. $Ma_{\rm c}$ increases with $Pr$ at moderate $Pr$. For the oblique wave, the amplitude of perturbation temperature may appear in the middle region of flow field, while the amplitudes of other two modes only exist on the free surface. The distribution of streamlines is almost symmetric at high $Pr$. The energy analysis shows that the work done by the basic flow changes from positive to negative when the elastic number increases. The work done by the perturbation stress may either dissipate or provide energy at small $Pr$, while the work done by the basic flow is negligible at high $Pr$. For the downstream streamwise wave, the perturbation velocity and the work done by perturbation stress fluctuate several times in the vertical direction. Comparing the droplet migration with thermocapillary liquid layers, it can be found that due to the differences of basic flow and boundary conditions, there are quite different between their preferred modes and perturbation flow fields.