In the present paper, the formation of an air bubble in a shear-thinning non-Newtonian fluid was investigated numerically. For modeling, an algebraic volume of fluid (VOF) solver of $$\hbox {OpenFOAM}^\circledR $$ was improved by applying a Laplacian filter and was evaluated using the experimental results from the literature. The enhanced solver could compute the surface tension force more accurately, and it was important especially at low capillary and Bond numbers due to the dominance of surface tension force relative to the other forces. The adiabatic bubble growth was simulated in an axisymmetric domain for $$\hbox {Bo}=0.05,0.1,0.5$$ and $$\hbox {Ca}=10^{-1},10^{-2},10^{-3},10^{-4}$$ , and the bubble detachment time and volume were examined. According to the results, for Newtonian fluids, there is a critical capillary number for a given Bond number, and for lower values of this critical number, no difference is observed between the bubble detachment volumes and also bubble detachment times. Similarly, the results indicated that for non-Newtonian fluids, if apparent capillary number (obtained by apparent viscosity) is less than the critical capillary number, the detachment volume is the same as the corresponding Newtonian case. The velocity of the bubble during its formation in Newtonian and shear-thinning fluids was also studied. Moreover, the bubble formation and detachment characteristics such as instantaneous contact angle and necking radius were investigated for Newtonian and non-Newtonian liquids, and the shear-thinning effect was examined as well. The results indicated that changing the ambient fluid to a non-Newtonian liquid has no effect on the trend of the contact angle; however, the minimum contact angle has a higher value when the shear-thinning effect increases. Also, the variation of neck radius with the time until detachment was presented by the power relation $${R}_{\mathrm{n}} =\left( {{t}_{\mathrm {det}} -{t}} \right) ^{{\eta }}$$ , and the effect of shear thinning on its exponent $${\eta }$$ was investigated.

中文翻译：

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低毛细管数和邦德数下剪切稀化液体中气泡形成及其分离的研究

在本文中，对剪切稀化的非牛顿流体中气泡的形成进行了数值研究。对于建模，$$\hbox {OpenFOAM}^\circledR $$ 的代数流体体积 (VOF) 求解器通过应用拉普拉斯滤波器进行了改进，并使用文献中的实验结果进行了评估。增强型求解器可以更准确地计算表面张力，由于表面张力相对于其他力占主导地位，因此在低毛细管数和邦德数时尤为重要。在轴对称域中模拟了 $$\hbox {Bo}=0.05,0.1,0.5$$ 和 $$\hbox {Ca}=10^{-1},10^{-2},10 的绝热气泡增长^{-3},10^{-4}$$ ，并检查气泡脱离时间和体积。根据结果​​，对于牛顿流体，对于给定的债券数，存在临界毛细管数，并且对于该临界数的较低值，气泡脱离体积和气泡脱离时间之间没有观察到差异。类似地，结果表明，对于非牛顿流体，如果表观毛细管数（由表观粘度获得）小于临界毛细管数，则脱离体积与相应的牛顿情况相同。还研究了气泡在牛顿流体和剪切稀化流体中形成过程中的速度。此外，研究了牛顿和非牛顿液体的气泡形成和分离特性，如瞬时接触角和颈缩半径，并研究了剪切稀化效应。结果表明，将环境流体改为非牛顿流体对接触角的趋势没有影响；然而，当剪切稀化效应增加时，最小接触角具有更高的值。此外，颈部半径随分离时间的变化由幂关系 $${R}_{\mathrm{n}} =\left( {{t}_{\mathrm {det}} -{t }} \right) ^{{\eta }}$$ ，研究了剪切稀化对其指数 $${\eta }$$ 的影响。