We investigate the two-dimensional flows of a viscoplastic fluid in symmetric channels with impermeable walls under no-slip boundary conditions. To characterise the mechanical response of the viscoplastic fluid we consider both the celebrated Bingham model and a very general class of its regularisations. In order to make the problem amenable to analysis, we assume that the aspect ratio of the channel is small so that the lubrication approximation can be used. This allows us to obtain analytical solutions, perform an asymptotic analysis of the regularised solutions and compare the results predicted by the Bingham model and its regularisations. We find that in the limit as the regularisation parameter tends to zero, the regularised flow tends to those predicted by the Bingham model only in plane channels. In channels with curved walls, the results are instead markedly different.

中文翻译：

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小纵横比通道中的粘塑性流动：宾厄姆模型与正则化模型

我们研究了无滑移边界条件下具有不渗透壁的对称通道中粘塑性流体的二维流动。为了表征粘塑性流体的机械响应，我们考虑著名的宾厄姆模型及其一类非常通用的正则化模型。为了使问题易于分析，我们假设通道的纵横比很小，以便可以使用润滑近似。这使我们能够获得解析解，对正则化解进行渐近分析，并比较 Bingham 模型及其正则化的预测结果。我们发现，在正则化参数趋于零的极限下，正则化流仅在平面通道中趋向于 Bingham 模型预测的流。在具有弯曲壁的通道中，结果却明显不同。