The flow of viscous fluid injected from a point source into the space between two horizontal plates initially filled with a second fluid of lesser density and different viscosity is studied theoretically and numerically. The volume of the dense input fluid increases with time in proportion to tα. When the fluid has spread far from the source, lubrication theory is used to derive the governing equations for the axisymmetric evolution of the interface between the fluids. The flow is driven by the combination of pressure gradients associated with buoyancy and pressure gradients associated with the input flux. The governing equation is integrated numerically, and we identify that with a constant input flux, the flow is self-similar at all times with the radius growing in proportion to t1/2. In the regimes of injection-dominated and gravity-dominated currents, we obtain asymptotic approximations for the interface shape, which are found to agree well with the numerical computations. For a decreasing input flux (0 1) with buoyancy dominating initially while the pressure associated with the injection dominates at late times. Finally, we consider the release of a fixed volume of fluid (α = 0). The current slumps under gravity and transitions from confined to unconfined, and we obtain asymptotic predictions for the interface shape in both regimes.

中文翻译：

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两个水平板之间的轴对称粘性流

从点源注入到两个水平板之间的空间中的粘性流体的流动，最初充满了密度较小和粘度不同的第二种流体，从理论上和数值上进行了研究。稠密输入流体的体积与 tα 成比例地随时间增加。当流体远离源头时，润滑理论用于推导出流体之间界面轴对称演化的控制方程。流动由与浮力相关的压力梯度和与输入通量相关的压力梯度的组合驱动。对控制方程进行数值积分，我们确定在输入通量恒定的情况下，流动始终自相似，半径与 t1/2 成比例增长。在注入主导和重力主导的电流状态下，我们获得了界面形状的渐近近似值，这与数值计算非常吻合。对于减小的输入通量 (0 1)，浮力最初占主导地位，而与注入相关的压力在后期占主导地位。最后，我们考虑释放固定体积的流体 (α = 0)。电流在重力作用下坍塌，从受限到无受限的转变，我们获得了两种状态下界面形状的渐近预测。我们考虑释放固定体积的流体 (α = 0)。电流在重力作用下坍塌，从受限到无受限的转变，我们获得了两种状态下界面形状的渐近预测。我们考虑释放固定体积的流体 (α = 0)。电流在重力作用下坍塌，从受限到无受限的转变，我们获得了两种状态下界面形状的渐近预测。