Abstract

We study an initial–boundary value problem with free boundary for one-dimensional equations of viscous gas dynamics. The problem models the motion of a crankshaft mechanism under gas pressure. It is assumed that the gas fills a cylinder, which is modeled by the interval \([0,1]\). A variable point \(a(t)\in[0,1]\) models a piston moving inside the cylinder. The piston is assumed to be connected to a planar three-link crankshaft mechanism. We also assume that a velocity distribution on the boundary of the cylinder and a density distribution on gas inflow segments are given. The gas motion is described by the one-dimensional Navier–Stokes equations of viscous compressible fluid dynamics. It is required to determine the joint motion of the gas and crankshaft mechanism. We prove that this problem has a weak renormalized solution.

中文翻译：

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粘性气体压力下曲轴机构的动力学

摘要

对于粘性气体动力学的一维方程，我们研究了具有自由边界的初边值问题。该问题模拟了在气压下曲轴机构的运动。假设气体填充了一个气缸，该气缸由间隔\（[0,1] \）建模。可变点\（a（t）\ in [0,1] \）模拟在气缸内移动的活塞。假设活塞连接到平面三连杆曲轴机构。我们还假定给出了圆柱边界上的速度分布和气体流入段的密度分布。气体运动由粘性可压缩流体动力学的一维Navier-Stokes方程描述。需要确定汽油和曲轴机构的联合运动。我们证明此问题的重新规范化解决方案很弱。