不可压缩流体和磁流体的压力测定
Pressure determinations for incompressible fluids and magnetofluids
摘要：
某些未解决的模糊性围绕不可压缩流的压力测定，纳维尔-斯托克斯和磁流体动力学。对于具有标准牛顿粘性项的均匀密度流体，取运动运动方程的发散，留下一个用于求解压力的泊松方程。但是泊松方程需要边界条件。对于矩形周期边界条件，用这种方法确定的压力是明确的.但在存在“无滑移”刚性壁的情况下，运动方程可用于推断压力P的狄利克雷和诺依曼边界条件，因此相当于超定。这有时被认为是一个问题，而壁面剪切流的数值处理通常已建立了一些相对特别的动力学方法来处理它，通常是一个似乎令人满意的“工作”。在这里，我们考虑一类螺线管速度场消失在无滑动壁，有所有的空间导数，但足够简单，明确的解析解P可以给出。满足两个边界条件分别给出两个压力，一个“诺依曼压力”和一个“狄利克雷压力”，它们在初始时刻甚至在任何动力学实现之前都有很大的不同。我们比较了这两种压力，并发现，特别是，它们导致不同的体积力附近的墙壁。这表明重新考虑无滑移边界条件，其中在无滑移壁处切向速度的消失由运动方程中的局部壁摩擦项代替。
英文摘要：
Certain unresolved ambiguities surround pressure determinations for incompressible flows, both Navier-Stokes and magnetohydrodynamic. For uniform-density fluids with standard Newtonian viscous terms, taking the divergence of the equation of motion leaves a Poisson equation for the pressure to be solved. But Poisson equations require boundary conditions. For the case of rectangular periodic boundary conditions, pressures determined in this way are unambiguous. But in the presence of "no-slip" rigid walls, the equation of motion can be used to infer both Dirichlet and Neumann boundary conditions on the pressure P, and thus amounts to an over-determination. This has occasionally been recognized as a problem, and numerical treatments of wall-bounded shear flows usually have built in some relatively ad hoc dynamical recipe for dealing with it, often one which appears to "work" satisfactorily. Here we consider a class of solenoidal velocity fields which vanish at no-slip walls, have all spatial derivatives, but are simple enough that explicit analytical solutions for P can be given. Satisfying the two boundary conditions separately gives two pressures, a "Neumann pressure" and a "Dirichlet pressure" which differ non-trivially at the initial instant, even before any dynamics are implemented. We compare the two pressures, and find that in particular, they lead to different volume forces near the walls. This suggests a reconsideration of no-slip boundary conditions, in which the vanishing of the tangential velocity at a no-slip wall is replaced by a local wall-friction term in the equation of motion.
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