Parametrization of general solutions for the Navier-Stokes equations
Parametrization of general solutions for the Navier-Stokes equations
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纳维-斯托克斯方程通解的参数化
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发表时间：
1994
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通讯作者：
K. B. Ranger
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作者：
K. B. Ranger
描述了一种构造控制粘性不可压缩液体运动的稳定二维纳维-斯托克斯方程通解的方法。流函数的解以包含两个任意复函数或四个任意实函数的隐式参数形式表示。介绍。为了确定粘性不可压缩液体稳态运动的物理量，例如力、扭矩、压力、边界涡度或剪切应力，需要满足纳维-斯托克斯方程的流体速度解。流动方程虽然是准线性和自主的，但通过流体速度和涡度的耦合，包含足够的非线性，因此只能通过使用参数函数隐式定义找到通用解；参见[2]，然而，这些解的隐式表示虽然不是最理想的形式，但并非没有意义，可以用来研究和简化基本流量及其导数的边界值。本文解决了构造根据参数方程隐式定义的一般类型解的问题。出发点是一个简洁的复杂变量公式；有关纳维-斯托克斯方程的最简单非简并形式，请参阅 [1]；即控制粘性不可压缩液体运动的稳态二维流动方程。该复杂方程包含流函数 y/ 和辅助实数函数 <> 作为因变量，具有拟线性、自治的优点，唯一的自变量为 ~z = x iy 。本分析中使用了所有三个属性。求解方法是将 y 和 参数化为 2, -, y/2, ，这是通过引入复杂的流函数来实现的，该函数提供 Riccati 类型的非线性方程并允许显式积分。复流函数没有明显的物理意义，被用作确定真实物理流函数的分析工具（//。一般来说，复流函数的实部和虚部都不是 1992 年 4 月 6 日接收的解。1991 数学学科分类。小学 76。©1994 布朗大学 335
A method is described for constructing general solutions of the steady two-dimensional Navier-Stokes equations governing the motion of a viscous incompressible liquid. The solution for the stream function is expressed in implicit parametric form containing two arbitrary complex functions or four arbitrary real functions. Introduction. To determine quantities of physical interest for the steady motion of a viscous incompressible liquid such as force, torque, pressure, boundary vorticity, or shear stress requires a solution for the fluid velocity satisfying the Navier-Stokes equations. The flow equations though quasi-linear and autonomous contain sufficient nonlinearity, through coupling of the fluid velocity and vorticity, in such a way that solutions of generality can only be found implicitly defined through the use of parametric functions; see [2], However, these implicit representations for the solutions, although not of the most desirable form, are not without interest and can be employed to investigate and simplify the boundary values of the basic flow quantities and their derivatives. The present paper addresses the problem of constructing general type solutions that are implicitly defined in terms of parametric equations. The starting point is a concise complex variable formulation; see [1] for the simplest nondegenerate form of the Navier-Stokes equations; namely, the steady two-dimensional flow equations governing the motion of viscous incompressible liquid. This complex equation contains the stream function y/ and an auxiliary real function <> as dependent variables and has the advantages of being quasi-linear, autonomous, with the only independent variable being ~z = x iy . All three properties are used in the present analysis. The method of solution is to parametrize y and in terms of 2, -, y/2, , and this is achieved by the introduction of complex stream functions which provide nonlinear equations of Riccati type and allow explicit integration. The complex stream functions have no obvious physical significance and are used as an analytical device to determine the real physical stream function (//. In general, neither the real nor the imaginary parts of the complex stream functions are solutions of the Received April 6, 1992. 1991 Mathematics Subject Classification. Primary 76. ©1994 Brown University 335