This paper deals with the construction of a computational approach based on B-spline functions for solving a nonlinear boundary value problem describing the electro-hydrodynamic flow (EHF) of a fluid in a circular cylindrical conduit. The radial dependence of the velocity field emerging in the EHF is computed. We study the effects of two relevant parameters, namely the Hartmann electric number H and the strength of the nonlinearity β, on the velocity field. Computational results show that the method is of sixth-order accuracy. It is shown that the Hartmann electric number (HEN) and the strength of the nonlinearity both have a profound impact on the velocity profile of EHF and that these effects can be understood from analytical considerations. In particular, quantitative results include: The velocity, taking its maximum at the center of the conduit, does not exceed the value $1/(1+\beta )$ (this confirms a previous result). At large HEN, a boundary layer develops near the outer radial boundary of the conduit ($r=1$). Its thickness is proportional to $1/(H\sqrt{1+\beta })$, being determined by both the HEN and the nonlinearity. Moreover, when a boundary layer is present, the flow velocity has a plug-like profile approaching the plateau value $1/(1+\beta )$ (from below) for r values smaller than those of the boundary layer. If both the nonlinearity and the HEN are too small for a boundary layer to develop, then the flow profile is essentially parabolic and describable via a modified Bessel function. The CPU time for our method is provided.

本文讨论了一种基于B样条函数的计算方法的构造，用于解决非线性边界值问题，该问题描述了圆柱管道中流体的电动流体流动（EHF）。计算了EHF中出现的速度场的径向相关性。我们研究了两个相关参数的影响，即哈特曼电数H和非线性β的强度，在速度场上。计算结果表明，该方法具有六阶精度。结果表明，哈特曼数（HEN）和非线性强度都对EHF的速度分布产生深远的影响，并且这些影响可以通过分析考虑来理解。特别是，定量结果包括：速度，在导管中心取其最大值，不超过该值$\mathrm{1个}/（\mathrm{1个}+\beta ）$（这确认了先前的结果）。在较大的HEN处，在导管的外部径向边界附近会形成边界层（$\mathrm{[R}=\mathrm{1个}$）。它的厚度与$\mathrm{1个}/（H\sqrt{\mathrm{1个}+\beta }）$由HEN和非线性决定。此外，当存在边界层时，流速具有接近平稳值的塞状轮廓$\mathrm{1个}/（\mathrm{1个}+\beta ）$（从下面开始）对于r值小于边界层的r值。如果非线性和HEN都太小而无法形成边界层，则流量曲线本质上是抛物线形的，并且可以通过修改后的Bessel函数进行描述。提供了我们方法的CPU时间。