应用拉普拉斯变换和留数法求解常见非稳态扩散情况下的菲克定律

发布时间：2018-02-03 17:04

  本文关键词： 菲克定律 无限扩散 有限扩散 拉普拉斯变换 复变函数　出处：《数学的实践与认识》2017年01期 　论文类型：期刊论文

【摘要】：介绍了三维和一维扩散下的菲克定律,以及两类涉及到扩散的实际问题,即求扩散粒子通过曲面的扩散通量和求解扩散粒子的浓度分布.通过拉普拉斯变换和复变函数相关数学理论,求解了菲克扩散定律在无限长介质和有限长介质两种非稳态扩散情况下的解.粒子在无限长介质中的非稳态扩散和浓度分布可通过方程φ(z,t)=Φ·erfc(z/2DT~(1/2))表示.方程为余补高斯误差函数.粒子在有限长介质中的非稳态扩散和浓度分布可通过方程φ(z,t)=Φ+Φ·4/π∑_(n=1)~(+∞)((-1)~n)/(2n-1)cos[z/L(n-1/2)π]e~((D_t)/(L~2)(n-1/2)~2π~2)表示.该方程为无限加和形式,当n≥100000时,φ可以精确到小数点后6位,在方程的图像上不再能观察出由n的取值造成的误差.从方程的图像可得到粒子在扩散介质中达到饱和的时间或粒子扩散到z=0处的时间等具有重要物理意义的参数.
[Abstract]:The Fick's law under three dimensional and one dimensional diffusion and two kinds of practical problems related to diffusion are introduced. In other words, the diffusion flux of diffusion particles through the surface and the concentration distribution of diffusion particles can be solved by means of Laplace transformation and mathematical theory of complex function. The solution of Fick's diffusion law in two kinds of unsteady diffusion of infinite medium and finite length medium is solved. The unsteady diffusion and concentration distribution of particles in infinite medium can be obtained by the equation 蠁 ~ (z). The equation is a complementary Gao Si error function. The unsteady-state diffusion and concentration distribution of particles in a finite medium can be obtained by the equation 蠁 _ (z). Tr = 桅 桅 路4 / 蟺 鈭，

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