标题： 一种多孔弹性材料中两种溶质传输的数学模型及其应用 显示英文标题

标题： A Mathematical Model for Two Solutes Transport in a Poroelastic Material and Its Applications

摘要： 利用弹性理论的已知数学基础，提出了在多孔弹性材料（软组织是一个典型例子）中两种溶质传输的数学模型。假设溶解在流体中且通过不同尺寸孔隙传输的分子具有本质上不同的尺寸。应力张量，导致材料变形的主要力，不仅以标准线性形式给出，还包含一个额外的非线性部分。该模型在1D空间中构建，由六个非线性方程组成。结果显示，在稳态情况下，控制方程是可积的，因此所有稳态解都被构造出来。所得到的解用于健康组织和肿瘤组织的示例中，特别是计算并比较了线性和非线性应力张量情况下参数取自实验数据时的组织位移。由于在非稳态情况下控制方程不可积，因此使用李对称分析来构造时间相关的精确解。根据控制方程中出现的参数，确定了几种具有非平凡李对称性的特殊情况。结果是构造了多参数精确解族，包括用特殊函数（超几何函数和贝塞尔函数）表示的解。展示了所得解的一个可能应用。 显示英文摘要

摘要： Using well-known mathematical foundations of the elasticity theory, a mathematical model for two solutes transport in a poroelastic material (soft tissue is a typical example) is suggested. It is assumed that molecules of essentially different sizes dissolved in fluid and are transported through pores of different sizes. The stress tensor, the main force leading to the material deformation, is taken not only in the standard linear form but also with an additional nonlinear part. The model is constructed in 1D space and consists of six nonlinear equations. It is shown that the governing equations are integrable in stationary case, therefore all steady-state solutions are constructed. The obtained solutions are used in an example for healthy and tumour tissue, in particular, tissue displacements are calculated and compared for parameters taken from experimental data in cases of the linear and nonlinear stress tensors. Since the governing equations are non-integrable in non-stationary case, the Lie symmetry analysis is used in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with non-trivial Lie symmetries are identified. As a result, multi-parameter families of exact solutions are constructed including those in terms of special functions(hypergeometric and Bessel functions). A possible application of the solutions obtained is demonstrated.