Theoretical analyses of solute concentration-gradient driven water flow and solute concentration variation in solution-saturated semi-permeable materials

Based on the commonly-used definition of the semi-permeable material(SPM), in which the solvent(such as water)can pass through but the solute cannot move freely, a novel constitutive equation is presented, in this paper, to theoretically express the intrinsic constitutive relationship between the porosity variation rate with time and the solute concentration variation rate with time. From this theoretical finding, a mathematical model is established to describe the solute concentrationgradient driven water flow and solute concentration variation problem in solution-saturated semi-permeable materials(SSSPMs). In particular, the solute concentration variation in the considered problem can be mathematically described as a linear homogeneous second-order partial differential equation with a variable coefficient in the front of the time term. A special mathematical transform is presented and used to solve this equation, so that the analytical solution for the solute concentration variation in the considered problem has been derived in a purely mathematical manner. The derived analytical solution is then used to provide some theoretical understanding of solute concentration variations in the SSSPM layer.