The principles of continuum mechanics applied to transport processes and deformation in plant tissue

Abstract

The principles of continuum mechanics provide a consistent framework for the derivation of mathematical statements describing transport and deformation in a continuous medium. In this manuscript we formulate mathematical expressions for the transport of water and solute along the direction of deformation in plant tissue. Mathematical expressions for the local tissue deformation are also obtained and are expressed in terms of the displacement velocities and axial strain. These derivations are based upon the explicit consideration of the tissue as a mathematical continuum composed of a cell wall matrix, water, and solute phases. The principles of mass conservation for each component phase are employed to construct individual continuity equations for the matrix, water, and solute phases respectively. The differential equation for the flow of water in tissue is found by combining the continuity equations for both the water and cell wall phases with the axial derivative of the constitutive relationship relating the motion of the water with respect to the cell wall matrix. The partial differential equation for solute flow in deforming tissue is found by combining the continuity statements for the solute and the cell wall phases with the axial derivative of the constitutive relationship for solute motion with respect to cell wall matrix. The mathematical description of tissue deformation is found from the balance of linear momentum. Momentum balance for plant tissue states that the time rate of change of momentum of the cell wall phase equals the forces acting on the matrix expressed as the axial gradients of mechanical stress and osmotic potential, respectively. This expression is then combined with the axial derivative of constitutive relations found from a generalized form of Hooke's law and the Boyle-Van't Hoff relationship. The resulting description of deformation is formed as a system of coupled, quasi-linear, first-order, partial differential equations for the local tissue displacement velocities and the longitudinal tissue strain. The deformation equations describe either elastic or plastic one dimensional tissue expansion or contraction. The overall mathematical statements of transport processes and deformation in plant tissue is given by four differential equations coupled through the gradient of the tissue displacement velocities which describe the local shrinking or swelling of the tissue. This coupling provides the explicit connection between the local tissue deformation and the instantaneous water potential and solute concentration of the tissue.