( 15) and discuss a cell which is expanding only in the direction of its long axis and which is in contact with water at only one end. For this one-dimensional system (Fig. 1, left) to a first approximation, the rate of volume increase (dV/dt) equals the rate of volumetric water flow (Q) into the cell. Furthermore, Q is proportional to the water flux, and, equivalently, to the growth velocity.d= Q=AJ=Ag (1) where A is the cross-sectional area of the cell face perpendicular to the flow, g is the cell extension rate (growth velocity), and J is the water flux (volumetric flow rate/unit cross-sectional area). Now, diverging from the earlier theory, a cell embedded in a continuum of expanding cells is considered (Fig. 1, right). For this cell, only water which is moving faster than wall 1 will cross the wall and enter the cell. In this case, the amount of water which flows into the cell equals the area of wall 1 times the velocity by which the water flux exceeds the rate of movement of the wall:Flow into the cell = Q, = A(J, -gi) where gi is the growth velocity at 1; similarly Flow out of the cell = Q2 = A(J2 -g2)Much of the large literature on plant water transport concerns flow through nongrowing tissues and deals with the often large +2 gradients in the soil-plant-atmosphere continuum. Recently, Molz and Boyer (13) made a pioneering study of *P distributions which would characterize a growing tissue. They showed that 'I would become more negative along a line extending from a xylem element into the cortex of an elongating soybean hypocotyl. Below, an inhomogeneous distribution of growth rates is considered and a two-dimensional pattern of which can sustain growth of a corn root is found.'This research was supported by National Science Foundation Grant PCM 78-23710.2 Abbreviations: 'P, water potential; A, area of plant element in plane perpendicular to flux; g, growth velocity vector; J, water flux vector; K, hydraulic conductivity tensor; L, relative elemental growth rate (local strain rate); 1, cell length; Q, volumetric flow rate; S, surface area of plant element; t, time; V, volume of plant element; x, distance; r, radial distance from root center line; z, longitudinal distance from root tip.(3) where J2 is the water flux at 2 and g2 is the growth velocity at 2.Since water is highly incompressible, its velocity does not change along the one-dimensional continuum. Therefore J1 = J2. The net flow of water into the expanding cell is equal to the difference between the inflow and the outflow so that the net flow (A Q) into the cell equals the cross-sectional area times the difference in growth rates between the apical and basal end of the cell: AQ = A(g2-gg).(4) As in the simpler example above, the rate of volume change (dV/dt) in the deformable cell is equal to the net volumetric water flow into the cell: where n is the unit normal to the surface. Equation 5a shows that the rate of volume change of the cell which is instantaneously occupying the fixed volume V at a 859 (2) www.plantphysiol.org on May 9, 2018 -Pub...