Adrian Bejan | Root shape, from Design in Nature

The video asks for the shape of a root that maximizes water production, treating the root as a solid body of revolution in wet soil, with the ground at the top and the vertical z-axis pointing upward. Water seeps from the ground into the root and then upward through a bundle of tubules, driven by a pressure difference and hindered by viscosity, while the pressure in the ground is higher than above ground. Inside the root, the average speed in the vertical direction and the transversal velocity in the radial direction are linked to permeability, so the mass flow rate at ground level follows from how the diameter at altitude z is drawn under a volume fixed constraint. With these pieces, the profile that carries more water emerges, and the conclusion is a conical root, along with the lesson that big roots convey high water flow rates out of the ground.

The drawing sets radial and vertical coordinates, pressure in the ground, pressure inside the root, and pressure above ground, then reads seepage upward with average speed in the vertical direction and transversal inflow from the wet soil, both expressed with permeability and viscosity, so the pressure difference along the height explains why water moves from high to low inside the root.

The root interior is a bundle of tubules, like very narrow straws, so water moves with great difficulty and leaves a trace of pressure that decreases upward. This picture supports using Darcy’s law in the vertical and radial directions, with permeability in the z direction greater than in the transverse direction, which matches the idea of a shell that channels water along the height.

A small slice between altitude z and z plus a small increment balances what enters from below and from the side with what exits above, using the wetted perimeter, lateral area, and cross-sectional area to write the mass flow rate in terms of the average speeds, so the relation between vertical and radial motions links shape and production at the top.

The fixed-volume condition resolves the problem, and the profile is assumed to be a power law in altitude, together with a power law for the pressure excess inside the root. Reading the exponents reveals that the diameter grows linearly with height. Hence, the shape is conical, and the constants follow from the same set of statements without changing the physical picture.

The mass flow rate at ground level rises with size, so big roots and big trees are big water pumps. The final message is that the conical root, with the given volume fixed and the pressure difference maximized, maximizes production, which explains why the shape and the flow belong in the same drawing.