Lava flow in tubes with elliptical cross sections

We develop a model of lava flow in a cylindrical tube with elliptical cross section. The lava is considered an isothermal, incompressible Newtonian fluid. We solve analytically the steady-state Navier–Stokes equation under a constant driving force, given by the component of gravity along the axis of the tube and obtain the velocity and stress field components in the fluid. The ratio between the flow rate of the elliptical tube and that of a circular tube, having the same cross sectional area, is found to be always less than 1 and to depend only on the value of eccentricity. The ratio decreases rapidly when the eccentricity becomes lower than about 0.5. The average flow velocity in a partially filled tube is calculated under the assumption of constant flow rate. In an elliptical tube, the shear traction is not uniform on the wall of the tube, but changes periodically with the position. It is maximum at the intersections with the minor axis and minimum at the intersections with the major axis, the ratio between the maximum and the minimum value being equal to the ratio between the lengths of the two axes. Assuming that the erosion rate of the wall of the tube is proportional to shear traction, we calculate the erosion of the wall as a function of time and find that its effect is such as to make the tube cross section closer to the circular shape.