Notes on the Arrow of Flow: Fluid Mechanics Using Navier–Stokes in Four-Dimensional Arrows

Abstract

The equations of Navier and Stokes are Newton's law for a fluid. Whether their solutions stay smooth, free of sudden jumps and infinite slopes, is a great open question of mathematics. In this paper, we write the fluid entirely in the arrow numbers. Then we ask the question in the interval calculus, where a derivative is a quotient over a finite step and therefore exists for every flow. Consequently, smoothness becomes a measurement: two neighbouring past derivatives must come together in proportion to the step, and the step at which they part is the flow's finest scale. Next, the arrow of flow, whose vector part is the velocity, has a length that sets a saturation velocity, as for the carriers of a semiconductor. Moreover, the convective term is the arrow of flow times its own gradient, in Lamb's form, and the Navier–Stokes equation is one exact four-dimensional arrow identity. Its friction is the square of the gradient, a damped wave equation with no new constant. Consequently, below the saturation velocity the energy of the flow stays finite at every time. We then restate the question of smoothness for a fluid of carriers and prove that its flow is smooth at every finite time; the classical question is the limit with no saturation velocity. Then an arrow walk, in which carriers step along their arrows, measures the Taylor–Green vortex. It reproduces the published dissipation and shows the vortex sheets thinning steadily, with no sign of a collapse. Finally, in the electron fluid of graphene, the measured viscosity, read as the friction's relaxation time with a shear speed of half the Fermi velocity, agrees with the collision time of the electrons within about 15%; near neutrality, a measurement near 0.4 to 1 THz would test the arrows' shorter relaxation time. Submitted to the Journal of Modern Physics. The checks archive contains the 41 symbolic and numerical check scripts, each with a control that must fail, their data, and the log of the full run.