Streaming currents in turbulent flows and metal capillaries. II. Theory (2). Charge transported by the flow of liquid

Abstract

The mean charge density which appears if a liquid flows through a tube is calculated. The following dimensionless numbers are used.

1°. κRh, i.e. the ratio of the tube's hydraulic radius Rh and the diffuse layer thickness 1/κ.

2°. 12νκ/υ*, i.e. the ratio of the laminar sublayer thickness 12ν/υ* and the diffuse layer thickness 1/κ, where ν is the kinematic viscosity of the liquid and υ* the friction velocity as defined in I.5.

3°. τυ*κ, i.e. the ratio of the distance traveled with velocity υ* in the relaxation time τ and the diffuse layer thickness 1/κ.

4°. , i.e. the ratio of the mean charge density in the liquid and the mean charge density ρ0 = − 2εξ/πRh2 which appears in a laminar flow if κRh 1, where ε is the dielectric constant of the liquid and ξ the wall potential.

5°. Reynolds number and friction number where s is the specific density of the liquid, the mean velocity across the tube cross section and dp/dl the pressure gradient along the tube axis. Hydrodynamics teaches that λ is a function of Re, which is drawn in fig. I-7.

The following relations are deduced for κRh 1. In a laminar flow = 1. In a turbulent flow, as long as 12νκ/υ* 1, = Re λ/64, but if 12νκ/υ* 1, =kRh/4. The complete function is given in fig. II-1, which illustrates also the situation κRh 1.

It is proved that is only slightly dependent on the shape of the charge distribution in the turbulent core of the flow (i.e. on the value of τυ*κ). In tubes of finite lenth L, is influenced by the relaxation effect, by a possible saturation of wall current and by the fact that the velocity profile close behind the tube entrance differs from that farther in the tube. The consequences for κRh 1, illustrated in fig. II-3, are the following.

If LRh is small: ~ L2 for small L/Rh and ~ L2/Rh for large L/Rh.

If LRh is large: ~ l/Rh for small L/Rh and ~l/Rh2 for large L/Rh.

Similar relations are deduced for κRh 1 and illustrated in fig. II-2. All results are valid for flow through round tubes as well as between parallel walls.