  We report a linear scaling law for an electrical voltage as a function of the
pressure drop in capillary pipes and ducts. This voltage is generated by a
process which is termed spin hydrodynamic generation (SHDG), a result of the
collective electron spin--coupling to the vorticity field in the laminar flow
in combination with an inverse spin-Hall effect. We study this phenomenon in
laminar duct flows with different width-to-height aspect ratios ranging from 1
(square ducts) to infinite (two dimensional channels). First, we analytically
solve the governing Valet-Fert spin diffusion equations for the SHDG by means
of the method of small parameters together with proper boundary conditions for
the set of inhomogeneous elliptic partial differential equations. Secondly, the
proposed linear scaling law is validated through a series of experiments using
capillary tubes with rectangular and square cross-sections. The experimental
results show a very good agreement to the analytically found scaling law. A
subsequent substitution of the bulk velocity of the laminar wall-bounded flows
by the pressure drop reveals a universal scaling law for the electrical voltage
that incorporates all pipe and duct geometries which we could study in our
experiments. Finally, the efficiency of the system is estimated for circular
pipes, rectangular and square ducts. This study shows that the efficiency of a
spin hydrodynamic generator is the same for a circular pipe and a square duct
with the same diameter and height, respectively. Hence, due to the ease of
manufacturing and the possibility to scale the experiments up to parallel
settings in a compact form, micro-channels with a square cross-section seem to
be the optimum for a spin hydrodynamic generator.
