  Steady-state circulation currents are predicted in tidally deformed binary
stars, which are believed to be progenitors of double black-hole merger events.
This work aims to quantitatively characterise the steady-state circulation
currents in components of a tidally locked binary system and to explore the
effects of such currents on numerical models. Previous results describing the
circulation in a single rotating star and a binary star are used to deduce a
new prescription for the internal circulation in tidally locked binaries. We
explore the effect of this prescription numerically for binary systems with
primary masses between 25 and 100 solar masses. When comparing circulation
velocities in the radial direction for the single rotating star and binary
star, it is found that the average circulation velocity in the binary star may
be described as an enhancement to the circulation velocity in a single rotating
star. This velocity enhancement is a simple function depending on the masses of
the binary components and amounts to a factor of approximately two when the
components have equal masses. It is found that the ehancement causes the
formation of double helium stars through efficient mixing to occur for systems
with higher initial orbital periods, lower primary masses and lower mass
ratios, compared to the standard circulation scenario. Taking into account
appropriate distributions for primary mass, initial period and mass ratio,
models with enhanced mixing predict 2.4 times more double helium stars being
produced in the parameter space than models without. We conclude that the
effects of companion-induced circulation have strong implications for the
formation of close binary black holes. Not only do the predicted detection
rates increase but double black-hole systems with mass ratios as low as 0.8 may
be formed when companion-induced circulation is taken into account.
