  We derive a set of nontrivial relations between second-order transport
coefficients which follow from the second law of thermodynamics upon
considering a regime close to uniform rotation of the fluid. We demonstrate
that extension of hydrodynamics by spin variable is equivalent to modifying
conventional hydrodynamics by a set of second-order terms satisfying the
relations we derived. We point out that a novel contribution to the heat
current orthogonal to vorticity and temperature gradient reminiscent of the
thermal Hall effect is constrained by the second law.
