The steady state thermodynamic properties of a binary-phase shear fluid are studied quantitatively using
the compressible lattice Boltzmann BGK theory with mesoscopic inter-particle potentials. For the
Newtonian van der Waals fluid, numerical calculation shows that the effect of boundary shear on steady
state phase diagram of immiscible phases is negligible when the fluid is not in the near-critical region.
Streamlines show no penetration of macroscopic flow through the interface to cause the mass density
shift even when the boundary shear velocities are significant. The deformation of the droplets depends
on the shear rate and interfacial energy but the change of phase diagram during deformation is negligible.
In the near critical region, however, shear causes significant derivation in the phase diagram