We perform a comprehensive numerical study of d-wave Fermi surface
deformations (dFSD) on a square lattice, the so-called d-wave Pomeranchuk
instability, including bilayer coupling. Since the order parameter
corresponding to the dFSD has Ising symmetry, there are two stacking patterns
between the layeres, (+,+) and (+,-). This additional degree of freedom gives
rise to a rich variety of phase diagrams. The phase diagrams are classified by
means of the energy scale Lambda_{z}, which is defined as the bilayer splitting
at the saddle points of the in-plane band dispersion. As long as Lambda_{z} ne
0, a major stacking pattern is usually (+,-), and (+,+) stacking is stabilized
as a dominant pattern only when the temperature scale of the dFSD instability
becomes much smaller than Lambda_z. For Lambda_{z}=0, the phase diagram depends
on the precise form of the bilayer dispersion. We also analyze the effect of a
magnetic field on the bilayer model in connection with a possible dFSD
instability in the bilyared ruthenate Sr_3Ru_2O_7.Comment: 18 pages, 7 figure