  The fluid models mentioned in the title are classified. All characteristics
of the fluid are expressed through a master potential, satisfying an ordinary
second order differential equation. Different constraints are imposed on this
core of relations, finding new solutions and deriving the classical results for
perfect fluids and dust as particular cases. Many uncharged and charged
anisotropic solutions, all conformally flat and some uniform density solutions
are found. A number of solutions with linear equation among the two pressures
are derived, including the case of vanishing tangential pressure.
