The stochastic gravitational-wave backgrounds (SGWBs) from the cosmological
    first-order phase transitions (FOPTs) serve as a promising probe for the new
    physics beyond the standard model of particle physics. When most of the bubble
    walls collide with each other long after they had reached the terminal wall
    velocity, the dominated contribution to the SGWBs comes from the sound waves
    characterized by the efficiency factor of inserting the released vacuum energy
    into the bulk fluid motions. However, the previous works of estimating this
    efficiency factor have only considered the simplified case of the constant
    sound velocities in both symmetric and broken phases, either for the bag model
    with equal sound velocities or $\nu$-model with different sound velocities in
    the symmetric and broken phases, which is unrealistic from a viewpoint of
    particle physics. In this paper, we propose to solve the fluid EoM with an
    iteration method when taking into account the sound-velocity variation across
    the bubble wall for a general and realistic equation of state (EoS) beyond the
    simple bag model and $\nu$-model. We have found a suppression effect for the
    efficiency factor of bulk fluid motions, though such a suppression effect could
    be negligible for the strong FOPT, in which case the previous estimation from a
    bag EoS on the efficiency factor of bulk fluid motions still works as a good

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