The equations of hydrodynamics and the resulting conservation laws are derived for a non-viscous fluid, in the post-Newtonian approximation of the Brans-Dicke theory, and applied to the problem of the dynamical instability of gaseous masses. It is found that the Brans-Dicke theory modifies the results of the theory of general relativity only by changing the numerical coefficients of the terms already present in the latter theory. And the dynamical instability predicted by general relativity is slightly suppressed in the Brans-Dicke theory
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Y. Nutku (1969) studied this question.