  The Navier-Stokes equations are paradigmatic equations describing
hydrodynamics of an interacting system with microscopic interactions encoded in
transport coefficients. In this work we show how the Navier-Stokes equations
arise from the microscopic dynamics of nearly integrable $1d$ quantum many-body
systems. We build upon the recently developed hydrodynamics of integrable
models to study the effective Boltzmann equation with collision integral taking
into account the non-integrable interactions. We compute the transport
coefficients and find that the resulting Navier-Stokes equations have two
regimes, which differ in the viscous properties of the resulting fluid. We
illustrate the method by computing the transport coefficients for an
experimentally relevant case of coupled $1d$ cold-atomic gases.
