  We consider initial-boundary value problems for the $\kappa$-dependent family
of chemotaxis-(Navier--)Stokes systems \begin{align*} \left\{
\begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c}
n_{t}&+&u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(n\nabla c),\ &x\in\Omega,&
t>0,\\ c_{t}&+&u\cdot\!\nabla c&=\Delta c-cn,\ &x\in\Omega,& t>0,\\
u_{t}&+&\kappa(u\cdot\nabla)u&=\Delta u+\nabla P+n\nabla\phi,\ &x\in\Omega,&
t>0,\\ &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*}
in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary and given
potential function $\phi\in C^{1+\beta}(\overline{\Omega})$ for some $\beta>0$.
It is known that for fixed $\kappa\in\mathbb{R}$ an associated initial-boundary
value problem possesses at least one global weak solution
$(n^{(\kappa)},c^{(\kappa)},u^{(\kappa)})$, which after some waiting time
becomes a classical solution of the system. In this work we will show that upon
letting $\kappa\to0$ the solutions $(n^{(\kappa)},c^{(\kappa)},u^{(\kappa)})$
converge towards a weak solution of the Stokes variant $(\kappa=0)$ of the
systems above with respect to the strong topology in certain Lebesgue and
Sobolev spaces.
  We thereby extend the recently obtained result on the Stokes limit process
for classical solutions in the two-dimensional setting to the more intricate
three-dimensional case.
