  We consider the system of equations describing the flow of incompressible
fluids in bounded domain. In the considered setting, the Cauchy stress tensor
is a monotone mapping and has asymptotically $(s-1)$-growth with the parameter
$s$ depending on the spatial and time variable. We do not assume any smoothness
of $s$ with respect to time variable and assume the log-H\"{o}lder continuity
with respect to spatial variable. Such a setting is a natural choice if the
material properties are instantaneously, e.g. by the switched electric field.
We establish the long time and the large data existence of weak solution
provided that $s\ge(3d+2)(d+2)$.
