We consider a model describing the unsteady motion of an incompressible, electro-rheological fluid. Due to the time-space-dependence of the power-law index, the analytical treatment of this system is involved. Standard results like a Poincare or a Korn inequality are not available. Introducing natural energy spaces, we establish the validity of a formula of integration-by-parts which allows to extend the classical
theory of pseudo-monotone operators to the framework of variable Bochner-Lebesgue spaces. This leads to generalised notions of pseudo-monotonicity and coercivity, the so-called Bochner pseudo-monotonicity and Bochner coercivity. With the aid of these notions and the established formula of integration-by-parts it is possible to prove an abstract existence result which immediately implies the weak solvability of the model describing the unsteady motion of an incompressible, electro-rheological fluid.