In this paper, we establish the large-data and long-time existence of a suitable weak solution to an initial and boundary value problem driven by a system of partial differential equations consisting of the Navier-Stokes equations with the viscosity $\nu$ polynomially increasing with a scalar quantity $k$ that evolves according to an evolutionary convection diffusion equation with the right hand side $\nu \left(k\right)|𝖣\left(\overrightarrow{v}\right){|}^2$ that is merely $L^1$-integrable over space and time. We also formulate a conjecture concerning regularity...