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A visco-elastic medium, such as a strongly coupled dusty plasma (SCDP), permits both the propagation of longitudinal dust acoustic mode due to compressibility and transverse shear mode due to elasticity. In the presence of finite velocity shear, these two modes get coupled with each other and eventually the coupled mode becomes unstable. In a non-Newtonian dust fluid, it has been found that the viscosity gradient has a modulating effect on this shear-driven instability under various parametric regimes. A detailed investigation has been carried out on the effect of viscosity gradient on the stability characteristics of a strongly coupled dusty plasma by using the conventional dust fluid equations; both analytically and numerically. These results can be helpful in understanding the phenomena associated with mechanical instabilities in highly viscous fluids; such as metallic glasses, Earth's mantle etc. [ABSTRACT FROM AUTHOR]