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A fractional differential equation for a MEMS viscometer used in the oil industry.

The motion of a novel MEMS viscometer for downhole oil-well logging is analysed by coupling exact solutions of the Navier-Stokes equations to a Hookeian spring model. The device is examined in both “forced” and “plucked” mode. We show that the device displacement satisfies a novel fractional differential equation (FDE), which is solved in closed-form using Mittag-Leffler functions. This shows that, contrary to previous industry assumptions, the motion decays not exponentially, but algebraically. This new result is confirmed numerically, and more complicated nonlinear applications are also discussed. This work has been used during experimental tests of the viscometer by Schlumberger.