The viscosity of some fluids changes when a strong electric field is applied on them. This phenomenon is known as the electrorheological (ER) effect, and fluids that exhibit such behavior are known as ER fluids. The Bingham plastic model for shear stress, which is expressed as \(\tau=\tau_{y}+\mu(d u / d y)\) is widely used to describe ER fluid behavior because of its simplicity. One of the most promising applications of ER fluids is the ER clutch. A typical multidisk ER clutch consists of several equally spaced steel disks of inner radius \(R_{1}\) and outer radius \(R_{2}, N\) of them attached to the input shaft. The gap \(h\) between the parallel disks is filled with a viscous fluid.

(a) Find a relationship for the torque generated by the clutch when the output shaft is stationary and \((b)\) calculate the torque for an ER clutch with \(N=11\) for \(R_{1}=50 \mathrm{~mm}, R_{2}=200 \mathrm{~mm}\), and \(\dot{n}=2400 \mathrm{rpm}\) if the fluid is SAE 10 with \(\mu=0.1 \mathrm{~Pa} \cdot \mathrm{s}\), \(\tau_{y}=2.5 \mathrm{kPa}\), and \(h=1.2 \mathrm{~mm}\).

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h = 1.2 mm -Shell R2 R Input shaft Plates mounted on input shaft Output shaft Plates mounted on shell Variable magnetic field