The viscosity of some fluids, called magnetorheological (MR) fluids, changes when a magnetic field is applied.

Such fluids involve micron-sized magnetizable particles suspended in an appropriate carrier liquid, and are suitable for use in controllable hydraulic clutches. See Fig. P2-128. The MR fluids can have much higher viscosities than the ER fluids, and they often exhibit shear-thinning behavior in which the viscosity of the fluid decreases as the applied shear force increases. This behavior is also known as pseudoplastic behavior, and can be successfully represented by HerschelBulkley constitutive model expressed as \(\tau=\tau_{y}+K(d u / d y)^{m}\). Here \(\tau\) is the shear stress applied, \(\tau_{v}\) is the yield stress, \(K\) is the consistency index, and \(m\) is the power index. For a Herschel-Bulkley fluid with \(\tau_{y}=900 \mathrm{~Pa}, K=58 \mathrm{~Pa} \cdot \mathrm{s}^{m}\), and \(m=0.82\),

(a) find a relationship for the torque transmitted by an MR clutch for \(N\) plates attached to the input shaft when the input shaft is rotating at an angular speed of \(\omega\) while the output shaft is stationary and

(b) calculate the torque transmitted by such a clutch with \(N=11\) plates for \(R_{1}=50 \mathrm{~mm}\), \(R_{2}=200 \mathrm{~mm}, \dot{n}=3000 \mathrm{rpm}\), and \(h=1.5 \mathrm{~mm}\).