Question #42125

A conducting incompressible fluid of density $\rho$, dynamic viscosity $\eta$, and electrical conductivity $\sigma$ flows with an average velocity $v$ through a non-conducting cylindrical pipe of diameter $d$. A uniform transverse magnetic field $B$ is applied. The induced eddy currents generate a local Lorentz body force. A student incorrectly models the eddy currents as closed magnetic dipoles and concludes that since the field is uniform, the net magnetic force is zero. In reality, the local Lorentz force acts as an additional velocity-dependent drag. Assuming the inertial force per unit volume is $f_i = \rho \frac{v^2}{d}$, the standard viscous force per unit volume is $f_v = \eta \frac{v}{d^2}$, and the magnetic drag force per unit volume is $f_m = \sigma v B^2$, the flow transitions to turbulence when the effective Reynolds number (ratio of inertial force to total drag force) reaches a critical value $Re_0$. What is the correct critical velocity $v_c$ for the onset of turbulence?