  We develop a quantum representation for Newtonian viscous fluid flows by
establishing a mapping between the Navier-Stokes equation (NSE) and the
Schr\"odinger-Pauli equation (SPE). The proposed nonlinear SPE incorporates the
two-component wave function and the imaginary diffusion. Consequently,
classical fluid flow can be interpreted as a non-Hermitian quantum spin system.
Using the SPE-based numerical simulation of viscous flows, we demonstrate the
quantum/wave-like behavior in flow dynamics. Furthermore, the SPE equivalent to
the NSE can facilitate the quantum simulation of fluid dynamics.
