  A new algorithm for solving the Navier-Stokes equations (NSE) on a quantum
device is presented. The non-linearity of fluid flow equations is managed by
using the stream function-vorticity formulation, while the lattice Boltzmann
method (LBM) is utilized for solving the corresponding system of equations
numerically. Following the nature of the lattice Boltzmann method, the proposed
quantum algorithm consists of five major sections: initialization, collision,
propagation, boundary condition implementation, and macros calculation. The
collision and boundary condition step is quantumly implemented by applying the
standard-form encoding approach, while the quantum walk procedure is applied
for the propagation step. The algorithm is implemented by using IBM's quantum
computing software development framework Qiskit, while for the verification
purposes two-dimensional (2D) cavity flow is simulated and compared with
classical code.
