The article uses a singular boundary integral method to solve some reverse flow problems of laminar isochoric viscous fluid motion. Eddy-velocity and eddy-velocity-pressure formulations are used to solve the steady motion of a fluid subject to the Navier-Stokes equation. By introducing vorticity, the calculation process is divided into kinetic and kinematic parts. The first is described by the transport equation for vorticity, and the second by an integral equation, known as the Biot-Savart law for the enclosed area. Linear and quadratic conformal boundary elements and internal cells are used to discretize the integral equations. All edge and area integrations are performed analytically to increase the accuracy and speed of the computer calculation. Some examples of reverse flow currents are evaluated and compared with other published results.