In this talk, we will introduce a boundary treatment for solving general convection-diffusion equations on time-varying domain on a fixed Cartesian mesh. This method can achieve high order accuracy, and has a unified form for pure convection, convection-dominated, convection-diffusion, diffusion-dominated and pure diffusion cases. We also extend the boundary treatment to the compressible Navier-Stokes equations. Numerical tests demonstrate that our boundary treatment is high order accurate for problems with smooth solutions and also performs well for problems involving interactions between viscous shocks and moving rigid bodies.