This paper presents a dynamical and generalized synchronization (GS) of two dependent chaotic nonlinear advection-diffusion-reaction (ADR) processes with forcing term, which is unidirectionally coupled in the master-slave configuration. By combining backward differentiation formula-Spline (BDFS) scheme with the Lyapunov direct method, the GS is studied for designing controller function of the coupled nonlinear ADR equations without any linearization. The GS behaviors of the nonlinear coupled ADR problems are obtained to demonstrate the effectiveness and feasibility of the proposed technique without losing natural properties and reduce the computational difficulties on capturing numerical solutions at the low value of the viscosity coefficient. This technique utilizes the master configuration to monitor the synchronized motions. The nonlinear coupled model is described by the incompressible fluid flow coupled to thermal dynamics and motivated by the Boussinesq equations. Finally, simulation examples are presented to demonstrate the feasibility of the synchronization of the proposed model.