# Computational methodologies for forced convection

 \begin{align} & \frac{\partial (\rho \varphi )}{\partial t}+\frac{\partial (\rho u\varphi )}{\partial x}+\frac{\partial (\rho v\varphi )}{\partial y}+\frac{\partial (\rho w\varphi )}{\partial z} \\ & =\frac{\partial }{\partial x}\left( \Gamma \frac{\partial \varphi }{\partial x} \right)+\frac{\partial }{\partial y}\left( \Gamma \frac{\partial \varphi }{\partial y} \right)+\frac{\partial }{\partial z}\left( \Gamma \frac{\partial \varphi }{\partial z} \right)+S \\ \end{align} (1)
where $\varphi$ is a general variable that can represent the directional components of velocity (u, v, or w), temperature, or mass concentration. The general diffusivity Γ can be viscosity, thermal conductivity, or mass diffusivity. S is the volumetric source term. Compared with the governing equation for heat conduction, the convection terms seem to be the only new terms introduced into eq. (4.200). As will become evident later, the contribution of the convection term on the overall heat transfer depends on the relative scale of convection over diffusion. Therefore, these two effects are always handled as one unit in the derivation of the discretization scheme (Patankar, 1980; Patankar, 1991). While analytical solutions of convection problems presented in the preceding sections based on boundary layer theory are few, the numerical solutions of convection based on boundary layer theory have been abundant (Tao, 2001; Minkowycz et al., 2006). With significant advancement of computational capability, the numerical solution of convection problems can now be performed by solving the full Navier-Stokes equation. The algorithms that will be discussed in this section are therefore based on the solution of the full Navier-Stokes equation and the energy equation, not on the boundary layer equations.