# Intro to Transport Equations ## Navier-Stokes Newton’s 2nd Law For constant mass: $ \boldsymbol{F}=m\boldsymbol{a} \longrightarrow \boldsymbol{F}=m\frac{d\boldsymbol{v}}{dt} $ For varying mass: $ \boldsymbol{F}=\frac{d(m\boldsymbol{v})}{dt} $ $ F_x=\frac{d(mv_x)}{dt} \quad F_y=\frac{d(mv_y)}{dt} \quad F_z=\frac{d(mv_z)}{dt} $ Formulated for a solid body, need to use [[Navier-Stokes equations]] for a fluid volume: $ \boldsymbol{F}=\frac{D(m\boldsymbol{U})}{Dt} $ where $\frac{D}{Dt}$ is the [[Material Derivative|material derivative]]. It is standard practice to divide both sides by the fluid volume: $ \boldsymbol{f}=\frac{D(\rho\boldsymbol{U})}{Dt} $ where $\rho$ is the fluid density and $\boldsymbol{f}$ is the sum of external forces per unit volume acting on the fluid. $ \frac{D}{Dt}=\frac{\partial}{\partial t}+\boldsymbol{U}\cdot\nabla $ $ \boldsymbol{f}=\frac{\partial}{\partial t}(\rho\boldsymbol{U})+\boldsymbol{U}\cdot\nabla(\rho\boldsymbol{U}) $ $ \boldsymbol{f}=\frac{\partial(\rho\boldsymbol{U})}{\partial t}+\nabla\cdot(\rho\boldsymbol{UU}) $ ### External Forces $ \frac{\partial(\rho\boldsymbol{U})}{\partial t}+\nabla\cdot(\rho\boldsymbol{UU})=\underbrace{-\nabla p}_\text{pressure}+\underbrace{\nabla\cdot\boldsymbol{\tau}}_\text{viscous shear}+\underbrace{\rho\boldsymbol{g}}_\text{gravity} $ Minus sign in front of pressure gradient because fluid accelerates in direction of decreasing pressure Difficult because LHS is non-linear in velocity $\boldsymbol{U}$ ## [[]] Thermal energy transport:[[Transport equations]] Thermal energy transport: $ \frac{\partial(\rho c_pT)}{\partial t}+\underbrace{\nabla\cdot(\rho c_p\boldsymbol{U}T)}_\text{convection}=\underbrace{\nabla\cdot(k\nabla T)}_\text{diffusion} +S $ where $c_p$ is specific heat capacity, $k$ is thermal conductivity, and $S$ is a heat source. Transport is mainly driven by [[Convection|convection]] and [[Diffusion|diffusion]]. Radiation is neglected here. Expanding the Expanding the diffusion term: $ \nabla\cdot(k\nabla T)=\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right)+\frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right)+\frac{\partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right) $ Expanding the convection term: $ \nabla\cdot(\rho c_p \boldsymbol{U} T)=\frac{\partial}{\partial x}(\rho c_p U_x T)+\frac{\partial}{\partial y}(\rho c_p U_y T)+\frac{\partial}{\partial z}(\rho c_p U_z T) $ Concentration transport: $ \frac{\partial (\rho C)}{\partial t}+\nabla\cdot(\rho \boldsymbol{U}C)=\nabla \cdot(D\nabla C)+S $ where $C$ is the concentration of a species and $D$ is the diffusivity of the species. # 1D Diffusion Equation For thermal energy: $ \frac{\partial(\rho c_pT)}{\partial t}+\nabla\cdot(\rho c_p\boldsymbol{U}T)=\nabla\cdot(k\nabla T) +S $ We neglect the unsteady term and convection: $ \cancel{\frac{\partial(\rho c_pT)}{\partial t}}+\cancel{\nabla\cdot(\rho c_p\boldsymbol{U}T)}=\nabla\cdot(k\nabla T) +S $