流体传热方程

Equation of Energy in terms of qq q q \pmb q

控制方程表达通式：

ρC^pDTDt=−∇⋅qq−∂lnρ∂lnTDpDt−ττ:∇vv ρ C ^ p D T D t = − ∇ ⋅ q q − ∂ l n ρ ∂ l n T D p D t − τ τ : ∇ v v

\rho \hat {C}_p \dfrac {DT}{Dt}=-\nabla \cdot \pmb q-\dfrac {\partial ln \rho}{\partial ln T} \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v

1.直角坐标系( x,y,z x , y , z x, y, z )

直角坐标系Cartesian coordinates ( x,y,z x,y,z \textit { x,y,z }):	NO.
ρC^p(∂T∂t+vx∂T∂x+vy∂T∂y+vz∂T∂z)=−[∂qx∂x+∂qy∂y+∂qz∂z]−(∂lnρ∂lnT)pDpDt−ττ:∇vv ρ C ^ p ( ∂ T ∂ t + v x ∂ T ∂ x + v y ∂ T ∂ y + v z ∂ T ∂ z ) = − [ ∂ q x ∂ x + ∂ q y ∂ y + ∂ q z ∂ z ] − ( ∂ l n ρ ∂ l n T ) p D p D t − τ τ : ∇ v v \rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_x \dfrac {\partial T}{\partial x} +v_y \dfrac {\partial T}{\partial y} + v_z \dfrac {\partial T}{\partial z}\right) = - \left [\dfrac {\partial q_x}{\partial x} +\dfrac {\partial q_y}{\partial y}+\dfrac {\partial q_z}{\partial z}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v	1-1

直角坐标系Cartesian coordinates ( x,y,z x,y,z \textit { x,y,z }):

NO.

ρC^p(∂T∂t+vx∂T∂x+vy∂T∂y+vz∂T∂z)=−[∂qx∂x+∂qy∂y+∂qz∂z]−(∂lnρ∂lnT)pDpDt−ττ:∇vv ρ C ^ p ( ∂ T ∂ t + v x ∂ T ∂ x + v y ∂ T ∂ y + v z ∂ T ∂ z ) = − [ ∂ q x ∂ x + ∂ q y ∂ y + ∂ q z ∂ z ] − ( ∂ l n ρ ∂ l n T ) p D p D t − τ τ : ∇ v v \rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_x \dfrac {\partial T}{\partial x} +v_y \dfrac {\partial T}{\partial y} + v_z \dfrac {\partial T}{\partial z}\right) = - \left [\dfrac {\partial q_x}{\partial x} +\dfrac {\partial q_y}{\partial y}+\dfrac {\partial q_z}{\partial z}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v

1-1

2.圆柱坐标系（ r,θ,z r , θ , z r,\theta, z)

圆柱坐标系Cylindrical coordinates coordinates ( r, θ, z r, θ , z \textit {r, $\theta$, z }):	NO.
ρC^p(∂T∂t+vr∂T∂r+vθr∂T∂θ+vz∂T∂z)=−[1r∂∂r(rqr)+1r∂qθ∂θ+∂qz∂z]−(∂lnρ∂lnT)pDpDt−ττ:∇vv ρ C ^ p ( ∂ T ∂ t + v r ∂ T ∂ r + v θ r ∂ T ∂ θ + v z ∂ T ∂ z ) = − [ 1 r ∂ ∂ r ( r q r ) + 1 r ∂ q θ ∂ θ + ∂ q z ∂ z ] − ( ∂ l n ρ ∂ l n T ) p D p D t − τ τ : ∇ v v \rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + v_z \dfrac {\partial T}{\partial z}\right) = - \left [\dfrac {1}{r} \dfrac {\partial}{\partial r} (r q_r) + \dfrac {1}{r} \dfrac {\partial q_\theta}{\partial \theta}+\dfrac {\partial q_z}{\partial z}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v	2-1

圆柱坐标系Cylindrical coordinates coordinates ( r, θ, z r, θ , z \textit {r, $\theta$, z }):

NO.

ρC^p(∂T∂t+vr∂T∂r+vθr∂T∂θ+vz∂T∂z)=−[1r∂∂r(rqr)+1r∂qθ∂θ+∂qz∂z]−(∂lnρ∂lnT)pDpDt−ττ:∇vv ρ C ^ p ( ∂ T ∂ t + v r ∂ T ∂ r + v θ r ∂ T ∂ θ + v z ∂ T ∂ z ) = − [ 1 r ∂ ∂ r ( r q r ) + 1 r ∂ q θ ∂ θ + ∂ q z ∂ z ] − ( ∂ l n ρ ∂ l n T ) p D p D t − τ τ : ∇ v v \rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + v_z \dfrac {\partial T}{\partial z}\right) = - \left [\dfrac {1}{r} \dfrac {\partial}{\partial r} (r q_r) + \dfrac {1}{r} \dfrac {\partial q_\theta}{\partial \theta}+\dfrac {\partial q_z}{\partial z}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v

2-1

3.球坐标系（ r,θ,ϕ r , θ , ϕ r, \theta, \phi )

球坐标系Spherical coordinates（ r, θ, ϕ r, θ , ϕ \textit {r, $\theta$, $\phi$ }):	NO.
ρC^p(∂T∂t+vr∂T∂r+vθr∂T∂θ+vϕrsinθ∂T∂ϕ)=−[1r2∂∂r(r2qr)+1rsinθ∂∂θ(qθsinθ)+1rsinθ∂qϕ∂ϕ]−(∂lnρ∂lnT)pDpDt−ττ:∇vv ρ C ^ p ( ∂ T ∂ t + v r ∂ T ∂ r + v θ r ∂ T ∂ θ + v ϕ r s i n θ ∂ T ∂ ϕ ) = − [ 1 r 2 ∂ ∂ r ( r 2 q r ) + 1 r s i n θ ∂ ∂ θ ( q θ s i n θ ) + 1 r s i n θ ∂ q ϕ ∂ ϕ ] − ( ∂ l n ρ ∂ l n T ) p D p D t − τ τ : ∇ v v \rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + \dfrac {v_\phi}{rsin\theta} \dfrac {\partial T}{\partial \phi}\right) = - \left [\dfrac {1}{r^2} \dfrac {\partial}{\partial r} (r^2 q_r) + \dfrac {1}{r sin\theta} \dfrac {\partial}{\partial \theta}(q_\theta sin\theta)+ \dfrac {1}{r sin\theta} \dfrac {\partial q_\phi}{\partial \phi}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v	3-1

球坐标系Spherical coordinates（ r, θ, ϕ r, θ , ϕ \textit {r, $\theta$, $\phi$ }):

NO.

ρC^p(∂T∂t+vr∂T∂r+vθr∂T∂θ+vϕrsinθ∂T∂ϕ)=−[1r2∂∂r(r2qr)+1rsinθ∂∂θ(qθsinθ)+1rsinθ∂qϕ∂ϕ]−(∂lnρ∂lnT)pDpDt−ττ:∇vv ρ C ^ p ( ∂ T ∂ t + v r ∂ T ∂ r + v θ r ∂ T ∂ θ + v ϕ r s i n θ ∂ T ∂ ϕ ) = − [ 1 r 2 ∂ ∂ r ( r 2 q r ) + 1 r s i n θ ∂ ∂ θ ( q θ s i n θ ) + 1 r s i n θ ∂ q ϕ ∂ ϕ ] − ( ∂ l n ρ ∂ l n T ) p D p D t − τ τ : ∇ v v \rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + \dfrac {v_\phi}{rsin\theta} \dfrac {\partial T}{\partial \phi}\right) = - \left [\dfrac {1}{r^2} \dfrac {\partial}{\partial r} (r^2 q_r) + \dfrac {1}{r sin\theta} \dfrac {\partial}{\partial \theta}(q_\theta sin\theta)+ \dfrac {1}{r sin\theta} \dfrac {\partial q_\phi}{\partial \phi}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v

3-1

注：黏度耗散项（−ττ:∇vv）（ − τ τ : ∇ v v ）（-\pmb \tau : \nabla \pmb v）很小，可以被忽略，除非速度的梯度非常大。另外对于恒定密度的流体 ∂lnρ∂lnT ∂ l n ρ ∂ l n T \dfrac {\partial ln \rho}{\partial ln T}项等于零。

参考文献

R. Byron Bird, Warren E. stewart, Edwin N. Lightfoot.* Transport phenomena:Revised second edition* John Wiely &Sons, Inc.