N-S方程（二）-各坐标系下N-S方程的形式

N-S方程

ρ d V d t = ρ g − ∇ p + μ ∇ 2 V \rho\frac{dV}{dt}=\rho g-\nabla p + \mu\nabla^2V ρdtdV​=ρg−∇p+μ∇2V

ρ ( ∂ V ∂ t + ( V ⋅ ∇ ) V ) = ρ g − ∇ p + μ ∇ 2 V \rho\big( \frac{\partial V}{\partial t} + (V\cdot \nabla)V \big)=\rho g-\nabla p + \mu\nabla^2V ρ(∂t∂V​+(V⋅∇)V)=ρg−∇p+μ∇2V

∂ V ∂ t + ( V ⋅ ∇ ) V = g − 1 ρ ∇ p + μ ρ ∇ 2 V \frac{\partial V}{\partial t} + (V\cdot \nabla)V = g-\frac{1}{\rho}\nabla p + \frac{\mu}{\rho}\nabla^2V ∂t∂V​+(V⋅∇)V=g−ρ1​∇p+ρμ​∇2V

d V d t = ∂ v i ∂ t + ( V ⋅ ∇ ) V = ∂ v i ∂ t + v j ∂ v i ∂ x j \frac{dV}{dt} = \frac{\partial v_i}{\partial t}+(V\cdot \nabla)V = \frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j} dtdV​=∂t∂vi​​+(V⋅∇)V=∂t∂vi​​+vj​∂xj​∂vi​​

笛卡爾坐标系下的N-S方程

∂ v i ∂ t + v j ∂ v i ∂ x j = g − 1 ρ ∂ p ∂ x i + μ ρ ∂ 2 v i ∂ x j 2 \frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j}=g-\frac{1}{\rho}\frac{\partial p}{\partial x_i}+\frac{\mu}{\rho}\frac{\partial^2v_i}{\partial x^2_j} ∂t∂vi​​+vj​∂xj​∂vi​​=g−ρ1​∂xi​∂p​+ρμ​∂xj2​∂2vi​​