Cauchy momentum equation
The Cauchy momentum equation is a vector partial differential equation put forth by Augustin-Louis Cauchy that describes the non-relativistic momentum transport in any continuum. == Main equation == In convective (or Lagrangian) form the Cauchy momentum equation is written as: D u D t = 1 ρ ∇ ⋅ σ + a {\displaystyle {\frac {D\mathbf {u} }{Dt}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {a} } where u {\displaystyle \mathbf {u} } is the flow velocity vector field, which depends on time and space, (unit: m / s {\displaystyle \mathrm {m/s} } ) t {\displaystyle t} is time, (unit: s {\displaystyle \mathrm {s} } ) D u D t {\displaystyle {\frac {D\mathbf {u} }{Dt}}} is the material derivative of u {\displaystyle \mathbf {u} } , equal to ∂ t u + u ⋅ ∇ u {\displaystyle \partial _{t}\mathbf {u} +\mathbf {u} \cdot \nabla \mathbf {u} } , (unit: m / s 2 {\displaystyle \mathrm {m/s^{2}} } ) ρ {\displaystyle \rho } is the density at a given point of the continuum (for which the continuity equation holds), (unit: k g / m 3 {\displaystyle \mathrm {kg/m^{3}} } ) σ {\displaystyle {\boldsymbol {\sigma }}} is the stress tensor, (unit: P a = N / m 2 = k g ⋅ m − 1 ⋅ s − 2 {\displaystyle \mathrm {Pa=N/m^{2}=kg\cdot m^{-1}\cdot s^{-2}} } ) a = [ f x f y f z ] {\displaystyle \mathbf {a} ={\begin{bmatrix}f_{x}\\f_{y}\\f_{z}\end{bmatrix}}} is a vector containing all of the body accelerations (sometimes simply gravitational acceleration), (unit: m / s 2 {\displaystyle \mathrm {m/s^{2}} } ) ∇ ⋅ σ = [ ∂ σ x x ∂ x + ∂ σ y x ∂ y + ∂ σ z x ∂ z ∂ σ x y ∂ x + ∂ σ y y ∂ y + ∂ σ z y ∂ z ∂ σ x z ∂ x + ∂ σ y z ∂ y + ∂ σ z z ∂ z ] {\displaystyle \nabla \cdot {\boldsymbol {\sigma }}={\begin{bmatrix}{\dfrac {\partial \sigma _{xx}}{\partial x}}+{\dfrac {\partial \sigma _{yx}}{\partial y}}+{\dfrac {\partial \sigma _{zx}}{\partial z}}\\{\dfrac {\partial \sigma _{xy}}{\partial x}}+{\dfrac {\partial \sigma _{yy}}{\partial y}}+{\dfrac {\partial \sigma _{zy}}{\partial z}}\\{\dfrac {\partial \sigma _{xz}}{\partial x}}+{\dfrac {\partial \sigma _{yz}}{\partial y}}+{\dfrac {\partial \sigma _{zz}}{\partial z}}\\\end{bmatrix}}} is the divergence of stress tensor.