4.1.12. Navier-Stokes Momentum Equation
\[\nu = {\mu \over \rho}\]
Mnemonic:
Kinematic viscosity \(\ll\) Dynamic viscosity, as kinematics deals with smaller things than dynamics.
Units:
\(\nu \ (m^2/s)\)
\(\mu \ (kg/m/s)\)
\(\rho \ (kg/m^3)\)
\(\rho = \text{constant} \longrightarrow \nabla \cdot \vec{u} = 0\) i.e. velocity field must be divergence free
\(\rho = \text{constant} \longrightarrow\) pressure and density are decoupled, i.e. pressure is no longer a state variable, it is a constant \(\longrightarrow\) require pressure-velocity coupling
Continuum hypothesis \(\longrightarrow Kn \ll 1\) and no mass-energy conversion
- Form of diffusive fluxes e.g.:
Newtownian \(\tau_{ij} = \mu \gamma_{ij}\)
Fourier’s Law \(\vec{q} = -k \nabla T\)
Viscosity model \(\mu = \mu(T)\) e.g. Sutherland’s Law
- Equation of state (but this concerns the solution)
Stokes assumption \({2 \over 3} \mu + \lambda = 0\)
Thermally perfect gas \(p=\rho RT\)
Calorcally perfect gas \(e=C_v T\)
\[\text{Total stress tensor = Normal stress tensor + Deviatoric stress tensor}\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} =
-p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} +
\overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau}\]
(Deviatoric stress tensor is also the shear stress tensor)
where the identity matrix is:
\[\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{I} =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}\end{split}\]
Total stress tensor:
Inviscid:
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau} = 0\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = -p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I}\]
Viscous:
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau} \ne 0\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = -p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} + \overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau}\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma}=-p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} + \overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau}\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau}= \mu \overset{\underset{\mathrm{\rightrightarrows}}{}}{\gamma}\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\gamma}= 2 \left( \overset{\underset{\mathrm{\rightrightarrows}}{}}{e} - {1 \over 3} (\nabla \cdot \vec{u}) \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \right)\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{e}= {1 \over 2} \left( \nabla \otimes \vec{u} + (\nabla \otimes \vec{u})^T \right)\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma}=-p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} +
\mu \left( \nabla \otimes \vec{u} + (\nabla \otimes \vec{u})^T - {2 \over 3} (\nabla \cdot \vec{u}) \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \right)\]
\[e_{ij} = {1 \over 2} \left( {{\partial u_j} \over {\partial x_i}} + {{\partial u_i} \over {\partial x_j}} \right)\]
\[\begin{split}e_{ij} =
\begin{bmatrix}
{{\partial u_1} \over {\partial x_1}} & {1 \over 2} \left( {{\partial u_2} \over {\partial x_1}} + {{\partial u_1} \over {\partial x_2}} \right) & {1 \over 2} \left( {{\partial u_3} \over {\partial x_1}} + {{\partial u_1} \over {\partial x_3}} \right) \\
{1 \over 2} \left( {{\partial u_1} \over {\partial x_2}} + {{\partial u_2} \over {\partial x_1}} \right) & {{\partial u_2} \over {\partial x_2}} & {1 \over 2} \left( {{\partial u_3} \over {\partial x_2}} + {{\partial u_2} \over {\partial x_3}} \right) \\
{1 \over 2} \left( {{\partial u_1} \over {\partial x_3}} + {{\partial u_3} \over {\partial x_1}} \right) & {1 \over 2} \left( {{\partial u_2} \over {\partial x_3}} + {{\partial u_3} \over {\partial x_2}} \right) & {{\partial u_3} \over {\partial x_3}}
\end{bmatrix}\end{split}\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{e}= {1 \over 2} \left( \nabla \otimes \vec{u} + (\nabla \otimes \vec{u})^T \right)\]
\[{D \over {Dt}} (\rho \vec{u}) = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
\[\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = -p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} +
\mu \left( \nabla \otimes \vec{u} + (\nabla \otimes \vec{u})^T - {2 \over 3} (\nabla \cdot \vec{u}) \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \right)\]
\[\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = \nabla \cdot (-p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I}) +
\mu \nabla \cdot \left( \nabla \otimes \vec{u} + (\nabla \otimes \vec{u})^T \right) -
{2 \over 3} \mu \nabla \cdot ( (\nabla \cdot \vec{u}) \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} )\]
\[\nabla \cdot (-p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I}) = -(\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{I})p -
(\overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \cdot \nabla)p = -\nabla p\]
\[\mu \nabla \cdot \left( \nabla \otimes \vec{u} + (\nabla \otimes \vec{u})^T \right) =
\mu \left(\nabla^2 \vec{u} + \nabla (\nabla \cdot \vec{u}) \right) =
\mu \nabla^2 \vec{u} + \mu \nabla (\nabla \cdot \vec{u})\]
\[-{2 \over 3} \mu \nabla \cdot ( (\nabla \cdot \vec{u}) \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} ) =
-{2 \over 3} \mu \left( (\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{I}) \nabla \cdot \vec{u} +
(\overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \cdot \nabla) \nabla \cdot \vec{u} \right) =
-{2 \over 3} \mu \nabla (\nabla \cdot \vec{u})\]
where: \(\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} = 0\) and \(\overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \cdot \nabla=\nabla\)
\[{D \over {Dt}} (\rho \vec{u}) = -\nabla p + \mu \nabla^2 \vec{u} + \mu \nabla (\nabla \cdot \vec{u}) -{2 \over 3} \mu \nabla (\nabla \cdot \vec{u}) + \rho \vec{g}\]
\[{D \over {Dt}} (\rho \vec{u}) = -\nabla p + \mu \left( \nabla^2 \vec{u} + {1 \over 3} \nabla (\nabla \cdot \vec{u}) \right) + \rho \vec{g}\]
\[{\partial \over {\partial t}} (\rho \vec{u}) + \nabla \cdot (\rho \vec{u} \otimes \vec{u} ) = -\nabla p + \mu \left( \nabla^2 \vec{u} + {1 \over 3} \nabla (\nabla \cdot \vec{u}) \right) + \rho \vec{g}\]
\[{\partial \over {\partial t}} (\rho \vec{u})\]
\[\nabla \cdot (\rho \vec{u} \otimes \vec{u} )\]
\[- \nabla p\]
\[\mu \nabla^2 \vec{u}\]
\[{1 \over 3} \mu \nabla (\nabla \cdot \vec{u})\]
\[\rho \vec{g}\]
\[{D \over {Dt}} (\rho \vec{u}) = {\partial \over {\partial t}} (\rho \vec{u}) + \nabla \cdot (\rho \vec{u} \otimes \vec{u} ) = -\nabla p + \mu \left( \nabla^2 \vec{u} + {1 \over 3} \nabla (\nabla \cdot \vec{u}) \right) + \rho \vec{g}\]
\[{\partial \over {\partial t}} (\rho u) +
{\partial \over {\partial x}} (\rho u^2) +
{\partial \over {\partial x}} (\rho uv) +
{\partial \over {\partial x}} (\rho uw) =
-{{\partial p} \over {\partial x}} +
\mu \left( {{\partial^2 u} \over {\partial x^2}} +
{{\partial^2 u} \over {\partial y^2}} +
{{\partial^2 u} \over {\partial z^2}} \right) +
\rho g_x\]
\[{\partial \over {\partial t}} (\rho v) +
{\partial \over {\partial x}} (\rho vu) +
{\partial \over {\partial x}} (\rho v^2) +
{\partial \over {\partial x}} (\rho vw) =
-{{\partial p} \over {\partial y}} +
\mu \left( {{\partial^2 v} \over {\partial x^2}} +
{{\partial^2 v} \over {\partial y^2}} +
{{\partial^2 v} \over {\partial z^2}} \right) +
\rho g_y\]
\[{\partial \over {\partial t}} (\rho w) +
{\partial \over {\partial x}} (\rho wu) +
{\partial \over {\partial x}} (\rho wv) +
{\partial \over {\partial x}} (\rho x^2) =
-{{\partial p} \over {\partial z}} +
\mu \left( {{\partial^2 w} \over {\partial x^2}} +
{{\partial^2 w} \over {\partial y^2}} +
{{\partial^2 w} \over {\partial z^2}} \right) +
\rho g_z\]
\[{1 \over 3} \nabla (\nabla \cdot \vec{u})\]
\[{1 \over 3} {\partial \over {\partial x}} \left(
{{\partial u} \over {\partial x}} +
{{\partial v} \over {\partial y}} +
{{\partial w} \over {\partial z}} \right)\]
\[{1 \over 3} {\partial \over {\partial y}} \left(
{{\partial u} \over {\partial x}} +
{{\partial v} \over {\partial y}} +
{{\partial w} \over {\partial z}} \right)\]
\[{1 \over 3} {\partial \over {\partial z}} \left(
{{\partial u} \over {\partial x}} +
{{\partial v} \over {\partial y}} +
{{\partial w} \over {\partial z}} \right)\]