Editing Navier–Stokes equations (section)

====Weak form====
In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation<ref name="Quarteroni" /> 
<math display="block">
\rho \frac{\partial \mathbf{u}}{\partial t}  - \mu \Delta \mathbf{u} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u} + \nabla p  = \mathbf{f} 
</math>
multiply it for a test function <math display="inline">
\mathbf{v}
</math>, defined in a suitable space <math display="inline">
V
</math>, and integrate both members with respect to the domain <math display="inline">
\Omega
</math>:<ref name="Quarteroni" />
<math display="block">
\int \limits_\Omega \rho \frac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v}  - \int \limits_\Omega \mu \Delta \mathbf{u} \cdot \mathbf{v} + \int \limits_\Omega \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v} + \int \limits_\Omega \nabla p \cdot \mathbf{v} = \int \limits_\Omega \mathbf{f} \cdot \mathbf{v}
</math>
Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:<ref name="Quarteroni" /> 
<math display="block">\begin{align}
-\int \limits_\Omega \mu \Delta \mathbf{u} \cdot \mathbf{v} &= \int_\Omega \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} - \int \limits_{\partial \Omega} \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} \cdot \mathbf{v} \\
\int \limits_\Omega \nabla p \cdot \mathbf{v} &= -\int \limits_\Omega p \nabla \cdot \mathbf{v} + \int \limits_{\partial \Omega} p \mathbf{v} \cdot {\hat{\mathbf{n}}}
\end{align}</math>

Using these relations, one gets:<ref name="Quarteroni" /> 
<math display="block">
\int \limits_\Omega \rho \dfrac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v}
 + \int \limits_\Omega \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v}
 + \int \limits_\Omega \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v}
 - \int \limits_\Omega p \nabla \cdot \mathbf{v}
= \int \limits_\Omega \mathbf{f} \cdot \mathbf{v}
 + \int \limits_{\partial \Omega} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}}\right) \cdot \mathbf{v}
 \quad \forall \mathbf{v} \in V.
</math>
In the same fashion, the continuity equation is multiplied for a test function {{mvar|q}} belonging to a space <math display="inline">
Q
</math> and integrated in the domain <math display="inline">
\Omega
</math>:<ref name="Quarteroni" />
<math display="block">\int \limits_\Omega q \nabla \cdot \mathbf{u} = 0. \quad \forall q \in Q. </math>
The space functions are chosen as follows: 
<math display="block">\begin{align}
V = \left[H_0^1(\Omega) \right]^d &= \left\{ \mathbf{v} \in \left[H^1(\Omega)\right]^d: \quad \mathbf{v} = \mathbf{0} \text{ on } \Gamma_D \right\}, \\
Q &= L^2(\Omega)
\end{align}</math>
Considering that the test function {{math|'''''v'''''}} vanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:<ref name="Quarteroni" /> 
<math display="block">
\int \limits_{\partial \Omega} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right) \cdot \mathbf{v} 
= 
\underbrace{
    \int \limits_{\Gamma_D} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right) \cdot \mathbf{v}
}_{ \mathbf{v} = \mathbf{0} \text{ on } \Gamma_D \ } 
+ 
\int \limits_{\Gamma_N} \underbrace{ \vphantom{\int \limits_{\Gamma_N} }
    \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right)
}_{= \mathbf{h} \text{ on } \Gamma_N} \cdot \mathbf{v}
= 
\int \limits_{\Gamma_N} \mathbf{h} \cdot \mathbf{v}.
</math>
Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as:<ref name="Quarteroni" /> 
<math display="block">\begin{align}
&\text{find } \mathbf{u} \in L^2 \left(\mathbb R^+\; \left[H^1(\Omega)\right]^d\right) \cap C^0\left(\mathbb R^+ \; \left[L^2(\Omega)\right]^d\right) \text{ such that: } \\[5pt]
&\quad\begin{cases}
\displaystyle \int \limits_{\Omega}\rho \dfrac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v} + \int  \limits_{\Omega} \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} + \int \limits_{\Omega} \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v} - \int \limits_{\Omega} p \nabla \cdot \mathbf{v} = \int \limits_{\Omega}\mathbf{f} \cdot \mathbf{v} +  \int \limits_{\Gamma_N} \mathbf{h} \cdot \mathbf{v} \quad \forall \mathbf{v} \in V, \\
\displaystyle  \int \limits_{\Omega} q \nabla \cdot \mathbf{u} = 0 \quad \forall q \in Q. 
\end{cases}\end{align}
</math>