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Diffstat (limited to 'source/know/concept/navier-stokes-equations/index.md')
| -rw-r--r-- | source/know/concept/navier-stokes-equations/index.md | 30 |
1 files changed, 15 insertions, 15 deletions
diff --git a/source/know/concept/navier-stokes-equations/index.md b/source/know/concept/navier-stokes-equations/index.md index fd26860..964acda 100644 --- a/source/know/concept/navier-stokes-equations/index.md +++ b/source/know/concept/navier-stokes-equations/index.md @@ -33,10 +33,10 @@ $$\begin{aligned} = \va{f^*} \end{aligned}$$ -$\mathrm{D}/\mathrm{D}t$ is the [material derivative](/know/concept/material-derivative/), -$\rho$ is the density, and $\va{f^*}$ is the effective force density, -expressed in terms of an external body force $\va{f}$ (e.g. gravity) -and the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $\hat{\sigma}$: +$$\mathrm{D}/\mathrm{D}t$$ is the [material derivative](/know/concept/material-derivative/), +$$\rho$$ is the density, and $$\va{f^*}$$ is the effective force density, +expressed in terms of an external body force $$\va{f}$$ (e.g. gravity) +and the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $$\hat{\sigma}$$: $$\begin{aligned} \va{f^*} @@ -51,8 +51,8 @@ $$\begin{aligned} = - p \delta_{ij} + \eta (\nabla_i v_j + \nabla_j v_i) \end{aligned}$$ -Where $\eta$ is the dynamic viscosity. -Inserting this, we calculate $\nabla \cdot \hat{\sigma}^\top$ in index notation: +Where $$\eta$$ is the dynamic viscosity. +Inserting this, we calculate $$\nabla \cdot \hat{\sigma}^\top$$ in index notation: $$\begin{aligned} \big( \nabla \cdot \hat{\sigma}^\top \big)_i @@ -62,7 +62,7 @@ $$\begin{aligned} &= - \nabla_i p + \eta \nabla_i \sum_{j} \nabla_j v_j + \eta \sum_{j} \nabla_j^2 v_i \end{aligned}$$ -Thanks to incompressibility $\nabla \cdot \va{v} = 0$, +Thanks to incompressibility $$\nabla \cdot \va{v} = 0$$, the middle term vanishes, leaving us with: $$\begin{aligned} @@ -70,7 +70,7 @@ $$\begin{aligned} = \va{f} - \nabla p + \eta \nabla^2 \va{v} \end{aligned}$$ -We assume that the only body force is gravity $\va{f} = \rho \va{g}$. +We assume that the only body force is gravity $$\va{f} = \rho \va{g}$$. Newton's second law then becomes: $$\begin{aligned} @@ -78,8 +78,8 @@ $$\begin{aligned} = \rho \va{g} - \nabla p + \eta \nabla^2 \va{v} \end{aligned}$$ -Dividing by $\rho$, and replacing $\eta$ -with the kinematic viscosity $\nu = \eta/\rho$, +Dividing by $$\rho$$, and replacing $$\eta$$ +with the kinematic viscosity $$\nu = \eta/\rho$$, yields the main equation: $$\begin{aligned} @@ -90,7 +90,7 @@ $$\begin{aligned} \end{aligned}$$ Finally, we can optionally allow incompressible fluids -with an inhomogeneous "lumpy" density $\rho$, +with an inhomogeneous "lumpy" density $$\rho$$, by demanding conservation of mass, just like for the Euler equations: @@ -115,11 +115,11 @@ $$\begin{aligned} } \end{aligned}$$ -Due to the definition of viscosity $\nu$ as the molecular "stickiness", -we have boundary conditions for the velocity field $\va{v}$: -at any interface, $\va{v}$ must be continuous. +Due to the definition of viscosity $$\nu$$ as the molecular "stickiness", +we have boundary conditions for the velocity field $$\va{v}$$: +at any interface, $$\va{v}$$ must be continuous. Likewise, Newton's third law demands that the normal component -of stress $\hat{\sigma} \cdot \vu{n}$ is continuous there. +of stress $$\hat{\sigma} \cdot \vu{n}$$ is continuous there. |