Cheat sheet
SI-Units
- Base Units
| Dimension | \(\mathrm{T}\) | \(\mathrm{L}\) | \(\mathrm{M}\) | \(\theta\) | \(\mathrm{N}\) | \(\mathrm{I}\) | \(I_v\) |
| Unit | \(\mathrm{s}\) | \(\mathrm{m}\) | \(\mathrm{kg}\) | \(\mathrm{K}\) | \(\mathrm{mol}\) | \(\mathrm{A}\) | \(\mathrm{K}\) |
- Derived units
| \(\mathrm{Hz}\) | \(\mathrm{s}^{-1}\) |
| \(\mathrm{N}\) | \(\mathrm{kg} \, \mathrm{m} \, \mathrm{s}^{-1}\) |
| \(\mathrm{Pa}\) | \(\mathrm{N} \, \mathrm{m}^{-2}\) |
| \(\mathrm{J}\) | \(\mathrm{N} \, \mathrm{m}\) |
| \(\mathrm{W}\) | \(\mathrm{J} \, \mathrm{s}^{-1}\) |
Characteristic numbers
| Reynold number | Re | \(\dfrac{\rho \, v \, L}{\nu}\) |
| Froude number | Fr | \(\dfrac{v}{\sqrt{gh}}\) |
| Shallowness parameter | \(\epsilon\) | \(\dfrac{H}{L}\) |
| Peclet number | Pe | \(\dfrac{L v}{\alpha}\) |
| Prandtl number | Pr | \(\dfrac{\nu}{\alpha}\) |
| Stefan number | Ste | \(\dfrac{c_p \Delta T}{L}\) |
| Strouhal number | Str | \(\dfrac{f L}{v}\) |
| Mach number | Ma | \(\dfrac{v}{c}\) |
| Lewis number | Le | \(\dfrac{\alpha}{D}\) |
Principles
- Material symmetry \[ \hat \sigma^{(\zeta)} (*) = \hat \sigma^{(\eta)} (* P). \]
- Material Isotropy \[ \hat {\sigma} (F) =\hat \sigma ( V \cdot Q) = \hat \sigma ( V \cdot Q \cdot P) = \hat \sigma ( V \cdot Q \cdot Q^T) = \hat \sigma ( V) \]
- Material objectivity \[ \sigma^{(\mathbf y)} = Q \cdot \sigma^{(\mathbf x)} \cdot Q^T \]
- Galilean invariance \[ \pmb{\zeta} = \mathbf{x} - \mathbf{v} t \]
Models
- Mass and momentum balance \[ \begin{aligned} \partial_t \rho + \nabla \cdot \left( \rho \mathbf v \right) &= 0 \\ \partial_t ( \rho \mathbf v ) + \nabla \cdot \left(\rho \mathbf v \otimes \mathbf v \right) &= - \nabla p + \nabla \cdot \pmb{\tau} + \rho \mathbf b \end{aligned} \]
- Incompressible Euler \[ \begin{aligned} \nabla \cdot \mathbf v & = 0\\ \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v &= - \frac{1}{\rho}\nabla p + \mathbf b \end{aligned} \]
- Incompressible Navier-Stokes \[ \begin{aligned} \nabla \cdot \mathbf{v}&= 0 \\ \partial_t \mathbf{v} + \mathbf{v} \cdot \nabla \mathbf{v} &= - \frac{1}{\rho}\nabla p + \nu \triangle \mathbf{v} + \mathbf{b} \end{aligned} \]
- Incompressible Navier-Stokes in cylindrical coordinates \[ \begin{aligned} \frac{\partial u_r}{\partial t} + \left(\vec{u}\cdot \vec\nabla \right) u_r - \frac{u_\theta^2}{r} & = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left(\nabla^2 u_r - \frac{u_r}{r^2} - \frac{2}{r^2} \frac{\partial u_\theta}{\partial \theta}\right) \\ \frac{\partial u_\theta}{\partial t} + \left(\vec{u}\cdot \vec\nabla \right) u_\theta + \frac{u_r u_\theta}{r} & = -\frac{1}{\rho r} \frac{\partial p}{\partial \theta} + \nu \left(\nabla^2 u_\theta + \frac{2}{r^2} \frac{\partial u_r}{\partial \theta} - \frac{u_\theta}{r^2} \right) \\ \frac{\partial u_z}{\partial t} + \left(\vec{u}\cdot \vec\nabla \right) u_z & = -\frac{1}{\rho} \frac{\partial p}{\partial z} + \nu \nabla^2 u_z \\ \frac{1}{r} \frac{\partial}{\partial r} \left( r u_r \right) + \frac{1}{r} \frac{\partial u_\theta}{\partial \theta} + \frac{\partial u_z}{\partial z} & = 0 \end{aligned} \]
- Heat equation \[ \partial_t T + \nabla \cdot \left( T \mathbf{v} \right) = \alpha \Delta T \]
- Incompresisble Navier-Stokes-Boussinesq-Fourier \[ \begin{aligned} & \nabla \cdot \mathbf{v} = 0 \\ & \partial_t \mathbf{v} + \mathbf{v} \cdot \nabla \mathbf{v} = - \frac{1}{\rho_0} \nabla \left( p - \rho_0 g z \right) + \nu \Delta \mathbf{v} - \mathbf{g} B \left(T-T_0\right) \\ & \partial_t \left(\rho c_p T \right) + \nabla \cdot \left( \rho c_p T \mathbf{v} \right) = \nabla \cdot \left( \kappa \nabla T \right) + \mathbf{S} \end{aligned} \]
Closure Relations
- Reiner-Rivilin fluid \[ \pmb{\tau} = \lambda \, \mathrm{tr} ( \mathbf{D}) \mathbf{I} + 2 \eta \mathbf{D} + \kappa \mathbf{D}^2 \]
- Newtonian fluid \[ \pmb{\tau} = \lambda (\nabla \cdot \mathbf v) \mathbf I + \eta \left( \nabla \mathbf v + \nabla \mathbf v^T \right), \]
- Hooke’s law \[ \pmb{\sigma} = \lambda \mathrm{tr} (\mathbf{D}) \mathbf{I} + 2 \mu \mathbf{D}, \]
- Boussinesq approximation \[ \rho = \rho_0 (1-B(T-T_0)) \]
- Stefan condition \[ \rho_s L \partial_t X_m(t) = \kappa \partial_x T (X_m^{-}(t),t) - \kappa \partial_x T (X_m^{+}(t),t) \]
Miscellaneous
- Shape factor \[ \alpha = \frac{\overline{u^2}}{\overline{u}^2}, \; \text{where} \; \bar \ast = (1/h) \int_b^s \ast \,dz \]
- Decomposition of the velocity profile \[ \mathbf{v}(\mathbf{x} + \mathbf{r}) = \mathbf{v}(\mathbf{x}) + \mathbf{w} \times \mathbf{r} + \mathbf{D} \cdot \mathbf{r} \] \[ \text{where} \;\; \mathbf{w} = \frac{1}{2} \nabla \times \mathbf{v}, \; \mathbf{D} = \frac{1}{2} \left( \nabla \mathbf{v} + \nabla \mathbf{v}^T \right), \; \mathbf{W} = \frac{1}{2} \left( \nabla \mathbf{v} - \nabla \mathbf{v}^T \right) \]
- Cylindrical coordinate transformation rules \[ \begin{aligned} \nabla f & = \frac{\partial f}{\partial r } \vec{e}_r + \frac{1}{r} \frac{\partial f}{\partial \theta} \vec{e}_\theta+ \frac{\partial f}{\partial z} \vec{e}_z, \\ \nabla \cdot \vec{A} & = \frac{1}{r}\frac{\partial}{\partial r }(r A_r) + \frac{1}{r} \frac{\partial A_\theta}{\partial \theta} + \frac{\partial A_z}{\partial z} \\ \nabla \times \vec{A} & = \left(\frac{1}{r}\frac{\partial A_z}{\partial \theta}-\frac{\partial A_\theta}{\partial z}\right)\vec{e}_r+ \left(\frac{\partial A_r}{\partial z}-\frac{\partial A_z}{\partial r}\right)\vec{e}_\theta+\frac{1}{r}\left(\frac{\partial}{\partial r}(r A_\theta)-\frac{\partial A_r}{\partial \theta}\right) \vec{e}_z \end{aligned} \]
- Material derivative \[ \frac{D}{Dt} f = \partial_t f + \mathbf{v} \cdot \nabla f \]
- Error function \[ \begin{aligned} \text{erf} (x) := \frac{2}{\sqrt{\pi}} \int_0^x e^{-y^2} dy, \;\;\;\; &\text{erfc} (x) := 1 - \text{erf}(x)\\ \partial_x \text{erf} (C x) = \frac{2}{\sqrt{\pi}} C e^{-(C x)^2}, \;\;\;\; &\partial_x \text{erfc} (C x) = -\frac{2}{\sqrt{\pi}} C e^{-(C x)^2}\\ \end{aligned} \]