4.19. Fluid dynamics in a non-inertial reference frame
Following the same approach as Mechanics:Relative Kinematics:Point, the position of a point \(P\) is determined by the position vectors w.r.t. an inertial reference frame - here defined by an origin and a set oc Cartesian coordinates - \(I: \, (O, x, y, z)\), and w.r.t. a generic reference frame - \(G: \, (O_1, x_1, y_1, z_1)\).
\[\begin{split}\begin{aligned}
(P - O) & = (O_1 - O) + (P - O_1) \\
\mathbf{r}_{P/O} & = \mathbf{r}_{O_1/O} + \mathbf{r}_{P/O_1}
\end{aligned}\end{split}\]
The unit vectors of the generic basis can be projected on the inertial basis as
\[\hat{\mathbf{e}}^1_i = \hat{\mathbf{e}}^1_i \cdot \hat{\mathbf{e}}_k \hat{\mathbf{e}}_k = \mathbb{R}^{0 \rightarrow 1} \cdot \hat{\mathbf{e}}_i \ ,\]
and their time derivatives read
\[\dfrac{d}{dt}\hat{\mathbf{e}}^1_i = \boldsymbol\omega_{1/} \times \hat{\mathbf{e}}^1_i \ .\]
Vector \(\mathbf{r}_{P/O}\) is written as \(\mathbf{r}\) for bervity, and \(\mathbf{r}_{P/O_1}\) is written as a reference rotated vector, so that
\[\mathbf{r} = \mathbf{r}_{O_1/O} + \mathbb{R}^{0\rightarrow 1} \cdot \mathbf{r}' \ ,\]
or explicitly writing the dependence from time and the material coordinates - label - along with the relation prescribing absolute time for every observer, to get the transformation \((\mathbf{r},t)(\mathbf{r}',t')\)
\[\begin{split}\begin{aligned}
& \mathbf{r}(\mathbf{r}_m, t) = \mathbf{r}_{O_1/O}(t) + \mathbb{R}^{0\rightarrow 1}(t) \cdot \mathbf{r}'(\mathbf{r}_m, t) \\
& t = t' \ .
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
& \mathbf{r}'(\mathbf{r}_m, t) = \mathbb{R}^T(t) \cdot \left[ \mathbf{r}(\mathbf{r}_m, t) - \mathbf{r}_{O_1}(t) \right] \\
& t' = t \ .
\end{aligned}\end{split}\]
Rule of differentiation of composite functions.
\[\begin{split}\begin{aligned}
\partial_{t'} \_
& = \partial_{t'} t \, \partial_t \_ + \partial_{t'} \mathbf{r} \cdot \partial_{\mathbf{r}} \_ \\
& = \partial_t \_ + \left[ \mathbf{v}_{O_1} + \dot{\mathbb{R}} \cdot \mathbf{r}' + \mathbb{R} \cdot \mathbf{v}' \right] \cdot \partial_{\mathbf{r}} \_ \\
\partial_{\mathbf{r}'} \_
& = \partial_{\mathbf{r}'} t \, \partial_t \_ + \partial_{\mathbf{r}'} \mathbf{r} \cdot \partial_{\mathbf{r}} \_ \\
& = \mathbb{R}^T \cdot \partial_{\mathbf{r}} \_
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\partial_{t} \_
& = \partial_{t} t' \, \partial_{t'} \_ + \partial_t \mathbf{r'} \cdot \partial_{\mathbf{r'}} \_ \\
& = \partial_{t'} \_ + \left[ \dot{\mathbb{R}}^T \cdot \left( \mathbf{r} - \mathbf{r}_{O_1} \right) - \mathbb{R}^T \cdot \mathbf{v}_{O_1} \right] \cdot \partial_{\mathbf{r}} \_ \\
\partial_{\mathbf{r}} \_
& = \partial_{\mathbf{r}} t \, \partial_t \_ + \partial_{\mathbf{r}} \mathbf{r}' \cdot \partial_{\mathbf{r}'} \_ \\
& = \mathbb{R} \cdot \partial_{\mathbf{r}'} \_
\end{aligned}\end{split}\]
\[\mathbf{v}(\mathbf{r}) = \mathbb{R} \cdot \mathbf{u}(\mathbf{r}) = \mathbf{u}(\mathbf{r}) \cdot \mathbb{R}^T\]
and using Cartesian coordinates, \(\mathbf{r} = x^i \hat{\mathbf{e}}_i\)
\[\mathbf{v} = v^i \hat{\mathbf{e}}_i = R^{ij} \hat{\mathbf{e}}_i \hat{\mathbf{e}}_j \cdot \left( x^k \hat{\mathbf{e}}_k \right) = R^{ij} x^j \hat{\mathbf{e}}_i\]
so that the gradient becomes
\[\partial_{\mathbf{r}} \mathbf{v} = \hat{\mathbf{e}}_i \partial_{x^{i}} \left( v^k(\mathbf{r}) \hat{\mathbf{e}}_k \right) = \partial_{x^i} v^k \hat{\mathbf{e}}_i \, \hat{\mathbf{e}}_k
= \partial_{x^i} \left( R^{kj} x^j \right) \hat{\mathbf{e}}_i \, \hat{\mathbf{e}}_k
= \left( R^{kj} \delta^{ij} \right) \hat{\mathbf{e}}_i \, \hat{\mathbf{e}}_k
= R^{ki} \, \hat{\mathbf{e}}_i \, \hat{\mathbf{e}}_k = \mathbb{R}^T \ .
\]
\[\mathbf{v}(\mathbf{r}) = \mathbb{R} \cdot \mathbf{r} = \mathbf{r} \cdot \mathbb{R}^T\]
and using Cartesian coordinates, \(\mathbf{r} = x^i \hat{\mathbf{e}}_i\)
\[\mathbf{v} = v^i \hat{\mathbf{e}}_i = R^{ij} \hat{\mathbf{e}}_i \hat{\mathbf{e}}_j \cdot \left( u^k(\mathbf{r}) \hat{\mathbf{e}}_k \right) = R^{ij} u^j(\mathbf{r}) \hat{\mathbf{e}}_i\]
so that the gradient becomes
\[\partial_{\mathbf{r}} \mathbf{v} = \hat{\mathbf{e}}_i \partial_{x^{i}} \left( v^k(\mathbf{r}) \hat{\mathbf{e}}_k \right) = \partial_{x^i} v^k \hat{\mathbf{e}}_i \, \hat{\mathbf{e}}_k
= \partial_{x^i} \left( R^{kj} u^j(x^\ell) \right) \hat{\mathbf{e}}_i \, \hat{\mathbf{e}}_k
= \left( R^{kj} \partial_i u^j \right) \hat{\mathbf{e}}_i \, \hat{\mathbf{e}}_k
= \nabla \mathbf{u} \cdot \mathbb{R}^T \ .
\]
\[\mathbf{a} \cdot \partial_{\mathbf{r}} \mathbf{v} = a^m \hat{\mathbf{e}}_m \cdot \hat{\mathbf{e}}_i \partial_{x^{i}} \left( v^k(\mathbf{r}) \hat{\mathbf{e}}_k \right) = \partial_{x^m} v^k \, \hat{\mathbf{e}}_k
= a^m \partial_{x^m} \left( R^{kj} u^j(x^\ell) \right) \, \hat{\mathbf{e}}_k
= a^m \left( R^{kj} \partial_m u^j \right) \, \hat{\mathbf{e}}_k
= \mathbf{a} \cdot \nabla \mathbf{u} \cdot \mathbb{R}
= \mathbb{R} \cdot \left( \mathbf{a} \cdot \nabla \mathbf{u} \right).
\]
If the tensor \(\mathbb{A}\) is independent from \(\mathbf{r}\),
\[\nabla \cdot \left( \mathbb{A} \cdot \mathbf{v}(\mathbf{r}) \right) = \partial_{x^i} \left( A^{ij} v^j \right) = \mathbb{A}^T : \nabla \mathbf{v} \ .\]
Velocity. Velocity of a material point reads
\[\begin{split}\begin{aligned}
\mathbf{u}(\mathbf{r}(\mathbf{r}_m,t), t)
& := \left.\partial_t \mathbf{r}\right|_{\mathbf{r}_m} = \\
& = \mathbf{v}_{O_1} + \dot{\mathbb{R}} \cdot \mathbf{r}' + \mathbb{R} \cdot \partial_{t} \mathbf{r}'|_{\mathbf{r}_m} = \\
& = \mathbf{v}_{O_1} + \dot{\mathbb{R}} \cdot \mathbf{r}' + \mathbb{R} \cdot \mathbf{v}' = \\
\end{aligned}\end{split}\]
Acceleration.
\[\begin{split}\begin{aligned}
\mathbf{a}
& = \left.\partial_t \mathbf{u}\right|_{\mathbf{r}_m} = \\
& = \left.\partial_t \mathbf{u}\right|_{\mathbf{r}} + \partial_{t} \mathbf{r}|_{\mathbf{r}_m} \cdot \partial_{\mathbf{r}} \mathbf{u} = \\
& = \left.\partial_t \mathbf{u}\right|_{\mathbf{r}} + \mathbf{u} \cdot \partial_{\mathbf{r}} \mathbf{u} \\
\end{aligned}\end{split}\]
Using diffferentiation rules for composite functions, the advection operator becomes
\[\begin{split}\begin{aligned}
\partial_t|_{\mathbf{r}} + \mathbf{u} \cdot \partial_{\mathbf{r}}
& = \partial_{t'} + \left[ \left( \mathbf{r} - \mathbf{r}_{O_1} \right) \cdot \dot{\mathbb{R}} - \mathbf{v}_{O_1} \cdot \mathbb{R} \right] \cdot \partial_{\mathbf{r}'} + \left[ \mathbf{v}_{O_1} + \mathbf{r}' \cdot \dot{\mathbb{R}}^T + \mathbf{v}' \cdot \mathbb{R}^T \right] \cdot \mathbb{R} \cdot \partial_{\mathbf{r}'} = \\
& = \partial_{t'} + \underbrace{\left[ (\mathbf{r}-\mathbf{r}_{O_1}) - \mathbf{r}' \cdot \mathbb{R}^T \right]}_{ = \mathbf{0} } \cdot \dot{\mathbb{R}} \cdot \partial_{\mathbf{r}'} + \mathbf{v}' \cdot \partial_{\mathbf{r}'} = \\
& = \left.\partial_{t'}\right|_{\mathbf{r}'} + \mathbf{v}' \cdot \partial_{\mathbf{r}'} \ ,
\end{aligned}\end{split}\]
and thus
\[\begin{split}\begin{aligned}
\left[ \partial_{t'}|_{\mathbf{r}'} + \mathbf{v}' \cdot \partial_{\mathbf{r}'} \right] \mathbf{u}
& = \left[ \partial_{t'}|_{\mathbf{r}'} + \mathbf{v}' \cdot \partial_{\mathbf{r}'} \right] \left( \mathbf{v}_{O_1}(t) + \dot{\mathbb{R}}(t') \cdot \mathbf{r}' + \mathbb{R}(t) \cdot \mathbf{v}'(\mathbf{r}', t') \right) \\
& = \left[ \partial_{t'}|_{\mathbf{r}'} + \mathbf{v}' \cdot \partial_{\mathbf{r}'} \right] \left( \mathbf{v}_{O_1}(t) + \boldsymbol\omega_\times(t') \cdot \mathbb{R}(t') \cdot \mathbf{r}' + \mathbb{R}(t) \cdot \mathbf{v}'(\mathbf{r}', t') \right) \\
& = \mathbf{a}_{O_1} + \boldsymbol\alpha_\times \cdot \mathbb{R} \cdot \mathbf{r}' + \boldsymbol\omega_\times \cdot \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{r}' + 2 \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{v}' + \mathbb{R} \cdot \partial_{t'} \mathbf{v}' + \mathbf{v}' \cdot \partial_{\mathbf{r}'} \left( \mathbb{R} \cdot \mathbf{v}' \right) = \\
& = \mathbf{a}_{O_1} + \boldsymbol\alpha_\times \cdot \mathbb{R} \cdot \mathbf{r}' + \boldsymbol\omega_\times \cdot \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{r}' + 2 \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{v}' + \mathbb{R} \cdot \left[ \partial_{t'} \mathbf{v}' + \mathbf{v}' \cdot \partial_{\mathbf{r}'} \mathbf{v}' \right] = \\
& = \mathbf{a}_{O_1} + \boldsymbol\alpha_\times \cdot \mathbb{R} \cdot \mathbf{r}' + \boldsymbol\omega_\times \cdot \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{r}' + 2 \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{v}' + \mathbb{R} \cdot \underbrace{\left[ \partial_{t'} \mathbf{v}' + \mathbf{v}' \cdot \nabla' \mathbf{v}' \right]}_{ =: \mathbf{a}'} \ ,
\end{aligned}\end{split}\]
having defined the acceleration of the material particles \(\mathbf{a}'(\mathbf{r}', t)\) as seen by the non-inertial observer.
Multiplying by \(\mathbb{R}^T \cdot\), all the terms are written w.r.t. the reference coordinates of the non-inertial reference frame. The very same procedure applies to the transformation ov balance equations.
4.19.1. Differential equations
4.19.1.1. Mass
Conservative form of differential mass equation reads
\[\begin{split}\begin{aligned}
0
& = \partial_t \rho + \nabla \cdot \left( \rho \mathbf{u} \right) = \\
& = \partial_t \rho + \mathbf{u} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{u} = \\
& = \partial_{t'} \rho + \mathbf{v}' \cdot \nabla' \rho + \rho \nabla \cdot \mathbf{u} \ ,
\end{aligned}\end{split}\]
Since
\[\begin{split}\begin{aligned}
\nabla \cdot \mathbf{u}
& = \nabla \cdot \left( \mathbf{v}_{O_1} + \dot{\mathbb{R}} \cdot \mathbf{r}' + \mathbb{R} \cdot \mathbf{v}' \right) = && \text{(see dropdown below)} \\
& = \nabla' \cdot \mathbf{v}' \ ,
\end{aligned}\end{split}\]
the mass equation in the non-inertial reference frame has the very same expression as the mass equation in the inertial reference frame,
\[\begin{split}\begin{aligned}
0
& = \partial_{t'} \rho + \mathbf{v}' \cdot \nabla' \rho + \rho \nabla' \cdot \mathbf{v}' = \\
& = \partial_{t'} \rho + \nabla' \cdot \left( \rho \mathbf{v}' \right) \ .
\end{aligned}\end{split}\]
For generic tensor \(\mathbb{A}\) and vector \(\mathbf{v}\),
\[\begin{split}\begin{aligned}
\nabla \cdot \left( \mathbb{A} \cdot \mathbf{v} \right)
& = \partial_{x^i} \left( A^{ij} v^j \right) = \\
& = \partial_{x^i} x^{' \ k} \partial_{x^{' \ k}} \left( A^{ij} v^j \right) = \\
& = \partial_{x^{' \ k}} v^j A^{ij} \partial_{x^i} x^{' \ k} \ .
\end{aligned}\end{split}\]
If \(\mathbb{A} = \mathbb{R}\), \(\mathbf{v} = \mathbf{v}'\), then
\[\nabla \cdot \left( \mathbb{R} \cdot \mathbf{v}' \right) = \partial_{x'^k} v'^j \underbrace{R^{ij} R^{ik}}_{\delta_{jk}} = \partial_{x'k} v'^k = \nabla' \cdot \mathbf{v}'(\mathbf{r}',t) \ .\]
If \(\mathbb{A} = \dot{\mathbb{R}}\), \(\mathbf{v} = \mathbf{r}'\), then
\[\nabla \cdot \left( \dot{\mathbb{R}} \cdot \mathbf{r}' \right) = \underbrace{\partial_{x'^k} r'^j}_{\delta^{jk}} \underbrace{\dot{R}^{ij} R^{ik}}_{ \left\{ - \boldsymbol\omega_\times \right\}_{jk} } = \delta_{jk} \varepsilon_{jlk} \omega_{l} = 0 \ ,\]
as Levi-Civita symbols are zero with repeated indices, and \(\dot{\mathbb{R}} = \boldsymbol\omega_\times \cdot \mathbb{R}\), and thus \(-\boldsymbol\omega_\times = \boldsymbol\omega_\times^T = \dot{\mathbb{R}}^T \cdot \mathbb{R}\)
4.19.1.2. Momentum
Convective form of differential momentum equation reads
\[\rho \left[ \partial_t + \mathbf{u} \cdot \nabla \right] \mathbf{u} = \rho \mathbf{g} + \nabla \cdot \mathbb{T} \ .\]
Using the expression of the acceleration above
\[\rho \mathbb{R} \cdot \mathbf{a}' = \rho \mathbf{g} + \nabla \cdot \mathbb{T} - \rho \left[ \mathbf{a}_{O_1} + \boldsymbol\alpha_\times \cdot \mathbb{R} \cdot \mathbf{r}' + \boldsymbol\omega_\times \cdot \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{r}' + 2 \, \boldsymbol\omega_\times \cdot \mathbb{R} \cdot \mathbf{v}' \right] \ .\]
multiplying by \(\mathbb{R}^T \cdot \) gives the expression of the momentum equation as seen by the non-inertial reference frame
\[\rho \mathbf{a}' = \rho \mathbf{g}' + \nabla' \cdot \mathbb{T}' - \rho \left[ \mathbf{a}'_{O_1} + \boldsymbol\alpha'_{\times} \cdot \mathbf{r}' + \boldsymbol\omega'_{\times} \cdot \boldsymbol\omega'_{\times} \cdot \mathbf{r}' + 2 \boldsymbol\omega'_\times \cdot \mathbf{v}' \right] \ ,\]
being \(\mathbf{g'} := \mathbb{R}^T \cdot \mathbf{g}\), \(\boldsymbol\alpha'_{\times} = \mathbb{R}^T \cdot \boldsymbol\alpha_\times \cdot \mathbb{R}\), and the transformation of the divergence of the stress tensor given in the following dropdown box.
\[\begin{aligned}
\partial_{x^i} T^{ij}(x^\ell) = \partial_{x^i} x'^{k} \partial_{x'^k} T^{ij}(x^\ell(x'^m)) \ ,
\end{aligned}\]
and thus
\[\mathbb{R}^T \cdot \nabla \cdot \mathbb{T} = R^{j \ell} R^{ik} \partial_{x'^k} T^{ij} = \partial_{x'^k} \left( R^{ik} T^{ij} R^{j \ell} \right) = \nabla \cdot \left( \mathbb{R}^T \cdot \mathbb{T} \cdot \mathbb{R} \right) = \nabla' \cdot \mathbb{T}' \ .\]
4.19.1.3. Kinetic energy
4.19.1.4. Total energy
4.19.2. Homogeneity and isotropy
4.19.3. Galilean relativity
With \(\mathbf{v}\) constant in time and independent from position in space
\[\begin{split}
\begin{cases}
\mathbf{r} = \mathbf{r}' + \mathbf{v} t \\
t = t'
\end{cases}
\qquad , \qquad
\begin{cases}
\mathbf{r}' = \mathbf{r} - \mathbf{v} t \\
t' = t
\end{cases}
\end{split}\]
and thus, the velocity of a point whose position in function of time \(\mathbf{r}(t)\) reads
\[\begin{split}
\mathbf{u} = \mathbf{u}' + \mathbf{v} \\
\end{split}\]
Using derivatives of composite functions
\[\begin{split}\begin{aligned}
\partial_t & = \partial_{t'} - \mathbf{v} \cdot \nabla' \\
\nabla & = \nabla'
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\partial_t f(\mathbf{r},t)
& = \partial_t f'(\mathbf{r}'(\mathbf{r},t), t'(\mathbf{r},t)) = \\
& = \partial_t \mathbf{r}' \cdot \partial_{\mathbf{r}'} f' + \partial_t t' \, \partial_{t'} f' = \\
& = - \mathbf{v} \cdot \nabla' f' + \partial_{t'} f' \ .
\end{aligned}\end{split}\]
\[\begin{aligned}
\nabla = \dots = \nabla'
\end{aligned}\]
Mass. Conservative form
\[\begin{split}\begin{aligned}
0
& = \partial_t \rho + \nabla \cdot \left( \rho \mathbf{u} \right) = \\
& = \partial_{t'} \rho - \mathbf{v} \cdot \nabla' \rho + \nabla' \cdot \left( \rho \left( \mathbf{u}' + \mathbf{v} \right) \right) = \\
& = \partial_{t'} \rho - \mathbf{v} \cdot \nabla' \rho + \nabla' \cdot \left( \rho \mathbf{u}' \right) + \mathbf{v} \cdot \nabla' \rho = \\
& = \partial_{t'} \rho + \nabla' \cdot \left( \rho \mathbf{u}' \right) \ .
\end{aligned}\end{split}\]
4.19.4. Integral equations
4.19.4.1. Galilean relativity
