A The equations of variable mass systems

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The discussion focuses on the formal proof of equations governing variable mass systems, which are often derived informally in existing literature. A detailed derivation of the local conservation of mass and linear momentum balance equations is presented, including the introduction of vector fields and mass density. The proof culminates in a theorem that describes the equation of linear momentum balance for a volume filled with continuous media. Additionally, the local equation of angular momentum balance is explored, with corrections made to ensure accuracy in the final formulations. The thread emphasizes the need for rigorous mathematical treatment in the study of variable mass systems.
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The equations of variable mass systems are usually deduced from some very informal argument. It is so at least for the books I know.

So I tried to construct a formal proof based on the continuous media equations.

Criticism, remarks etc are welcomed.

Let ##D\subset \mathbb{R}^q## be an open and bounded domain with ##C^1-##smooth boundary ##\partial D##. The cases ##q=1,2,3## are physically reasonable.

The space ##\mathbb{R}^q## is endowed with an inertial frame of reference ##Ox^1,\ldots,x^q##, where ## x=(x^1,\ldots,x^q)## are the standard Cartesian coordinates:
$$\boldsymbol{x}=x^i\boldsymbol{e}_i\in \mathbb{R}^q.$$ A theorem we formulate below has an invariant form but we use the Cartesian coordinates just for convenience.

Let ##(\cdot,\cdot)## denote the standard inner product in ##\mathbb{R}^q##.

Note also that in the Cartesian frame there is no need to distinguish co- and contravariant components of tensors.

Let $$\boldsymbol w(t,x)=w^i(t,x)\boldsymbol e_i,\quad w^i\in C^1(\mathbb{R}^{q+1})$$ stand for a vector field in ##\mathbb{R}^q## and ##g^t_{t_0}(x)## be its flow:
$$\frac{d}{dt}g^t_{t_0}(x)=\boldsymbol w(t,g^t_{t_0}(x)),\quad g^{t_0}_{t_0}(x)=x.$$ We assume that ##g^t_{t_0}(x)## is defined for all real ##t,t_0## and for all ##x\in \mathbb{R}^q##.
Introduce a notation $$D(t)=g^t_{t_0}(D).$$
Loosely speaking ##\boldsymbol w## is a velocity of the volume ##D(t)##.

Assume that the domain ##D(t)## is filled with a continuous media with a mass density ##\rho(t,x)## and a flow velocity field ##\boldsymbol{v}(t,x)=v^i(t,x)\boldsymbol{e}_i##;
$$v^i,\rho\in C^1(\overline\Omega),\quad \Omega=\{(t,x)\mid x\in D(t),\quad t\in\mathbb{R}\}\subset\mathbb{R}^{q+1}.$$
The local conservation of mass equation is
$$\rho_t+\mathrm{div}(\rho\boldsymbol{v})=0$$ or
$$
\rho_t+\frac{\partial (\rho v^i)}{\partial x^i}=0.\qquad(1)$$
Let $$\boldsymbol{F}(t,x)=F^i(t,x)\boldsymbol{e}_i,\quad F^i\in C(\overline\Omega)$$ stand for a force per unit mass. So that the whole matter in ##D(t)## experiences a net force $$\boldsymbol{G}(t)=\int_{D(t)}\rho(t,x)\boldsymbol{F}(t,x)dV,$$ where ##dV## is the volume element.
We denote the stress tensor by ##p^{ij}(t,x)##. The boundary ##\partial D(t)## experiences the following external contact net force
$$\boldsymbol{P}(t)=\int_{\partial D(t)}\boldsymbol{p}_n dS,\quad \boldsymbol{p}_n=p^{ij}n_j\boldsymbol{e}_i,$$
where ##\boldsymbol{n}=n^i\boldsymbol{e}_i## is the outer unit normal to ##\partial D(t)##; ##dS## is the area element of the surface;
$$p^{ij}\in C^1(\overline\Omega).$$
The local equation of linear momentum balance is
$$
\rho\Big(v^k_t+\frac{\partial v^k}{\partial x^i}v^i\Big)=\rho F^k+\frac{\partial p^{kj}}{\partial x^j}.\qquad (2)$$
Let
$$\boldsymbol{Q}(t)=\int_{D(t)}\boldsymbol{v}(t,x)\rho(t,x) dV$$stand for the linear momentum of ##D(t)##.

THEOREM. The equation of linear momentum balance for the volume ##D(t)## is as follows
$${\boldsymbol{\dot Q}}=\boldsymbol{G}+\boldsymbol{P}+\boldsymbol{R},$$where
$$\boldsymbol{R}=-\int_{\partial D(t)}\boldsymbol{v}(\boldsymbol{v}-\boldsymbol{w},\boldsymbol{n})\rho dS.$$PROOF. By the well-known theorem from the integral calculus and due to (2) it follows that
$$\frac{d}{dt} Q^k=\int_{D(t)}\big( v_t^k\rho+ v^k\rho_t+\mathrm{div}\,(\rho v^k\boldsymbol w) \big)dV$$
$$=\int_{D(t)}\Big[ \Big(F^k-\frac{\partial v^k}{\partial x^j}v^j\Big)\rho+\frac{\partial p^{kl}}{\partial x^l}+ v^k\rho_t\Big] dV$$
$$+\int_{D(t)}\mathrm{div}\,(\rho v^k\boldsymbol w)dV.$$
Integration by parts gives
$$\int_{D(t)}\frac{\partial v^k}{\partial x^j}v^j \rho dV=\int_{\partial D(t)} v^kv^j n_j\rho ds-\int_{D(t)} v^k\mathrm{div}(\rho\boldsymbol{v})dV$$
$$\int_{D(t)}\frac{\partial p^{kl}}{\partial x^l} dV=\int_{\partial D(t)} p^{kl}n_ldS,$$
$$
\int_{D(t)}\mathrm{div}\,(\rho v^k\boldsymbol w)dV=\int_{\partial D(t)}\rho v^kw^jn_jdS.$$
To finish the proof it remains to employ (1).

The Theorem is proved.

Other theorems (the angular momentum balance and the energy balance) can be deduced by the same way.
 
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As an exercise I tried to determine the equation of angular momentum balance; I wondered if my working looks correct to you? The local equation of angular momentum balance is ##p^{ij} = p^{ji}##. Let\begin{align*}
\boldsymbol{L}(t) &= \int_{D(t)} \boldsymbol{r} \times \boldsymbol{v}(\boldsymbol{r},t) \rho(\boldsymbol{r},t) dV \\

L^i(t) &= \int_{D(t)} \epsilon_{ijk} x^j v^k \rho dV
\end{align*}and also let\begin{align*}
\boldsymbol{M}(t) &= \int_{D(t)} \boldsymbol{r} \times \boldsymbol{F}(\boldsymbol{r},t) \rho(\boldsymbol{r},t)dV \\

M^i(t) &= \int_{D(t)} \epsilon_{ijk} x^j F^k \rho dV
\end{align*}Take the time derivative of ##L^i##, using the local momentum balance equation as well as ##\epsilon_{ijk} v^j v^k = 0##,\begin{align*}

\dfrac{d}{dt} L^i &= \int_{D(t)} \epsilon_{ijk} \left( v^j v^k \rho + x^j v^k_t \rho + x^j v^k \rho_t + \mathrm{div}(\rho x^j v^k \boldsymbol{w}) \right) dV \\

&= \int_{D(t)} \epsilon_{ijk} \left( x^j \left[ \left( F^k-\frac{\partial v^k}{\partial x^l}v^l\right)\rho+\frac{\partial p^{kl}}{\partial x^l} \right] + x^j v^k \rho_t \right) dV \\

&\hspace{35pt} + \int_{D(t)} \epsilon_{ijk} \mathrm{div}(\rho x^j v^k \boldsymbol{w}) dV \\ \\

&= \int_{D(t)} \epsilon_{ijk} \left( - x^j \frac{\partial v^k}{\partial x^l}v^l \rho + x^j\frac{\partial p^{kl}}{\partial x^l} + x^j v^k \rho_t \right) dV + M^i \\

&\hspace{35pt} + \int_{D(t)} \epsilon_{ijk} \mathrm{div}(\rho x^j v^k \boldsymbol{w}) dV

\end{align*}Using partial integration, and the relation ##\rho_t + \mathrm{div}(\rho \boldsymbol{v})= 0##, we may write\begin{align*}
\int_{D(t)} \epsilon_{ijk} x^j\frac{\partial v^k}{\partial x^l}v^l \rho dV - \int_{\partial D(t)} \epsilon_{ijk} x^j v^kv^l n_l\rho ds &=-\int_{D(t)} \epsilon_{ijk} v^k \dfrac{\partial}{\partial x^l} ( x^j v^l \rho)dV \\

&= -\int_{D(t)} \epsilon_{ijk} v^k (v^j \rho + x^j \mathrm{div}(\rho\boldsymbol{v}))dV \\

&= -\int_{D(t)} \epsilon_{ijk} x^j v^k \mathrm{div}(\rho\boldsymbol{v})dV \\

&= + \int_{D(t)} \epsilon_{ijk} x^j v^k \rho_t dV

\end{align*}as well as\begin{align*}
\int_{D(t)} \epsilon_{ijk} x^j \dfrac{\partial p^{kl}}{\partial x^l} dV &= \int_{\partial D(t)} \epsilon_{ijk} x^j p^{kl} n_l ds - \int_{D(t)} \epsilon_{ijk} p^{kj} dV \\

&= \int_{\partial D(t)} \epsilon_{ijk} x^j p^{kl} n_l ds
\end{align*}where in the final equality the symmetry of ##p^{ij} = p^{ji}## was used. The time derivative of ##L^i## can hence be rewritten as\begin{align*}
\dfrac{d}{dt} L^i &=
\int_{\partial D(t)} \epsilon_{ijk} x^j (p^{kl} - v^k v^l \rho) n_l ds + M^i \\

&\hspace{35pt} + \int_{D(t)} \epsilon_{ijk} \mathrm{div}(\rho x^j v^k \boldsymbol{w}) dV
\end{align*}Does this equation look correct? It can probably be simplified further.
 
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I think you just missed ##\rho## and the formulas must be as follows
Screenshot from 2021-06-24 16-27-32.png
 
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Thanks! I missed out a ##\rho## when writing my final equation [it is now corrected]. Then\begin{align*}

\dfrac{d}{dt} L^i &=

\int_{\partial D(t)} \epsilon_{ijk} x^j (p^{kl} - v^k v^l \rho) n_l ds + M^i \\
&\hspace{35pt} + \int_{D(t)} \epsilon_{ijk} \mathrm{div}(\rho x^j v^k \boldsymbol{w}) dV \\ \\&=
\int_{\partial D(t)} \epsilon_{ijk} x^j p^{kl} n_l ds + M^i \\
&\hspace{35pt} + \int_{D(t)} \epsilon_{ijk} \rho x^j v^k (w^l - v^l) n_l ds

\end{align*}Multiplying by ##\boldsymbol{e}_i## yields\begin{align*}
\dfrac{d}{dt} \boldsymbol{L} = \int_{\partial D(t)} \boldsymbol{r} \times \boldsymbol{p}_n ds + \boldsymbol{M} + \int_{\partial D(t)} \boldsymbol{r} \times \boldsymbol{v} (\boldsymbol{w} - \boldsymbol{v}, \boldsymbol{n}) ds
\end{align*}where ##(\boldsymbol{p}_n)^i = p^{ij} n_j##.
 
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