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Statement of the balance of mass

The balance of mass can be expressed as:

 \dot{\rho} + \rho~\boldsymbol{\nabla} \bullet \mathbf{v} = 0 

where \rho(\mathbf{x},t) is the current mass density, \dot{\rho} is the material time derivative of ρ, and \mathbf{v}(\mathbf{x},t) is the velocity of physical particles in the body Ω bounded by the surface \partial{\Omega}.


We can show how this relation is derived by recalling that the general equation for the balance of a physical quantity f(\mathbf{x},t) is given by

 \cfrac{d}{dt}\left[\int_{\Omega} f(\mathbf{x},t)~\text{dV}\right] = \int_{\partial{\Omega}} f(\mathbf{x},t)[u_n(\mathbf{x},t) - \mathbf{v}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)]~\text{dA} + \int_{\partial{\Omega}} g(\mathbf{x},t)~\text{dA} + \int_{\Omega} h(\mathbf{x},t)~\text{dV} ~.

To derive the equation for the balance of mass, we assume that the physical quantity of interest is the mass density \rho(\mathbf{x},t). Since mass is neither created or destroyed, the surface and interior sources are zero, i.e., g(\mathbf{x}, t) = h(\mathbf{x},t) = 0. Therefore, we have

 \cfrac{d}{dt}\left[\int_{\Omega} \rho(\mathbf{x},t)~\text{dV}\right] = \int_{\partial{\Omega}} \rho(\mathbf{x},t)[u_n(\mathbf{x},t) - \mathbf{v}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)]~\text{dA}~.

Let us assume that the volume Ω is a control volume (i.e., it does not change with time). Then the surface \partial{\Omega} has a zero velocity (un = 0) and we get

 \int_{\Omega} \frac{\partial \rho}{\partial t}~\text{dV} = - \int_{\partial{\Omega}} \rho~(\mathbf{v}\cdot\mathbf{n})~\text{dA}~.

Using the divergence theorem

 \int_{\Omega} \boldsymbol{\nabla} \bullet \mathbf{v}~\text{dV} = \int_{\partial{\Omega}} \mathbf{v}\cdot\mathbf{n}~\text{dA}

we get

 \int_{\Omega} \frac{\partial \rho}{\partial t}~\text{dV} = - \int_{\Omega} \boldsymbol{\nabla} \bullet (\rho~\mathbf{v})~\text{dV}.


 \int_{\Omega} \left[\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \bullet (\rho~\mathbf{v})\right]~\text{dV} = 0 ~.

Since Ω is arbitrary, we must have

 \frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \bullet (\rho~\mathbf{v}) = 0 ~.

Using the identity

 \boldsymbol{\nabla} \bullet (\varphi~\mathbf{v}) = \varphi~\boldsymbol{\nabla} \bullet \mathbf{v} + \boldsymbol{\nabla} \varphi\cdot\mathbf{v}

we have

 \frac{\partial \rho}{\partial t} + \rho~\boldsymbol{\nabla} \bullet \mathbf{v} + \boldsymbol{\nabla} \rho\cdot\mathbf{v} = 0 ~.

Now, the material time derivative of ρ is defined as

 \dot{\rho} = \frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \rho\cdot\mathbf{v} ~.


 { \dot{\rho} + \rho~\boldsymbol{\nabla} \bullet \mathbf{v}= 0 ~. }


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