Continuum mechanics/Balance of mass

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Statement of the balance of mass[edit]

The balance of mass can be expressed as:


where is the current mass density, is the material time derivative of , and is the velocity of physical particles in the body bounded by the surface .

Proof[edit]

We can show how this relation is derived by recalling that the general equation for the balance of a physical quantity is given by

To derive the equation for the balance of mass, we assume that the physical quantity of interest is the mass density . Since mass is neither created or destroyed, the surface and interior sources are zero, i.e., . Therefore, we have

Let us assume that the volume is a control volume (i.e., it does not change with time). Then the surface has a zero velocity () and we get

Using the divergence theorem

we get

or,

Since is arbitrary, we must have

Using the identity

we have

Now, the material time derivative of is defined as

Therefore,