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Although the idea of mass is not a kinematical one, it is convenient to introduce it here and to obtain the continuity equation. Let be the mass per unit volume of a homogeneous fluid at position and time . Then the mass of any finite material volume is
If is a material volume, that is, if it is composed of the same particles, and there are no sources or sinks in the medium we take it as a principle that the mass does not change. By inserting in Reynolds' transport theorem we have
This is true for an arbitrary material volume and hence the integrand itself must vanish everywhere. It follows that
which is the equation of continuity.
A fluid for which the density is constant is called incompressible . In this case the equation of continuity becomes
and the motion is isochoric or the velocity field solenoidal.
Combining the equation of continuity with Reynolds' transport theorem for a function we have
This equation is useful for deriving the conservation equation of a quantity that is expressed as specific to a unit of mass, e.g., specific internal energy and species mass fraction.
The motion of fluids differs from that of rigid bodies in the deformation or strain that occurs with motion. Material coordinates give a reference frame for this deformation or strain.
Consider two nearby points and with material coordinates and . At time they are to be found at and . Now
where represents terms of order and higher which will be neglected from this point onward. Thus the small displacement vector has now become
where
It is clear from the quotient rule (since is arbitrary) that the nine quantities are the components of a tensor. It may be called the displacement gradient tensor and is basic to the theories of elasticity and rheology. For fluid motion, its material derivative is of more direct application and we will concentrate on this.
If is the velocity, the relative velocity of two particles and has components
However, by inverting the above relation, we have
expressing the relative velocity in terms of current position. Again it is evident that the are components of a tensor, the velocity gradient tensor , for which we need to obtain a sound physical feeling.
We first observe that if the motion is a rigid body translation with a constant velocity ,
and the velocity gradient tensor vanishes identically. Secondly, the velocity gradient tensor can be written as the sum of symmetric and antisymmetric parts,
We have seen that a relative velocity related to the relative position by an antisymmetric tensor , i.e., , represents a rigid body rotation with angular velocity . In this case
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