0.3 The kinematics of fluid motion  (Page 7/9)

Conservation of mass and the equation of continuity

Although the idea of mass is not a kinematical one, it is convenient to introduce it here and to obtain the continuity equation. Let $\rho (\mathbf{x},t)$ be the mass per unit volume of a homogeneous fluid at position $\mathbf{x}$ and time $t$ . Then the mass of any finite material volume $V\left(t\right)$ is

If $V$ 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 $\Im =\rho$ 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 $\rho$ 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 $\Im =\rho F$ 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.

Deformation and rate of strain

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 $P$ and $Q$ with material coordinates $\xi$ and $\xi +d\xi$ . At time $t$ they are to be found at $\mathbf{x}(\xi ,t)$ and $\mathbf{x}(\xi +d\xi ,t)$ . Now

where $O\left(d^2\right)$ represents terms of order $d{\xi }^2$ and higher which will be neglected from this point onward. Thus the small displacement vector $d\xi$ has now become

where

It is clear from the quotient rule (since $d\xi$ is arbitrary) that the nine quantities $\partial x_i/\partial {\xi }_j$ 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 $\mathbf{v}=D\mathbf{x}/Dt$ is the velocity, the relative velocity of two particles $\xi$ and $\xi +d\xi$ 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 $(\partial v_i/\partial x_j)$ 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 $\mathbf{u}$ ,

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 $d{v}_i$ related to the relative position $d{x}_j$ by an antisymmetric tensor ${\Omega }_{ij}$ , i.e., $d{v}_i={\Omega }_{ij}d{x}_j$ , represents a rigid body rotation with angular velocity $\mathbf{\omega }=-vec\mathbf{\Omega }$ . In this case