Continuity equation

Consider the divergence of the product p (r)v(r). According to the definition of the divergence this is the flux of mass per unit time issuing from an infinitesimal volume per unit volume. If the box from which the mass is issuing is fixed in space (and there are no sources of mass inside), the mass inside the box has to be changing:

dM ff loss of mass/time = —— = ®(pv) ■ dS (9.68)

where M is the mass inside the fixed box. Dividing by the volume of the box and letting it shrink to zero, we obtain:

The minus sign takes into account that the flux out of the box represents a negative rate of change of mass in the box. Note that we used the partial derivative in the last formula because we are referring to a fixed position for our box. Rearranging we have the Eulerian form of the equation of continuity:

[Euler form of the equation of continuity]. (9.70)

By expanding the divergence of the product we can write it in another form:

This last equation has a very special meaning if we regroup the first two terms

[Lagrangian form of the equation of continuity] (9.72)

-i- = -p V • v Dt where the differential operator

[material derivative] (9.73)

Dt d t is called the material derivative, and we will return to it in the next section.

An alternative and perhaps a more physical derivation of the continuity equation (in two dimensions for simplicity) can be conducted as follows. Consider a small

Figure 9.16 Schematic view of a box element in motion at velocity v. The box might extend its width or height during the motion, but mass must be conserved.

Figure 9.16 Schematic view of a box element in motion at velocity v. The box might extend its width or height during the motion, but mass must be conserved.

rectangular box with sides dx and dy, whose area is dx dy. The mass density of material in the box is p giving the mass in the box as M = p dx dy = constant since mass will be conserved along the path of the box. The box might be distorted due to the differential motions of the fluid (e.g., shear). We can write:

dM d

— = - {p(x,y, t) • (xr(t) — xl(t))(yu(t) — yL(t))} (9.74)

where xR (t) represents the location of the right hand edge of the box as it moves, xL(t) similarly represents the location of the left hand edge (see Figure 9.16). The same notation goes for the upper and lower edges of the box. Note that xR (t) might be moving at a different speed from xL (t) and therefore the box might be stretched or squeezed in that direction. As we take the derivative through the expression we obtain dM dp

where by du we mean (dx/dt)(R) — (dx/dt)(L), and by dv we mean (dy/dt)(U) — (dy/dt)(L). In the last equation we have to recognize that the derivative of p is along the motion, as was the derivative of the mass. After dividing through by the area dx dy and taking the limit:

dp du dv

dt dx dy

Now with the fancy notation:

In the last equation we used the material derivative for the rate of change of p since it is taken along the motion. Note that this is the same statement as in (9.72). The generalization to three dimensions is straightforward.

As a parcel moves, its temperature or some other property might change along the path of the parcel. On the other hand, some properties are conserved along the path, for example, the potential temperature in adiabatic flows. The rate of change of a field along the fluid's motion is an important point of view to take because many physical laws are most easily expressed in this form. For example, the rate of change of the momentum in reaction to an imposed force is to be taken along the path of the parcel (Newton's Second Law).

Take a small rectangular parallelepiped of dimension dx, dy, dz. Let the center of the figure move with the local velocity of the fluid in which it is embedded, v (r, t). The velocity field v(r, t) might be changing in both space and time. Consider the temperature T(r, t) as an example. If we know the value of the temperature at a certain point (r, t), say T (r, t), what can we say about its value at neighboring points in space-time? The total differential can be used to make an estimate. We can write the total differential for the temperature field as

This is the change in the temperature due to a displacement from point r to r + dr and from t to t + dt. Now divide through by dt

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