Chapter 5

The continuity equation

5.1 The equation of continuity

It is evident that in a certain region of space the matter entering it must be equal to the matter leaving it. Let us consider an infinitesimal volume of rectangular parallelepiped form with sides δx,δy and δz (Fig 5.1). The fluid mass that enters the elementary volume in the unit time through the surface of equation x − δx/2=0 is δx δx δx δMx−δx/2 = ρ x− ,y,z,t u x− ,y,z,t δyδz =(ρu) x− ,y,z,t δyδz,  2   2   2  where the last step serves only to introduce a more concise notation. Similarly, the fluid mass that leaves the volume in the same time through the surface of equation x + δx/2=0 is δx δMx+δx/2 =(ρu) x + ,y,z,t δyδz.  2  Hence, the total mass entering the volume in the unit time in the x-direction is equal to

δMx = δMx−δx/2 − δMx+δx/2

27 28 Franco Mattioli (University of Bologna)

...... δz ...... − . . • . . (ρu)(x δx/2,y,z) . . . . (ρu)(x + δx/2,y,z) ...... (x,y,z)...... δy ...... δx

Fig. 5.1: The flux of matter entering the infinitesimal parallepiped can be de- composed into the sum of three independent contributions along the coordinate directions.

δx δx ∂(ρu) = (ρu) x− ,y,z,t − (ρu) x+ ,y,z,t δyδz = − δx δyδz,   2   2  " ∂x # where in the last step we have divided and multiplied by δx. Similar contributions arise in the y and z-directions, so that the mass of the matter entering from all the directions becomes ∂(ρu) ∂(ρv) ∂(ρw) δMT = δMx + δMy + δMz = − + + δxδyδz. " ∂x ∂y ∂z # On the other hand, the same mass can be evaluated on the basis of the variation of the density of the parcel ∂ρ δMT = δxδyδz. ∂t

Equating the two expressions of MT and dividing by the elementary volume δxδyδz, we obtain ∂ρ ∂(ρu) ∂(ρv) ∂(ρw) + + + =0, ∂t ∂x ∂y ∂z which in vector form can also be written as ∂ρ + ∇· (ρu)=0. (5.1) ∂t Principles of Fluid Dynamics (www.fluiddynamics.it) 29

This equation is known as the continuity equation, or also law of the mass con- servation. The continuity equation can be written in another equivalent form by ex- panding the divergence term. We obtain ∂ρ + u ·∇ρ + ρ∇· u =0, ∂t that is, dρ + ρ∇· u =0. (5.2) dt In an incompressible fluid, for which dρ/dt = 0, this relationship is further sim- plified, thus reducing to ∇· u =0. (5.3)

Problem 5.1 Use the continuity equation to derive the meaning of the divergence term already seen in problem [D.4]. Solution. Consider a parcel of mass δm = ρ δV . Since the mass of the parcel does not vary in time, one has

d d dρ d δm = (ρ δV ) = δV + ρ δV =0, dt dt dt ! dt from which, dividing by δV , one obtains

dρ 1 d + ρ δV =0. dt δV dt

A comparison with the continuity equation shows that the divergence of the velocity field in a given point at a given instant is proportional to the fractional variation of volume of the parcels passing through the point in that instant, as in (D.2).

5.2 Integral form of the continuity equation

Let us integrate the continuity equation (5.1) over a certain volume V, bounded by the surface S. We obtain ∂ρ dV + ∇· (ρu) dV =0. (5.4) V V Z ∂t Z 30 Franco Mattioli (University of Bologna)

In the first term the time derivative can be moved outside the integral symbol. Since it applies to a quantity that does not depend on space variables, then it becomes an ordinary derivative of the first kind (see section [C.1]). By further applying Gauss’ theorem to the second term, the equation can be rewritten as

d ρ dV + ρu · dS =0. (5.5) V S dt Z I The first integral clearly represents the mass enclosed within the volume V. The second integral represents the flux of matter per unit time (see Fig. 5.2).

...... n ...... α ...... u ...... δS ...... udt ......

Fig. 5.2: After an interval of time dt, the fluid that crossed the infinitesimal surface dS occupies an oblique cylinder of length udt and base dS. The scalar product u,dt cos α represents the height of the cylinder, so that dtu · n dS rep- resents its volume. The multiplication of this quantity by the density provides the mass that enters or leaves the volume V through the surface dS in the unit time.

Therefore, (5.5) states that the variation of the total mass within a certain finite volume equals the quantity of matter flowing through its delimiting surface. This is equivalent to saying that there cannot exist either sources nor sinks of matter. Let us extend in (5.5) the volume integral to all the space available to the fluid, assuming that its boundaries are either rigid and motionless or placed at infinity. Along such boundaries in the former case the normal velocity and in the latter the velocity itself are zero. Thus, the surface integral in (5.5) vanishes, and it follows that the total quantity of matter is conserved over time. The integral formulation (5.5) is equivalent to the differential formulation (5.1). In fact, from (5.5) we can derive (5.4), and this equation must hold what- ever the volume. Therefore it must hold also in any infinitesimal region of the space, where the integrand can be assumed as constant. This implies the differ- ential equation (5.1). Principles of Fluid Dynamics (www.fluiddynamics.it) 31

Indeed, there is a subtle difference. In the integral formulation the fields can also be discontinuous. The step between (5.5) and (5.4) requires the continuity and differen- tiability of the integrand with respect to time. Thus the integral formulation is slightly more general than the differential formulation. On the other hand the transport theorem (4.7) allows us to write (5.4) as

d dM ρ dV = =0, V dt Z dt where V(t) is the volume occupied by a certain set of parcels of total mass M(t) as they move in the space. In this form the meaning of the continuity equation as the law of the conservation of the mass is absolutely clear.

5.3 Particularly simple flows

In a fluid with a density that is constant in time and uniform over each horizontal plane and moves with a horizontal velocity equally uniform over each horizontal plane, all the terms of the continuity equation separately vanish. In fact, in this case, the local derivative of the density is zero because of stationarity, the first two components of the momentum do not depend on the horizontal coordinates, while the third one is zero because the vertical velocity is zero. Such flows, defined only by the vertical density stratification and by the horizontal components of the velocity as a function of the time and the vertical coordinate, are particularly simple to study. They can be dealt with by not considering the continuity equation, which is automatically satisfied because all of its terms vanish. Such flows are briefly called plane-parallel flows. Another kind of flow satisfying a similar property is a stationary radially symmetric flow. In this case, both the density and the horizontal velocity depend only on the vertical coordinate z and on the distance r = (x2 + y2)1/2 from the vertical coordinate axis. We have shown in problem [E.2] that such a flow is not divergent. Furthermore, since the circular motion of the parcels is not able to modify the structure of the density field, the total derivative of density also vanishes.

5.4 Historical notes and essential bibliography

The continuity equation for the particularly simple case of the flow in a chan- nel was qualitatively understood by many authors of the XVI and XVII century, but a differential form of the equation appeared much later in a work by Jean 32 Franco Mattioli (University of Bologna)

Le Rond d’Alembert [12] in 1747 and in the basic work by Leonhard Euler [17] in 1755.