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a curve. Is the circulation positive or negative for the curve C in gure (7.10) for

the indicated sense of integration?

Problem b: Consider now the surface S shown in gure (7.11). Show that the circulation

satis es I Z

v dr = ! dS (7.21)

C S

where ! is the vorticity. (See the sections (5.2)-(5.4).)

This expression implies that whenever lift is generated by the circulation along the contour

C around the wing, the integral of the vorticity over a surface that envelopes the wingtip

is nonzero. The vorticity depends on the derivative of the velocity. Since the ow is

relatively smooth along the wing, the derivative of the velocity eld is largest near the

wingtips. Therefore, expression (7.21) implies that vorticity is generated at the wingtips.

As shown in section (5.3) the vorticity is a measure of the local vortex strength. A wing

can only produce lift when the circulation along the curve C is nonzero. The above

reasoning implies that wingtip vortices are unavoidably associated with the lift produced

by an airfoil.

7.6. WINGTIPS VORTICES 71

C

S

^

n

A

B

Figure 7.11: Geometry of the surface S and the wingtip vortex for an aircraft seen from

above.

Problem c: Consider the wingtip vortex shown in gure (7.11). You have obtained the

H

sign of the circulation C v dr in problem a. Does this imply that the wingtip vortex

rotates in the direction A of gure (7.11) or in the direction B? Use equation (7.21)

in your argumentation. You may assume that the vorticity is mostly concentrated

at the trailing edge of the wingtips, see gure (7.11).

Problem d: The wingtip-vortex obviously carries kinetic energy. As such it entails an

undesirable loss of energy for a moving aircraft. Why do aircraft such as the Boeing

747-400 have wingtips that are turned upward? (These are called \winglets.")

Problem e: Just like aircraft, sailing boats su er from energy loss due a vortex that is

generated at the upper part of the sail, see the discussion of Marchaj 37]. (A sail

can be considered to be a \vertical wing.") Consider the two boats shown in gure

(7.12). Suppose that the sails have the same area. If you could choose one of these

boats for a race, would you choose the one on the left or on the right? Use equation

(7.21) to motivate your choice.

CHAPTER 7. THE THEOREM OF STOKES

72

Figure 7.12: Two boats carrying sails with a very di erent aspect ratio.

Chapter 8

Conservation laws

In physics one frequently handles the change of a property with time by considering

properties that do not change with time. For example, when two particles collide, the

momentum and the energy of each particle may change. However, this change can be

found from the consideration that the total momentum and energy of the system are

conserved. Often in physics, such conservation laws are main ingredients for describing a

system. In this section we deal with conservation laws for continuous systems. These are

systems where the physical properties are a continuous function of the space coordinates.

Examples are the motion in a uid or solid, the temperature distribution in a body. The

introduced conservation laws are not only of great importance in physics, they also provide

worthwhile exercises of the vector calculus introduced in the previous sections.

8.1 The general form of conservation laws

In this section a general derivation of conservation laws is given. Suppose we consider

a physical quantity Q. This quantity could denote the mass density of a uid, the heat

content within a solid or any other type of physical variable. In fact, there is no reason

why Q should be a scalar, it could also be a vector (such as the momentum density) or a

higher order tensor. Let us consider a volume V in space that does not change with time.

This volume is bounded by a surface @V . The total amount of Q within this volume is

R

given by the integral V QdV . The rate of change of this quantity with time is given by

@ R QdV .

R

@t V

In general, there are two reason for the quantity V QdV to change with time. First,

the eld Q may have sources or sinks within the volume V , the net source of the eld Q

per unit volume is denoted with the symbol S. The total source of Q within the volume

R

is simply the volume integral V SdV of the source density. Second, it may be that the

quantity Q is transported in the medium. With this transport process, a current J is

associated.

R QdVan example one can the uid Q the volume. This total mass can change because

As think of being the mass density of a uid. In that case

is the total mass of in

V

there is a source of uid within the volume (i.e. a tap or a bathroom sink), or the total

mass may change becauseR of the ow through the boundary of the volume.

The rate of change of V QdV by the current is given by the inward ux of the current

73

CHAPTER 8. CONSERVATION LAWS

74

J through the surface @V . If we retain the convention that the surface element dS points

H

out o the volume, the inward ux is given by @V J dS. Together with the rate of

;

change due to the source density S within the volume this implies that the rate of change

of the total amount of Q within the volume satis es:

@ Z QdV = I J dS+ Z SdV : (8.1)

@t V ;

@V V

Using Gauss' law (6.1), the surface integral in the right hand side can be written as

R (r J)dV , so that the expression above is equivalent with

V

;

@ Z QdV + Z (r J)dV = Z SdV : (8.2)

@t V V V

Since the volume V is assumed to be xed with time, the time derivative of the volume

@R R

integral is the volume integral of the time derivative: @t V QdV = V @Q dV . It should

@t

be noted that expression (8.2) holds for any volume V . If the volume is an in nitesimal

volume, the volume integrals in (8.2) can be replaced by the integrand multiplied with

the in nitesimal volume. Using these results, one nds that expression (8.2) is equivalent

with:

@Q + (r J) = S : (8.3)

@t

This is the general form of a conservation law in physics, it simply states that the

rate of change of a quantity is due to the sources (or sinks) of that quantity and due to

the divergence of the current of that quantity. Of course, the general conservation law

(8.3) is not very meaningful as long as we don't provide expressions for the current J

and the source S. In this section we will see examples where the current and the source

follow from physical theory, but we will also encounter examples where they follow from

an \educated" guess.

Equation (8.3) will not be completely new to you. In section (6.4) the probability

density current for a quantum mechanical system was derived.

Problem a: Use the derivation of this section to show that expression (6.15) can be

written as

@ 2 + (r J) = 0 (8.4)

@t j j

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