Incompressible Inviscid Fluid Dynamics 77

5 Incompressible Inviscid Fluid Dynamics

5.1 Introduction

This chapter introduces some of the fundamental concepts that arise in the theory of incompressible, inviscid (or, tobe more exact, high Reynolds number) fluid motion.

5.2 Streamlines, Stream Tubes, and Stream Filaments

A line drawn in a fluid such that its tangent at each point is parallel to the local fluid velocity is called a streamline. Theaggregate of all the streamlines at a given instance in time constitutes the instantaneous flow pattern. The streamlinesdrawn through each point of a closed curve constitute a stream tube. Finally, a stream filament is defined as a streamtube whose cross-section is a curve of infinitesimal dimensions.

When the flow is unsteady then the configuration of the stream tubes and filaments changes from time to time.However, when the flow is steady then the stream tubes and filaments are stationary. In the latter case, a stream tubeacts like an actual tube through which the fluid is flowing. This follows because there can be no flow across the walls,and into the tube, since the flow is, by definition, always tangential to these walls. Moreover, the walls are fixed inspace and time, since the motion is steady. Thus, the motion of the fluid within the tube would be unchanged were thewalls replaced by a rigid frictionless boundary.

Consider a stream filament of an incompressible fluid whose motion is steady. Suppose that the cross-sectionalarea of the filament is sufficiently small that the fluid velocity is the same at each point on the cross-section. Moreover,let the cross-section be everywhere normal to the direction of this common velocity. Suppose that v1 and v2 are theflow speeds at two points on the filament at which the cross-sectional areas are S 1 and S 2, respectively. Consider thesection of the filament lying between these points. Since the fluid is incompressible, the same volume of fluid mustflow into one end of the section, in a given time interval, as flows out of the other, which implies that

v1 S 1 = v2 S 2. (5.1)

This is the simplest manifestation of the equation of fluid continuity discussed in Section 2.9. The above result isequivalent to the statement that the product of the speed and cross-sectional area is constant along any stream filamentof an incompressible fluid in steady motion. Thus, a stream filament within such a fluid cannot terminate unless thevelocity at that point becomes infinite. Leaving this case out of consideration, it follows that stream filaments insteadily flowing incompressible fluids are either closed loops, or terminate at the boundaries of the fluid. The same is,of course, true of streamlines.

5.3 Bernoulli’s Theorem

In its most general form, Bernoulli’s theorem—which was discovered by Daniel Bernoulli (1700–1783)—states that,in the steady flow of an inviscid fluid, the quantity

pρ+ T (5.2)

is constant along a streamline, where p is the pressure, ρ the density, and T the total energy per unit mass.The proof is straightforward. Consider the body of fluid bounded by the cross-sectional areas AB and CD of the

stream filament pictured in Figure 5.1. Let us denote the values of quantities at AB and CD by the suffixes 1 and 2,respectively. Thus, p1, v1, ρ1, S 1, T1 are the pressure, fluid speed, density, cross-sectional area, and total energy perunit mass, respectively, at AB, etc. Suppose that, after a short time interval δt, the body of fluid has moved such that itoccupies the section of the filament bounded by the cross-sections A′B′ and C′D′, where AA′ = v1 δt and CC′ = v2 δt.Since the motion is steady, the mass m of the fluid between AB and A′B′ is the same as that between CD and C′D′, sothat

m = S 1 v1 δt ρ1 = S 2 v2 δt ρ2. (5.3)

78 FLUID MECHANICS

A′

B′

CC ′

DD′

B

A

Figure 5.1: Bernoulli’s theorem.

Let T denote the total energy of the section of the fluid lying between A′B′ and CD. Thus, the increase in energy ofthe fluid body in the time interval δt is

(mT2 + T ) − (mT1 + T ) = m (T2 − T1). (5.4)

In the absence of viscous energy dissipation, this energy increase must equal the net work done by the fluid pressuresat AB and CD, which is

p1 S 1 v1 δt − p2 S 2 v2 δt = m(p1ρ1

−p2ρ2

). (5.5)

Equating expressions (5.4) and (5.5), we find thatp1ρ1+ T1 =

p2ρ2+ T2, (5.6)

which demonstrates that p/ρ + T has the same value at any two points on a given stream filament, and is thereforeconstant along the filament. Note that Bernoulli’s theorem has only been proved for the case of the steady motion ofan inviscid fluid. However, the fluid in question may either be compressible or incompressible.

For the particular case of an incompressible fluid, moving in a conservative force-field, the total energy per unitmass is the sum of the kinetic energy per unit mass, (1/2) v2, and the potential energy per unit mass, Ψ , and Bernoulli’stheorem thus becomes

pρ+12v2 + Ψ = constant along a streamline. (5.7)

If we focus on a particular streamline, 1 (say), then Bernoulli’s theorem states that

pρ+12v2 + Ψ = C1, (5.8)

where C1 is a constant characterizing that streamline. If we consider a second streamline, 2 (say), then

pρ+12v2 + Ψ = C2, (5.9)

where C2 is another constant. It is not generally the case that C1 = C2. If, however, the fluid motion is irrotationalthen the constant in Bernoulli’s theorem is the same for all streamlines (see Section 5.7), so that

pρ+12v2 + Ψ = C (5.10)

Incompressible Inviscid Fluid Dynamics 79

S2

A

B

S1

Figure 5.2: A vortex filament.

throughout the fluid.

5.4 Vortex Lines, Vortex Tubes, and Vortex Filaments

The curl of the velocity field of a fluid, which is generally termed vorticity, is usually represented by the symbol ω:i.e.,

ω = ∇ × v. (5.11)

A vortex line is a line whose tangent is everywhere parallel to the local vorticity vector. The vortex lines drawnthrough each point of a closed curve constitute the surface of a vortex tube. Finally, a vortex filament is a vortex tubewhose cross-section is of infinitesimal dimensions.

Consider a section AB of a vortex filament. The filament is bounded by the curved surface that forms the filamentwall, as well as two plane surfaces, whose vector areas are S1 and S2 (say), which form the ends of the section atpoints A and B, respectively. See Figure 5.2. Let the plane surfaces have outward pointing normals that are parallel (oranti-parallel) to the vorticity vectors, ω1 and ω2, at points A and B, respectively. Gauss’s theorem (see Section A.20),applied to the section, yields ∮

ω · dS =∫∇ · ω dV, (5.12)

where dS is an outward directed surface element, and dV a volume element. However,

∇ · ω = ∇ · ∇ × v ≡ 0 (5.13)

[see Equation (A.173)], implying that ∮ω · dS = 0. (5.14)

Now, ω · dS = 0 on the curved surface of the filament, since ω is, by definition, tangential to this surface. Thus, theonly contributions to the surface integral come from the plane areas S1 and S2. It follows that∮

ω · dS = S 2 ω2 − S 1 ω1 = 0. (5.15)

This result is essentially an equation of continuity for vortex filaments. It implies that the product of the magnitude ofthe vorticity and the cross-sectional area, which is termed the vortex intensity, is constant along the filament. It followsthat a vortex filament cannot terminate in the interior of the fluid. For, if it did, the cross-sectional area, S , would have

80 FLUID MECHANICS

to vanish, and, therefore, the vorticity, ω, would have to become infinite. Thus, a vortex filament must either form aclosed vortex ring, or must terminate at the fluid boundary.

Since a vortex tube can be regarded as a bundle of vortex filaments whose net intensity is the sum of the intensitiesof the constituent filaments, we conclude that the intensity of a vortex tube remains constant along the tube.

5.5 Circulation and Vorticity

Consider a closed curve C situated entirely within a moving fluid. The vector line integral (see Section A.14)

ΓC =

∮Cv · dr, (5.16)

where dr is an element of C, and the integral is taken around the whole curve, is termed the circulation of the flowaround the curve. The sense of circulation (e.g., either clockwise or counter-clockwise) is arbitrary.

Let S be a surface having the closed curve C for a boundary, and let dS be an element of this surface (see Sec-tion A.7) with that direction of the normal which is related to the chosen sense of circulation aroundC by the right-handcirculation rule (see Section A.8). According to Stokes’ theorem (see Section A.22),

ΓC =

∮Cv · dr =

∫Sω · dS. (5.17)

Thus, we conclude that circulation and vorticity are intimately related to one another. In fact, according to the aboveexpression, the circulation of the fluid around loop C is equal to the net sum of the intensities of the vortex filamentspassing through the loop and piercing the surface S (with a filament making a positive, or negative, contribution to thesum depending on whether it pierces the surface in the direction determined by the chosen sense of circulation aroundC and the right-hand circulation rule, or in the opposite direction). One important proviso to (5.17) is that the surfaceS must lie entirely within the fluid.

5.6 Kelvin Circulation Theorem

According to the Kelvin circulation theorem, which is named after Lord Kelvin (1824–1907), the circulation aroundany co-moving loop in an inviscid fluid is independent of time. The proof is as follows. The circulation around a givenloop C is defined

ΓC =

∮Cv · dr. (5.18)

However, for a loop that is co-moving with the fluid, we have dv = d(dr/dt) = d(dr)/dt. Thus,

dΓCdt=

∮C

dvdt· dr +

∮Cv · dv. (5.19)

However, by definition, dv/dt = Dv/Dt for a co-moving loop (see Section 2.10). Moreover, the equation of motion ofan incompressible inviscid fluid can be written [see Equation (2.79)]

DvDt= −∇

(pρ+ Ψ

), (5.20)

since ρ is a constant. Hence,dΓCdt= −

∮C∇(pρ−12v2 + Ψ

)· dr = 0, (5.21)

since v · dv = d(v2/2) = ∇(v2/2) · dr (see Section A.18), and p/ρ − v2/2 + Ψ is obviously a single-valued function.One corollary of the Kelvin circulation theorem is that the fluid particles that form the walls of a vortex tube at a

given instance in time continue to form the walls of a vortex tube at all subsequent times. To prove this, imagine aclosed loop C that is embedded in the wall of a vortex tube but does not circulate around the interior of the tube. See

Incompressible Inviscid Fluid Dynamics 81

C ′

C

Figure 5.3: A vortex tube.

Figure 5.3. The normal component of the vorticity over the surface enclosed byC is zero, since all vorticity vectors aretangential to this surface. Thus, from (5.17), the circulation around the loop is zero. By Kelvin’s circulation theorem,the circulation around the loop remains zero as the tube is convected by the fluid. In other words, although the surfaceenclosed by C deforms, as it is convected by the fluid, it always remains on the tube wall, since no vortex filamentscan pass through it.

Another corollary of the circulation theorem is that the intensity of a vortex tube remains constant as it is convectedby the fluid. This can be proved by considering the circulation around the loop C′ pictured in Figure 5.3.

5.7 Irrotational Flow

Flow is said to be irrotational when the vorticity ω has the magnitude zero everywhere. It immediately follows, fromEquation (5.17), that the circulation around any arbitrary loop in an irrotational flow pattern is zero (provided that theloop can be spanned by a surface that lies entirely within the fluid). Hence, from Kelvin’s circulation theorem, if aninviscid fluid is initially irrotational then it remains irrotational at all subsequent times. This can be seen more directlyfrom the equation of motion of an inviscid incompressible fluid which, according to Equations (2.39) and (2.79), takesthe form

∂v∂t+ (v · ∇) v = −∇

(pρ+ Ψ

), (5.22)

since ρ is a constant. However, from Equation (A.171),

(v · ∇) v = ∇(v2/2) − v ×ω. (5.23)

Thus, we obtain∂v∂t= −∇

(pρ+12v2 + Ψ

)+ v ×ω. (5.24)

Taking the curl of this equation, and making use of the vector identities∇×∇φ ≡ 0 [see Equation (A.176)],∇·∇×A ≡ 0[see Equation (A.173)], as well as the identity (A.179), and the fact that ∇ ·v = 0 in an incompressible fluid, we obtainthe vorticity evolution equation

DωDt= (ω · ∇) v. (5.25)

Thus, if ω = 0, initially, then Dω/Dt = 0, and, consequently,ω = 0 at all subsequent times.

82 FLUID MECHANICS

Suppose that O is a fixed point, and P an arbitrary movable point, in an irrotational fluid. Let O and P be joined bytwo different paths, OAP and OBP (say). It follows that OAPBO is a closed curve. Now, since the circulation aroundsuch a curve in an irrotational fluid is zero, we can write

∫OAP

v · dr +∫PBO

v · dr = 0, (5.26)

which implies that ∫OAP

v · dr =∫OBP

v · dr = −φP (5.27)

(say). It is clear that φP is a scalar function whose value depends on the position of P (and the fixed point O), but noton the path taken between O and P. Thus, if O is the origin of our coordinate system, and P an arbitrary point whoseposition vector is r, then we have effectively defined a scalar field φ(r) = −

∫ PO v · dr.

Consider a point Q that is sufficiently close to P that the velocity v is constant along PQ. Let η be the positionvector of Q relative to P. It then follows that (see Section A.18)

Incompressible Inviscid Fluid Dynamics 83

P

C

B

A

Figure 5.4: Two-dimensional flow.

whereC(t) is uniform in space, but can vary in time. In fact, the time variation of C(t) can be eliminated by adding theappropriate function of time (but not of space) to the velocity potential, φ. Note that such a procedure does not modifythe instantaneous velocity field v derived from φ. Thus, the above equation can be rewritten

pρ+12v2 + Ψ −

∂φ

∂t= C, (5.36)

where C is constant in both space and time. Expression (5.36) is a generalization of Bernoulli’s theorem (see Sec-tion 5.3) that takes non-steady flow into account. However, this generalization is only valid for irrotational flow. Forthe special case of steady flow, we get

pρ+12v2 + Ψ = C, (5.37)

which demonstrates that for steady irrotational flow the constant in Bernoulli’s theorem is the same on all streamlines.(See Section 5.3.)

5.8 Two-Dimensional Flow

Fluid motion is said to be two-dimensionalwhen the velocity at every point is parallel to a fixed plane, and is the sameeverywhere on a given normal to that plane. Thus, in Cartesian coordinates, if the fixed plane is the x-y plane then wecan express a general two-dimensional flow pattern in the form

v = vx(x, y, t) ex + vy(x, y, t) ey. (5.38)

Let A be a fixed point in the x-y plane, and let ABP and ACP be two curves, also in the x-y plane, that join A to anarbitrary point P. See Figure 5.4. Suppose that fluid is neither created nor destroyed in the region, R (say), bounded bythese curves. Since the fluid is incompressible, which essentially means that its density is uniform and constant, fluidcontinuity requires that the rate at which the fluid flows into the region R, from right to left across the curve ABP, isequal to the rate at which it flows out the of the region, from right to left across the curve ACP. Now, the rate of fluidflow across a surface is generally termed the flux. Thus, the flux (per unit length parallel to the z-axis) from right toleft across ABP is equal to the flux from right to left across ACP. Since ACP is arbitrary, it follows that the flux fromright to left across any curve joining points A and P is equal to the flux from right to left across ABP. In fact, once thebase point A has been chosen, this flux only depends on the position of point P, and the time t. In other words, if wedenote the flux by ψ then it is solely a function of the location of P and the time. Thus, if point A lies at the origin, and

84 FLUID MECHANICS

P2

A

P1

Figure 5.5: Two-dimensional flow.

point P has Cartesian coordinates x, y, then we can write

ψ = ψ(x, y, t). (5.39)

The function ψ is known as the stream function. Moreover, the existence of a stream function is a direct consequenceof the assumed incompressible nature of the flow.

Consider two points, P1 and P2, in addition to the fixed point A. See Figure 5.5. Let ψ1 and ψ2 be the fluxes fromright to left across curves AP1 and AP2. Now, using similar arguments to those employed above, the flux across AP2 isequal to the flux across AP1 plus the flux across P1P2. Thus, the flux across P1P2, from right to left, is ψ2 − ψ1. Now,if P1 and P2 both lie on the same streamline then the flux across P1P2 is zero, since the local fluid velocity is directedeverywhere parallel to P1P2. It follows that ψ1 = ψ2. Hence, we conclude that the stream function is constant along astreamline. The equation of a streamline is thus ψ = c, where c is an arbitrary constant.

Let P1P2 = δs be an infinitesimal arc of a curve that is sufficiently short that it can be regarded as a straight-line.The fluid velocity in the vicinity of this arc can be resolved into components parallel and perpendicular to the arc. Thecomponent parallel to δs contributes nothing to the flux across the arc from right to left. The component perpendicularto δs contributes v⊥ δs to the flux. However, the flux is equal to ψ2 − ψ1. Hence,

v⊥ =ψ2 − ψ1δs

. (5.40)

In the limit δs→ 0, the perpendicular velocity from right to left across ds becomes

v⊥ =dψds. (5.41)

Thus, in Cartesian coordinates, by considering infinitesimal arcs parallel to the x- and y-axes, we deduce that

vx = −∂ψ

∂y, (5.42)

vy =∂ψ

∂x. (5.43)

These expressions can be combined to give

v = ez × ∇ψ = ∇z × ∇ψ. (5.44)

Incompressible Inviscid Fluid Dynamics 85

Note that when the fluid velocity is written in this form then it immediately becomes clear that the incompressibilityconstraint ∇ · v = 0 is automatically satisfied [since ∇ · (∇A×∇B) = 0—see Equations (A.175) and (A.176)]. It is alsoclear that the stream function is undefined to an arbitrary additive constant.

The vorticity in two-dimensional flow takes the form

ω = ωz ez, (5.45)

whereωz =

∂vy

∂x−∂vx

∂y. (5.46)

Thus, it follows from Equations (5.42) and (5.43) that

ωz =∂2ψ

∂x2+∂2ψ

∂y2= ∇2ψ. (5.47)

Moreover, irrotational two-dimensional flow is characterized by

∇2ψ = 0. (5.48)

When expressed in terms of cylindrical coordinates (see Section C.3), Equation (5.44) yields

v = vr(r, θ, t) er + vθ(r, θ, t) eθ, (5.49)

where

vr = −1r∂ψ

∂θ, (5.50)

vθ =∂ψ

∂r. (5.51)

Moreover, the vorticity is ω = ωz ez, where

ωz =1r∂

∂r

(r∂ψ

∂r

)+

1r 2

∂2ψ

∂θ 2. (5.52)

5.9 Two-Dimensional Uniform Flow

Consider a steady two-dimensional flow pattern that is uniform: i.e., such that the fluid velocity is the same everywherein the x-y plane. For instance, suppose that the common fluid velocity is

v = V0 cos θ0 ex + V0 sin θ0 ey, (5.53)

which corresponds to flow at the uniform speed V0 in a fixed direction that subtends a (counter-clockwise) angle θ0with the x-axis. It follows, from Equations (5.42) and (5.43), that the stream function for steady uniform flow takesthe form

ψ(x, y) = V0 (sin θ0 x − cos θ0 y) . (5.54)

When written in terms of cylindrical coordinates, this becomes

ψ(r, θ) = −V0 r sin(θ − θ0). (5.55)

Note, from (5.54), that ∂2ψ/∂x2 = ∂2ψ/∂y2 = 0. Thus, it follows from Equation (5.47) that uniform flow isirrotational. Hence, according to Section 5.7, such flow can also be derived from a velocity potential. In fact, it iseasily demonstrated that

φ(r, θ) = −V0 r cos(θ − θ0). (5.56)

86 FLUID MECHANICS

y

S

x

Figure 5.6: Streamlines of the flow generated by a line source coincident with the z-axis.

5.10 Two-Dimensional Sources and Sinks

Consider a uniform line source, coincident with the z-axis, that emits fluid isotropically at the steady rate of Q unitvolumes per unit length per unit time. By symmetry, we expect the associated steady flow pattern to be isotropic, andeverywhere directed radially away from the source. See Figure 5.6. In other words, we expect

v = vr(r) er. (5.57)

Consider a cylindrical surface S of unit height (in the z-direction) and radius r that is co-axial with the source. In asteady state, the rate at which fluid crosses this surface must be equal to the rate at which the section of the sourceenclosed by the surface emits fluid. Hence,

∫Sv · dS = 2π r vr(r) = Q, (5.58)

which implies that

vr(r) =Q2π r

. (5.59)

According to Equations (5.50) and (5.51), the stream function associated with a line source of strength Q that iscoincident with the z-axis is

ψ(r, θ) = −Q2π

θ. (5.60)

Note that the streamlines, ψ = c, are directed radially away from the z-axis, as illustrated in Figure 5.6. Note, also, thatthe stream function associated with a line source is multivalued. However, this does not cause any particular difficulty,since the stream function is continuous, and its gradient single-valued.

Note, from Equation (5.60), that ∂ψ/∂r = ∂2ψ/∂θ 2 = 0. Hence, according to (5.52), ωz = −∇2ψ = 0. In otherwords, the steady flow pattern associated with a uniform line source is irrotational, and can, thus, be derived from avelocity potential. In fact, it is easily demonstrated that this potential takes the form

φ(r, θ) = −Q2π

ln r. (5.61)

Incompressible Inviscid Fluid Dynamics 87

A uniform line sink, coincident with the z-axis, which absorbs fluid isotropically at the steady rate of Q unitvolumes per unit length per unit time has an associated steady flow pattern

v = −Q2π r

er, (5.62)

whose stream function isψ(r, θ) =

Q2π

θ. (5.63)

This flow pattern is also irrotational, and can be derived from the velocity potential

φ(r, θ) =Q2π

ln r. (5.64)

Consider a line source and a line sink of equal strength, which both run parallel to the z-axis, and are located asmall distance apart in the x-y plane. Such an arrangement is known as a dipole or doublet line source. Suppose thatthe line source, which is of strength Q, is located at r = d/2 (where r is a position vector in the x-y plane), and thatthe line sink, which is also of strength Q, is located at r = −d/2. Let the function

ψQ(r) = −Q2π

θ = −Q2π

tan−1(y/x) (5.65)

be the stream function associated with a line source of strength Q located at r = 0. Thus, ψQ(r − r0) is the streamfunction associated with a line source of strength Q located at r = r0. Furthermore, the stream function associatedwith a line sink of strength Q located at r = r0 is −ψQ(r − r0). Now, we expect the flow pattern associated with thecombination of a source and a sink to be the vector sum of the flow patterns generated by the source and sink taken inisolation. It follows that the overall stream function is the sum of the stream functions generated by the source and thesink taken in isolation. In other words,

ψ(r) = ψQ(r − d/2) − ψQ(r + d/2) � −d · ∇ψQ(r), (5.66)

to first order in d/r. Hence, if d = d (cos θ0 ex + sin θ0 ey) = d [cos(θ − θ0) er − sin(θ − θ0) eθ], so that the line joiningthe sink to the source subtends a (counter-clockwise) angle θ0 with the x-axis, then

ψ(r, θ) = −D2π

sin(θ − θ0)r

, (5.67)

where D = Qd is termed the strength of the dipole source. Note that the above stream function is antisymmetric acrossthe line θ = θ0 joining the source to the sink. It follows that the associated dipole flow pattern,

vr(r, θ) =D2π

cos(θ − θ0)r 2

, (5.68)

vθ(r, θ) =D2π

sin(θ − θ0)r 2

, (5.69)

is symmetric across this line. Figure 5.7 shows the streamlines associated with a dipole flow pattern characterized byD > 0 and θ0 = 0. Note that the flow speed in a dipole pattern falls off like 1/r 2.

A dipole flow pattern is necessarily irrotational since it is a linear superposition of two irrotational flow patterns.The associated velocity potential is

φ(r, θ) =D2π

cos(θ − θ0)r

. (5.70)

5.11 Two-Dimensional Vortex Filaments

Consider a vortex filament of intensity Γ that is coincident with the z-axis. By symmetry, we expect the associatedflow pattern to circulate isotropically around the filament. See Figure 5.8. In other words, we expect

v = vθ(r) eθ. (5.71)

Incompressible Inviscid Fluid Dynamics 89

x

y

Figure 5.8: Streamlines of the flow generated by a line vortex coincident with the z-axis.

The solution that is well-behaved at r = 0, and continuous (up to its first derivative) at r = a, is

ψ(r, θ) ={(Γ/4π) (r2/a2 − 1) r ≤ a(Γ/2π) ln(r/a) r > a

. (5.77)

Note that this expression is equivalent to (5.74) (apart from an unimportant additive constant) outside the filament, butdiffers inside. The associated circulation velocity, vθ(r) = ∂ψ/∂r, is

vθ(r) ={(Γ/2π) (r/a2) r ≤ a(Γ/2π) (1/r) r > a

, (5.78)

whereas the circulation, Γr(r) = 2π r vθ(r), is written

Γr(r) ={Γ (r/a)2 r ≤ aΓ r > a

. (5.79)

Thus, we conclude that the flow pattern associated with a straight vortex filament is irrotational outside the filament,but has finite vorticity inside the filament. Moreover, the non-zero internal vorticity generates a constant net circulationof the flow outside the filament. In the limit in which the radius of the filament tends to zero, the vorticity within thefilament tends to infinity (in such a way that the product of the vorticity and the cross-sectional area of the filamentremains constant), and the region of the fluid in which the vorticity is non-zero becomes infinitesimal in extent.

Let us determined the pressure profile in the vicinity of a vortex filament of finite radius. Assuming, from symme-try, that p = p(r), Equation (2.149), yields

dpdr= ρ

v 2θr, (5.80)

which can be integrated to give

p = p∞ − ρ∫ ∞

r

v 2θ

rdr, (5.81)

where p∞ is the pressure at infinity. Making use of expression (5.78), we obtain

p(r) ={p∞ − (ρ/2) (Γ/2π a)2 (2 − r2/a2) r ≤ ap∞ − (ρ/2) (Γ/2π a)2 (a/r)2 r > a

. (5.82)

90 FLUID MECHANICS

It follows that the minimum pressure occurs at the center of the vortex (r = 0), and takes the value

p0 = p∞ − ρ(Γ

2π a

)2. (5.83)

Now, under normal circumstances, the pressure in a fluid must remain positive, which implies that a vortex filamentof intensity Γ, embedded in a fluid of density ρ and background pressure p∞, has a minimum radius of order

amin �(ρ

p∞

)1/2 (Γ

2π

). (5.84)

Finally, since the flow pattern outside a straight vortex filament is irrotational, it can be derived from a velocitypotential. In fact, it is easily demonstrated that the appropriate potential takes the form

φ(r, θ) = −Γ

2πθ. (5.85)

Note that the above potential is multivalued. However, this does not cause any particular difficulty, since the potentialis continuous, and its gradient single-valued.

5.12 Two-Dimensional Irrotational Flow in Cylindrical Coordinates

As we have seen, in a two-dimensional flow pattern, we can automatically satisfy the incompressibility constraint,∇ · v = 0, by expressing the pattern in terms of a stream function. Suppose, however, that, in addition to beingincompressible, the flow pattern is also irrotational. In this case, Equation (5.47) yields

∇2ψ = 0. (5.86)

In cylindrical coordinates, since ψ = ψ(r, θ, t), this expression implies that (see Section C.3)

1r∂

∂r

(r∂ψ

∂r

)+

1r 2

∂2ψ

∂θ 2= 0. (5.87)

Let us search for a separable steady-state solution of Equation (5.87) of the form

ψ(r, θ) = R(r)Θ(θ). (5.88)

It is easily seen thatrRddr

(rdRdr

)= −

1Θ

d2Θdθ 2

, (5.89)

which can only be satisfied if

rddr

(rdRdr

)= m2 R, (5.90)

d2Θdθ 2

= −m2Θ, (5.91)

where m2 is an arbitrary (positive) constant. The general solution of Equation (5.91) is a linear combination ofexp( im θ) and exp(−im θ) factors. However, assuming that the flow extends over all θ values, the function Θ(θ)must be single-valued in θ, otherwise ∇ψ—and, hence, v—would not be be single-valued (which is unphysical). Itfollows that m can only take integer values (and that m2 must be a positive, rather than a negative, constant). Now, thegeneral solution of Equation (5.90) is a linear combination of rm and r−m factors, except for the special case m = 0,when it is a linear combination of r0 and ln r factors. Thus, the general stream function for steady two-dimensionalirrotational flow (that extends over all values of θ) takes the form

ψ(r, θ) = α0 + β0 ln r +∑m>0

(αm rm + βm r−m) sin[m (θ − θm)], (5.92)

Incompressible Inviscid Fluid Dynamics 91

where αm, βm, and θm are arbitrary constants. We can recognize the first few terms on the right-hand side of the aboveexpression. The constant term α0 has zero gradient, and, therefore, does not give rise to any flow. The term β0 ln r isthe flow pattern generated by a vortex filament of intensity 2π β0, coincident with the z-axis. (See Section 5.11.) Theterm α1 r sin(θ− θ1) corresponds to uniform flow of speed α1 whose direction subtends a (counter-clockwise) angle θ1with the minus x-axis. (See Section 5.9.) Finally, the term β1 sin(θ − θ1)/r corresponds to a dipole flow pattern. (SeeSection 5.10.)

The velocity potential associated with the irrotational stream function (5.92) satisfies [see Equations (5.29) and(5.44)]

∂φ

∂r=

1r∂ψ

∂θ, (5.93)

1r∂φ

∂θ= −

∂ψ

∂r. (5.94)

It follows thatφ(r, θ) = α0 − β0 θ +

∑m>0

(αm rm − βm r−m) cos[m (θ − θ0)]. (5.95)

5.13 Inviscid Flow Past a Cylindrical Obstacle

Consider the steady flow pattern produced when an impenetrable rigid cylindrical obstacle is placed in a uniformlyflowing, incompressible, inviscid fluid, with the cylinder orientated such that its axis is normal to the flow. For instance,suppose that the radius of the cylinder is a, and that its axis corresponds to the line x = y = 0. Furthermore, let theunperturbed fluid velocity be of magnitude V0, and be directed parallel to the x-axis. Now, we expect the flow patternto remain unperturbed very far away from the cylinder. In other words, we expect v(r, θ) → V0 ex as r/a → ∞, whichcorresponds to a boundary condition on the stream function of the form (see Section 5.9)

ψ(r, θ) → −V0 r sin θ as r/a→ ∞. (5.96)

Given that the fluid velocity field a large distance upstream of the cylinder is irrotational (since we have alreadyseen that the flow pattern associated with uniform flow is irrotational—see Section 5.9), it follows from the Kelvincirculation theorem (see Section 5.6) that the velocity field remains irrotational as it is convected past the cylinder.Hence, according to Section 5.8, the stream function of the flow satisfies Laplace’s equation,

∇2ψ = 0. (5.97)

The appropriate boundary condition at the surface of the cylinder is simply that the normal fluid velocity there bezero, since the fluid must stay in contact with the cylinder, but cannot penetrate its surface. Hence, vr(a, θ) ≡−(1/a) ∂ψ/∂θ|r=a = 0, which implies that

ψ(a, θ) = 0, (5.98)

since ψ is undetermined to an arbitrary additive constant. It follows that we are searching for the most general solutionof (5.97) that satisfies the boundary conditions (5.96) and (5.98). Comparison with Equation (5.92) reveals that thissolution takes the form

ψ(r, θ) = V0 a[−γ ln

( ra

)−

( ra−ar

)sin θ

], (5.99)

whereγ = −

Γ

2π a V0, (5.100)

and Γ is the circulation of the flow around the cylinder. (Note that the velocity field can be irrotational, but still possessnonzero circulation around the cylinder, because a loop that encloses the cylinder cannot be spanned by a surface lyingentirely within the fluid. Thus, zero fluid vorticity does not necessarily imply zero circulation around such a loop fromStokes’ theorem.) Let us assume that γ ≥ 0, for the sake of definiteness.

92 FLUID MECHANICS

F i g u r e 5 . 9 : v 0e 0F i g u r e 5 . 9 – 5 . 1 1 s h o w s t r e a m l i n e s o f t h e fl o w c a l c u l a t e d f o r v a r i o u s d i e r e n t v a l u e s o f t h e n o r m a l i z e d c i r c u l a t i o n , . F o r 2 t h e r e e x i s t a p a i r o f p o i n t s o n t h e s u r f a c e o f t h e c y l i n d e r a t w h i c h t h e fl o w s p e e d i s z e r o . T h e s e a r e k n o w n a s, a n d c a n b e l o c a t e d i n F i g u r e s 5 . 9 a n d 5 . 1 0 a s t h e p o i n t s a t w h i c h s t r e a m l i n e s i n t e r s e c t t h e s u r f a c e o f t h e c y l i n d e r a t r i g h t - a n g l e s . N o w , t h e t a n g e n t i a l fl u i d v e l o c i t y a t t h e s u r f a c e o f t h e c y l i n d e r i s ()( )Ἡ

ἠ

0( 2 s i n ) ( 5 . 1 0 1 ) T h e s t a g n a t i o n p o i n t s c o r r e s p o n d t o t h e p o i n t s a t w h i c h 0 ( s i n c e t h e n o r m a l v e l o c i t y i s a u t o m a t i c a l l y z e r o a t t h es u r f a c e o f t h e c y l i n d e r ) . T h u s , t h e s t a g n a t i o n p o i n t s l i e a t sin1( 2 ) . W h e n 2 t h e s t a g n a t i o n p o i n t s c o a l e s c e andmoveot h e s u r f a c e o f t h e c y l i n d e r , a s i l l u s t r a t e d i n F i g u r e 5 . 1 1 ( t h e s t a g n a t i o n p o i n t c o r r e s p o n d s t o t h e p o i n t a t w h i c h t w o s t r e a m l i n e s c r o s s a t r i g h t - a n g l e s ) .T h e i r r o t a t i o n a l f o r m o f B e r n o u l l i ’ s t h e o r e m , ( 5 . 3 7 ) , c a n b e c o m b i n e d w i t h t h e b o u n d a r y c o n d i t i o n ḓ0a s Ḝ , a s w e l l a s t h e f a c t t h a t яi s c o n s t a n t i n t h e p r e s e n t c a s e , t o g i v e

0 1

2Ⱦ 20 2

( 5 . 1 0 2 ) w h e r e 0i s t h e c o n s t a n t s t a t i c fl u i d p r e s s u r e a l a r g e d i s t a n c e f r o m t h e c y l i n d e r . I n p a r t i c u l a r , t h e fl u i d p r e s s u r e o n t h e s u r f a c e o f t h e c y l i n d e r i s

( ) ( ) 0 1

2Ⱦ 20 2

1 Ⱦ 20(c o s 2 2 sin)( 5 . 1 0 3 ) w h e r e 1 0 ( 1 2 ) Ⱦ 20( 1 2) . T h e n e t f o r c e p e r u n i t l e n g t h e x e r t e d o n t h e c y l i n d e r b y t h e fl u i d h a s t h e C a r t e s i a n c o m p o n e n t s

я c o s ( 5 . 1 0 4 )

я sin ( 5 . 1 0 5 )

Incompressible Inviscid Fluid Dynamics 93

− 5

− 4

− 3

− 2

− 10

1

2

3

4

5

y / a −5− 4− 3− 2− 1 0 1 2 3 4 5

x/aF i g u r 5 . 1 0 :S t r e a m l i n e s o f t h e flo w g e n e r a t e d b y a c y l i n d r i c a l o b s t a c l e o f r a d i u s a , w h o s e a x i s r u n s a l o n g t h e z - a x i s , placedintheuniformflo w fie l d =V0ex. T h e n o r m a l i z e d c i r c u l a t i o n i s γ=1.T h u s , i f l w s f r m ( . 1 0 3 ) a Fx=0, . 1 0 6 ) Fy=2π γ ρ V20a=ρ V0(− Γ ). . 1 0 7 ) N w , o m p o n n f t o r w h a o v e e r o n a n o b s t , l a n t a h , i n r i o n a r a l t t a o f t e n e r u r w i u s u a c a d d r a g . O n t h e o t h e r h a n t h e c o m p o n n o f t h e f o r w h i c h t h e fl u i d e e r i n d i r e i o n p e r n c u l a r t o h a o f h e u n e r u r w i s u s u a l c a d lift. H e n h e a b o e u a i o n i m p l a i y l n c a o b s t s l n a u n f o r l fl o w n n v s c idfluidthentherszerodrag. O n t h e o t h e r h a a s n a s e r s e t r c u l a o n o f t h e w r u n e c l i n r t e l i f t i s n n z e r . N o w , l i f t i s g n r a e d b u s e ( n g a i e ) c r c u l a i o n t n t i n s e t h e fl u i d s e e d r l a v e , n d t d e s e i d i r l b e l w , e c y l n d e r . T u s , f r m B e r u l ’ s t e o r e m e fl u i d r e s s u r e i s d e s e d a o v e , n n s e w , t e l n r g i v t a n u p w a r d o r ( i . e . , a f o r i n h e +y- d i r i o n ) . S u p p o s e h a t h e y l n r i s p l i n a u i d w h i c h i s n i a t r s t a t h a t h e fl u i d s u n f o r fl o w e l o c t y , V0,i s h e n r s o w l y r e d u p i n c h a a r h a n o r i i t y i s i n n t h e p s t r e fl o w t i n n i t y ) . S i n t e w p a t t r i s i n i t i a i r t a i o n a l a n d s i n c e e w p a t t r w l l u p s r a o f h e c l i r s a s t o r m a n i r t a i o n a l t e K l i n i r c u l a i o n e o r e m i n d i c t s t h a t h e w p a t t e r a r u n l i r l s o r m a i t t o n l . ConsidethetimeoltionofecrculaionΓ=

∮Cv·dr, u n s o m e fix e d c u r Ct a li s en tir wit in thefl u i d , n n o s s h e c l n r . W e h a v e dΓC

d t =

∮C∂v

∂t·dr=∮C[− ∇

(p

ρ+12v2)+v× ω]·dr=

∮Cv× ω·dr, . 1 0 8 ) w e r u s e a s e e n m a o f ( . 2 4 ) ( t Ψa s s u m e c o n t n . H o w v r ω=ωzezi n t w - m e i o n l fl o , a dr× ez= dS, r e dSi s a o u t w a r d s u r f a c e l m e n u n t d p t h ( n h e z- d i r e i o n s u r f a w o s n r m a l i e s i n tex-yp l , a h a t s h e x-yp l a C. n t h r r d s , dΓC

d t = −∮Sωzv·dS. . 1 0 9 ) W e , h u s , c o n d a e r a e o f c h a f t e c i c u l a t n u n Ci s e u a t m i u s t e fl u x o f e v o r i a s s S[ s s u m i n g h a o r i c i t i s c o n e t h e o w i c h f o l l f m ( 5 . 2 ) , t e f a h a ω=ωzez, n t e f a

94 FLUID MECHANICS

− 5

− 4

− 3

− 2

− 10

1

2

3

4

5y / a − 5− 4− 3− 2− 1 0 1 2 3 4 5

x/aF i g u r e 5 . 1 1 : S t r e a m l i n e s o f t h e flo w g e n e r a t e d b y a c y l i n d r i c a l o b s t a c l e o f r a d i u s a , w h o s e a x i s r u n s a l o n g t h e z - a x i s ,

placedintheuniformflo w fie l d v=V0ex. T h e n o r m a l i z e d c i r c u l a t i o n i s γ=2.5.

t h a t ∂ / ∂ z=0 i n t w o - d i m e n s i o n a l fl o w ] . H o w e v e r , w e h a v e a l r e a d y s e e n t h a t t h e fl o w fi e l d s u r r o u n d i n g t h e c y l i n d e r isi r r o t a t i o n a l (i . e . , s u c h t h a t ωz=0 ) . I t f o l l o w s t h a t ΓCisc o n s t a n t i n t i m e . M o r e o v e r , s i n c e ΓC=0 o r i g i n a l l y , b e c a u s e t h e fl u i d s u r r o u n d i n g t h e c y l i n d e r w a s i n i t i a l l y a t r e s t , w e d e d u c e t h a t ΓC=0 a t a l l s u b s e q u e n t t i m e s . H e n c e , w e c o n c l u d e t h a t , i n a n i n v i s c i d fl u i d , i f t h e c i r c u l a t i o n o f t h e fl o w a r o u n d t h e c y l i n d e r i s i n i t i a l l y z e r o t h e n i t r e m a i n s z e r o . I t f o l l o w s , f r o m t h e a b o v e a n a l y s i s , t h a t , i n s u c h a fl u i d , z e r o d r a g f o r c e a n d z e r o l i f t f o r c e a r e e x e r t e d o n t h e c y l i n d e r a s a c o n s e q u e n c e o f t h e fl u i d fl o w . T h i s r e s u l t i s k n o w n a s d ’ A l e m b e r t ’ s p a r a d o x , a f t e r t h e F r e n c h s c i e n t i s t J e a n - B a p t i s t e l e R o n d d ’ A l e m b e r t ( 1 7 1 7 – 1 7 8 3 ) . D ’ A l e m b e r t ’ s r e s u l t i s p a r a d o x i c a l b e c a u s e i t w o u l d s e e m , a t fi r s t s i g h t , t o b e a r e a s o n a b l e a p p r o x i m a t i o n t o n e g l e c t v i s c o s i t y a l l t o g e t h e r i n h i g h R e y n o l d s n u m b e r fl o w . H o w e v e r , i f w e d o t h i s t h e n w e e n d u p w i t h t h e n o n s e n s i c a l p r e d i c t i o n t h a t a h i g h R e y n o l d s n u m b e r fl u i d i s i n c a p a b l e o f e x e r t i n g a n y f o r c e o n a n o b s t a c l e p l a c e d i n i t s p a t h . 5.14InviscidFlowPastaSemi-InfiniteWedge

Considerthesituation,illustratedinFigure5.12,inwhichincompressib l e i r r o t a t i o n a l fl o w i s i n c i d e n t o n a i m p e n e - t r a b l e r i g i d w e d g e w h o s e a p e x s u b t e n d s a n a n g l e α π . L e t t h e c r o s s - s e c t i o n o f t h e w e d g e i n t h e x-yp l a n e b e b o t h z- i n d e p e n d e n t a n d s y m m e t r i c a b o u t t h ex- a x i s . F u r t h e r m o r e , l e t t h e a p e x o f t h e w e d g e l i e a t x=y=0.Finally,lettheu p s t r e a m fl o w a l a r g e d i s t a n c e f r o m t h e w e d g e b e p a r a l l e l t o t h ex- a x i s . S i n c e t h e fl o w i s t w o - d i m e n s i o n a l , i n c o m p r e s s i b l e , a n d i r r o t a t i o n a l , i t c a n b e r e p r e s e n t e d i n t e r m s o f a s t r e a m f u n c t i o n t h a t s a t i s fi e s L a p l a c e ’ s e q u a t i o n . M o r e o v e r , i n c y l i n d r i c a l c o o r d i n a t e s , t h i s e q u a t i o n t a k e s t h e f o r m ( 5 . 8 7 ) . T h e b o u n d a r y c o n d i t i o n s o n t h e s t r e a m f u n c t i o n a r e ψ(r,απ/2 ) =ψ(r,2π−α π / 2 ) =ψ(r,π)=0.( 5 . 1 1 0 ) T h e fi r s t t w o b o u n d a r y c o n d i t i o n s e n s u r e t h a t t h e n o r m a l v e l o c i t y a t t h e s u r f a c e o f t h e w e d g e i s z e r o . T h e t h i r d b o u n d a r y c o n d i t i o n f o l l o w s f r o m t h e o b s e r v a t i o n t h a t , b y s y m m e t r y , t h e s t r e a m l i n e t h a t m e e t s t h e a p e x o f t h e w e d g e s p l i t s i n t w o , a n d t h e n fl o w s a l o n g i t s t o p a n d b o t t o m b o u n d a r i e s , c o m b i n e d w i t h w e l l - k n o w n r e s u l t t h a t ψi s c o n s t a n t o n a s t r e a m l i n e . I t i s e a s i l y d e m o n s t r a t e d t h a t ψ(r,θ)=A

1+mr

1+m

sin[(1+m) ( π−θ) ] ( 5 . 1 1 1 )

Incompressible Inviscid Fluid Dynamics 95

y

xα π

Figure 5.12: Inviscid flow past a wedge.

is a solution of (5.87). Moreover, this solution satisfies the boundary conditions provided (1 + m) (1 − α/2) = 1, or

m =α

2 − α. (5.112)

Since, as is well-known, the solutions to Laplace’s equation (for problems with well-posed boundary conditions) areunique, we can be sure that (5.111) is the correct solution to the problem under investigation. According to thissolution, the tangential velocity on the surface of the wedge is given by

vt(r) = A rm, (5.113)

where m ≥ 0. Note that the tangential velocity is zero at the apex of the wedge. Since the normal velocity is also zeroat this point, we conclude that the apex is a stagnation point of the flow. Figure 5.13 shows the streamlines of the flowfor the case α = 1/2.

5.15 Inviscid Flow Over a Semi-Infinite Wedge

Consider the situation illustrated in Figure 5.14 in which an incompressible irrotational fluid flows over an impene-trable rigid wedge whose apex subtends an angle απ. Let the cross-section of the wedge in the x-y plane be bothz-independent and symmetric about the y-axis. Furthermore, let the apex of the wedge lie at x = y = 0. Finally, let theupstream flow a large distance from the wedge be parallel to the x-axis.

Since the flow is two-dimensional, incompressible, and irrotational, it can be represented in terms of a streamfunction that satisfies Laplace’s equation. The boundary conditions on the stream function are

ψ (r, [3 − α] π/2) = ψ (r,−[1 − α] π/2) = 0. (5.114)

These boundary conditions ensure that the normal velocity at the surface of the wedge is zero. It is easily demonstratedthat

ψ(r, θ) = −A

1 − mr1−m cos [(1 − m) (θ − π/2)] (5.115)

is a solution of Laplace’s equation, (5.87). Moreover, this solution satisfies the boundary conditions provided that(1 − m) (1 − α/2) = 1/2, or

m =α′

1 + α′, (5.116)

96 FLUID MECHANICS

−3−2−10123y−3−2−10 1 2 3 xFigure 5.13:S t r e a m l i n e s o f i n v i s c i d i n c o m p r e s s i b l e i r r o t a t i o n a l flo w p a s t a 9 0 ◦wedge.xy

α π α′πFigure 5.14:I n v i s c i d flo w o v e r a w e d g e .

Incompressible Inviscid Fluid Dynamics 97

−3−2−10123y

−3−2−1 0 1 2 3

xɧ ʊ ʈ e ɖ ɏ ɒ ɖ ɛ S trea mlin es of inviscidinco mp ressib leirrotatio na lflowoveraɚɑ◦we d ge.

ʘ ʆ eα′=ɒ−αɏɴʊ ʄeʕ e ʔʐl ʊ ʔʅʐɭ ʑl ʄe’ʔ ʆʒ ʅʊ ʂ e ʊʒ ʘʆ ʏ ʈ ʊ ʃʆʔ e ʅʉʂʅ ɉɖɏɒɒɖɊʊʔ ʅʉʆ ʀ beʄd

ʔʐʍʅɿ ʅ ʅe ʑb _b ʍ ʆ ʅʆ ʊ ʆ ʕʂʅ ɏ ɢʄʄ_ dʊʈʅ ʅiʔʐʍʅ ɍʅʉʆʅʂ ʈʆ ʅʊʂʍʗʆʍʐʄʊʕʚ ʅʉʆʄ fʂ e ʉʆ

ʘʅʈʆ ʊʈ ʗʆɿ ʃʚ

vtɉrɊ=A r −m,ɉɖɏɒɒɘɊ

ʘʆem≥ɑɏɯʕʆʕʉʂʅʕʉʆʕʂʈʆʅʊʂʍʗʆʍʐʄʊʕʚɍ ɿʅ eʄe ʕʉʆ˾ʀʘ ʄʑeɍ ʔʏ˽ɿʊʕʆ ʂʅʅʉʆʑeʙ ʀʇ ʅʉʆ ʘʆʅʈʆɩʀʘʆʗʆɍ

ʅʉʔʏʈʓʕʚʏʅʉ ˾ʀʘʄʂʏʃ ʆʎʊʂʅʆʚʄʈʉʅʍʚʃ_eʅʊʏʈʅʉ ʂʑeʙ

ɏɧʊʈʆʃʆɖɏɒɖ ʄʉʀʈʔ ʅʉʆʄʅʓeʂʎʍʊʆ ʀʇʅʉʆ

ʈʇʀ ʅʉʆ ʄʂʔʆα=ɒ/ɓɏ ɢ ʔ ʘ ʆ ʉ ʂ ʗ ʆ ʄ ʆ ɍ ʅ ʘ ʊ ʎ ʆ ʄ ʊ ʐ ʂ ʍ ʆ ʐ ʄ ʊ ʅ ʚ ˽ ʆ ʍ ʅ ʊ ɿ ʘ ʉ ʊ ʄ ʉ ʕ ʉ ʆ ˾ ʀ ʘ ʊ ʄ ʆ ʗ ʆ ʓ ʚ ʘ ʉ

ʆeʓʍʍʆʍʕʀ ʅexɎyʑlɿʆ ʂʅ ʕee

ʊɿʀʓʂʅʊɿʂʍɿʅʉezɎʅbeʄdʊʅʂʌʆʔʅʉʆʇʎ=vxɉx,y,tɊx

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ɴ

ʆ

ʅ

ʊ

ɖ

ɏ

ɘ

ɏ

Ɋ

ɩ

e

ʄ

e

ɍ

v

x

=

−

∂

φ

∂x

,ɉɖɏɒɒɚɊvy=−

∂φ

∂y

.ɉɖɏɒɓɑɊɰɿʅʉ ʆ ʐ ʅʉ ʆ ʉ ɿ ʅ ɍʊ ʅʉ ʆ ˾ʐʘʊin c o mp re ssib le

ʅʉʆ ∇· =ɑʊʔʂʖ ʅʐ ɾʂʕʊʄʂʍʍʚ ʄʂʅʊ ˽ʆ ʊʈʓʊʅʊ ʈ =∇z×∇ψɍʘʉʆʃʆ

ψɉx,y,tɊ ʊ ʄ ʅ ʆ ʓ ʎ ʆ ʅ ʕ ʉ stream function

ɏɉɴʆ ɴʆʅʊ ɖɏəɏɊ ɩeʄeɍvx=−

∂ψ

∂y

,ɉɖɏɒɓɒɊvy=

∂ψ

∂x

.ɉɖɏɒɓɓɊ