Vorticity and Vortex Dynamics 2011 Part 15 ppsx

A.12a,bObviously, we also have A.2.1 Generalized Gauss Theorem and Stokes Theorem First, let V be a volume, having closed boundary surface ∂V with outward unit normal vector n and ◦ deno

698 A Vectors, Tensors, and Their Operations

The particular importance of the permutation tensor in vorticity andvortex dynamics lies in the fact that

ω i = ijk u k,j = ijk Ω jk , (A.11a)due to (A.3) and (A.6) Inversely, by using (A.9b) it is easily seen that

Ω jk= 1

This pair of intimate relations between vorticity vector and spin tensor showthat they have the same nonzero components and hence can represent, or aredual to, each other Note that (A.11b) also indicates that the inner product of

a vector and an antisymmetric tensor can always be conveniently expressed asthe cross-product of the former and the dual vector of the latter; for instance,

a · Ω = 1

2ω × a, Ω · a = 1

2a × ω. (A.12a,b)Obviously, we also have

A.2.1 Generalized Gauss Theorem and Stokes Theorem

First, let V be a volume, having closed boundary surface ∂V with outward

unit normal vector n and ◦ denote any permissible differential operation of the

gradient operator ∇ on a tensor F of any rank Then the generalized Gauss theorem states that ∇ ◦ F dV must be a total differentiation, and its integral

can be cast to the surface integral of n ◦ F dS over the boundary surface ∂V

of V , where n is the unit outward normal vector:



∇ ◦ F dV =



In particular, ifF is constant, (A.14) yields a well-known result

i.e., the integral of vectorial surface element dS = n dS over a closed surface

must vanish Moreover, for the total dilatation and vorticity in V we have

One often needs to consider integrals in two-dimensional flow In this case

the volume V can be considered as a deck on the flow plane of unit thickness.

Then (A.14) and some of the volume-integral formulas below remain the same

in both two and three dimensions, but care is necessary since in n-dimensions

δ ii = n is n-dependent Some formulas for n = 3 need to be revised or do not

exist at all; see Sect A.2.4 for issues special in two dimensions

Next, let S be a surface with unit normal n, then without leaving S only

tangential derivatives can be performed and have chance to be integrated out,

expressed by line integrals over the boundary loop ∂S The tangent differential

operator is naturally n × ∇, and the line element of ∂S has an intrinsic

direction along its tangent, dx = t ds, where t is the tangent unit vector and ds the arc element The directions of the normal n of S and t obey the

right-hand rule Then, as the counterpart of (A.14), the generalized Stokes

theorem states that on any open surface S any (n × ∇) ◦ FdS must be the

total differentiation, and its surface integral can be cast to the line integral of

Thus, if F is constant, (A.17) shows that the integral of element dx over a

closed line must vanish

700 A Vectors, Tensors, and Their Operations

with (A.15) as its special case since a closed surface has no boundary Ingeneral, (A.17) implies



S

(n × ∇) ◦ F dS = 0 on closed S. (A.20)

The most familiar application of (A.17) to fluid mechanics is the relation

between total vorticity flux through a surface and circulation along the

boun-dary of the surface Since (n × ∇) · u = n · (∇ × u), there is

A.2.2 Derivative Moment Transformation on Volume

The Gauss and Stokes theorems permit the construction of useful identities forintegration by parts In particular, we need to generalize the one-dimensional

to various integrals of a vector f over a volume or surface, so that they are cast to the integrals of proper moments of the derivatives of f plus bound-

ary integrals We call this type of transformations the derivative moment transformation.

We first use the generalized Gauss theorem (A.14) to cast the volume

integral of f to the moments of its divergence and curl Since

(f i x j),i = f j + f i,i x j ,

 ijk  jlm(fm x k),l = ijk(jlm f m,l)xk + ijk  jkm f m ,

where x is the position vector, by (A.14) we find a pair of vector identities

flow, (A.22) still holds if f is on the plane (e.g., velocity), but becomes trivial

if f is normal to the plane (e.g., vorticity).2

2

This can be verified by considering a deck-like volume of unit thickness Whenthe vector is normal to the deck plane, one finds 0 = 0 from (A.22)

Then, we need to cast the first vector moment x ×f to the second moments

of its curl, say F = ∇ × f When n = 3, F has three second moments x2F ,

If we make a Helmholtz–Hodge decomposition f = f ⊥ + f , see (2.87) and

associated boundary conditions (2.98a) or (2.98b), then we can replace f by f 

on the right-hand side of (A.22) Namely, the integral of a vector is expressible solely by the derivative-moment integrals of its longitudinal part However,

(A.23) is not simply a counterpart of this result in terms of the transverse

part of the vector Rather, as long as n × f  = 0 on ∂V , a boundary coupling

with the longitudinal part must appear For some relevant discussions see Wuand Wu (1993)

A.2.3 Derivative Moment Transformation on Surface

By similar procedure, we may use the Stokes theorem (A.17) to cast surfaceintegrals of a vector to that of its corresponding derivative moments plusboundary line integrals To this end we first decompose the vector to a normal

vector φn and a tangent vector n ×A, since they obey different transformation

rules Then for the normal vector we find a surface-integral identity effective

702 A Vectors, Tensors, and Their Operations

(A.25) is also a special case of (A.23) with f = ∇φ Note that the

cross-product on the right-hand side of (A.26) can be replaced by inner cross-product

Then, for both n = 2 and 3, the integral of the first moment x × nφ can

be transformed to the following alternative forms:

Here, since what matters in x ×(n×A) is only the tangent components of A,

we may well drop its normal component n k A k Hence, subtracting the second

identity from 1/2 times the first yields an integral of x ×(n×A) Using (A.17)

to cast the left-hand side to line integral then leads to the desired identity

is a tensor depending on x only.

As a general comment of derivative moments, we note that, since in(A.22),(A.23), and (A.25)–(A.27) the left-hand side is independent of the

choice of the origin of x, so must be the right-hand side In general, if I

represents any integral operator (over volume or surface or a sum of both),than the above independence requires

I{(x0+ x) ◦ F} = I{x ◦ F}

for any constant vector x0 Thus, we have

x0◦ I{F} + I{x ◦ F} = I{x ◦ F}, which implies that, due to the arbitrariness of x0, there must be

Namely, if we remove x from the right-hand side of (A.22), (A.23), and (A.25)–

(A.27), the remaining integrals must vanish It is easily seen that this condition

is precisely the Gauss and Stokes theorem themselves

A.2.4 Special Issues in Two Dimensions

The preceding integral theorems and identities are mainly for three-dimensionaldomain, with some of them also applicable to two-dimensional domain A fewspecial issues in two dimensions are worth discussing separately

In many two-dimensional problems it is convenient to convert a plane

vector aex +bey to a complex number z = x+iy by replacing ez × by i = √ −1

(Milne-Thomson 1968), so that

e y = e z × e x=⇒ ie x (A.31a)and hence

ae x + bey = (a + bez ×)e x=⇒ e x(a + ib). (A.31b)

Then the immaterial ex can be dropped Thus, denoting the complex

conju-gate of z by ¯ z = x − iy, for derivatives there is

∂ x = ∂ z + ∂¯, ∂ y = i(∂ z − ∂¯), (A.32a)

2∂z = ∂x − i∂ y , 2∂¯= ∂x + i∂y , (A.32b)

so by (A.31)

∇ =⇒ 2e x ∂¯, ∇2=⇒ 4∂ z ∂¯. (A.33)The replacement rule (A.31) cannot be extended to tensors of higher ranks

If in a vector equation one encounters the inner product of a tensor S and a

vector a that yields a vector b, then (A.31) can be applied after b is obtained

by common real operations For example, consider the inner product of atrace-free symmetric tensor

S = e x e x S11+1

2(ex e y + ey e x)S12+ ey e y S22, S11+ S22= 0,

704 A Vectors, Tensors, and Their Operations

and a vector a = e x a1+ e y a2=⇒ e x (a1+ ia2) After obtaining a · S = S · a

by real algebra, we use (A.31) to obtain

the second formula being the complex conjugate of the first Milne-Thomson

(1968) calls this result the area theorem.

The two-dimensional version of the derivative-moment transformation onsurface, i.e., the counterpart of (A.25) and (A.26), also needs special care

We proceed on the real (x, y)-plane Let C be an open plane curve with end

points a and b, and es and n be the unit tangent and normal vectors along

C so that (n, e s .e z) form a right-hand orthonormal triad Then since

simply leads to a trivial result What we can find is only a scalar moment

A.3 Curvilinear Frames on Lines and Surfaces

In the above development we only encountered Cartesian components ofvectors and tensors In some situations curvilinear coordinates are more conve-nient, especially when they are orthonormal Basic knowledge of vector analy-sis in a three-dimensional orthonormal curvilinear coordinate system can befound in most relevant text books (e.g., Batchelor 1967), where the coordi-nate lines are the intersections of a set of triply orthogonal surfaces But,

the Dupin theorem (e.g., Weatherburn 1961) of differential geometry requires that in such a system the curves of intersection of every two surfaces must

be the lines of principal curvature on each While the concept of principal curvatures of a surface will be explained later, here we just notice that the theorem excludes the possibility of studying flow quantities on an arbitrary

curved line or surface and in its neighborhood by a three-dimensional normal curvilinear coordinate system These lines and surfaces, however, areour main concern Therefore, in what follows we construct local coordinateframes along a single line or surface only and as intrinsic as possible, with anarbitrarily moving origin thereon

ortho-A.3.1 Intrinsic Line Frame

If we are interested in the flow behavior along a smooth line C with length element ds, say a streamline or a vorticity line, the intrinsic coordinate frame

with origin O(x) on C has three orthonormal basis vectors: the tangent vector

t = ∂x/∂s, the principal normal n (toward the center of curvature), and the

binormal b = t × n, see Fig A.2 This (t, n, b) frame can continuously move

along C and is known as intrinsic line frame The key of using this frame is to know how the basis vectors change their directions as s varies This is given

by the Frenet–Serret formulas, which form the entire basis of spatial curve

theory in classical differential geometry (e.g., Aris 1962):

curve deviates from a plane curve, i.e., it is the curvature of the projection of

C onto the (n, b) plane For a plane curve τ = 0 and we have a (t, n) frame

as already used in deriving (A.36)–(A.38)

706 A Vectors, Tensors, and Their Operations

Fig A.2 Intrinsic triad along a curve

Now, let the differential distances from O along the directions of n and b

be dn and db, respectively Then

which involves curves along n and b directions that have their own curvature

and torsion Then one might apply the Frenet–Serret formulas to these curves

as well to complete the gradient operation But due to the Dupin theorem we

prefer to leave the two curves orthogonal to C undetermined.

For example, if C is a streamline such that u = qt, then the continuity

equation for incompressible flow reads

Therefore, the second term of ∇ × t must be along the t direction, with the

magnitude

ξ ≡ t · (∇ × t) = b · ∂t

∂n − n · ∂t

The scalar ξ is known as the torsion of neighboring vector lines (Truesdell

1954) Thus, using this notation we obtain

Thus, ξ0 if ω · u = 0 Note that in a three-dimensional orthonormal frame

there must be ξ ≡ 0, so by the Dupin theorem a curve with ξ = 0 cannot be

the principal curvature line of any orthogonally intersecting surfaces

A.3.2 Intrinsic operation with surface frame

Derivatives of tensors along a curved surface S can be made simple by an

intrinsic use of an intrinsic surface frame, which is more complicated than theintrinsic line frame since now there are two independent tangential directions

on S.

Covariant Frame

At a given time, a two-dimensional surface S in a three-dimensional space is

described by the position vector x of all points on S, which is a function of

two independent variables, say u α with α = 1, 2 Then

r α(u1, u2)≡ ∂x

define two nonparallel tangent vectors (not necessarily orthonormal) at each

point x ∈ S, see Fig A.3 Note that by convention when an upper index

appears in the denominator it implies a lower index in the numerator, and

708 A Vectors, Tensors, and Their Operations

Fig A.3 Covariant and contravariant frames on a surface

vise versa r α are used as the covariant tangent basis vectors characterized

by lower indices Their inner products

g αβ ≡ r α · r β , α, β = 1, 2, (A.47)form a 2× 2 matrix which gives the covariant components of the so-called

metric tensor and completely determine the feature of r α Note that the area

of the parallelogram spanned by r1and r2 is|r1× r2| = √g = det{g αβ }.

From rαone can obtain the unit normal vector

f = r α f α + f3n, since then f · r β = fβ would equal rα · r β f α = gαβ f α, which is possible only if gαβ = δαβ, but in general this is not the case Thus,

a covariant frame alone is insufficient ; one has to construct another frame

conjugate to it This is similar to the situation in complex domain, where

a complex basis-vector set, say a i, needs be complemented by its complex

conjugate a ∗ i with a i · a ∗

j = δ ij , such that if f · a i = f i then there should

be f = f i a ∗

We therefore introduce the second pair of tangent vectors r α with upperindex, by requiring the hybrid components of the metric tensor be a unitmatrix

Then obviously r α · r β = 0 for α = β, and by (A.48) r α · r β = 1 for α = β as

desired Moreover, by using identity

(a × b) × (c × d) = b[a · (c × d)] − a[b · (c × d)] (A.51)there is

710 A Vectors, Tensors, and Their Operations

This time the result is correct

f · r γ = f α δ α γ = f α , f · r γ = f β δ β γ = fγ Here we see a hybrid use of the two frames Accordingly, the summation

convention over the repeated index is implied only if one index is upper andthe other is lower

Moreover, by (A.57) we have

f α = r α · f = r α · r β f β = g αβ f β ,

f β = r β · f = r β · r α f α = g βα f α (A.58)

Thus, raising or lowering the index of a component can be achieved by using

the metric tensor g αβ or g αβ, respectively Actually, in (A.56) we have donethis for the metric tensor itself

A tensor of higher rank can be similarly decomposed by its repeated innerproducts with basis vectors Depending on the choice of basis vectors, thecomponents of a tensor of rank 2 can be covariant, contravariant, or hybrid

For example, if a tensor T has only tangent component, we can write

T = T αβ r α r β = T α ·β · r α r β = T ·β α · r α r β = T αβ r α r β ,

where the dots indicate the order of components which cannot be exchangedunless the tensor is symmetric with respect to relevant indices

Tangent Derivatives of a Vector

The most important differential operations on a surface S is the tangent

deriv-atives of vectors and tensors We split the gradient vector into

∇ = ∇ π + n ∂

∂n , ∇ π(·) = r α ∂( ·)

∂u α = r α(·) ,α , (A.59)

where a hybrid use has been made: r α is not created from u α but from u α

Now, for the tangent derivatives of a vector f , there is

∇ π f = r α (r β f β + fn n) ,α

= r α (r β ,α f β + r β f β,α + n,α f n + nfn,α).

Here, the derivatives of f β and f n are merely that of numbers; the key is the

derivatives of basis vectors r α , r β , and n These vectors are not coplaner, and

hence their derivatives with respect to u α can be expressed by their linear

combinations The result is given by the classic Gauss formulas

r β,α = Γ αβ λ r λ + bαβ n, r ,α β =−Γ β

αλ r λ + b β α n, Γ αβ λ = Γ βα λ , (A.60a,b,c)

and Weingarten formulas

n ,α=−b β

α r β=−b αβ r β , b αβ = bβα = b γ β g αγ (A.61a,b)

The six coefficients Γ αβ λ = Γ βα λ in (A.60) are called Christoffel symbols of the

second kind They do not form a tensor since r α are artificially chosen In

contrast, three coefficients b αβ = g γα b α

β = b βα are covariant components of

the symmetric curvature tensor, defined by the intrinsic operation

K≡ −∇ π n = −r α n ,α = r α r β b β α = r α r β b αβ (A.62)

Intrinsic Operation on Surface

In view of the complicated involvement of nontensorial Γ λ

αβin the derivatives

of r α or rβ, it is desired to bypass these operations We call this kind of

oper-ation intrinsic operoper-ation To start, we look at some more intrinsic properties

of the normal vector n First, the mean curvature κ is defined as half of the

two-dimensional divergence

2κ ≡ −∇ π · n = b β

α δ α β = g αβ b αβ = b α (A.63)This result directly comes from the contraction of (A.62 Second, by (A.61a,b)there is

∇π× n = r α × n ,α=−b αβ r α × r β = 0. (A.64)Finally, (A.61a) implies

From the derivation of (A.64 we can already see the main feature of the

intrinsic operation: (1) The operation is in vector form rather than component form; and (2) the tangent basis vectors are carried along in the operation but their derivatives do not appear The final result will be automatically free from

these vectors The following examples further show the strategy

First, we compute the jump of normal vorticity across a vortex sheet

[[ω n ]] = n · [[∇ × u]] = (n × ∇) · (n × γ), where γ is the sheet strength Using the surface frame, we have

[[ωn]] = (n × r α)· (n × γ ,α + n,α × γ),

which amounts to common vector algebra Thus, by (A.55),

[[ω n ]] = (n · n)(r α · γ ,α)− (n · γ ,α )(r α · n) +(n · n )(r α · γ) − (n · γ)(r α · n ),

712 A Vectors, Tensors, and Their Operations

where n · γ = 0, r α · n = 0, and we have (A.65) Thus, we simply obtain

(Saffman 1992; where no proof is given)

in which the first two terms vanish due to (A.65) and assumed feature of A.

The last term gives (A.67)

Third, (n × ∇) × A is a tangent-derivative operation and we develop it to

several fundamental constituents There is

(n × ∇) × A = (n × r α)× A ,α = r α (A,α · n) − n(r α · A ,α)

The second term comes from the tangent divergence of A, while the first term must be related to the normal component of A; but even if An= 0, the surfacecurvature may cause∇πA to have normal components In fact, we can further

split the first term of (A.68) to

of which the proof requires the knowledge of Γ λ

αβ that is beyond our presentconcern

Orthonormal Surface Frame

If in the above surface-moving frame the covariant tangent basis vectors r1and r2are already orthogonal, then by (A.50) there is r α = r α / √

g, implying

that there is no need to distinguish covariant and contravariant tangent basisvectors Thus, like Cartesian tensors, it suffices to use one pair of tangentbasis vectors and denote the components by subscripts only

The orthogonality of basis vector implies that

g ii = h2i δ ii (no summation with respect to i),

where

h i=√

g ii=√

r i · r i (no summation with respect to i) (A.71)

is the length of ri, called scale coefficients or Lam´ e coefficients They are still

functions of the moving point x Nevertheless, we can now introduce a set of

orthonormal basis vectors

e α= r α

h α

(no summation with respect to α), e3= n, (A.72)

which form an orthonormal surface frame For a given curved surface S, hα

depends on its intrinsic geometric feature as well as the orientation of rα, the

latter can be arbitrarily chosen, not restricted by the Dupin theorem as long

as the frame is only defined on a single surface S.

With the ei frame and using the notation

714 A Vectors, Tensors, and Their Operations

Here, b αβ = b βα (α, β = 1, 2) are redefined by

b αβ = e α · K · e β or K = b αβ e α e β (A.75)

instead of (A.62) From bαβ one can construct two intrinsic scalar curvatures

which describe the wall geometry and are independent of the choice of e1and

the total curvature Moreover, except some isolated points a curved surface

has a pair of orthogonal principal directions If e1and e2 coincide with these

directions then bαβ = 0 for α = β In this case b11 = K1 and b22 = K2

are the principal curvatures, which are the greatest and least bαβ among all

orientations of the tangent vectors The total curvature is simply K = K1K2

On a sphere any tangent direction is a principal direction

It is convenient to express bαβby principal curvatures since the latter are

independent of the choice of (e1, e2) Denote the unit tangent vectors along

the principal directions by p1 and p2 (they define the curvature lines of thesurface), then by (A.75) there is

K = p1p1K1+ p2p2K2.

Thus, if (p1, p2, e3) form a right-handed frame and β is the angle by which

the (p1, p2) pair rotates to the (e1, e2) pair in counterclockwise sense, thereis

b11= K1cos2β + K2sin2β, (A.78a)

b12=−1

2(K1− K2) sin 2β, (A.78b)

b22= K1sin2β + K2cos2β. (A.78c)

Hence, for a given surface, b αβ depend solely on a single parameter β Because

reversing the direction of (p1, p2) does not affect K, without loss of generality

define a pair of on-surface curvatures of coordinate lines x1 and x2,

respec-tively (they are the geometric curvatures of these lines if K = 0) Therefore,

(A.74) can be written in a geometrically clearer form:

Note that since in (A.71) h α (α = 1, 2) are functions of x1and x2, operators

∂ α and ∂β for α = β are not commutative Instead, there is

(∂1− κ2)∂2= (∂2− κ1)∂1 or ∂1∂2− ∂2∂1= κ1∂1− κ2∂2, (A.81)

but we still have ∂α ∂3= ∂3∂ α.

The orthogonality of x1-lines and x2-lines implies that the variation of κ1

and κ2 are not independent In fact, by (A.81) and using (A.80), there is

∂2κ1= ∂2[(∂1e1)· e2]

= (∂1∂2e1)· e2+ (∂1e1)· (∂2e2) + κ21+ κ22

=−∂1κ2+ b11b22− b2

12+ κ21+ κ22, from which it follows a differential identity

∂2κ1+ ∂1κ2= κ21+ κ22+ K. (A.82)Finally, by using (A.80), the components of the gradient of any vector

f = e i f i in the (x1, x2, x3)-frame read

716 A Vectors, Tensors, and Their Operations

A.4 Applications in Lagrangian Description

Tensor analysis is necessary in studying the transformation of physical tities and equations between the physical space and the reference space (inLagrangian description) In this section, we present some materials which aredirectly cited in the main text

quan-A.4.1 Deformation Gradient Tensor and its Inverse

Consider the deformation gradient tensor and associated Jacobian in the

ref-erence space, defined by (2.3) and (2.4)

F =∇ X x or F αi = xi,α , (A.84)

 ijk J =  αβγ x i,α x j,β x k,γ (A.86b)

Next, keeping the labels of particles, any variation of J can only be caused by

that of x Using (A.86), an infinitesimal change of J is then given by

Then, owing to (2.8), (2.3) is invertible, and hence F has inverse tensor

defined in the physical space

F−1=∇X, or F −1

which satisfies F· F −1 = F−1 · F = I, i.e.,

x i,α X β,i = δαβ , X α,i x j,α = δij (A.89)

which has expressions symmetrical to (A.86a,b) For example, we have

 ijk J −1 =  αβγ X α,i X β,j X γ,k (A.91)Similar to (A.87), there is

δJ −1 = J −1 δXα,α = J −1 δXα,i x i,α = J −1 ∇ X · δX. (A.92)

The deformation tensor F and its inverse F−1 have one index in physicalspace and one in the reference space This property can be used to transform

a physical-space vector to its dual or image in the reference space, or vise

versa

A.4.2 Images of Physical Vectors in Reference Space

We are particularly interested in seeking the image of vorticity ω in the space To this end consider a general vector f first Multiply both sides of

X-(A.86b) by fk,j /J and notice that

The mapping between∇ × f and ∇ X × (F · f) is one-to-one, and has three

features: (1) they are identical at t = τ = 0; (2) if one vanishes, so must the

other; and (3) they are divergenceless in their respective spaces We therefore

identify the latter as the image of the former in X-space Besides, for dimensional vector field f = (f1, f2, 0) it can be shown that (A.94) is simplified

718 A Vectors, Tensors, and Their Operations

Then by (A.94), the image of ω is

which we call the Lagrangian vorticity By using (A.89) and mass conservation

(2.40), the transformation between ω and Ω is given by

and, if in addition the flow is incompressible we simply have Ω = ω.

Note that the images of u and ω have opposite structures In the former F

is used but in the latter, in addition to the factor J , it is F −1 This is related

to the fact that u is a true vector (polar vector) but ω is a pseudo-vector

(axial vector) One might switch the use of F and F−1 for true and pseudo

vectors; which however does not lead to useful result unless U is to be written

as the curl of another vector, which will be used once below

Once we introduced the Lagrangian vorticity, we may study vorticity

kine-matics in physical space by that of Ω in reference space First, since

is the image of∇×a Therefore, the rate of change of the image of the curl of

velocity (Lagrangian vorticity) equals the image of the curl of acceleration We

stress that this result is not true for Lagrangian velocity U and acceleration

A: in (A.101) there is an extra gradient term The implication of this difference

has been made clear in Sect 3.6

As an application of utilizing the vector images in reference space, weextend the content of Sect 3.6.2 by showing that the circulation preserving issufficient but not necessary for having a Bernoulli integral Rewrite (A.101)as

∂U

∂τ = A

where Ψ is defined in (3.152) and A
is the rotational part of A Similar to

the approach leading to (3.148), assume there exists a family of material

sur-faces defined by two parameters, say ξ1(X) and ξ2(X), such that the normal

∇ X ξ1× ∇ X ξ2 is along A

A
× (∇ X ξ1× ∇ X ξ2) = 0. (A.104)Then a Bernoulli integral like (3.155a) along these surfaces can be obtained

An important example is inviscid baroclinic flow without shock waves, for

which Ds/Dt = 0 or s = s(X) in the X-space Hence, any material

sur-faces are isotropic Note that the existence of these sursur-faces is ensured by the Crocco–Vazsonyi equation (2.163), in which the rotational term T ∇s is a

complex-lamellar vector field according to the definition in Sect 3.3.1 Then,

by using (A.96) there is

A
(X, τ ) = F · T ∇s,

which is normal to isentropic surfaces Therefore, on these surfaces we haveBernoulli integral (3.155) Correspondingly, those conservation theoremsinvolves material integrals can survive along these surfaces For example,instead of volume integrals (3.132) and (3.134) we may have a similar sur-face integrals; and the Kelvin circulation theorem (3.130c) will hold if theloopC is on one of such surfaces.

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