Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)

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50 2 Conservation Equations in Continuum Mechanics



(

2.3.7)

where

T is the skew-symmetric part of the tensor T derived from the fol-

lowing relationship



TTTTTTT  

(

2.3.8)

where

T is the symmetric part of the tensor T .

Equation (2.3.6) implies the fact that A is the vector of the tensor T ,

indicating that A can be obtained from the tensor T and conversely

can be found from the pseudovector A as follows

T:İ A

(

2.3.9)

and

A İT

(

2.3.10)

Employing Eqs. (2.3.3) and (2.3.5) to the integral equation of Eq. (2.3.2),

we have the resultant integral equation

³³

u

x Tේ

(2.3.11)

The volume integral vanishes identically since the volume is arbi-

trary, so that

u T

(2.3.12)

The left hand side vanishes identically by Cauchy’s equation of motion

(see given by Eq. (2.2.6)), the conservation law of linear momentum, con-

sequently, we reach the conclusion that A =0. This implies from Eq.

(2.3.10), that the skew-symmetric part of the stress tensor T vanishes, so

that the stress is written

TT or

jiij

(2.3.13)

2.3 Angular Momentum Conservation 51

Considering the angular momentum of the linear momentum equation

(Cauchy’s equation of motion), for a non-polar fluid it can be concluded

that the stress tensor is symmetric. If, in other words, the stress tensor of a

continuum medium in motion is symmetric, the angular momentum of a

linear momentum equation is always conserved so that the motion of fluid

can only be determined by the linear momentum equation.

In treating a polar fluid, however, an angular momentum due to a long-

range force may be exerted on a fluid particle, and likewise for the body

force per unit mass from distant surroundings. As displayed schematically

in Fig. 2.4, for example, the extra angular momentum we introduce to a

polar fluid may be a body couple f

in addition to the body torque

, i.e. f per unit mass. In similar fashion, a surface couple



per

unit surface may also be introduced to the surface of a fluid particle, as a

surface traction couple, due to a short-range force, and likewise for the sur-

face torque



tx u . The total angular momentum L of a fluid particle is

considered in a certain way that L may consist of the sum of the moment

of linear momentum ux

u per unit mass and an internal angular momen-

tum (or intrinsic angular momentum s ) per unit mass, which accounts for

the local spin field of a material element. Thus, in consideration of the bal-

ance of total angular momentum, we have



 



³³

uu

u

dSdV

Ctxfgx

sux

(

2.3.14)

In Eq. (2.3.14),



t can be given by Cauchy’s stress formula, as seen earlier

in Eq. (1.6.8), i.e.



 nt

. Analogously,



C can also be found by a

similar expression given below, since



C arises from diffusive transport

of internal angular momentum where

c is called the couple stress tensor.



 nC

(2.3.15)

Introducing Eqs. (1.6.8) and (2.3.15) to Eq. (2.3.14) yields, after tensor

calculus likewise deriving Eq. (2.3.5), we have



^`

uu

u

Tc xgxf

sux

(

2.3.16)

52 2 Conservation Equations in Continuum Mechanics

Applying the Reynolds’ transport theorem and the continuity equation to

the left hand side of Eq. (2.3.16), and vanishing the volume integration due

to an arbitrary volume, we have

 

Axgxfsux uu u Tc

UUU

(

2.3.17)

Equation (2.3.17) is an equation of the total angular momentum of general

form for polar fluids. In order to exploit the conservation of angular mo-

mentum, Eq. (2.3.17) can be further reduced to simpler form with the fol-

lowing procedure. That is, first taking the vector product of

x to

Cauchy’s equation of motion, we can obtain

Tu u ේxgx

(

2.3.18)

and then using the relationship



DtDDtD uxux u

, Eq. (2.3.17) can

be reduced to the following expression after subtracting Eq. (2.3.18) from

(2.3.17) as follows

 cේ

(

2.3.19)

This is a resultant equation for the internal angular momentum for a polar

fluid.

In the case of a polar fluid, the skew-symmetric part of the stress ten-

sor

T from Eq. (2.3.10) would be generated by an effect of the body cou-

ple

and the diffusion of the surface couple c



 to the net change of

the internal angular momentum



DtDs

. Equation (2.3.19) is a non-

conservation form. The conservation form of the equation for the internal

angular momentum can be reduced to the following form, after the mass

conservation is taken in account, so as to yield the following, where

is called the spin flux.

Afus

 

)(

(

2.3.20)

2.4 Energy Conservation

The energy conservation of a continuum medium can be considered from the

first law of thermodynamics, when the law is applied to the thermodynamic

2.4 Energy Conservation 53

system of a particle. The first law of thermodynamics in a dynamic system

implies the conservation of thermal energy and work, which means that

WQukd

  )(

(2.4.1)

where W

and Q

are the work done by the system, and the heat sup-

plied to the system respectively.

and

in Eq. (2.4.1) are the kinetic en-

ergy and the internal energy of the system respectively. When the system is

at equilibrium, Eq. (2.4.1) can be written with a unit of power by

ukd







)(

(2.4.2)

where W



and



are the work output by the system and the heat input to

the system respectively.

In consideration of the first law of thermodynamics as applied to a sys-

tem of a certain fluid particle as depicted in Fig. 2.5, we may be able to ob-

tain the work output



in the first place, taking a dot product to the forces

of the fluid particle as follows



³³



dVdSW ugut



(

2.4.3)

The first term of the right hand side is the work output by a surface force

and the second term is the work output by a body force. While applying a

dot product of

to Cauchy’s equation of motion from Eq. (2.2.6), we can

have the following expression

uguu

ugu





ේ)(ේ

)ේ(

:TT

(

2.4.4)

Equation (2.4.4) can also be written by a volume integral form as

dVdV

³³





uuu

)(

ේ)(

(

2.4.5)

54 2 Conservation Equations in Continuum Mechanics

Fig. 2.5 The first law of thermodynamics to a fluid particle

Denoting Cauchy’s stress formula from Eq. (1.6.8), i.e.



 nt

and

applying Gauss’ divergence theorem to the surface integral in Eq. (2.4.3),

we can write the work output W





dVdVW

ugu 

³³



(2.4.6)

Thus, equating the right hand side of Eq. (2.4.5) with Eq. (2.4.6), we can

newly express the work output





dVdV

³³

 uuu ේ



(

2.4.7)

Equation (2.4.7) indicates that the work output of a system of a fluid parti-

cle can be divided into two parts; the change of kinetic energy and the rate

at which the internal stresses do work.

Let us examine further the scenario where

T is not symmetric. Denot-

ing again



TTT 

in Eq. (2.3.8), and Ȧe  uේ in Eq. (1.1.18), the

work output due to the internal stress can be written as



Ȧ:Te:T

Ȧe:TT:T





ේ u

(

2.4.8)

2.4 Energy Conservation 55

second term of Eq. (2.4.8) from Eq. (1.1.29) for the spin tensor

Ȧee 



ˆˆ

(

2.4.9)

Furthermore,

T as given by Eq. (2.3.10), utilizing these identities, we can

reduce the term to

AȦ

ȦA



pkkp

pkijpijk

ZHH

İİȦ:T ᧶

(

2.4.10)

As a result, Eq. (2.4.10) indicates that the skew-symmetric part of the

stress tensor does produce an output work, owing to the vorticity. However,

as easily demonstrated when the stress is symmetric, the work output due

to the internal stress is simply shown by the deformation as

e:TT

uේ:

(

2.4.11)

The heat input of the system of a fluid particle is conceived to consist of

heat transferred to the system through the surface and heat generated in the

system, so that



can be written as

³³



bdVdQ



(

2.4.12)

Here, q is the heat flux vector; the negative sign is assigned toward the

surface, i.e. opposite to the surface direction

. Moreover, b is the

amount of heat generated per unit mass in the system. Equation (2.4.12)

can be converted into a volume integral by applying Gauss’ divergence

theorem as follows





dVbQ



(

2.4.13)

The total change of the system energy expressed with

and

in Eq.

(2.4.2) can be written as

Thus, we have decomposed u into the symmetric and skew-symmetric

parts so that other products vanish identically. Let us further consider the

56 2 Conservation Equations in Continuum Mechanics



dVu

ukd

෾





(

2.4.14)

As we can see,

is the internal energy per unit mass. The energy pos-

sessed by the system may include spin energy if the continuum has an in-

ternal structure and field energy derived from an externally imposed filed,

depending upon the circumstance and property of the continuum in motion.

Generally speaking, within the continuum mechanics we write the energy

equation of a fluid, by substituting Eqs. (2.4.7) and (2.4.13) together with

Eq. (2.4.14) into Eq. (2.4.2), we can obtain the equation of the energy con-

servation as



dVbdVu

෾෾

ේේ

 qu᧩:T

(

2.4.15)

Note that the power W



is chosen as to the work input to the system in Eq.

(2.4.15), where the sign of plus is assigned. After vanishing the volume in-

tegral from both sides of equation (2.4.15) for an arbitrary volume and

with Reynolds’ transport theorem to the right hand side of Eq. (2.4.15), we

can obtain the resultant equation to yield



 

uq ේ:ේේ T᧩

(2.4.16)

This is the conservation equation of energy, which is called the Neumann

energy equation in the conservation form. The equation can also be re-

duced to the non-conservation form, as practiced previously by consider-

ing the equation of mass continuity, which yield the form

 uq ේ:ේ T᧩

(

2.4.17)

The Neumann energy equation given by Eq. (2.4.16) or (2.4.17) is an ex-

pression derived from the first law of thermodynamics. The equations con-

tain thermodynamic properties, such as

u and

, so that the equations can

be further expanded thermodynamically in order to define the state of con-

tinuum undergoing thermal process.

2.5 Thermodynamic Relations

The state of a thermodynamic system can be determined by its thermody-

namic properties, which are connected by its relationship to the general term

2.5 Thermodynamic Relations 57

0 ),,( Tvpf

(

2.5.1)

As such,

is the thermodynamic pressure, or simply the pressure,

the specific volume

1 v

and

is the absolute temperature. Equation

(2.5.1) is called the equation of state, where its functional form depends

upon the state of the thermodynamic properties of the substance contained

in the system. Any one of the three variables in Eq. (2.5.1) can be ex-

pressed as a function of the other two by solving Eq. (2.5.1). This means

that the thermodynamic state is completely determined by two remaining

thermodynamic properties. An important concept to note here is the state

of equilibrium, which we can determine through the thermodynamic state

from Eq. (2.5.1). The state of equilibrium is that property which does not

vary over time when the external conditions remain unchanged.

In some situations, when a continuum is in motion with chemical

reaction, a relaxation process or in a large temperature gradient, that is a

process that results in the inability of the system to reach the state of

equilibrium in the time available, some processes have to be considered by

the states of non-equilibrium. However, the majority of processes in

engineering fluid mechanics are in the state of equilibrium, and the system

undergoes the reversible process where the process is connected only

between those initial and final states which are states of equilibrium.

As introduced in Eq. (2.4.1), the first law of thermodynamics in a dy-

namic system of a continuum, the internal energy

u can be regarded as in-

dependent of the kinematics of the motion of flow in the limit of the equi-

librium thermodynamics (thermostatics) as follows

WQdu



(2.5.2)

The first law of thermodynamics, demonstrated by Eq. (2.5.2), gives the

conservation of energy in quantity, but does not have any information on

the quality of the energy. The work done by the system

and the heat

supplied to the system

are not thermodynamic properties, which can-

not be determined by being given two equilibrium states between a trans-

formation process. However,

may be determined by a known reversi-

ble process of work transfer, considering

and

at two given

equilibrium points of states as follows

pdvW

(2.5.3)

It is the

that can not be determined by any other known thermody-

namic properties, but only by the thermodynamic property

s, the entropy.

The second law of thermodynamics gives a corollary that there exists a

58 2 Conservation Equations in Continuum Mechanics

thermodynamic property of a system such that a change in its value from

state 1 to 2 is equal to



(

2.5.4)

For any reversible process,

can be written by the change (the differen-

tiation) of the entropy as

TdsQ

(2.5.5)

Thus, Eq. (2.5.2) can be written with Eqs. (2.5.3) and (2.5.5) as follows

pdvTdsdu 

(2.5.6)



pdTdsdu

(2.5.7)

Obtaining a new thermodynamic property s, we have the following

thermodynamic relationship between the thermodynamic properties of

sTvp ,,, :



෕

(

2.5.8)

෕

(2.5.9)

෕

(2.5.10)



෕

(2.5.11)

Equations (2.5.8) to (2.5.11) are called the Maxwell equations, which

form the basis for obtaining further important thermodynamic relationships

which may be utilized for evaluation of thermal properties of continuum

substance.

2.5 Thermodynamic Relations 59

Among others, an important thermal property is the specific heat,

which is a quantity that gives the heat supplied to the system when the

temperature difference is given, so that

cdTQ

(2.5.12)

where c is the specific heat. Substituting Eq. (2.5.12) to the first law of

thermodynamics Eq. (2.5.2) and denoting

pdvW

from Eq. (2.5.3), we

have the following relationships;

pdvducdT 

(2.5.13)

vdpdhcdT 

(

2.5.14)

Such that h is defined as

pvuh 



(

2.5.15)

In Eq. (2.5.15) h is the enthalpy per unit mass, which is a specific energy

function. From Eqs. (2.5.13) and (2.5.14), therefore, we can obtain two

kinds of specific heat:

(2.5.16)

and

(

2.5.17)

denotes the specific heat evaluated at constant volume (constant den-

sity) and

denotes the specific heat evaluated at constant pressure.

Considering the thermodynamic relations, we are now in position to

expand the Neumann energy equation of Eq. (2.4.17) by decomposing the

total stress tensor

T into Ĳ and

as described in Eq. (1.6.13). Denoting

that

in Eq. (1.6.13) is regarded as the thermodynamic pressure in the

state of equilibrium, so that Eq. (2.4.17) becomes

 uĲuq ᧶

(

2.5.18)