KUNDU Fluid Mechanics 2 Episode 3 ppt

In a small time interval, a small spherical element in this flow would become an ellipsoid oriented at 45" to the XI x2-coordinate system.. Using polar coordinates rr H, the velocity i

axis i 1 and compressed along the principal axis &

vorticity) The average value does not depend on which two mutually perpendicular

elements in the x1 x2-plane are chosen to compute it

In contrast, the components of strain rate do depend on the orientation of the

element From Eq (3.1 l), the strain rate tensor of an element such as ABCD, with

the sides parallel to the XI XZ-axes, is

0 i y o

e = [ i y 0 0 1 ,

which shows that there are only off-diagonal elements of e Therefore, the element

ABCD undergoes shear, but no normal strain As discussed in Chapter 2, Section 12

and Example 2.2, a symmetric tensor with zero diagonal elements can be diagonalized

by rotating the coordinate system through 45" It is shown there that, along these

principal axes (denoted by an overbar in Figure 3.13), the strain rate tensor is

e =

so that there is a linear extension rate of Z1 1 = y / 2 , a linear compression rate of

E= = - y / 2 , and no shear This can be understood physically by examining the

deformation of an element PQRS oriented at 45", which deforms to P'Q'R'S' It is

clear that the side PS elongates and the side PQ contracts, but the angles between the

sides of the clement remain 90' In a small time interval, a small spherical element in

this flow would become an ellipsoid oriented at 45" to the XI x2-coordinate system

Smmurizing, the element ABCD in a parallel shear flow undergoes only shear

but no normal strain, wherear the element PQRS undergoes only normal but no shear

strain Both of these elements rotate at the same angular velocity

1 I Kinetnacic: Conxideratiom OJ Vorlm Flows

Flows in circular paths arc callcd vot-texflows, some ba,ic forms of which are described

in what lollows

Solid-Body Rotation

Consider first the case in which the velocity is proportional to thc radius of the stream- lines Such a flow can be generated by steadily rotating a cylindrical tank containing

a viscous fluid and waiting until the transients die out Using polar coordinates (rr H),

the velocity in such a flow is

where 041 is a constant equal to thc angular vclocity of revolution of each particle

about the origin (Figure 3.14) We shall scc shortly that fN is also equal to the angular specd of roturion of each particle about its own center The vorticity cornponcnts of

a fluid clcment in polar coordinates are given in Appendix B The component about thc z-axis is

(3.28)

1 a 1 au,

r ar r aH

w: = (rue) - = 2 ~ ,

whcrc wc' havc used thc vclocity distribution cquation (3.27) This shows that the

angular velocity or each fluid element about its own centcr is a constant and qual

to wg This is evident in Figure 3.14, which shows the location af element ABCD at two succcssivc timcs It is sccn that thc two mutually perpcndicular Ruid lincs AD and AB both rotatc countcrclockwisc (about the center ofthe elcment) w i h speed q-,

Figure 3.14 Solid-hody rotation Muid clcmcnls arc spinning about thcir own ccnkrs while they revolvc around the origin There is no dcli)rmalion or the elements

66 Kinemutiwv

The time period for one mtation of the particle about its own center equals the time period for one revolution around the origin It is also clear that the deformation of the

fluid elements in this flow is zero, as each fluid particle retains its location relative

to other particles A flow defined by ue = w r is called a sok-body rotation as the

fluid elements behave as in a rigid, rotating solid

The circulation around a circuit of radius r in this flow is

2 I’ = s u = d s = 1” uerde = 2arus = 2nr 00: (3.29)

which shows that circulation equals vorticity 200 times area It is easy to show (Exercise 12) that this is true of m y contour in the fluid, regardless of whether or not it contains the center

Irrotational vortex

Circular streamlines, however, do not imply that a flow should have vorticity every- where Consider the flow around circular paths in which the velocity vector is tan- gential and is inversely proportional to the radius of the streamline That is,

This shows that the vorticity is zero everywhere except at the origin, where it canuot

be determined from this expression However, the vorticity at the origin can be deter- mined by considering the circulation around a circuit enclosing the origin Around a contour of radius r , the circulation is

I’ = 6” uer d e = 2 a C

This shows that r is constant, independent of the radius (Compare this with the case

of solid-body rotation, for which Eq (3.29) shows that I‘ is proportional to r2.) In

fact, the circulation around a circuit of any shape that encloses the origin is ~ J c C

Now consider the implication of Stokes’ theorem

(3.31) for a contour enclosing the origin The left-hand side of Eq (3.31) is nonzero, which implies that o must be nonzero somewhere within the area enclosed by the contour Because r in this flow is independent of r , we can shrink the contour without altering

the left-hand side of Eq (3.31) In the limit the area approaches zero, so that the

vorticity at the origin must be infinite in order that o SA may have a finite nonzero limit at the origin We have therefore demonstrated that thcjhw represented by

u

e- r

Figure 3.15

where e: se

Irrotational vortex Vorticity of a Ruid element is iniinite at the origin and zero every-

ue = C / r is irrotutional everywhere except at th.e origin, where the vortici1.y is iqlinire Such a flow is called an imtatianul or potentiul vortex

Although the circulation around a circuit containing the origin in an irrotational

vortex is nonzero, that around a circuit not contaiajng the originis zero The circulation around any such conlour ABCD (Figure 3.15) is

Because thc linc intcgrals of u ds around BC and DA are 72~0, wc obtain

FAUCI) = -uor A0 + (ug + Auo)(r + Ar) A Q = 0,

where we have noted that thc line integral along AB is negative bccause u and ds arc oppositcly directed, and we have used ugr = const A zero circulation around ABCD is expected becausc of Stokes' theorem, and the fact that vorticity vanishes everywhere within ABCD

a behavior is called the Runkine vortex, in which the vorticity is assumed uniform

within a corc ol'radius R and zero outside the core (Figurc 3.16b)

Figure 3.16 Vclocity and vorticiy distributions in a rcal vortex and a Rankine v o r h : (a) real vorm;

(b) Rankine vortex

A truly one-dimensional Jlow is one in which all flow characteristics vary in one direction only Few real flows are strictly one dimensional Consider the flow in a

conduit (Figure 3.17a) The flow characteristics here vary both along the direction

of flow and over the cross section However, for somc purposes, the analysis can

be simpliiied by assuming that the flow variables are uniform over the cross section (Figure 3.1 7b) Such a simplification is called a one-dimensional approximation, and

is satisfactory if one is interested in the overall effects at a cross section

A t wo-dimensional or plane flow is one in which the variation of flow charac-

teristics occurs in two Cartesian directions only The flow past a cylinder of arbikary cross section and infinite length is an example of plane flow, (Note that in this contcxt the word “cylinder” is used for describing any body whosc shape is invariant along the length of the body It can have an arbitrary cross section A cylinder with a circlslar

The D/.; signifies that a specific fluid particle is followed, so thal the volume of a particle is inversely proportional to its density Subslituling 6 T o( p-’ , we obtain

(3.32)

This is called the C C J ~ Z U ~ Q equation because it assumes that the fluid flow has no voids in it; the name is somewhat mislcading because all laws of continuum mechanics makc this assumption

The density of Ruid particles docs not change appreciably along the fluid path

under certain conditions, the most importanl of which is that the flow spccd should be small compared with the spccd of sound in the medium This is callcd the Boussinesq approximation and is discussed in more detail in Chapter 4, Section 18 The condition holds in most flows of liquids, and in flows of gases in which the speeds are less than

two dimensional und steady (Exercisc 2).)

The streamlines of the flow are given by

which says that d @ = 0 along a streamline The instanlaneous streamlines in a flow

are therefore givcn by the curves @ = const., a different value of the constant giving

a different streamline (Figure 3.18)

Consider an arbitrary line element d x = ( d x , d y ) in the flow of Figure 3.18

Here we have shown a case in which both d x and d y are positive The volume rate

of flow across such a line element is

showing that the volume flow rate between a pair of streamlines is numerically equal

to the difference in their + values Thc sign oF $ is such that, facing the direction

of motion, II increases to the left This can also be seen h m the defmition equation

(3.34), according to which the dcrivative of @ in a certain direction gives the vclocity

Figure 3.18 Flow thmugh pair of streamlines

component in a direction 90" clockwise from the direction of differentiation This

requires that e in Figure 3.18 must increase downward if the flow is from right

The pair of simultaneous equations in u and u can be combined into a single equation

by defining a streamfunction, when the momentum equation (3.36) becomes

a+ a2$ a+a2$ a3$

ay a x a y ax ay2 = v- ay3

We now have a single unknown function and a single differential equation The continuity equation (3.37) has been satisfied automatically

Sum.m.arizing, a streamfunction can be defined whenever the continuity equation

consists of two m s The flow can otherwise be completely general, for example,

it can be rotational, viscous, and so on The lines $ = C are thc instantaneous streamlines, and the flow rate between two streamlines equals d @ This concept will

be generalized following our derivation of mass conservation in Chapter 4, Section 3

14 I'olur Cmrdinatca

It is sometimes easier to work with polar coordinates, especially in problems involv- ing circular boundaries In fact, we often select a coordinate system to conform to the shape of the body (boundary) It is customary to consult a reference source for expressions of various quantities in non-Cartesian coordinates, and this practice is

perfectly satisfactory However, it is good to know how an equation can be trans-

formed from Cartesian into other coordinates Here, we shall illustrate the procedure

by transforming the Laplace equation

to plane polar coordinates

Cartesian and polar coordinatcs are related by

(Z)* = (:)y (E)() + ( $ ) x ($)/

Omitting parentheses and subscripts, we obtain

(3.39)

Figure 3.19 shows that ug = vcose - u sine, so that Eq (3.39) implies a$/&-

= -u6 Similarly, we can show that a$/% = Tur Themfore, the polar velocity

components are related to the streamfunction by

This is in agreement with our previous observation that the derivative of $ gives the

velocity component in a direction 9 0 clockwise € o m the direction of differentiation

Now let us write the Laplace equation in polar coordinatcs The chain rule gives

Hpre 3.19 Relation d vclocily components in Cartesian and plane polar coordinates

Tn a similar manncr,

(3.41)

The addition of Eqs (3.40) and (3.41) leads to

which completes the transformation

2 Consider a steady axisymmetric flow of a compressible fluid The equation

of continuity in cylindrical coordinates (R, p: x) is

74 KkUWlUtXC#

Show how we can define a skamfunction so that the equation of continuity is satisfied automatically

3 Tf a velocity field given by u = ay, compute h e circulation around a circle of

radius r = 1 about the origin Check the result by using Stokes’ theorem

4 Consider a plane Couette flow of a viscous fluid confined between two flat plates at a distance b apart (see Figure 9.4~) At steady stale the velocity distribution is

u = U y / b li = w = 0,

where the upper plate at y = b is moving parallel to itself at speed U , and the lower plate is held stationary Find the rate of linear strain, the rate of shear strain, and vorticity Show that the streamfunction is given by

U Y 2

$ = + const

5 Show that thc vorticity for a plane flow on the xy-plane is given by

Using this expression, find the vorticity for the flow in Exercise 4

6 The velocity components in an unsteady plane flow are given by

Describe the path lines and the strcamlines Note that path lines are found by following the motion of each particle, that is, by solving the merentia1 equations

dxldt = u(x, t ) and dyldt = v ( x , t ) ,

subject lo x = q at t = 0

t h a t u o = U a t r = R

right-hand si& of Stokes’ theorem

7 Determine an expression for $ for a Rankine vortex (Figure 3 la), assuming

8 Take a planc polar elemcnt of fluid of dimensions dr and r de Evaluate the

and thcreby show that the expression for vorticity in polar coordinates is

Also, find the cxpressions for w, and we in polar coordinates in a similar manner

9 The velocity field of a certain flow is givcn by

2

u = z r 1 2 + 2xz2, 2: = x 2 y , w = x z

Consider the fluid region inside a spherical volume x 2 + y2 + z2 = u2 Verify the validity of Gauss’ theorem

by inlegrating over the sphere

10 Show that the vorticity field for any flow satisfies

v * w = o

1 1 A flow field on h e xy-plane has the velocity components

u = 3 x + y u = 2 x - 3 y

Show that the circulation around the circle (x - 1)2 + (y - 6)2 = 4 is 417

12 Consider the solid-body rotation

ue = q r u, =o

Take a polar element of dimension r d 0 and dr, and verify that the circulation is

vorticity times area (Tn Section 11 we performed such a verification for a circular

element surrounding the origin.)

13 Using thc indicial notation (and without using any vector identity) show that the acceleration of a fluid particle is given by

where q is the magnitude of velocity u and w is the vorticity

14 The definition of the streamfunction in vector notation is

Wxndtl, L and 0 C rietjens (1934) Fundurnmruls aJ’Hydm- and Aeromechanics, K e w York: Dovcr

Puhlications (Chqlcr V contains a simplc but useful treatmcnl or kinematics.)

Prandt.1, L and 0 G Ticljcns (1934) Applied Hydm- andAaromechanics, New York Dovcr Publications (This volumc contains classic photographs from Prandrs laboratory.)

Chapter 4

Conservation Laws

1 l r 1 ~ 1 1 c h n 76

2 ‘ h e lleril.ulhm ~ ~ h l u r n e lrikgmh 77

Cerierd ( b c 77

Fixed V’olii~nc 78

Matwid X’olurnc 78

3 (:onsercUtivri c$Mm.s 79

4 Stmin@mhns: Rciisitcxi m d C w i c n i l i d 8 1 5 Ol+’ri vf biirwrx in Fhd 82

6 Stnx.sa~alhbit 84

8 .W omnhrn Pririujil(?Ji)r (1 Ikcd 7 C o n w n w h n rfMomerilum 86

Ih!.urne 88

Examplc4.1 89

F h d K h n e 92

Kxamplt: 4.2 93

Fluid 94

Nori-R’cwtnnian Fluids 97

1 I :Vm*k4ok(!.!.u KqimLori 97

Cnmmrxitlls or1 the hcouii X:im 98

12 Ikituhg I k m e 99

L&xq of Coriolis Forc.c: 103

9 Angular :Mommtum I’rincipleJi, r (I IO Gn,s iru&ice r;i t;) n.$r :Vewtoniun Effm or C m i h g d I h t ~ 102

13 .V ecAu&l Phergy f i p d o n 104

Viscous IXYsipatiori 105

Coiicq’ot of Ilcforrnahn Wnrk and Equalion ~ I I Tcims ol‘l’otrmtinl 1:qilotiori for a I?wd R(:gion 107

14 t M 1 Iau cf‘lhcrmoc&runnic.s: Ilimnd hketgy Equalion 108

15 Second IAW os Il‘/umw+mic.s: I:nlropy l h d i t d o r i 109

16 Bcrmulli l ? ( p i h n 1 1 0 Steady Flow 112

l!nstcaciy Im)tarionul Flmv 113

Equation 114

Orifice hi ai Tank 115

I8 ~ o l o u u s i n e ~ q A ~ ~ ) N ~ ~ ~ a & i o r i 117

Coriiiriuig lkpitjori 118

Monlcntiim Equation 119

lieat Equatiori 119

19 Boundwy (!i)ndiLorLs 121

Fxcmkw 122

I & a d m C d d 124

Supplwnmlal Ikziding 124

EncTgy 106

I 7 1 7 Applica&iorLs r!/’h’emoulli‘.s pitot mIc ? 114

1 Inlmduction

All fluid rncchanics is based on the conservation laws for mass momentum, and energy These laws can be staled in the di#ere.nriul form applicable at a point They can also be stated in thc integral form applicable to an cxtended rcgion In thc integral

76

2 Tune l k r i c a ~ e # ff Volume lnhbml#

form, the expressions of thc laws depend on whether they relate to a volumefied in

space, or to a material volume, which consists of the same fluid particles and whose

bounding surface moves with the fluid Both types of volumes will be considered

in t h i s chapter; c i $xed region will be denoted by V and a material volume will he

&rwted by ”Ir In engineering literature a fixed region is called a control volume,

whose surfaces are called control suTfaces

Thc integral and differential forms can be derived from each other As we shall

see, during the derivation surface integrals frequently need to be converted to volume

integrals (or vice versa) by means of the divergence theorem of Gauss

77

(4.1) where F ( x , t ) is a tensor of m y rank (including vectors and scalars), V is either a

fixed volume or a material volume, and A is its boundary surface Gauss’ theorem

was presented in Section 2.13

2 Time Deriuatives of Volume Inlcgrab

Tn deriving the conservation laws, one rrequently faces the problem of finding h e

time derivative of integrals such as

where F ( x , t ) is a tensor of any order, and V ( t ) is any region, which may be fixed or

move with the fluid The d / d t sign (in contrast to alar) has been written because only

a function of time remains after performing the integration in space The different

possibilities are discussed in what follows

General Case

Consider the general case in which V ( t ) is neither a fixed volume nor a material

volume The surfaces of the volume are moving, but not with the local fluid veloc-

ity The rule for diffcrentiating an integral becomes clear at once if we consider a

one-dimensional ( 1 D) analogy Tn books on calculus,

dt s”‘” X = U ( t ) d x + -F(b, dt t ) - - F ( u , dt t ) (4.2)

This is called the kihniz theorem, and shows how to differentiate an integral whose

integrand F as well as the limits of integration are functions of the variable with

respect to which we are diffcrentiating A graphical illustration of the three terms on

the right-hand sidc of the Leibniz theorem is shown in Figure 4.1 The continuous

line shows the integral S F d x at time t , and the dashed line shows the integral at time

t + dr The first tcrm on the right-hand side in Eq (4.2) is the integral of aF/at over

the region, the second term is due to the gain of F at the outer boundary moving at a

rate d b / d t , and the third term is due to the loss of F at the inner boundary moving at

d u / d t

Figure 4.1 Graphical illustrhon of Lcibnkr’s theorem

Generalizing the Leibniz theorem, we write

where un is the velocity of the boundary and A(b) is Ihc surface of V ( t ) The surface integral in Eq (4.3) accounts €or both “inlets” and “outlcts,” so that separale terms as

in Eq (4.2) are not necessary

Fixed Volume

For a fixed volume we have UA = 0, for which Q (4.3) becomes

(4.4)

which shows that thc time derivative can be simply taken inside the integral sign if

the boundary is fixed This merely reflects thc fact that the “limit of inlegration” V is not a function of time in this case

Material Volume

For a material volume V(f) the surfaces move with the fluid, so that UA = u, where

u is the fluid velocity Then Eq (4.3) becomes

This is sometimes called the Reynolds transport theorem Although no1 necessary,

we have used the D / D b symbol here to emphasize that we are following a material volume

Another form of the transport theorem is derived by using the mass conservation relation Eq (3.32) derived in the last chapter Using Gauss' theorem, the transport theorem Eq (4.5) becomes

Now define a new function f such that F = p f , where p is the fluid density Then the prcccding becomes

Using thc continuity equation

an integral form for a fixed region and then deduce the differential form Consider a volume fixed in space (Figure 4.2) The rate of increase of mass inside it is the volume integral

The time derivative has been taken inside the integral on the right-hand side because the volume is fixed and Eq (4.4) applies Now the rate of mass flow out of the volume

is the surface integral

pu dAl

Figorc 4.2 Mass conscrvatim of a volume fixed in space

because pu d A is the outward flux through an area element d A (Throughout the

book, we shall write dA for n d A , where n is the unit outward normal to the surlace

Vector dA therefore has a magnitude d A and a direction along the outward normal.)

The law of conservation of mass states that the rate of increase of mass within a fixed volume must equal the rate of i d o w through the boundaries Therefore,

(4.7)

which is the integral form of the law for a volume fixed in space

The differential fonn can be obtained by transforming the surface integral on the right-hand side of Eq (4.7) to a volume integral by means of the divergence theorem, which gives

pu - d A = V ( p u ) d V

Equation (4.7) then becomes

l [ z + V - ( p u ) d V = 0 1

The forementioned relation holds far any volume, which can be possible only if the

intcgrand vanishes at cvery point (Tf the integrand did not vanish at every point, then

we could choose a small volume around that point and obtain a nonzero integral.)

This requires

which is called thc continmi0 eyuutiun and cxpresses the differential form of thc

principlc of conscrvation of mass

The equation can bc written in scveral other forms Rewriting the divergence term in Eq (4.8) as

the equation of continuity becomes

The derivative Dp/Dt is the rate of change of density following a fluid particle; it can bc nonzero because of changes in pressure, temperature, or composition (such

as salinity i n sca water) A fluid is usually called incompressible if its density does not change with pressure Liquids are almost incompressible Although gases are

comprcssible, for speeds 5 100 m/s (that is, for Mach numbers 4 3 ) the fractional

change of absolute prcssure in the flow is small Tn this and scveral other cases the density changes in thc Flow are also small The neglect of p-.' D p / D t in the continuity equation is part of a scrics of simplifications grouped under thc Boussinesq approximation, discussed in Section 18 In such a case the continuity equation (4.9)

rcduccs to the incompressible form

(4.10)

whether or not thc flow is steady

4 ,'j~mamfrui.dii~ttionu= Kecisikd and ~;C!ni!rwlixrcd

Consider the steady-state form of mass conservation from Eq (4.8),

In Exercisc 10 of Chapter 2 we showcd that the divergence of the curl of any vector field is identically zcro Thus we can reprcscnt the mass flow vector as the curl of a vwtor potential

Figure 4.3 Ed@ v i m of two members of cach of two €milies of streadunctions Contour C is the boundary of surracc arca A : C = a A

x = b, = cy $ = d The intersections shown as darkened dots in Figure 4.3 are the streamlines coming out of the paper We calculate the mass per time through a

surface A bounded by the four streamfunctions with element dA having n out of the paper By Stokes' theorem,

= / ( x d I / ' + dq5) = / x d @ = b(d - c ) + U ( C - d ) = ( b - a)(d - c)

Here we have used the vector identity Vq5 ds = dq5 and recognized that integration around a closed path of a single-valued function results in zero The mass per time through a surface bounded by adjacent members of the two families of streamfunc-

tions is just the product of the differences of the numerical values of the respective streamfunctions As a very simple special case, consider flow in a z = constant plane (described by x and y coordinates) Because all the streamlines lie in z = constant planes, z is a streamfunction Define x = -z, where the sign is chosen to obey the usual convention Then V x = -k (unit vector in the z direction), and

PU = -k x ve; PU = ae/ay, PV = a*/a.T,

in conformity with Chapter 3, Exercise 14

Similarly, in cyclindrical polar coordinates as shown in Figure 3.1 flows, sym- metric with respect to rotation about the x-axis, that is, those for which = 0, have streamlines in q5 = constant planes (through the x-axis) For those axisymmetric flows, x = -q5 is one streamfunction:

1

pu = -j$ x v*,

then gives pRu, = &,+pit, ~ R U R = -a@/ax We note herc that if the density may

be taken as a constant, mass conservation reduces to V u = 0 (steady or not) and the entire preceding discussion follows for u rather than pu with the interpretation of streamfunction in terms of volumetric rather than mass flux

Before we can proceed further with the conservation laws, it is necessary to classify the various types of forces on a fluid mass The forces acting on a fluid element can

3 Ot*$ti a/ Fama iti I.Yuid 83

be divided conveniently into three classes, namely, body ~orces, surface forces, and

linc forces These arc: described as follows:

Body juxes: Body forces are those that arise from “action at a distance,” with-

out physical contact They result from the medium being placed in a certain

SorceJiefd, which can bc gravitational, magnetic, electrostatic, or electromag-

netic in origin They are distributed throughout the mass of the fluid and are

proportional to the mass Body forces are expressed either per unit mass or per

unjt volume In this book, the body force per unit mass will bc dcnoted by g

Body forces can be conservative or nonconservative Conservative body

fobrces arc those that can be expressed as the gradient of a potential function:

where n is called thefimepotentiul All forces directed cenfrully from a sourcc

are conservativc Gravity, clcclrostatic and magnetic forces are conservative

For example, the gravity force can be written as the gradient of the potential

function

n = gz,

where g is the acceleration due to gravity and z points vertically upward To

verify this, Eq (4.13) gives

a a

g = -V(gz) = - I- + J- + k- (gz) = -kg,

which is the gravity force per unit mass The negative sign in font of kg

cnsures that g is downward, along the negative z direction The exprcssion

ll = gz also shows that the jiirce potential equals the potential energy per

unit muss Forces satisfying Eq (4.13) are called “conservative” becausc rhc

resulting motion conserves the sum of kinetic and potential energies, if there

are no dissipative processes

Surfacejorces: Surface forces are thosc that are exerted on an arca elcmcnt by

the surroundings through direct contact They arc proportional to the extent

ofthe area and are convcniently expressed per unit of m a Surface forces can

be ~ s o l v c d into components normal and tangential to the arca Consider an

element of area d A in a fluid (Figurc 4.4) The force dF on lhe element can

be rcsolved into a component dF,, normal to the area and a component dF,

tangcntial to the area The normal and shear stress on the element are defincd,

rcspectively as,

t = - r = -

n - d A 5 - d A ’

Thcse are scalar definitions of stress components Note that the component of

forcc tangential to the surfacc is a two-dimensional (2D) vector in thc surrace

Thc state of stress at a point is, in fact, specified by a stress tensor, which has

nine componcnk This was explaincd in Section 2.4 and is again discussed in

the following section

Figure 4 Normal and shear forces on an m clcrnent

( 3 ) Lineforues: Surface tension forces are called h e f o x e s because they act along

a line (Figure 1.4) and have a magnitude proportional to the extent of the line They appear at the interface between a liquid and a gas, or at the interface between two immiscible liquids Surface tension forces do not appear directly

in the equations of motion, but enter only in the boundary conditions

6 Shww at a &in1

It waq explaincd in Chapter 2, Section 4 that the stress at a point can be completely specified by the nine components of the stress tensor 'c Consider an infinitesimal rect-

angular parallelepiped with faces perpendicular to the coordinate axes (Figure 4.5)

On each face there is a normal stress and a shear stress, which can be further resolved into two components in the directions of the axes The figure shows the directions of

positive stresses on four of the six faces; those on the remaining two faces are omitted for clarity The h t index of t i j indicates the direction of the normal to the surface on

which the stress is considered, and the second index indicates the direction in which the stress acts The diagonal elements til, t 2 2 , and t 3 3 of the stress matrix are the normal stresses, and the off-diagonal elements are the tangential or shear stresses Although a cube is shown, the figure really shows the stresses on four of the six

orthogonal planes passing through a point; the cube may be imagined to shrink to

a point

We shall now prove that the stress tensor is symmetric Consider the toque on

an element about a centroid axis parallel to xg (Figure 4.6) This torque is generated only by the shear stresses in the X I xz-plane and is (assuming dx3 = 1)

M e r canceling terms, this gives

T = ( t l ~ - tz1) dxl dx2

The rotational equilibrium of the element requires that T = Zh3, where h3 is the angular acceleration o€ the element and I is its moment of inertia For the rectan- gular element considered, it is easy to show that I = d x l dxz(dx: + dxz)p/12 The