# 0.4 Stress in fluids  (Page 2/4)

 Page 2 / 4
$\underset{d\to 0}{lim}\frac{1}{{d}^{2}}\underset{s}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}{\mathbf{t}}_{\left(n\right)}dS=0$

or, the stresses are locally in equilibrium .

## The stress tensor

To elucidate the nature of the stress system at a point $P$ we consider a small tetrahedron with three of its faces parallel to the coordinate planes through $P$ and the fourth with normal $n$ (see Fig. 5.1 of Aris). If $dA$ is the area of the slanted face, the areas of the faces perpendicular to the coordinate axis Pi is

$d{A}_{i}={n}_{i}dA.$

The outward normals to these faces are $-{\mathbf{e}}_{\left(i\right)}$ and we may denote the stress vector over these faces by $-{\mathbf{t}}_{\left(i\right)}$ . ( ${\mathbf{t}}_{\left(i\right)}$ denotes the stress vector when $+{\mathbf{e}}_{\left(i\right)}$ is the outward normal.) Then applying the principle of local equilibrium to the stress forces when the tetrahedron is very small we have

$\begin{array}{c}\hfill {\mathbf{t}}_{\left(n\right)}dA-{\mathbf{t}}_{\left(1\right)}d{A}_{1}-{\mathbf{t}}_{\left(2\right)}d{A}_{2}-{\mathbf{t}}_{\left(3\right)}d{A}_{3}\\ \hfill =\left({\mathbf{t}}_{\left(n\right)}-{\mathbf{t}}_{\left(1\right)}{n}_{1}-{\mathbf{t}}_{\left(2\right)}{n}_{2}{\mathbf{t}}_{\left(3\right)}{n}_{3}\right)dA=0\end{array}$

Now let ${T}_{ji}$ denote the ${i}^{th}$ component of ${\mathbf{t}}_{\left(j\right)}$ and ${t}_{\left(n\right)i}$ the ${i}^{th}$ component of ${\mathbf{t}}_{\left(n\right)}$ so that this equation can be written

${t}_{\left(n\right)i}={T}_{ji}{n}_{j}.$

However, ${\mathbf{t}}_{\left(n\right)}$ is a vector and $\mathbf{n}$ is a unit vector quite independent of the ${T}_{ji}$ so that by the quotient rule the ${T}_{ji}$ are components of a second order tensor $\mathbf{T}$ . In dyadic notation we might write

${\mathbf{t}}_{\left(n\right)}=\mathbf{T}•\mathbf{n}.$

This tells us that the system of stresses in a fluid is not so complicated as to demand a whole table of functions ${\mathbf{t}}_{\left(n\right)}\left(\mathbf{x},\mathbf{n}\right)$ at any given instant, but that it depends rather simply on $\mathbf{n}$ through the nine quantities ${T}_{ji}\left(\mathbf{x}\right)$ . Moreover, because these are components of a tensor, any equation we derive with them will be true under any rotation of the coordinate axis.

Inserting the tensor expression for the stress into the momentum balance and using the equation of continuity and Green's theorem we have

$\begin{array}{ccc}\hfill \frac{D}{Dt}\underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\rho \mathbf{v}dV& =& \underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\rho \mathbf{f}dV+\underset{s}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}{\mathbf{t}}_{\left(n\right)}dS\hfill \\ & =& \underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\rho \mathbf{f}dV+\underset{s}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}\mathbf{n}•\mathbf{T}dS\hfill \\ & =& \underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\left(\rho \mathbf{f}+\nabla •\mathbf{T}dV\hfill \\ \hfill \frac{D}{Dt}\underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\rho \mathbf{v}dV& =& \underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\left[\frac{D}{Dt},\left(\rho \mathbf{v}\right),+,\rho ,\mathbf{v},\left(\nabla •\mathbf{v}\right)\right]dV\hfill \\ & =& \underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\left[\rho ,\frac{D\mathbf{v}}{Dt},+,\mathbf{v},\left(\frac{D\rho }{Dt},+,\rho ,\nabla ,•,\mathbf{v}\right)\right]dV\hfill \\ & =& \underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\rho \frac{D\mathbf{v}}{Dt}dV\hfill \end{array}$

Since all the integrals are now volume integrals, they can be combined as a single integrand.

$\underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\left(\rho \frac{d\mathbf{v}}{Dt}-\rho \mathbf{f}-\nabla •\mathbf{T}\right)dV=0$

However, since $V$ is an arbitrary volume this equation is satisfied only if the integrand vanishes identically.

$\begin{array}{ccc}\hfill \rho \frac{D\mathbf{v}}{Dt}& =& \rho \mathbf{f}+\nabla •\mathbf{T}\hfill \\ & =& \rho \mathbf{a}\hfill \\ & \text{or}& \\ \hfill \rho \frac{D{v}_{i}}{Dt}& =& \rho {f}_{i}+{T}_{ji,j}\hfill \\ & =& \rho {\alpha }_{i}\hfill \end{array}$

This is Cauchy's equation of motion and $\mathbf{a}$ is the acceleration. It holds for any continuum no matter how the stress tensor $\mathbf{T}$ is connected with the rate of strain.

## The symmetry of the stress tensor

A polar fluid is one that is capable of transmitting stress couples and being subject to body torques, as in magnetic fluids. In case of a polar fluid we must introduce a body torque per unit mass in addition to the body force and a couple stress in addition to the normal stress ${\mathbf{t}}_{\left(n\right)}$ . The stress for polar fluids is discussed by Aris.

A fluid is nonpolar if the torques within it arise only as the moments of direct forces. For the nonpolar fluid we can make the assumption either that angular momentum is conserved or that the stress tensor is symmetric. We will make the first assumption and deduce the symmetry.

Return now to the integral linear momentum balance with the internal force expressed as a surface integral. If we assume that all torques arise from macroscopic forces, then not only linear momentum but also the angular momentum $\mathbf{x}×\left(\rho \mathbf{v}\right)$ are expressible in terms of $\mathbf{f}$ and ${\mathbf{t}}_{\left(n\right)}$ .

short run AC curves?
what is short run AC curves?
Jasmin
what is short run curves?
Jasmin
what is short run curves?
Jasmin
nooo am not from India why!?
h
Hamid
Godwin which level of education are you please
Millionaires
millionaires am in SHS 2
Godwin
vnsgu BBA ki first sem ki all subjects ki koi app he ?
mudasir
all subjects ki koi nhi he ?
Modi
for accounts u can download accounts complete course app ... step by step Al the topics r mentioned
mudasir
Muje BBA ke first semester ke liye chahiye.... Vnsgu
Modi
there is a app for bba
named as college tutor on play store
isme sab subject ki books aa jayegi ?
Modi
han g
baqi main bad main batata hun abi online paper ho raha ha
okk....
Modi
Apki knsi uni ha
aik aur app ha but wo indian ha
Named as bba books
okk
Modi
collage tutor mein join teacher wala hi aata he or kuch nhi aata he
Modi
Mera Vnsgu he
Modi
asslam alaikum
Hamid
BBA books app ki photo send kijiye naa please
Modi
photo kesy send ho skty ha yahan sy?
I mean iska option kahan ha
haa.. sorry
Modi
collage tutor mein join teacher wala hi aata he or kuch nhi aata he
Modi
ok
Mera Vnsgu he
Modi
Ap play store py ja ky waha search ker lain wahan sy asani sy ye app mil jay ga apko
ab books kaha lau ?
Modi
oh sorry
main apko sham rak batata hun abi paper ho raha ha☺☺
kya search maru ?
Modi
yar ap ki books ka name kia ha?
modi
Hamid
who know accounting
Hamid
Hmare pass koi bhi subject ki book nhi he
Modi
abi 1st semester ha
ha... First semester
Modi
Yes sir
Modi
mujy book pary ha pdf ma accounting economic statistics Kay book ha mery pass
Hamid
pdf ma h
Hamid
to send karo naa
Modi
ya pa kasy kro
Hamid
kya ?
Modi
whatsp number do
Hamid
number bejo whatsp ka
Hamid
jaldi
Hamid
technical error bta rha he
Modi
number send nhi ho paa rha he
Modi
fb I'd name batio
Hamid
mery Hamid Ali shaikh ha
Hamid
mein nai use krti
Modi
fb I'd ka name bejo
Hamid
aap kaha se ho ?
Modi
fr kiya hoga
Hamid
ap kiya Kiya chlty ho batio
Hamid
mtlb ?
Modi
kiya use krty h wo Batio ma koxhes krta h fr
Hamid
ma pakistan sa h
Hamid
fb dowled Kro ap fr kam hiGa ap ka
Hamid
Krishi Modi iss fb id pe bhejo
Modi
Modi
profile kon sy h
Hamid
Modi are you from india
yr privacy ha Aik I'd pa req nhi jaty
Hamid
who was the father of economic ?why?
who was the known as father of economic?why?
Mahesh
Rationing and hoarding
how do the size of a country's population affect labour force
a mixed economic system
What are the types of price elasticity of demand
what are massures to promote geographical mobility of labor?
Ngong
Is to make sure that a labourer to know more about his salary to earn before going to the direction
shehu
Trade is a basic economic concept involving the buying and selling of goods and services, with compensation paid by a buyer to a seller, or the exchange of goods or services between parties. Trade can take place within an economy between producers and consumers.
Miss
what is fisical policy?
fisical policy or fiscal policy?
Miss
what are.the characteristics of economic goods
Hamis
what are the importance of labour market?
Rachael
how discrib the rural development and their four stages
ye economics se related ha
Sheikh
1..traditional stage..no science and technology is applied hence poor productionuu.2..the take off stage..some development strategies are initiated eg transport system is improved but the traditional cultural belief still remain .3..the prematurely stage..technological methods of production are appl
President
applied leading to higher GDP..4..stage of mass consumption..
President
What is Easiest Formula For National Income?
national income/ agrrigate net value
Sheikh
what do you mean by the supply of goods
supply of good refer to the total unit of production which is ready to sell at a given price
Tenzin
what is implicit cost
Yeah
MOHAMED
any cost that has already occurred but not necessarily shown or reported as a separate expense.
President
The links don't seem to be working
Got questions? Join the online conversation and get instant answers! By Brooke Delaney By Janet Forrester By Ali Sid By Anh Dao By OpenStax By OpenStax By Rhodes By OpenStax By OpenStax By Megan Earhart