<< Chapter < Page Chapter >> Page >
v ρ a d V = v ρ f d V + s t ( n ) d S v ρ ( x × a ) d V = v ρ ( x × f ) d V + s ( x × t ( n ) d S

The surface integral has as its i t h component

s ( x × t ( n ) d S = ε i j k x j T k p n p d S = v ε i j k ( x j T p k ) , p d V

by Green's theorem. However, since x j , p = δ j p , this last integrand is

ε i j k ( x j T p k ) , p = ε i j k x j T p k , p + ε i j k T j k or = x × ( T ) + T ×

where T is the vector ε i j k T j k .

Since v × v = 0 , d ( x × v ) / d t = x × a , applying the transport theorem to the angular momentum we have

D D t v ρ ( x × v ) d V = v ρ ( x × a ) d V

Substituting back into the equation for the angular momentum and rearranging gives

V v × ( ρ a - ρ f - T ) d V = V T × d V

However, the left-hand side vanishes for an arbitrary volume and so

T × 0 .

The components of T × are ( T 23 - T 32 ) , ( T 31 - T 13 ) , and ( T 12 - T 21 ) and the vanishing of these implies

T i j = T j i

so that T is symmetric for nonpolar fluids.

Hydrostatic pressure

If the stress system is such that an element of area always experiences a stress normal to itself and this stress is independent of orientation, the stress is called hydrostatic . All fluids at rest exhibit this stress behavior. It implies that n T is always proportional to n and that the constant of proportionality is independent of n . Let us write this constant - p , then

n i T i j = - p n j , hydrostatic stress n T = p n

However, this equation means that any vector is a characteristic vector or eigenvector of T . This implies that the hydrostatic stress tensor is spherical or isotropic. Thus

T i j = - p δ i j , hydrostatic stress T = - p I

for the state of hydrostatic stress.

For a compressible fluid at rest, p may be identified with the classical thermodynamic pressure. On the assumption that there is local thermodynamic equilibrium even when the fluid is in motion this concept of stress may be retained. For an incompressible fluid the thermodynamic, or more correctly thermostatic, pressure cannot be defined except as the limit of pressure in a sequence of compressible fluids. We shall see later that it has to be taken as an independent dynamical variable.

The stress tensor for a fluid may always be written

T i j = p δ i j + P i j T = p I + P

and P i j is called the viscous stress tensor . The viscous stress tensor of a fluid vanishes under hydrostatic conditions.

If the external or body force is conservative (i.e., gradient of a scalar) the hydrostatic pressure is determined up to an arbitrary constant from the potential of the body force.

T = - ρ f , static conditions f = g = g z , uniform gravity field T = - p , hydrostatic conditions p = ρ Φ Φ ( p ) = p d p ρ + C 1 = g z + C 2 p = ρ g z + C 3 , if ρ is constant

Buoyancy (deen, 1998)

A consequence of the fact that pressure increases with depth in a static fluid is that the pressure exerts a net upward force on any submerged object. To calculate this force, consider an object of arbitrary shape submerged in a constant-density fluid as shown in the figure. The net force on this object, F p , is given by

F p = - s p n d S
Consisting of two subfigures. The left figure is labeled (a) and consist an oval with a solid border and gray shading on the inside. Inside the oval is phrase Solid ρ=ρ_s, and above the figure is the phrase Fluid ρ= ρ_t. The right figure is labeled (b) and consist of an oval with a dashed border. Inside the oval is the phrase Fluid ρ= ρ_t and above the  oval is  the phrase Fluid ρ= ρ_t.

where the minus sign reflects the fact that positive pressure are compressive (i.e., pressure acts in the - n direction). The pressure force is evaluated most easily by considering the situation in the figure where the solid has been replaced by an identical volume of fluid. Because the pressure in the fluid depends on depth only, this replacement of the solid does not affect the pressure distribution in the surrounding fluid. In particular, the pressure distribution on the fluid control surface in (b) must be identical to that on the solid surface in (a). Situation (b) has the advantage that we can apply the divergence theorem to the integral in the last equation, because p is a continuous function of position within the volume V; this is not necessarily true for (a), because we have said nothing about the meaning of p within a solid. Applying the divergence theorem and using the hydrostatic pressure field, we find that

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Transport phenomena' conversation and receive update notifications?

Ask