<< Chapter < Page Chapter >> Page >

We will consider the case of single-phase flow with conservative body forces (e.g., gravitational) and density a single valued function of pressure. The pressure and potential from the body force can be combined into a single potential.

f - 1 ρ p = - Ω w h e r e Ω = p d p ρ - g z

If the change in density is small enough, the potential can be approximated by potential that has the units of pressure.

Ω P ρ , small change in density where P = p - ρ g z

Suppose that the flow is characterized by a certain linear dimension, L , a velocity U , and a density ρ . For example, if we consider the steady flow past an obstacle, L may be it's diameter and U and ρ the velocity and density far from the obstacle. We can make the variables dimensionless with the following

v * = v U , x * x L , t * = U L t , P * = P ρ U 2 * = L , * 2 = L 2 2

The conservative body force, Navier-Stokes equation is made dimensionless with these variables.

ρ D v D t = - P + ( λ + μ ) Θ + μ 2 v ρ U 2 L D v * D t * = - ρ U 2 L * P * + μ U L 2 ( λ / μ + 1 ) * Θ * + μ U L 2 * 2 v * ρ U L μ D v * D t * + * P * = ( λ / μ + 1 ) * Θ * + * 2 v * N R e D v * D t * + * P * = ( λ / μ + 1 ) * Θ * + * 2 v * where N R e = ρ U L μ = ρ U 2 μ U / L

The Reynolds number partitions the Navier -Stokes equation into two parts. The left side or inertial and potential terms, which dominates for large NRe and the right side or viscous terms, which dominates for small NRe. The potential gradient term could have been on the right side if the dimensionless pressure was defined differently, i.e., normalized with respect to ( μ U ) / L , the shear stress rather than kinetic energy. Note that the left side has only first derivatives of the spatial variables while the right side has second derivatives. We will see later that the left side may dominate for flow far from solid objects but the right side becomes important in the vicinity of solid surfaces.

The nature of the flow field can also be seen form the definition of the Reynolds number. The second expression is the ratio of the characteristic kinetic energy and the shear stress.

The alternate form of the dimensionless Navier-Stokes equation with the other definition of dimensionless pressure is as follows.

N R e D V * D t * = - * P * * + ( λ / μ + 1 ) * Θ * + * 2 v * P * * = P μ U / L

Dissipation of energy by viscous forces

If there was no dissipation of mechanical energy during fluid motion then kinetic energy and potential energy can be exchanged but the change in the sum of kinetic and potential energy would be equal to the work done to the system. However, viscous effects result in irreversible conversion of mechanical energy to internal energy or heat. This is known as viscous dissipation of energy. We will identify the components of mechanical energy in a flowing system before embarking on a total energy balance.

The rate that work W is done on fluid in a material volume V with a surface S is the integral of the product of velocity and the force at the surface.

d W d t = s v t ( n ) d S = s v T n d S = v ( v T ) d V

The last integrand is rather complicated and is better treated with index notation.

( v i T i j ) , j = T i j v i , j + v i T i j , j = T i j v i , j + v i ρ D v i D t - ρ f i = T i j v i , j + 1 2 ρ D v 2 D t - ρ f i v i ( v t ) = T : v + 1 2 ρ D v 2 D t - ρ f v

We made use of Cauchy's equation of motion to substitute for the divergence of the stress tensor. The integrals can be rearranged as follows.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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
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