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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
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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what school?
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anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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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.
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Damian Reply
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Akash Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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