# 0.5 Equations of motion and energy in cartesian coordinates  (Page 2/12)

 Page 2 / 12

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.

$\begin{array}{c}\mathbf{f}-\frac{1}{\rho }\nabla p=-\nabla \Omega \hfill \\ where\hfill \\ \Omega ={\int }^{p}\frac{dp}{\rho }-gz\hfill \end{array}$

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

$\begin{array}{c}\Omega \approx \frac{P}{\rho },\phantom{\rule{4.pt}{0ex}}\text{small}\phantom{\rule{4.pt}{0ex}}\text{change}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{density}\phantom{\rule{4.pt}{0ex}}\hfill \\ \text{where}\hfill \\ P=p-\rho gz\hfill \end{array}$

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

$\begin{array}{ccc}\hfill {\mathbf{v}}^{*}& =& \frac{\mathbf{v}}{U},\phantom{\rule{0.277778em}{0ex}}{\mathbf{x}}^{*}\frac{\mathbf{x}}{L},\phantom{\rule{0.277778em}{0ex}}{t}^{*}=\frac{U}{L}t,\phantom{\rule{0.277778em}{0ex}}{P}^{*}=\frac{P}{\rho {U}^{2}}\hfill \\ \hfill {\nabla }^{*}& =& L\nabla ,\phantom{\rule{0.277778em}{0ex}}{\nabla }^{*2}={L}^{2}{\nabla }^{2}\hfill \end{array}$

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

$\begin{array}{ccc}\hfill \rho \frac{D\mathbf{v}}{Dt}& =& -\nabla P+\left(\lambda +\mu \right)\nabla \Theta +\mu {\nabla }^{2}\mathbf{v}\hfill \\ \hfill \rho \frac{{U}^{2}}{L}\frac{D{\mathbf{v}}^{*}}{D{t}^{*}}& =& -\rho \frac{{U}^{2}}{L}{\nabla }^{*}{P}^{*}+\frac{\mu \phantom{\rule{0.277778em}{0ex}}U}{{L}^{2}}\left(\lambda /\mu +1\right){\nabla }^{*}{\Theta }^{*}+\frac{\mu \phantom{\rule{0.277778em}{0ex}}U}{{L}^{2}}{\nabla }^{*2}{\mathbf{v}}^{*}\hfill \\ \hfill \frac{\rho UL}{\mu }\left[\frac{D{\mathbf{v}}^{*}}{D{t}^{*}},+,{\nabla }^{*},{P}^{*}\right]& =& \left(\lambda /\mu +1\right){\nabla }^{*}{\Theta }^{*}+{\nabla }^{*2}{\mathbf{v}}^{*}\hfill \\ \hfill {N}_{Re}\left[\frac{D{\mathbf{v}}^{*}}{D{t}^{*}},+,{\nabla }^{*},{P}^{*}\right]& =& \left(\lambda /\mu +1\right){\nabla }^{*}{\Theta }^{*}+{\nabla }^{*2}{\mathbf{v}}^{*}\hfill \\ & \text{where}& \\ \hfill {N}_{Re}& =& \frac{\rho U\phantom{\rule{0.277778em}{0ex}}L}{\mu }=\frac{\rho {U}^{2}}{\mu U/L}\hfill \end{array}$

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 $\left(\mu U\right)/L$ , 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.

$\begin{array}{c}{N}_{Re}\frac{D{\mathbf{V}}^{*}}{D{t}^{*}}=-{\nabla }^{*}{P}^{**}+\left(\lambda /\mu +1\right){\nabla }^{*}{\Theta }^{*}+{\nabla }^{*2}{\mathbf{v}}^{*}\\ {P}^{**}=\frac{P}{\mu U/L}\hfill \end{array}$

## 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.

$\begin{array}{ccc}\hfill \frac{dW}{dt}& =& \underset{s}{∯}\mathbf{v}•{\mathbf{t}}_{\left(n\right)}\phantom{\rule{0.277778em}{0ex}}dS\hfill \\ & =& \underset{s}{∯}\mathbf{v}•\mathbf{T}•\mathbf{n}\phantom{\rule{0.277778em}{0ex}}dS\hfill \\ & =& \underset{v}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\nabla •\left(\mathbf{v}•\mathbf{T}\right)\phantom{\rule{0.277778em}{0ex}}dV\hfill \end{array}$

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

$\begin{array}{ccc}\hfill {\left({v}_{i}{T}_{ij}\right)}_{,j}& =& {T}_{ij}{v}_{i,j}+{v}_{i}{T}_{ij,j}\hfill \\ & =& {T}_{ij}{v}_{i,j}+{v}_{i}\left[\rho ,\frac{D{v}_{i}}{Dt},-,\rho ,{f}_{i}\right]\hfill \\ & =& {T}_{ij}{v}_{i,j}+\frac{1}{2}\rho \frac{D{v}_{2}}{Dt}-\rho {f}_{i}{v}_{i}\hfill \\ \hfill \nabla •\left(\mathbf{v}•\mathbf{t}\right)& =& \mathbf{T}:\nabla \mathbf{v}+\frac{1}{2}\rho \frac{D{v}^{2}}{Dt}-\rho \mathbf{f}•\mathbf{v}\end{array}$

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.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!