# 0.7 Laminar flows with dependence on one dimension  (Page 3/6)

 Page 3 / 6
$\begin{array}{c}-\nabla p+\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{f}=-\nabla P\hfill \\ \mathrm{where}\hfill \\ P=p+\rho \phantom{\rule{0.166667em}{0ex}}g\phantom{\rule{0.166667em}{0ex}}h\hfill \end{array}$

The product $gh$ is the gravitational potential, where $g$ is the acceleration of gravity and $h$ is distance upward relative to some datum. The pressure, $p$ , is also relative to a datum, which may be the datum for $h$ .

The primary spatial dependence is in the direction normal to the plane of the plates. If there is no dependence on one spatial direction, then the flow is truly one-dimensional. However, if the velocity and pressure gradients have components in two directions in the plane of the plates, the flow is not strictly 1-D and nonlinear, inertial terms will be present in the equations of motion. The significance of these terms is quantified by the Reynolds number. If the flow is steady, and the Reynolds number negligible, the equations of motion are as follows.

$\begin{array}{c}0=-\frac{\partial P}{\partial {x}_{j}}-\frac{\partial {\tau }_{j3}}{\partial {x}_{3}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2\hfill \\ 0=-\frac{\partial P}{\partial {x}_{3}}-0\hfill \\ 0=-\frac{\partial P}{\partial {x}_{j}}+\mu \frac{{\partial }^{2}{v}_{j}}{\partial {x}_{3}^{2}},\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{1.em}{0ex}}\mathrm{Newtonian}\phantom{\rule{0.277778em}{0ex}}\mathrm{fluid}\hfill \end{array}$

Let $h$ be the spacing between the plates and the velocity is zero at each surface.

${v}_{j}=0,\phantom{\rule{1.em}{0ex}}{x}_{3}=0,\phantom{\rule{0.277778em}{0ex}}h\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2$

The velocity profile for a Newtonian fluid in plane-Poiseuille flow is

${v}_{j}=\frac{{h}^{2}}{2\mu }\frac{\partial P}{\partial {x}_{j}}\left[{\left(\frac{{x}_{3}}{h}\right)}^{2}-\frac{{x}_{3}}{h}\right],\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{1.em}{0ex}}0\le {x}_{3}\le h$

The average velocity over the thickness of the plate can be determined by integrating the profile.

${\overline{v}}_{j}=-\frac{{h}^{2}}{12\mu }\phantom{\rule{0.166667em}{0ex}}\frac{\partial P}{\partial {x}_{j}},\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2$

This equation for the average velocity can be written as a vector equation if it is recognized that the vectors have components only in the $\left(1,2\right)$ directions.

$\overline{\mathbf{v}}=-\frac{{h}^{2}}{12\mu }\phantom{\rule{0.166667em}{0ex}}\nabla P,\phantom{\rule{1.em}{0ex}}\overline{\mathbf{v}}=\overline{\mathbf{v}}\left({x}_{1},\phantom{\rule{0.166667em}{0ex}}{x}_{2}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{1.em}{0ex}}\nabla P=\nabla P\left({x}_{1},\phantom{\rule{0.166667em}{0ex}}{x}_{2}\right)$

If the flow is incompressible, the divergence of velocity is zero and the potential, $P$ , is a solution of the Laplace equation except where sources are present. If the strength of the sources or the flux at boundaries are known, the potential, $P$ , can be determined from the methods for the solution of the Laplace equation.

We now have the result that the average velocity vector is proportional to a potential gradient. Thus the average velocity field in a Hele-Shaw flow is irrotational. If the fluid is incompressible, the average velocity field is also solenoidal can can be expressed as the curl of a vector potential or the stream function. The average velocity field of Hele-Shaw flow is an physical analog for the irrotational, solenoidal, 2-D flow described by the complex potential. It is also a physical analog for 2-D flow of incompressible fluids through porous media by Darcy's law and was used for that purpose before numerical reservoir simulators were developed.

## Poiseuille flow

Poiseuille law describes laminar flow of a Newtonian fluid in a round tube (case 1). We will derive Poiseuille law for a Newtonian fluid and leave the flow of a power-law fluid as an assignment. The equation of motion for the steady, developed (from end effects) flow of a fluid in a round tube of uniform radius is as follows.

$\begin{array}{c}0=-\frac{\partial P}{\partial r}\hfill \\ 0=-\frac{\partial P}{\partial z}-\frac{1}{r}\frac{\partial }{\partial r}\left(r{\tau }_{rz}\right),\phantom{\rule{1.em}{0ex}}0

The boundary conditions are symmetry at $r=0$ and no slip at $r=R$ .

$\begin{array}{c}{\left({\tau }_{rz}|}_{r=0}=-\mu {\left(\frac{\partial {v}_{z}}{\partial r}|}_{r=0}=0\hfill \\ {v}_{z}=0,\phantom{\rule{1.em}{0ex}}r=R\hfill \end{array}$

From the radial component of the equations of motion, $P$ does not depend on radial position. Since the flow is steady and fully developed, the gradient of $P$ is a constant. The $z$ component of the equations of motion can be integrated once to derive the stress profile and wall shear stress .

#### Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
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!