# 0.8 Multidimensional laminar flow

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Chapter 4 in BSL, Transport Phenomena

## Lubrication and film flow

We already had two examples of flow in gaps that could be a thin film; Couette flow, and the steady, draining film. Here we will see that when the dimension of the gap or film thickness is small compared to other dimensions of the system, the Navier-Stokes equations simplify relatively and simple, classical solutions are possible. In Chapter 6, we saw that when the dimension of the gap or film in the ${x}_{3}$ direction is small and the Reynolds number is small, the equations of motion reduce to the following.

$\begin{array}{c}0=-\frac{\partial P}{\partial {x}_{3}}+O{\left(\frac{{h}_{o}}{L}\right)}^{2}+O\left(\mathrm{Re}\right)\hfill \\ 0=-{\nabla }_{12}P+\mu \frac{{\partial }^{2}{v}_{12}}{\partial {x}_{3}^{2}}+O{\left(\frac{{h}_{o}}{L}\right)}^{2}+O\left(\mathrm{Re}\right)\hfill \end{array}$

In the above equations, the subscript, 12, denote components in the plane of the gap or film. When the thickness is small enough, one wall of the gap or film can be treated as a plane even if it is curved with a radius of curvature that is large compared to the thickness.

Lubrication flow with slider bearings. (Ockendon and Ockendon, 1995) Bearings function preventing contact between two moving surfaces by the flow of the lubrication fluid between the surfaces. The generic example of lubrication flow is illustrated with the slider bearing.

A two-dimensional bearing is shown in which the plane of $y=0$ moves with constant velocity $U$ in the $x$ -direction and the top of the bearing (the slider) is fixed. The variables are nondimensionalised with respect to $U$ , the length $L$ of the bearing, and a characteristic gap-width, ${h}_{o}$ , so that the position of the slider is given in the dimensionless variables. Again, referring back to Chapter 6, the dimensionless variables for this problem may be the following.

$\begin{array}{c}x*=\frac{x}{L},\phantom{\rule{1.em}{0ex}}y*=\frac{y}{{h}_{o}},\phantom{\rule{1.em}{0ex}}h\left(x\right)=\frac{H\left(x\right)}{{h}_{o}}\hfill \\ u*=\frac{u}{U},\phantom{\rule{1.em}{0ex}}v*=\frac{v}{U}\frac{L}{{h}_{o}}\hfill \\ P*=\frac{{h}_{o}^{2}\phantom{\rule{0.166667em}{0ex}}p}{\mu \phantom{\rule{0.166667em}{0ex}}L\phantom{\rule{0.166667em}{0ex}}U}\hfill \end{array}$

Henceforth, the variables will be dimensionless with the $*$ dropped. The boundary conditions are as follows.

$\begin{array}{c}u=1,\phantom{\rule{1.em}{0ex}}v=0,\phantom{\rule{1.em}{0ex}}y=0\hfill \\ u=0,\phantom{\rule{1.em}{0ex}}v=0,\phantom{\rule{1.em}{0ex}}y=h\left(x\right)\hfill \\ P=0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}x=0,\phantom{\rule{0.277778em}{0ex}}1\hfill \end{array}$

The pressure can not suddenly equal the ambient pressure as assumed here because entrance and exit effects, but these will be neglected here. In reality, there may be a high pressure at the entrance of the bearing as a result of the liquid being scraped from the surface. The low pressure at the exit of the bearing may result in gas flowing in to equalize the pressure or cavitation may occur.

The dimensionless equations of motion and continuity equation are now as follows.

$\begin{array}{c}0=-\frac{\partial P}{\partial y},\phantom{\rule{1.em}{0ex}}⇒P=P\left(x\right)\hfill \\ 0=-\frac{dP}{dx}+\frac{{\partial }^{2}u}{\partial {y}^{2}}\hfill \\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\hfill \end{array}$

Integration of the equation of motion over the gap-thickness gives the velocity profile for a particular value of $x$ .

$u=\frac{{h}^{2}}{2}\frac{dP}{dx}\left[{\left(\frac{y}{h}\right)}^{2}-\frac{y}{h}\right]+1-\frac{y}{h}$

Notice that this profile is a combination of a profile due to forced flow (pressure gradient) and that due to induced flow (movement of wall). The velocity may pass through zero somewhere in the profile if the two contributions are in opposite directions. This is illustrated in the following figure.

Integration of the continuity equation over the thickness gives,

$\begin{array}{ccc}\hfill 0& =& {\int }_{0}^{h}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)\phantom{\rule{0.166667em}{0ex}}dy\hfill \\ & =& {\int }_{0}^{h}\frac{\partial u}{\partial x}\phantom{\rule{0.166667em}{0ex}}dy+{\left(v|}_{0}^{h}\hfill \\ & =& {\int }_{0}^{h}\frac{\partial u}{\partial x}\phantom{\rule{0.166667em}{0ex}}dy\hfill \end{array}$

The latter integral can be expressed as follows.

$\begin{array}{ccc}\hfill {\int }_{0}^{h}\frac{\partial u}{\partial x}\phantom{\rule{0.166667em}{0ex}}dy& =& \frac{d}{dx}{\int }_{0}^{h\left(x\right)}u\phantom{\rule{0.166667em}{0ex}}dy-{\left(\phantom{\rule{0.166667em}{0ex}}\frac{dh}{dx}u|}^{y=h}\hfill \\ & =& \frac{d}{dx}{\int }_{0}^{h\left(x\right)}u\phantom{\rule{0.166667em}{0ex}}dy\hfill \\ & =& 0\hfill \end{array}$

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
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
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Damian
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Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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Harper
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s.
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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
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
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Cied
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Porter
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Cesar
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Uday
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preparation of nanomaterial
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what is system testing
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Stotaw
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Azam
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Prasenjit
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Azam
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Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
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Azam
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Uday
I'm interested in Nanotube
Uday
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Prasenjit
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