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[edit] Internal and External Incompressible Viscous Flows
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Flows completely bounded by solid surfaces are called internal flows. External flows are flows over bodies immersed in an unbounded fluid[1]. Internal flows might be laminar or turbulent. The state of the flow regime is dependent on Re. There might be an analytical solution for laminar flows but not for turbulent flows. |
[edit] Laminar and Turbulent Channel and Pipe flows
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At fully developed state the velocity profile becomes parabolic for laminar flow. The average velocity at any cross section is: ![]() For the same flow value i.e. The reason is the turbulent eddies, which causes more momentum loss to the wall i.e. higher velocity gradients close to the wall. Note that such a direct comparison is only valid at the same |
[edit] Concept of Fully Developed Flow
Consider the flow in a channel between two plates having a height of
and an infinite depth in
direction. Starting from the entrance, the boundary layers develop due to the no-slip condition on the wall and, in turn, the momentum loss to the wall. At a finite distance, the boundary layers merge and the inviscid core (field with no velocity gradient in
direction) vanishes. The flow becomes fully viscous. The flow in
direction adjusts slightly further until
and it no longer changes with
direction. This state of the flow is called fully-developed. At that state:

Because of the two-dimensional nature of the flow, no gradient of the flow quantities in
direction is expected starting from the entrance.


Hence,

The entrance lengths for laminar pipe and channel flows are, respectively[4]:
![\frac{Le_{pipe}}{D} = \left[(0.619)^{1.6} + (0.0567Re)^{1.6}\right]^{\frac{1}{1.6}}](http://upload.wikimedia.org/wikiversity/en/math/0/7/b/07b7e37a7bf918da5c876dbe0c763df5.png)
![\frac{Le_{channel}}{D} = \left[(0.631)^{1.6} + (0.0442Re)^{1.6}\right]^{\frac{1}{1.6}}](http://upload.wikimedia.org/wikiversity/en/math/3/8/5/38520f6a449dc2b001494b700126a5ed.png)
[edit] Fully Developed Laminar Flow Between Inifnite Parallel Plates
Consider the fully developed laminar flow between two infinite plates.
Consider the continuity equation and momentum equation in
direction for an incompressible steady flow between two infinite plates as shown.
[edit] Continuity Equation

Since
,
because it is a fully-developed and two dimensional flow. Hence,
reads

As
is zero on the walls, it should be zero in the whole fully develooped region, i.e.

[edit] Momentum Equation in j-direction

in
direction


Consider term A:
![\rho U_{i}\frac{\partial U_{1}}{\partial x_{i}} = \rho \left[U_{1}\underbrace{\frac{\partial U_{1}}{\partial x_{1}}}_{= 0\ Fully-developed} + \underbrace{U_{2}}_{= 0}\frac{\partial U_{1}}{\partial x_{2}} + \underbrace{U_{3}}_{=0\ fully-developed\ 2D }\underbrace{\frac{\partial U_{1}}{\partial x_{3}}}_{= 0\ 2D}\right]=0](http://upload.wikimedia.org/wikiversity/en/math/b/e/a/bea4ea655220101738dfd02073d63de4.png)
Consider term B:

hence
.
Thus the momentum equation in
direction reads:

This equation should be valid for all
and
. This requires that
= constant.
Remember
is the stress in
direction on a face normal to
direction.

thus,


Thus the momentum equation reads:

This equation can be obtained also by using the Reynold's transport equations for a differential volume.
The momentum equation in
direction,

The flux term becomes zero since for fully-developed flow incoming flux is equal to the outgoing flux. Thus,

That is:




Finally, the governing equation of this kind of flow becomes:

with the following boundary conditions:
and 
Integrating the equation once results in a linear function of
:

The second integration reads:

The integration constants is obtained by using the boundary conditions:


Finally, the velocity profile reads:
![\begin{array}{lll}
U_{1} &=& \frac{1}{2\mu}\left(\frac{\partial P}{\partial x_{1}}x_{2}^{2}\right) - \frac{1}{2\mu}\left(\frac{\partial P}{\partial x_{1}}\right)D{x_{2}}\\
&=&\frac{D^{2}}{2\mu}\left(\frac{\partial P}{\partial x_1}\right) \left[\left(\frac{x_{2}}{D}\right)^{2} - \left(\frac{x_{2}}{D}\right)\right]
\end{array}](http://upload.wikimedia.org/wikiversity/en/math/6/7/f/67f2d55d01ae818c748b5731424227b2.png)
Note that the velocity profile is parabolic!
The shear stress becomes:
![\begin{array}{lll}
\tau_{21} &=& \left(\frac{\partial P}{\partial x_{1}}\right)x_{2} - \frac{1}{2}\left(\frac{\partial P}{\partial x_{1}}\right)D \\
&=& D\left(\frac{\partial P}{\partial x_{1}}\right)\left[\frac{x_{2}}{D} - \frac{1}{2}\right]
\end{array}](http://upload.wikimedia.org/wikiversity/en/math/c/5/7/c57c57ddcfce3639ec7f7bdf9e7db3c6.png)
at the wall i.e. at
= 0 and
= D


Note that
is maximum near the wall, i.e. momentum loss is maximum near the wall. This is due to the maximum velocity gradient
near the wall!
The volume flow rate is,

where
is the depth of the channel.
Thus the volume flow rate per depth
is given by:


Note that
should be constant for the fully developed flow. Hence, for a channel with a finite length
:

Where
is the pressure drop along L.

or the pressure drop can be calculated from:

For the same flow rate, increasing the height of the channel would cause a drastic reduction in the pressure drop.
The average velocity
is:

The maximum velocity occurs when:

Hence, at
, 

The velocity profile can be written as functions of bulk velocity
or maximum velocity
by replacing their value the velocity profile equation:
![\begin{array}{lll}
U_{1} &=& -4 U_{1max} \left[\left(\frac{x_{2}}{D}\right)^{2} - \left(\frac{x_{2}}{D}\right)\right]\\
&=&-6 \overline{U} \left[\left(\frac{x_{2}}{D}\right)^{2} - \left(\frac{x_{2}}{D}\right)\right]
\end{array}](http://upload.wikimedia.org/wikiversity/en/math/9/a/9/9a93b081dd6a34df7183cad36795e592.png)
Same problem can be solved by using moving plates.
[edit] Example
Consider the hydrolic control valve coprising a piston, fitted to a cylinder with a mean radial clearance of 0,005mm. Determine the leakage flow rate. The fluid is SAE low oil (
= 932
,
=0.018
at 55ºC). The flow can be assumed to be laminar, steady, incompressible, fully-developed flow. 
Since
= 5000 the flow in the clearance can be accepted to be 2-D, with the depth
, thus:



Check the Reynolds number to ensure that laminar flow assumption is correct.


Re
, i. e. the flow is laminar.
[edit] Layered Channel Flow
This channel flow contains two different and non miscible fluids. Fluids A and B flow at the same time through a channel, which is bounded two flat plates. They both occupy the half height of the channel. The fluid A has a viscosity
, a density
and the mass flow
. Fluid B, which is located above fluid A, has a viscosity
, a density
and the mass flow
. The following differential equations correspond to the molecular momentum
for each Fluid.
and
.
With
yields the velocity field:
and 
After integration of both equations we obtain:
and 
As boundary condition we consider that shear stress on the interface between A and B is the same. Therefore we obtain:

Then,

After the integration for the velocity field:

and

The second boundary condition turns out to be on the interface:
, i.e. 
therefore,
. The integration constants can be calculated with the following boundary conditions:
At
:

At
:

Therefore we obtain for the velocity distribution in the fluids A and B:
![U_{1}^{A} = -\frac{D^{2}}{2\mu_{A}}\frac{d\Pi}{dx_{1}} \left[+\frac{2\mu_{A}}{(\mu_{A} + \mu_{B})} + \left(\frac{\mu_{A} - \mu_{B}}{\mu_{A}+ \mu_{B}}\right)\left(\frac{x_{2}}{D}\right) - \left(\frac{x_{2}}{D}\right)^{2}\right] \ \ \ \ \](http://upload.wikimedia.org/wikiversity/en/math/d/6/3/d63de477ba7a2c6622fd8bade0c99726.png)
and
![U_{1}^{B} = -\frac{D^{2}}{2\mu_{B}}\frac{d\Pi}{dx_{1}} \left[+\frac{2\mu_{B}}{(\mu_{A} + \mu_{B})} + \left(\frac{\mu_{A} - \mu_{B}}{\mu_{A}+ \mu_{B}}\right)\left(\frac{x_{2}}{D}\right) - \left(\frac{x_{2}}{D}\right)^{2}\right]](http://upload.wikimedia.org/wikiversity/en/math/3/a/c/3acc9981f22a3ad89273f0cd50f3a243.png)
For the distribution of the shear stress we get:
![\tau_{21} = D\frac{d\Pi}{dx_{1}}\left[\left(\frac{x_{2}}{D}\right) - \frac{1}{2} \left(\frac{\mu_{A} - \mu_{B}}{\mu_{A}+ \mu_{B}}\right)\right]](http://upload.wikimedia.org/wikiversity/en/math/d/c/a/dca4190b800bf5c6f422a3c9fb2ca665.png)
If we choose
,
![U_{1} = \frac{-D^{2}}{2\mu_{A}}\frac{d\Pi}{dx_{1}}\left[1 - \left(\frac{x_{2}}{D}\right)^{2}\right]](http://upload.wikimedia.org/wikiversity/en/math/9/5/e/95ed56492c59f79131e00d70d78dafbc.png)

The solution gives that of the channel flow. In other words, velocity has a parabolic profile with the peak in the middle of the channel and a linear shear stress distribution
, where
at the channel's centerline.
If
, the position where the maximal velocity occurs can be calculated by introducing
on the velocity profile equation:

The shear stress on the upper plate is:
![\tau_{W}^{B} = \frac{D}{2}\frac{d\Pi}{dx_{1}}\left[\frac{\mu_{A} + 3\mu_{B}}{\mu_{A} + \mu_{B}}\right]](http://upload.wikimedia.org/wikiversity/en/math/6/9/d/69db437e48bde514a6d8a7cd74b4e6fa.png)
and the shear stress on the lower plate reads:
![\tau_{W}^{A} = -\frac{D}{2}\frac{d\Pi}{dx_{1}}\left[\frac{3\mu_{A} + \mu_{B}}{\mu_{A} + \mu_{B}}\right]](http://upload.wikimedia.org/wikiversity/en/math/2/d/0/2d04cf94d5ee398792c8da8c70854dc4.png)
The average velocities of the fluids A and B are:

and

Hence the respectively mass flow rates are:

and

[edit]
A change of variables on the Cartesian equations will yield[5] the following equations of momentum in r,
, and z directions:
The continuity equation is:
[edit] Fully Developed Pipe Flow
Applying the RTT to the infinitesmall cyclindrical CV along the symmetry axis of horizontal pipe, in which the flow is fully developed, the conservation of mass and the transport side of the conservation of momentum equation drops. Only remaining term governing this kind of flow is the balance of the forces on the CV in
direction.





in a laminar flow!

integrating,

The boundary condition is:

Thus
can be calculated from the boundary condition.

or
![U = -\frac{R^{2}}{4\mu}\left(\frac{\partial P}{\partial x}\right)\left[1 - \left(\frac{r}{R}\right)^{2}\right]](http://upload.wikimedia.org/wikiversity/en/math/5/8/2/5822462a5d05bffb19f775385fa701c9.png)
Knowing the velocity profile we can evaluate relevant quantities. The shear stress profile will look like:

at r = 0 
at r = R 
The volume flow rate would read



When we approximate 
![Q = -\frac{\pi R^{4}}{8\mu}\left[\frac{-\Delta P}{L}\right] = \frac{\pi \Delta PR^{4}}{8\mu L} = \frac{\pi \Delta PD^{4}}{128\mu L}](http://upload.wikimedia.org/wikiversity/en/math/b/4/d/b4d91fd845cf33eea2d2fdc95ed07c75.png)

Increase radius to create drastic reduktion in the pressure drop.
The mean velocity is:

The location where maximum velocity occurs can be found be setting:

at r = 0
U =
.

Note that in a channel was
.
can be written as a function of
i.e.
![U = \underbrace{-\frac{R^{2}}{4\mu}\left(\frac{\partial P}{\partial x}\right)}_{U_{max}}\left[1 - \left(\frac{r}{R}\right)^{2}\right]](http://upload.wikimedia.org/wikiversity/en/math/8/a/1/8a1849fc481709abd71938be1d170c54.png)
![\frac{U}{U_{max}} = \left[1 - \left(\frac{r}{R}\right)^{2}\right]](http://upload.wikimedia.org/wikiversity/en/math/c/9/9/c998dfd83829d0032b46b635180bb941.png)
Again, the velocity profile becomes parabolic.
[edit] References
- ↑ Fox, R.W. and McDonald, A.T., “Introduction to Fluid Mechanics”, John Willey and Sons.
- ↑ M. Nishi. PhD Thesis Friedrich-Alexander-Universität Erlangen-Nürnberg, 2009.
- ↑ M. Nishi, B. Ünsal, F. Durst, and G. Biswas. J. Fluid Mech., 614:425–446, 2008.
- ↑ Durst, F., Ray, S., Unsal, B., and Bayoumi, O. A., 2005, “The Development Lengths of Laminar Pipe and Channel Flows,” J. Fluids Eng., 127, pp. 1154– 1160.
- ↑ Acheson, D.J.: Elementary fluid dynamics, Clarendon Press, 1990.

, the fully developed turbulent pipe flow, would have higher velocity close to the wall and lower velocity at the center.
.
![r:\;\;\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} + u_x \frac{\partial u_r}{\partial x} - \frac{u_{\phi}^2}{r}\right) =
-\frac{\partial p}{\partial r} +
\mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_r}{\partial \phi^2} + \frac{\partial^2 u_r}{\partial x^2}-\frac{u_r}{r^2}-\frac{2}{r^2}\frac{\partial u_\phi}{\partial \phi} \right] + \rho g_r](http://upload.wikimedia.org/wikiversity/en/math/3/2/6/326d8e07ddc1526a1cd70104233cadd6.png)
![\phi:\;\;\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + u_x \frac{\partial u_{\phi}}{\partial x} + \frac{u_r u_{\phi}}{r}\right) =
-\frac{1}{r}\frac{\partial p}{\partial \phi} +
\mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_{\phi}}{\partial \phi^2} + \frac{\partial^2 u_{\phi}}{\partial z^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \phi} - \frac{u_{\phi}}{r^2}\right] + \rho g_{\phi}](http://upload.wikimedia.org/wikiversity/en/math/8/3/0/83005f646a4cf900917c729d44069769.png)
![x:\;\;\rho \left(\frac{\partial u_x}{\partial t} + u_r \frac{\partial u_x}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_x}{\partial \phi} + u_x \frac{\partial u_x}{\partial x}\right) =
-\frac{\partial p}{\partial x} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_x}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_x}{\partial \phi^2} + \frac{\partial^2 u_x}{\partial x^2}\right] + \rho g_x.](http://upload.wikimedia.org/wikiversity/en/math/c/0/6/c06d898a0537c0c6772a89139322d4e2.png)
