Team:SDU-Denmark/project-m

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(Difference between revisions)
(2. The real system)
(3. Description of model)
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where η(=8.94•10<sup>-4</sup> Pa•s for water) is the viscosity of the fluid in which the bacterium is swimming, r(=0.4•10<sup>-6</sup> m) is the radius of the bacterium and v(=50•10<sup>-6</sup> m/s) is the velocity.
where η(=8.94•10<sup>-4</sup> Pa•s for water) is the viscosity of the fluid in which the bacterium is swimming, r(=0.4•10<sup>-6</sup> m) is the radius of the bacterium and v(=50•10<sup>-6</sup> m/s) is the velocity.
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The dragforce and thereby the force created by the flagellabundle of one e. coli is
The dragforce and thereby the force created by the flagellabundle of one e. coli is
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The system we are trying to model consists of a lot of bacteria stuck to a wall, but the part we are interested in is really the flagella and the forces they create. So we ignore the bodies of the bacteria, and instead consider the flagella as being stuck directly to the wall, with one end glued to the wall and the other pointing out into the fluid.  
The system we are trying to model consists of a lot of bacteria stuck to a wall, but the part we are interested in is really the flagella and the forces they create. So we ignore the bodies of the bacteria, and instead consider the flagella as being stuck directly to the wall, with one end glued to the wall and the other pointing out into the fluid.  
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Next step is to figure out what kind of flowfield, such a pointforce creates. The flowfield created by a pointforce in a fluid with no walls or other obstructions near it can be calculated using the corresponding Green's function known as the stokeslet.
Next step is to figure out what kind of flowfield, such a pointforce creates. The flowfield created by a pointforce in a fluid with no walls or other obstructions near it can be calculated using the corresponding Green's function known as the stokeslet.
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If the pointforce is placed near a wall modifications must be made, since the flowvelocity has to go to zero at the boundary (known as the no-slip condition). This is done by using the Oseen-Blake tensor. Simply described what the Oseen-Blake tensor does, is to create a mirror image of our force on the other side of the wall, thus cancelling the flow near the wall. (This is not a completely accurate description, but rather an intuitive one.)
If the pointforce is placed near a wall modifications must be made, since the flowvelocity has to go to zero at the boundary (known as the no-slip condition). This is done by using the Oseen-Blake tensor. Simply described what the Oseen-Blake tensor does, is to create a mirror image of our force on the other side of the wall, thus cancelling the flow near the wall. (This is not a completely accurate description, but rather an intuitive one.)
In our case the flagella are stuck to a wall, so we'll be using the Oseen-Blake tensor. A flowfield corresponding to this is shown below.
In our case the flagella are stuck to a wall, so we'll be using the Oseen-Blake tensor. A flowfield corresponding to this is shown below.
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In our case the system we are trying to describe is a microtube. This means that the width of the tube is so small, that the forces created by the flagella are not only close to the wall to which the flagella are stuck, but also close to the opposite wall. This presents an interesting problem. Since the Oseen-Blake tensor works by creating a mirrorpoint of the real force on the opposite side of the wall, we will need a mirrorpoint behind the other wall if we are to uphold the no-slip condition. But the mirror forces also affect the flow near the other wall. In order to cancel this effect one could create another mirrorforce, corresponding to each of the mirror forces, but of course these would obstruct each other to, requirering yet more mirror forces. In the end we decided, to see how precise the system would be for one mirrorpoint behind each wall. The flowfield corresponding to this is shown below.
In our case the system we are trying to describe is a microtube. This means that the width of the tube is so small, that the forces created by the flagella are not only close to the wall to which the flagella are stuck, but also close to the opposite wall. This presents an interesting problem. Since the Oseen-Blake tensor works by creating a mirrorpoint of the real force on the opposite side of the wall, we will need a mirrorpoint behind the other wall if we are to uphold the no-slip condition. But the mirror forces also affect the flow near the other wall. In order to cancel this effect one could create another mirrorforce, corresponding to each of the mirror forces, but of course these would obstruct each other to, requirering yet more mirror forces. In the end we decided, to see how precise the system would be for one mirrorpoint behind each wall. The flowfield corresponding to this is shown below.
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[[Image:Team-SDU-Denamrk-Pointforces_notxt.jpg|thumb|center|580px|All 3 images show the flowfield created by a pointforce at (0.000007,0.000007) with an angle of 45 degrees with the x-axis. In the first image there are no walls. In the 2nd a wall is placed at y=0. In the 3rd one wall is placed at y=0 and one at y=0.000015]]  
[[Image:Team-SDU-Denamrk-Pointforces_notxt.jpg|thumb|center|580px|All 3 images show the flowfield created by a pointforce at (0.000007,0.000007) with an angle of 45 degrees with the x-axis. In the first image there are no walls. In the 2nd a wall is placed at y=0. In the 3rd one wall is placed at y=0 and one at y=0.000015]]  
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As the above figures show, there is quite a difference depending on how many walls you take in to consideration. We decided to keep working with both the single-wall and the double-wall flowfields.
As the above figures show, there is quite a difference depending on how many walls you take in to consideration. We decided to keep working with both the single-wall and the double-wall flowfields.

Revision as of 17:03, 16 October 2010