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Steady axisymmetric motion of a small bubble in a tube with flowing liquid
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| Zeitschriftentitel: | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
|---|---|
| Personen und Körperschaften: | |
| In: | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466, 2010, 2114, S. 549-562 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
The Royal Society
|
| Schlagwörter: |
| author_facet |
Feng, James Q. Feng, James Q. |
|---|---|
| author |
Feng, James Q. |
| spellingShingle |
Feng, James Q. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Steady axisymmetric motion of a small bubble in a tube with flowing liquid General Physics and Astronomy General Engineering General Mathematics |
| author_sort |
feng, james q. |
| spelling |
Feng, James Q. 1364-5021 1471-2946 The Royal Society General Physics and Astronomy General Engineering General Mathematics http://dx.doi.org/10.1098/rspa.2009.0288 <jats:p> The steady axisymmetric behaviour of a relatively small bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> and Reynolds number <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> . Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> , the velocity of a bubble increases with bubble size for large values of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> but decreases with bubble size for small values of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . At large values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> , however, a bubble of large size appears to move at a slower velocity for any given value of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . When <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> is large (e.g. <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> = 100) and <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> > 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> makes a small bubble of radius greater than a quarter of the tube radius to deform into a noticeable oblate shape as <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> increases from very small value, and then to be elongated into a bullet shape with further increasing <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> after <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> reaches an intermediate value. Even very small bubbles (e.g. of radius equal to one-tenth of the tube radius) can still be significantly deformed provided that the value of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> is adequately large. Despite significant shape deformations that may still occur, bubbles of radius less than a quarter of that of the tube almost always move at the same velocity as that of the local liquid flow at the tube centreline (i.e. twice that of the average liquid velocity), regardless the values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> and <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . This fact suggests that very small bubbles are basically carried by the local liquid flow. </jats:p> Steady axisymmetric motion of a small bubble in a tube with flowing liquid Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
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| imprint |
The Royal Society, 2010 |
| imprint_str_mv |
The Royal Society, 2010 |
| issn |
1364-5021 1471-2946 |
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1364-5021 1471-2946 |
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2010 |
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The Royal Society |
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
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49 |
| title |
Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_unstemmed |
Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_full |
Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_fullStr |
Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_full_unstemmed |
Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_short |
Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_sort |
steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| topic |
General Physics and Astronomy General Engineering General Mathematics |
| url |
http://dx.doi.org/10.1098/rspa.2009.0288 |
| publishDate |
2010 |
| physical |
549-562 |
| description |
<jats:p>
The steady axisymmetric behaviour of a relatively small bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
and Reynolds number
<jats:italic>R</jats:italic>
<jats:italic>e</jats:italic>
. Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of
<jats:italic>R</jats:italic>
<jats:italic>e</jats:italic>
, the velocity of a bubble increases with bubble size for large values of
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
but decreases with bubble size for small values of
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
. At large values of
<jats:italic>R</jats:italic>
<jats:italic>e</jats:italic>
, however, a bubble of large size appears to move at a slower velocity for any given value of
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
. When
<jats:italic>R</jats:italic>
<jats:italic>e</jats:italic>
is large (e.g.
<jats:italic>R</jats:italic>
<jats:italic>e</jats:italic>
= 100) and
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
> 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large
<jats:italic>R</jats:italic>
<jats:italic>e</jats:italic>
makes a small bubble of radius greater than a quarter of the tube radius to deform into a noticeable oblate shape as
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
increases from very small value, and then to be elongated into a bullet shape with further increasing
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
after
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
reaches an intermediate value. Even very small bubbles (e.g. of radius equal to one-tenth of the tube radius) can still be significantly deformed provided that the value of
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
is adequately large. Despite significant shape deformations that may still occur, bubbles of radius less than a quarter of that of the tube almost always move at the same velocity as that of the local liquid flow at the tube centreline (i.e. twice that of the average liquid velocity), regardless the values of
<jats:italic>R</jats:italic>
<jats:italic>e</jats:italic>
and
<jats:italic>C</jats:italic>
<jats:italic>a</jats:italic>
. This fact suggests that very small bubbles are basically carried by the local liquid flow.
</jats:p> |
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| description | <jats:p> The steady axisymmetric behaviour of a relatively small bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> and Reynolds number <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> . Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> , the velocity of a bubble increases with bubble size for large values of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> but decreases with bubble size for small values of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . At large values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> , however, a bubble of large size appears to move at a slower velocity for any given value of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . When <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> is large (e.g. <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> = 100) and <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> > 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> makes a small bubble of radius greater than a quarter of the tube radius to deform into a noticeable oblate shape as <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> increases from very small value, and then to be elongated into a bullet shape with further increasing <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> after <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> reaches an intermediate value. Even very small bubbles (e.g. of radius equal to one-tenth of the tube radius) can still be significantly deformed provided that the value of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> is adequately large. Despite significant shape deformations that may still occur, bubbles of radius less than a quarter of that of the tube almost always move at the same velocity as that of the local liquid flow at the tube centreline (i.e. twice that of the average liquid velocity), regardless the values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> and <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . This fact suggests that very small bubbles are basically carried by the local liquid flow. </jats:p> |
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| spelling | Feng, James Q. 1364-5021 1471-2946 The Royal Society General Physics and Astronomy General Engineering General Mathematics http://dx.doi.org/10.1098/rspa.2009.0288 <jats:p> The steady axisymmetric behaviour of a relatively small bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> and Reynolds number <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> . Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> , the velocity of a bubble increases with bubble size for large values of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> but decreases with bubble size for small values of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . At large values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> , however, a bubble of large size appears to move at a slower velocity for any given value of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . When <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> is large (e.g. <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> = 100) and <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> > 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> makes a small bubble of radius greater than a quarter of the tube radius to deform into a noticeable oblate shape as <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> increases from very small value, and then to be elongated into a bullet shape with further increasing <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> after <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> reaches an intermediate value. Even very small bubbles (e.g. of radius equal to one-tenth of the tube radius) can still be significantly deformed provided that the value of <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> is adequately large. Despite significant shape deformations that may still occur, bubbles of radius less than a quarter of that of the tube almost always move at the same velocity as that of the local liquid flow at the tube centreline (i.e. twice that of the average liquid velocity), regardless the values of <jats:italic>R</jats:italic> <jats:italic>e</jats:italic> and <jats:italic>C</jats:italic> <jats:italic>a</jats:italic> . This fact suggests that very small bubbles are basically carried by the local liquid flow. </jats:p> Steady axisymmetric motion of a small bubble in a tube with flowing liquid Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
| spellingShingle | Feng, James Q., Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Steady axisymmetric motion of a small bubble in a tube with flowing liquid, General Physics and Astronomy, General Engineering, General Mathematics |
| title | Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_full | Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_fullStr | Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_full_unstemmed | Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_short | Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_sort | steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| title_unstemmed | Steady axisymmetric motion of a small bubble in a tube with flowing liquid |
| topic | General Physics and Astronomy, General Engineering, General Mathematics |
| url | http://dx.doi.org/10.1098/rspa.2009.0288 |