View source for Squirmer

{{Short description|Model in fluid dynamics}}
[[File:Shaker pusher.png|thumb|160px|Spherical microswimmer in Stokes flow]]

The '''squirmer''' is a model for a spherical [[microswimmer]] swimming in [[Stokes flow]]. The squirmer model was introduced by [[James Lighthill]] in 1952 and refined and used to model [[Paramecium]] by John Blake in 1971.<ref name="Lighthill1952">{{cite journal|last1=Lighthill|first1=M. J.|title=On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers|journal=Communications on Pure and Applied Mathematics|volume=5|issue=2|year=1952|pages=109–118|issn=0010-3640|doi=10.1002/cpa.3160050201}}</ref>
<ref name="Blake1971">{{cite journal|last1=Blake|first1=J. R.|title=A spherical envelope approach to ciliary propulsion|journal=Journal of Fluid Mechanics|volume=46|issue=1|year=1971|pages=199–208|issn=0022-1120|doi=10.1017/S002211207100048X|bibcode=1971JFM....46..199B|s2cid=122519123 }}</ref>
Blake used the squirmer model to describe the flow generated by a carpet of beating short filaments called [[cilia]] on the surface of Paramecium. Today, the squirmer is a standard model for the study of [[self-propelled particles]], such as [[Janus particles]], in Stokes flow.<ref name="BickelMajee2013">{{cite journal|last1=Bickel|first1=Thomas|last2=Majee|first2=Arghya|last3=Würger|first3=Alois|title=Flow pattern in the vicinity of self-propelling hot Janus particles|journal=Physical Review E|volume=88|issue=1|year=2013|issn=1539-3755|doi=10.1103/PhysRevE.88.012301|arxiv=1401.7311|bibcode=2013PhRvE..88a2301B|pmid=23944457|page=012301|s2cid=36558271 }}</ref>

== Velocity field in particle frame ==

Here we give the flow field of a squirmer in the case of a non-deformable [[Rotational symmetry#Rotational symmetry with respect to any angle|axisymmetric]] spherical squirmer (radius <math>R</math>).<ref name="Lighthill1952" /><ref name="Blake1971" /> These expressions are given in a [[spherical coordinate system]].

<math>
 u_r(r,\theta)=\frac 2 3 \left(\frac{R^3}{r^3} -1\right)B_1P_1(\cos\theta)+\sum_{n=2}^{\infty}\left(\frac{R^{n+2}}{r^{n+2}}-\frac{R^n}{r^n}\right)B_nP_n(\cos\theta)\;,
</math>
<br />
<math>
 u_{\theta}(r,\theta)=\frac 2 3 \left(\frac{R^3}{2r^3}+1\right)B_1V_1(\cos\theta)+\sum_{n=2}^{\infty}\frac 1 2\left(n\frac{R^{n+2}}{r^{n+2}}+(2-n)\frac{R^n}{r^n}\right)B_nV_n(\cos\theta)\;.
</math>

Here <math>B_n</math> are constant coefficients, <math>P_n(\cos\theta)</math> are [[Legendre polynomials]], and <math>V_n(\cos\theta)=\frac{-2}{n(n+1)}\partial_{\theta}P_n(\cos\theta)</math>. <br />
One finds <math>P_1(\cos\theta)=\cos\theta, P_2(\cos\theta)=\tfrac 1 2 (3\cos^2\theta-1), \dots, V_1(\cos\theta)=\sin\theta, V_2(\cos\theta)= \tfrac{1}{2} \sin 2\theta, \dots</math>.<br />
The expressions above are in the frame of the moving particle. At the interface one finds <math>u_{\theta}(R,\theta)=\sum_{n=1}^{\infty} B_nV_n</math> and <math>u_r(R,\theta)=0</math>.

{| style="float: left; border: 1px solid #BBB; margin: .46em 0 0 .2em;"
|-
| [[File:Shaker pusher.png|thumb|Shaker, <math>\beta=-\infty</math>|100px]] || [[File:Pusher squirmer, lab frame.png|thumb|Pusher, <math>\beta=-5</math>|100px]] || [[File:Neutral squirmer, lab frame.png|thumb|Neutral, <math>\beta=0</math>|100px]]|| [[File:Puller squirmer, lab frame.png|thumb|Puller, <math>\beta=5</math>|100px]] || [[File:Shaker puller.png|thumb|Shaker, <math>\beta=\infty</math>|100px]] || [[File:Passive particle, lab frame.png|thumb|Passive particle|100px]]
|-
| [[File:Shaker pusher.png|thumb|Shaker, <math>\beta=-\infty</math>|100px]] || [[File:Pusher squirmer, swimmer frame.png|thumb|Pusher, <math>\beta=-5</math>|100px]] || [[File:Neutral squirmer, swimmer frame.png|thumb|Neutral, <math>\beta=0</math>|100px]] || [[File:Puller squirmer, swimmer frame.png|thumb|Puller, <math>\beta=5</math>|100px]]|| [[File:Shaker puller.png|thumb|Shaker, <math>\beta=\infty</math>|100px]] || [[File:Passive particle, particle frame.png|thumb|Passive particle|100px]]
|-
| colspan="6" style="text-align: center;" | Velocity field of squirmer and passive particle (top row: lab frame, bottom row: swimmer frame, <math> \beta = B_2/|B_1| </math> ).
|}

{{clear}}

== Swimming speed and lab frame ==

By using the [[Stokes flow#Lorentz reciprocal theorem|Lorentz Reciprocal Theorem]], one finds the velocity vector of the particle <math>\mathbf{U}=-\tfrac{1}{2} \int \mathbf{u}(R,\theta)\sin\theta\mathrm{d}\theta=\tfrac 2 3 B_1 \mathbf{e}_z</math>. The flow in a fixed lab frame is given by <math>\mathbf{u}^L=\mathbf{u}+\mathbf{U}</math>:

<math>
 u_r^L(r,\theta)=\frac{R^3}{r^3}UP_1(\cos\theta)+\sum_{n=2}^{\infty}\left(\frac{R^{n+2}}{r^{n+2}}-\frac{R^n}{r^n}\right)B_nP_n(\cos\theta)\;,
</math>
<br />
<math>
 u_{\theta}^L(r,\theta)=\frac{R^3}{2r^3}UV_1(\cos\theta)+\sum_{n=2}^{\infty}\frac 1 2\left(n\frac{R^{n+2}}{r^{n+2}}+(2-n)\frac{R^n}{r^n}\right)B_nV_n(\cos\theta)\;.
</math>

with swimming speed <math>U=|\mathbf{U}|</math>. Note, that <math>\lim_{r\rightarrow\infty}\mathbf{u}^L=0</math> and <math>u^L_r(R,\theta)\neq 0</math>.

== Structure of the flow and squirmer parameter ==

The series above are often truncated at <math>n=2</math> in the study of far field flow, <math>r\gg R</math>. Within that approximation, <math>u_{\theta}(R,\theta)=B_1\sin\theta+\tfrac 1 2 B_2 \sin 2 \theta</math>, with squirmer parameter <math>\beta=B_2/|B_1|</math>. The first mode <math>n=1</math> characterizes a hydrodynamic source dipole with decay <math>\propto 1/r^3</math> (and with that the swimming speed <math>U</math>). The second mode <math>n=2</math> corresponds to a hydrodynamic [[Faxén's law|stresslet]] or force dipole with decay <math>\propto 1/r^2</math>.<ref name="HappelBrenner1981">{{cite book|title=Low Reynolds number hydrodynamics|last1=Happel|first1=John|last2=Brenner|first2=Howard|series=Mechanics of fluids and transport processes |year=1981|volume=1 |issn=0921-3805|doi=10.1007/978-94-009-8352-6|isbn=978-90-247-2877-0 }}</ref> Thus, <math>\beta</math> gives the ratio of both contributions and the direction of the force dipole. <math>\beta</math> is used to categorize microswimmers into pushers, pullers and neutral swimmers.<ref name="DowntonStark2009">{{cite journal|last1=Downton|first1=Matthew T|last2=Stark|first2=Holger|title=Simulation of a model microswimmer|journal=Journal of Physics: Condensed Matter|volume=21|issue=20|year=2009|pages=204101|issn=0953-8984|doi=10.1088/0953-8984/21/20/204101|pmid=21825510 |bibcode=2009JPCM...21t4101D|s2cid=35850530 }}</ref>

{| class="wikitable"
|-
| '''Swimmer Type''' || pusher || neutral swimmer || puller || shaker || passive particle
|-
| '''Squirmer Parameter''' || <math>\beta<0</math> || <math>\beta=0</math> || <math>\beta>0</math> || <math>\beta=\pm\infty</math> ||
|-
| '''Decay of Velocity Far Field''' || <math>\mathbf{u}\propto 1/r^2</math> || <math>\mathbf{u}\propto 1/r^3</math> || <math>\mathbf{u}\propto 1/r^2</math> || <math>\mathbf{u}\propto 1/r^2</math> || <math>\mathbf{u}\propto 1/r</math>
|-
| '''Biological Example''' || ''[[E.Coli]]'' || ''[[Paramecium]]'' || ''[[Chlamydomonas reinhardtii]]'' || ||
|}

The above figures show the velocity field in the lab frame and in the particle-fixed frame. The hydrodynamic dipole and quadrupole fields of the squirmer model result from surface stresses, due to beating cilia on bacteria, or chemical reactions or thermal non-equilibrium on Janus particles. The squirmer is force-free. On the contrary, the velocity field of the passive particle results from an external force, its far-field corresponds to a "stokeslet" or hydrodynamic monopole. A force-free passive particle doesn't move and doesn't create any flow field.  

==See also==
* [[Protist locomotion]]

== References ==
{{Reflist}}

[[Category:Fluid dynamics]]