Stokesian dynamics

<h2 id="hydrodynamic-interaction">Hydrodynamic interaction</h2>
<p>When the motion on the particle scale is such that the particle Reynolds number is small, the hydrodynamic force exerted on the particles in a suspension undergoing a bulk linear shear flow is:
</p>

F
          
          
            
              H
            
          
        
        =
        −
        
          
            R
          
          
            
              F
              U
            
          
        
        (
        
          U
        
        −
        
          
            U
          
          
            ∞
          
        
        )
        +
        
          
            R
          
          
            
              F
              E
            
          
        
        :
        
          
            E
          
          
            ∞
          
        
        .
      
    
    {\displaystyle \mathbf {F} ^{\mathrm {H} }=-\mathbf {R} _{\mathrm {FU} }(\mathbf {U} -\mathbf {U} ^{\infty })+\mathbf {R} ^{\mathrm {FE} }:\mathbf {E} ^{\infty }.}

<p>Here, 
  
    
      
        
          
            U
          
          
            ∞
          
        
      
    
    {\displaystyle \mathbf {U} ^{\infty }}
  
 is the velocity of the bulk shear flow evaluated at the particle 
center, 
  
    
      
        
          
            E
          
          
            ∞
          
        
      
    
    {\displaystyle \mathbf {E} ^{\infty }}
  
 is the symmetric part of the velocity-gradient tensor;  
  
    
      
        
          
            R
          
          
            
              F
              U
            
          
        
      
    
    {\displaystyle \mathbf {R} _{\mathrm {FU} }}
  
 and 
  
    
      
        
          
            R
          
          
            
              F
              E
            
          
        
      
    
    {\displaystyle \mathbf {R} _{\mathrm {FE} }}
  
 are the configuration-dependent resistance matrices that give the hydrodynamic force/torque on the particles due to their motion relative to the fluid (
  
    
      
        
          
            R
          
          
            
              F
              U
            
          
        
      
    
    {\displaystyle \mathbf {R} _{\mathrm {FU} }}
  
) and due to the imposed shear flow (
  
    
      
        
          
            R
          
          
            
              F
              E
            
          
        
      
    
    {\displaystyle \mathbf {R} _{\mathrm {FE} }}
  
). Note that the subscripts on the matrices indicate the coupling   between kinematic (
  
    
      
        
          U
        
      
    
    {\displaystyle \mathbf {U} }
  
) and dynamic (
  
    
      
        
          F
        
      
    
    {\displaystyle \mathbf {F} }
  
) quantities.
</p><p>One of the key features of Stokesian dynamics is its handling of the hydrodynamic interactions, which is fairly accurate without being computationally inhibitive (like <a href="/facts/Boundary_element_method/FMIb4mLs">boundary integral methods</a>) for a large number of particles. Classical Stokesian dynamics requires 
  
    
      
        O
        (
        
          N
          
            3
          
        
        )
      
    
    {\displaystyle O(N^{3})}
  
 operations where <i>N</i> is the number of particles in the system (usually a periodic box). Recent advances have reduced the computational cost to about 
  
    
      
        O
        (
        
          N
          
            1.25
          
        
        
        log
        ⁡
        N
        )
        .
      
    
    {\displaystyle O(N^{1.25}\,\log N).}
  
<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="brownian-force">Brownian force</h2>
<p>The stochastic or Brownian force 
  
    
      
        
          
            F
          
          
            
              B
            
          
        
      
    
    {\displaystyle \mathbf {F} ^{\mathrm {B} }}
  
 arises from the thermal fluctuations in the fluid and is characterized by:
</p>

⟨
          
            
              F
            
            
              
                B
              
            
          
          ⟩
        
        =
        0
      
    
    {\displaystyle \left\langle \mathbf {F} ^{\mathrm {B} }\right\rangle =0}

⟨
          
            
              
                F
              
              
                
                  B
                
              
            
            (
            0
            )
            
              
                F
              
              
                
                  B
                
              
            
            (
            t
            )
          
          ⟩
        
        =
        2
        k
        T
        
          
            R
          
          
            
              F
              U
            
          
        
        δ
        (
        t
        )
      
    
    {\displaystyle \left\langle \mathbf {F} ^{\mathrm {B} }(0)\mathbf {F} ^{\mathrm {B} }(t)\right\rangle =2kT\mathbf {R} _{\mathrm {FU} }\delta (t)}

<p>The angle brackets denote an ensemble average, 
  
    
      
        k
      
    
    {\displaystyle k}
  
 is the Boltzmann constant, 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is the absolute temperature and 
  
    
      
        δ
        (
        t
        )
      
    
    {\displaystyle \delta (t)}
  
 is the delta function. The amplitude of the correlation between the Brownian forces at time 
  
    
      
        0
      
    
    {\displaystyle 0}
  
 and at time 
  
    
      
        t
      
    
    {\displaystyle t}
  
 results from the fluctuation-dissipation theorem for the N-body system.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Immersed_boundary_method/U60MK5sB">Immersed boundary methods</a></li>
<li><a href="/facts/Stochastic_Eulerian_Lagrangian_method/pL3BVT3c">Stochastic Eulerian Lagrangian methods</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Brady, John; Bossis, Georges (1988). "Stokesian Dynamics". Annu. Rev. Fluid Mech. 20: 111–157. Bibcode:1988AnRFM..20..111B. doi:10.1146/annurev.fl.20.010188.000551. <a href="https://resolver.caltech.edu/CaltechAUTHORS:BRAjfm88" target="_blank">https://resolver.caltech.edu/CaltechAUTHORS:BRAjfm88</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Seto, Ryohei; Romain Mari (2013). "Discontinuous Shear Thickening of Frictional Hard-Sphere Suspensions". Phys. Rev. Lett. 111 (21): 218301. arXiv:1306.5985. Bibcode:2013PhRvL.111u8301S. doi:10.1103/PhysRevLett.111.218301. PMID 24313532. S2CID 35020010. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Brady, John; Sierou, Asimina (2001). "Accelerated Stokesian Dynamics simulations" (PDF). Journal of Fluid Mechanics. 448 (1): 115–146. Bibcode:2001JFM...448..115S. doi:10.1017/S0022112001005912. S2CID 119505431. <a href="https://authors.library.caltech.edu/1580/1/SIEjfm01.pdf" target="_blank">https://authors.library.caltech.edu/1580/1/SIEjfm01.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Banchio, Adolfo J.; John F. Brady (2003). "Accelerated Stokesian dynamics: Brownian motion" (PDF). Journal of Chemical Physics. 118 (22): 10323. Bibcode:2003JChPh.11810323B. doi:10.1063/1.1571819. <a href="http://authors.library.caltech.edu/1841/1/BANjcp03.pdf" target="_blank">http://authors.library.caltech.edu/1841/1/BANjcp03.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Stokesian dynamics open-in-new

Stokesian dynamics