page_title: Stokes' law
section_title: Transversal Flow around a sphere
hierarchical_section_title: Stokes' law / Derivation / Transversal Flow around a sphere
caption_attribution_description: English: Flow around sphere
context_page_description: In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
context_section_description: For the case of a sphere in a uniform far field flow, it is advantageous to use a cylindrical coordinate system ( r , φ , z ). The z–axis is through the centre of the sphere and aligned with the mean flow direction, while r is the radius as measured perpendicular to the z–axis. The origin is at the sphere centre. Because the flow is axisymmetric around the z–axis, it is independent of the azimuth φ.
  In this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream function ψ, depending on r and z:
  with uᵣ and uz the flow velocity components in the r and z direction, respectively. The azimuthal velocity component in the φ–direction is equal to zero, in this axisymmetric case. The volume flux, through a tube bounded by a surface of some constant value ψ, is equal to 2π ψ and is constant.
  For this case of an axisymmetric flow, the only non-zero component of the vorticity vector ω is the azimuthal φ–component ωφ
  The Laplace operator, applied to the vorticity ωφ, becomes in this cylindrical coordinate system with axisymmetry:
  From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity u in the z–direction and a sphere of radius R, the solution is found to be
  The solution of velocity in cylindrical coordinates and components follows as:
  The solution of vorticity in cylindrical coordinates follows as:
  The solution of pressure in cylindrical coordinates follows as: 
  The solution of pressure in spherical coordinates follows as: 
  The formula of pressure is also called dipole-potential in analogues to electrostatics. 
  A more general formulation, with arbitrary far-field velocity-vector , in cartesian coordinates  follows with:
  In this formulation the non-conservative term represents a kind of so called Stokeslet. The Stokeslet is the Greens' function of the Stokes-Flow-Equations. The conservative term is equal to the dipole-gradient-field. The formula of vorticity is a kind of Biot-Savart-Formula, which is also used in electromagnetism.
  The following formula describes the viscous-stress-tensor for the special case of stokes-flow. It's needed in the calculation of the force acting on the particle. In cartesian coordinates the vector-gradient  is identical to the jacobian-matrix. The matrix  represents the identity-matrix. 
  The force acting on the sphere is to calculate by surface-integral, where  represents the radial unit-vector of spherical-coordinates:
caption: Stokes-Flow around sphere with parameters of Far-Field velocity 
    
      
        
          
            
              u
            
            
              ∞
            
          
          =
          
            
              
                (
                
                  
                    
                      6
                    
                    
                      0
                    
                    
                      6
                    
                  
                
                )
              
            
            
              T
            
          
          
            m/s
          
        
      
      {\displaystyle \mathbf {u} _{\infty }={\begin{pmatrix}6&0&6\end{pmatrix}}^{T}{\text{m/s}}}
    
  , radius of sphere 
    
      
        
          R
          =
          1
          
          
            m
          
        
      
      {\displaystyle R=1\;{\text{m}}}
    
  , viscosity of water (T = 20°C) 
    
      
        
          μ
          =
          1
          
          
            mPa
          
          ⋅
          
            s
          
        
      
      {\displaystyle \mu =1\;{\text{mPa}}\cdot {\text{s}}}
    
  . Shown are the field-lines of velocity-field and the amplitudes of velocity, pressure and vorticity with pseudo-colors.
exif: {}