Stokes’ drag law for spherical particles

    The acceleration of a small particle \mathbf{a}_p in a fluid medium can be described by,

    \displaystyle \mathbf{a}_p = \frac{\mathbf{F}_p}{m_p}

    where F_p are the forces acting on the particle and m_p is the mass of the particle. For spherical particles, we consider a drag force due the the velocity difference of the particle and the fluid medium.

    \displaystyle \mathbf{F}_d = C_d r_p \mu\left(\mathbf{u}_f - \mathbf{v}_p\right),

    where C_d = 6\pi, the Stokes drag coefficient, r_p the particle radius , \mu is the fluid medium’s dynamic viscosity and \left(\mathbf{u}_f - \mathbf{v}_p\right) is the volocity difference between the undisturbed fluid flow and the particle. Furthermore, the acceleration of gravity (G) is considered to act on the particle.

    \displaystyle \mathbf{F}_p = \mathbf{F}_d + \mathbf{F}_G,

    \displaystyle \mathbf{F}_G = G \times V_p \left(\rho_p - \rho_f\right),

    with V_p the volume of the particle. This gives a formulation for one-way coupled partile-laden flow. It can be casted in an inertial particle formalism.


    It can be interesting to compute “inertial-particles” paths, alongside flow tracer-particles. Therfore, it is chosen to reuse the names of tracer-particle members for the mass density (u2.x) and the radius (u2.y). Note if tracer-particles.h are also used, that headerfile should be included first. Finally, if u2.y is not set u2.z can be used to set the relaxation time scale \tau_p. By default it is computed from using u2.y as r_p

    \displaystyle \tau = \frac{\rho_p r_p^2}{18\mu}

    #ifndef ADD_PART_MEM
    #define ADD_PART_MEM coord u; coord u2; long unsigned int tag;
    #define PA_INP particle * pp; double dt;
    #define INP    (PA_inp){p(), &p(), 2.*dt}
    #include "inertial-particles.h"

    For two-way couling, the forces on the particles are stored in a field.

    extern vector u;        //Fluid medium flow
    extern face vector mu;  //Fluid medium dynamic viscosity
    extern scalar rho;      //Fluid medium density
    coord G;                //Gravity acceleration vector (should it be external) ?
    coord p_acc (PA_inp inp) {
      particle pa = inp.p;
      double xp = pa.x; double yp = pa.y; double zp = pa.z;

    Implicity time discretization for v_p

    Since the flow-adjustment timescale \tau can be smaller than the solver time step, it makes sense to use an implicit equation for v_p,

    \displaystyle \frac{v_p^{n+1}-v_p^n}{\mathtt{dt}} = \frac{1}{\tau}\left(u - v_p^{n+1}\right) + g',

    which yields a weighted averaged between v_p^n and the ternimal velocity for v_p^{n+1}.

    \displaystyle v_p^{n+1} =\frac{\mathtt{dt}^{-1}v_p^n + \tau^{-1}u + g'}{\mathtt{dt}^{-1} + \tau^{-1}}.

    Rather than computing the acceleration, the velocity is directly updated in this function.

      double muc = is_constant (mu.x) ? constant(mu.x) : interpolate (mu.x, xp, yp, zp);
      double rp = pa.u2.y;
      double rhop = pa.u2.x;
      double tau  = rp ? (muc > 0 ? rhop * sq(2*rp)/(18*muc) : HUGE) : pa.u2.z ? pa.u2.z : HUGE;
      double itau = tau > 0 ? 1/tau : HUGE;
      double idt  = inp.dt > 0 ? 1./inp.dt: HUGE;
        inp.pp->u.x = (idt*pa.u.x + itau*interpolate(u.x, xp, yp ,zp))/(idt + itau);
      // Gravity
      double rhof = is_constant(rho) ? constant(rho) : interpolate (rho, xp, yp, zp);
        inp.pp->u.x += rhop ? (G.x*(rhop - rhof)/rhop)/(idt + itau) : 0;
      // No additional acceleration
      return (coord){0., 0., 0.};
    #if TWO_WAY
    event defaults (i = 0) {  
      if (is_constant(a.x)) {
        a = new face vector;
          a.x[] = 0.;
        boundary ((scalar *){a});
    void add_force (Particles P, vector F) {
      particle_boundary (P);
      double rhof = constant(rho);
      foreach_particle_in(P) {
        double muc = is_constant (mu.x) ? constant(mu.x) : interpolate (mu.x, x, y, z);
        double rp = p().u2.y;
        double pref = 6*pi*muc*rp;
        Point point = locate (x,y,z);
        double rhop = p().u2.x;
          F.x[] += pref*(p().u.x - interpolate(u.x, p().x, p().y ,p().z));
        pref = 4./3.*pi*cube(rp)*(rhop-rhof);
    event acceleration (i++) {
      face vector av = a;
      vector Fp[];
          Fp.x[] = 0;
      foreach_P_in_list (inertial_particles)
        add_force (P, Fp);  
      double rhof = constant(rho);
      boundary ((scalar*){Fp});
        av.x[] += face_value(Fp.x, 0)/(dv()*rhof);




    • Examplify with usages
    • Two-way coupling with the centered solver.
    • Test if inertial-particles can indeed co-exist with tracer-particles
    • Critical tests