A Non-inertial Model for Particle Aggregation Under Turbulence

We consider an abstract non-inertial model of aggregation under the influence of a Gaussian white noise with prescribed space-covariance, and prove a formula for the mean collision rate <i>R</i>, per unit of time and volume. Specializing the abstract theory to a non-inertial model obtained by an inertial one, with physical constants, in the limit of infinitesimal relaxation time of the particles, and the white noise obtained as an approximation of a Gaussian noise with correlation time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi> <mml:mi>η</mml:mi></mml:msub> </mml:math> , up to approximations the formula reads <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi> <mml:mo>∼</mml:mo> <mml:msub><mml:mi>τ</mml:mi> <mml:mi>η</mml:mi></mml:msub> <mml:mfenced> <mml:msup> <mml:mfenced><mml:msub><mml:mi>Δ</mml:mi> <mml:mi>a</mml:mi></mml:msub> <mml:mi>u</mml:mi></mml:mfenced> <mml:mn>2</mml:mn></mml:msup> </mml:mfenced> <mml:mi>a</mml:mi> <mml:mo>·</mml:mo> <mml:msup><mml:mi>n</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:mrow> </mml:math> where <i>n</i> is the particle number per unit of volume and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:msup> <mml:mfenced><mml:msub><mml:mi>Δ</mml:mi> <mml:mi>a</mml:mi></mml:msub> <mml:mi>u</mml:mi></mml:mfenced> <mml:mn>2</mml:mn></mml:msup> </mml:mfenced> </mml:math> is the square-average of the increment of random velocity field <i>u</i> between points at distance <i>a</i>, the particle radius. If we choose the Kolmogorov time scale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>τ</mml:mi> <mml:mi>η</mml:mi></mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mfenced><mml:mfrac><mml:mi>ν</mml:mi> <mml:mi>ε</mml:mi></mml:mfrac> </mml:mfenced> <mml:mrow><mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:msup> </mml:mrow> </mml:math> and we assume that <i>a</i> is in the dissipative range where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:msup> <mml:mfenced><mml:msub><mml:mi>Δ</mml:mi> <mml:mi>a</mml:mi></mml:msub> <mml:mi>u</mml:mi></mml:mfenced> <mml:mn>2</mml:mn></mml:msup> </mml:mfenced> <mml:mo>∼</mml:mo> <mml:mfenced><mml:mfrac><mml:mi>ε</mml:mi> <mml:mi>ν</mml:mi></mml:mfrac> </mml:mfenced> <mml:msup><mml:mi>a</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:mrow> </mml:math> , we get Saffman-Turner formula for the collision rate <i>R</i>.