On the long-time persistence of hydrodynamic memory

Eur. Phys. J. E (2021)44:141 https://doi.org/10.1140/epje/s10189-021-00151-5 THE EUROPEAN PHYSICAL JOURNAL E Regular Article - Flowing Matter On the long-time persistence of hydrodynamic memory Miguel Villegas Dı́aza Departamento de fı́sica Aplicada, Facultad de Ingenierı́a, Universidad Central de Venezuela, Caracas, Venezuela Received 24 August 2021 / Accepted 16 November 2021 © The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract The Basset–Boussinesq–Oseen (BBO) equation correctly describes the nonuniform motion of a spherical particle at a low Reynolds number. It contains an integral term with a singular kernel which accounts for the diffusion of vorticity around the particle throughout its entire history. However, if there are any departures in either rigidity or shape from a solid sphere, besides the integral force with a singular kernel, the Basset history force, we should add a second history force with a non-singular kernel, related to the shape or composition of the particle. In this work, we introduce a fractional generalized Basset– Boussinesq–Oseen equation which includes both history terms as fractional derivatives. Using the Laplace transform, an integral representation of the solution is obtained. For a driven single particle, the solution shows that memory effects persist indefinitely under rather general driving conditions. 1 Introduction The study of viscous particle motion was initiated by Stokes [1], who determined the force acting on a small fixed particle that is subjected to a uniform fluid velocity at a low Reynolds number. Boussinesq [2] and Basset [3] independently extended the work of Stokes by considering the case where a spherical particle accelerates through the fluid due to a constant gravitational force but still neglecting nonlinear effects. They found that the hydrodynamic force F acting on a spherical particle undergoing arbitrary time-dependent motion in an otherwise quiescent fluid is 2 R2 1 du(t) F(t) = −6 πμR u(t) − π ρ R3 − 6πμR( )2 3 dt πν  t 1 du(τ ) √ dτ, (1) t − τ dτ 0 where ρ is the density, μ and ν = μρ are the dynamic and kinematic viscosities, respectively, u(t) is the particle velocity, R is the particle radius and t represents the time. The first term is the pseudo-steady Stokes drag. The second term, a purely inertial contribution, is the so-called added mass term. It represents the additional mass the particle appears to have due to the resistance to the acceleration of the surrounding fluid. The third term is the Basset memory integral, which depends on the history of particle motion. It is a combination of both viscous and inertial contributions to the force in a e-mail: (corresponding author) 0123456789().: V,-vol that it depends on both the viscosity of the fluid and the acceleration of the particle. Oseen [4] extended the work of Boussinesq and Basset, including the effects of higher Reynolds number on the equations. Due to the original contributions of Boussinesq, Basset, and Oseen, the particle equation of motion with a constant forcing (the gravity term) is sometimes referred to as the BBO equation. Lawrence and Weinbaum examined the force on a slightly nonspherical [5] solid body and for a spheroidal rigid body of arbitrary aspect ratio [6] in a timedependent uniform flow at a low Reynolds number. They concluded that there is a second memory integral term, in addition to the Basset-like term, when the body is nonspherical. For an oblate spheroid with semiaxis: a = b(1 + ), the solution for general motion is 2πμa3 am 3v  1 t 6 π 2 μ a2 du 1 du √ − dτ aB 1 dt t − τ dτ v2 0  t 1    6 π 2 μ a2 − Im aLW exp β (t − τ ) erfc 1 v2 0   12  du  dτ , (2) β (t − τ ) dτ F = −6πμ a as u − where the coefficients as , am , aB , and aLW are given by  37 2  1 as = 1 +  +  , 5 175  26 2  1 aB = 1 +  +  , 5 175  81 2  2  , am = 1 +  + 5 175 1 8 2 ( πβ) 2 , (3) aLW = 175 123 141 Page 2 of 10 √   where β = 32 1 + i 3 , v = 43 π (1 + )2 is the particle’s volume and erfc is the error function complementary. Following the pioneering works [5,6], several articles have shown [7–12] that the history force given by the Basset memory integral is only valid for the particular case of a rigid sphere. Recently, [13] studying the BBO equation describing the motion of a driven single particle discovered numerically that memory effects persist indefinitely under rather general driving conditions, thus showing that neglecting the history force can lead to qualitatively incorrect particle transport under general nonequilibrium conditions. An accurate numerical investigation analyzing spherical Brownian particles was carried out in [14] and concluded that hydrodynamic memory effects can be very profound in driven nonlinear diffusion processes. One of the earliest theoretical and experimental attempts to try to elucidate the significance of the Basset history force was carried out by Leichtberg et al. [15]. They examined the gravitational–hydrodynamical interaction between three or more spheres falling along a common axis and concluded that for slowly changing multiparticle gravitational motions. The Basset force is the most relevant inertial effect at low but nonzero Reynolds numbers. Experiments to try to understand the importance of the history force in the presence of gravity were carried out in [16], who consider particle motion in a fluid at rest. On the other hand, recently experimental observations of power-law temporal response for spheres in the transient regime of low Reynolds number flow under the effect of gravity were provided by [17]. This result is consistent with the generalized fractional-order Basset force proposed in Ref. [18]. Experiments on bubble dynamics in a standing wave were reported in [19,20]. However, [19,20] only consider the non-singular memory integral in their studies. On the other hand, [16] considered a spherical particle. Analytical efforts were carried out by [21,22], but also for spherical particles, thus neglecting the possible effects of the non-singular memory integral force. Here we are interested in studying the influence of both memory integral terms in the dynamics of a particle. The problem of modifying the BBO equation for the case of a uniform but time-dependent free-stream flow field was examined by Tchen [23]. Tchen’s equation is valid for rigid spherical particles, and very small bubbles without surface motion, in the limit of infinitesimal Reynolds number. The resulting equation relates the transient acceleration of the particle to the time-dependent free-stream or background flow velocity. Maxey and Riley [24] extended the BBO equation to conditions where the flow far from the particle was other than uniform. The equation is a second-order, implicit integrodifferential equation with a singular kernel. The resulting equation is valid for spherical solid particles in the limit of infinitesimal Reynolds numb (...truncated)