An experimental study on stationary history effects in high-frequency Stokes flows

Abstract

We report results of a series of detailed experiments designed to unveil the dynamics of a particle of radius $a$ moving in high-frequency, low-Reynolds-number oscillatory flow. The fundamental parameters in the problem are the Strouhal ($\hbox{\it Sl}$) and the particle Reynolds numbers ($\hbox{\it Re}_p$), as well as the fluid-to-particle density ratio $\alpha$. The experiments were designed to cover a range of $\hbox{\it Sl} \hbox{\it Re}_p$ from 0.015 to 5 while keeping $\hbox{\it Re}_p < 0.5$ and $\hbox{\it Sl} > 1$. The primary objective of the experiments is to investigate stationary history effects associated with the Basset drag, which are maximized when the viscous time scale $a^2/\nu$ is of the same order of the flow time scale $9/\Omega$, where $9$ is a geometrical factor for the sphere, $\nu$ is the kinematic viscosity and $\Omega$ is the angular frequency of the background flow. The theoretically determined behaviour of stationary history effects is confirmed unequivocally by the experiments, which also validate the fractional derivative behaviour (of order $1/2$) of the history drag for the range of parameters under study.

On the variety of particle accumulation structures under the effect of g-jitters

“…The latter authors, in particular, found the relative importance of history force for 1 to be particularly significant for St v 0.3 (more specifically, it was found to attain a maximum value for a dimensional frequency approaching the value 9/ 2 R and displacement amplitudes b smaller than the radius R of the particle, conditions that using the present reference units would read: 2PrSt -1 =1.6x10 5 and / 2 (9St/2) 1/2 2x10 -2 ). Albeit the importance of the Basset term was observed to decrease rapidly as the value of St v approached 0 for all values of  (as confirmed later in the experiment by Coimbra et al, 2004), however, they found the ratio of the Basset force to the overall force experienced by the particle (virtual mass + stokes drag + history term) to be 0.2 (see Figure 1 of their work) for a value of their control parameter S=0.06, which corresponds to the present 10 4 . As the studies by Vojir and Michaelides (1994) and Coimbra and Rangel (2001), both indicate that the Basset term may play a potentially significant role for the conditions considered in the present study when >5x10 3 , it has been therefore deemed necessary to assess expressly its influence on PAS for relatively high frequencies (the results of this study will be discussed in Sect.…”

Fractional model for malaria transmission under control strategies

“…Important applications of FC can be found in many areas of science, from electrochemistry [20], to physics [21,22], fluid mechanics [23], mechanical systems [24], other areas of engineering [25][26][27][28], and biology [29][30][31].…”

Dynamics and Control of Nonlinear Variable Order Oscillators

“…The increased interest in fractional systems in the past few decades is due mainly to a large body of physical evidence describing fractional order behavior in diverse areas such as fluid mechanics, mechanical systems, rheology, electromagnetism, quantitative finances, electrochemistry, and biology. Fractional order modeling provides exceptional capabilities for analysing memory-intense and delay systems and it has been associated with the exact description of complex transport phenomena such as fractional history effects in the unsteady viscous motion of small particles in suspension (Coimbra et al 2004, L'Esperance et al 2005. Although fractional order dynamical and control systems were studied only marginally until a few decades ago, the recent development of effective mathematical methods of integration of non-integer order differential equations (Charef et al (1992); Coimbra & Kobayashi (2002), Diethelm et al (2002); Momany (2006), Diethelm et al (2005)) has resulted in a number of control schemes and algorithms, many of which have shown better performance and disturbance rejection compared to other traditional integer-order controllers (Podlubni (1999); Hartly & Lorenzo (2002), Ladaci & Charef (2006), among others).…”