Variable Order Modeling of Diffusive-convective Effects on the Oscillatory Flow Past a Sphere

Abstract

This work advances our understanding of the drag force acting on a particle due to the oscillatory flow of a viscous fluid with finite Reynolds and Strouhal numbers. The drag force is is determined using the novel concept of variable order (VO) calculus, where the order of derivative can vary with the parameters and variables, according to the dynamics of the flow. Using the VO formulation we determine: (i) The region of validity of Tchen's equation for oscillatory flow, (ii) the region where the order of the derivative is fractional but constant, and (iii) the region where the strong non-linearity of the flow requires a variable order derivative to account for the increased complexity of the flow.

Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation

“…Still, the generalized solution is not required to be from the functional space (32) that, together with the relations (25)- (27), means that u k , k = 1, 2, . .…”

Variable-order fractional differential operators in anomalous diffusion modeling

“…Finally, the behavior of these diffusion or transport processes in response to system parameter changes can be better described using VO elements rather than time or space varying coefficients. Some applications of fractional operator also imply that the derivative order perhaps is not a constant, but a function of system parameters [27].…”

“…Ingman et al employed the time dependent VO operator to model the viscoelastic deformation process [25,26]. Pedro et al studied the motion of particles suspended in a viscous fluid with drag force is determined using the VO calculus [27]. Kobelev et al investigated the statistical and dynamical systems with fixed and variable memories, the fractal dimension of considered system is variable with time and coordinate [28].…”

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

“…Moreover, a physical experimental study of variable-order operators has been considered in [31]. Variable-order fractional derivatives can be used in modeling anomalous diffusion, as they can depict the time-dependent diffusion process more efficiently than fractional derivatives of constant order, see [22,32]. The effect of differences between using constant and variable-order fractional derivatives has been discussed in [33].…”

“…They introduced the study of fractional integration and differentiation when the order is a function rather than a constant of arbitrary order [20,21]. Afterward, many efforts in the field of mathematics and physics have been devoted to study physical problems described by the variable-order derivative (see, for example, [22][23][24][25]). …”