Fractional Transport Model Research Articles

Performance Analysis of Fully Intuitionistic Fuzzy Multi-Objective Multi-Item Solid Fractional Transportation Model

An investigation is conducted in this paper into a performance analysis of fully intuitionistic fuzzy multi-objective multi-item solid fractional transport model (FIF-MMSFTM). It is to be anticipated that the parameters of the conveyance model will be imprecise by virtue of numerous uncontrollable factors. The model under consideration incorporates intuitionistic fuzzy (IF) quantities of shipments, costs and profit coefficients, supplies, demands, and transport. The FIF-MMSFTM that has been devised is transformed into a linear form through a series of operations. The accuracy function and ordering relations of IF sets are then used to reduce the linearized model to a concise multi-objective multi-item solid transportation model (MMSTM). Furthermore, an examination is conducted on several theorems that illustrate the correlation between the FIF-MMSFTM and its corresponding crisp model, which is founded upon linear, hyperbolic, and parabolic membership functions. A numerical example was furnished to showcase the efficacy and feasibility of the suggested methodology. The numerical data acquired indicates that the linear, hyperbolic, and parabolic models require fewer computational resources to achieve the optimal solution. The parabolic model has the greatest number of iterations, in contrast to the hyperbolic model which has the fewest. Additionally, the elapsed run time for the three models is a negligible amount of time: 0.2, 0.15, and 1.37 s, respectively. In conclusion, suggestions for future research are provided.

Analysis of anomalous transport with temporal fractional transport equations in a bounded domain

Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations. The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α (α ∈ (0,1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result, if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.

Modelling fungal growth with fractional transport models

Due to the network-shaped colony formed by the filamentous fungi, fractional operators are likely to capture the time-evolution of their biomass distribution. In this paper, a generalised fractional transport model is developed to simulate the colony growth of a wood-rot fungus, Postia placenta. The colony is described by two variables, active and inactive biomass. The active biomass corresponds to the tips and the “active points”, responsible for biomass formation, while the inactive component represents the remaining biomass, which is assumed immobile. The fractional in time and space derivatives are applied to the active biomass to represent the spatial colonisation (i.e., tip movement). The proliferation of biomass (local densification of the network due to branching) is driven by a source term, while a portion of active biomass becomes inactive over time. The model is solved using an extended finite volume discretisation scheme and the accuracy is confirmed by comparison to an analytic solution obtained for the case where the source term is assumed linear. Next, the model parameters are identified for an optimized solution of the model that matches experimental observations well, indicating the suitability of the fractional operators for modelling biomass diffusion in a fungal colony. An interesting finding is that the best results are obtained when only anomalous diffusion in space (not time) is considered, which is probably related to the fractal dimension of mycelia. The spatial fractional–order derivative provides a unique ability to capture the non-local biomass diffusion of the fungal colony. We postulate that the fractional indices may provide a form of biological marker for the growth characteristics of different fungal species.

Field and Numerical Investigation of Transport Mechanisms in a Surface Storage Zone

In‐stream surface storage zones (SSZs) caused by lateral recirculation areas play a significant role in the transport and fate of contaminants in rivers. Lateral recirculating areas have long residence times that favor nutrient uptake, accumulation of pollutants, and interactions with reactive sediments. In watersheds affected by acid‐mine drainage, SSZs have profound effects on biogeochemical processes, controlling the local concentration and distribution of toxic elements along the channel. Despite the importance of turbulent flow dynamics on these processes, limited work has been carried out to analyze mass transport in natural SSZs with complex geometries. In this investigation we study a SSZ in the Lluta River, located in a high‐altitude environment in northern Chile, by coupling field measurements and 3‐D numerical simulations to understand the transport mechanisms with the main channel. We measure the velocity field using an acoustic Doppler velocimeter (ADV) and large‐scale particle image velocimetry (LSPIV), extracting the bathymetry from digital image processing. Using these data, we perform detached‐eddy simulations (DES) to analyze the mean flow, turbulence statistics, and the dynamics of large‐scale coherent structures. From this detailed description of the turbulent flow, we study the mass exchange and the time evolution of the mean concentration of a passive scalar in the SSZ by testing three upscaled models: a classical linear transport model, a two‐storage formulation, and a fractional transport model. The analysis integrates temporal and spatial scales to provide a new perspective on the turbulent flow in SSZs and their effects on global mass transport in rivers.

Comparison of a radial fractional transport model with tokamak experiments

A radial fractional transport model [Kullberg et al., Phys. Rev. E 87, 052115 (2013)], that correctly incorporates the geometric effects of the domain near the origin and removes the singular behavior at the outer boundary, is compared to results of off-axis heating experiments performed in the Rijnhuizen Tokamak Project (RTP), ASDEX Upgrade, JET, and DIII-D tokamak devices. This comparative study provides an initial assessment of the presence of fractional transport phenomena in magnetic confinement experiments. It is found that the nonlocal radial model is robust in describing the steady-state temperature profiles from RTP, but for the propagation of heat waves in ASDEX Upgrade, JET, and DIII-D the model is not clearly superior to predictions based on Fick's law. However, this comparative study does indicate that the order of the fractional derivative, α, is likely a function of radial position in the devices surveyed.

Directional transport of fractional asymmetric coupling system in symmetric periodic potential

Based on the fractional calculus theory, in the absence of external driving force, the fractional transport model of asymmetric coupling particle chain in symmetric periodic potential is established. Using the method of fractional difference, the model is solved numerically and the influences of the various system parameters on directional transport velocity are discussed. Numerical results show that in the case without external force and noise-driven, the fractional asymmetric coupling system can still generate directional transport, and the transport velocity increases as fractional order increases. When the fractional order is fixed, the average velocity of the particle chain varies non-monotonically with coupling strength and barrier height. In the case with noise, the generalized stochastic resonance phenomenon occurs. Besides, we can make the noise not affect the system or even promote directional transport by adjusting other parameters.

Spontaneous formation and degradation of pool‐riffle morphology and sediment sorting using a simple fractional transport model

Many gravel bed streams have a typical bed morphology consisting of pool‐riffle sequences, which provides important habitat diversity both in terms of flow and substrate. A complete explanation of pool‐riffle genesis and self‐maintenance remains elusive and, despite advances in understanding the effects of flow spatial and temporal variability, the key sediment processes have been only marginally explored. Here we use a 1D unsteady multi‐fraction morphodynamic model to explain the formation and degradation of pool‐riffle sequences. Using a 1‐year time series of measured flows below bankfull on a stream in which we have removed initial bedforms and sediment sorting our model spontaneously generates pools with finer substrate at narrow sections and riffles with coarser sediment at wider sections, closely resembling the natural bed morphology. Additional experiments show that under our modelling assumptions a variable flow regime is fundamental for development and self‐maintenance of the longitudinal grain sorting characteristic of pool‐riffle sequences, which could not be obtained or maintained with discharges held constant over relatively long periods.

A nonlocal theory of sediment transport on hillslopes

Hillslopes are typically shaped by varied processes which have a wide range of event‐based downslope transport distances, some of the order of the hillslope length itself. We hypothesize that this can lead to a heavy‐tailed distribution of displacement lengths for sediment particles. Here, we propose that such a behavior calls for a nonlocal computation of the sediment flux, where the sediment flux at a point is not strictly a function (linear or nonlinear) of the gradient at that point only but is an integral flux taking into account the upslope topography (convolution Fickian flux). We encapsulate this nonlocal behavior in a simple fractional diffusive model which involves fractional derivatives, with the order of differentiation (1 < α ≤ 2) dictating the degree of nonlocality (α = 2 corresponds to linear diffusion and strictly local dependence on slope). The model predicts an equilibrium hillslope profile which is parabolic close to the ridgetop and transits, at a short downslope distance, to a power law with an exponent equal to the parameter α of the fractional transport model. Hillslope profiles reported in previously studied sites support this prediction. Furthermore, we show that the nonlocal transport model gives rise to a nonlinear dependency on local slope and that variable upslope topography leads to widely varying rates of sediment flux for a given local hillslope gradient. Both of these results are consistent with available field data and suggest that nonlinearity in hillslope flux relationships may arise in part from nonlocal transport effects in which displacement lengths increase with hillslope gradient. The proposed hypothesis of nonlocal transport implies that field studies and models of sediment fluxes should consider the size and displacement lengths of disturbance events that mobilize hillslope colluvium.

Parameter estimation for fractional transport: A particle‐tracking approach

Space‐fractional advection‐dispersion models provide attractive alternatives to the classical advection‐dispersion equation for model applications that exhibit early arrivals and plume skewness. This paper develops a flexible method for estimating the parameters of the fractional transport model on the basis of spatial plume snapshots or temporal breakthrough curve data. A particle‐tracking approach provides error bars for the parameter estimates and a general method for model fitting and comparison via optimal weighted least squares. A simple model of concentration variance, based on the particle‐tracking approach, identifies the optimal weights.

Coupling Bank Stability and Bed Deformation Models to Predict Equilibrium Bed Topography in River Bends

Attempts to include river bank erosion predictors within morphological models are becoming increasingly common, but uncertainty surrounds the procedures used to couple bed deformation and bank erosion submodels in a way that maintains the mass continuity of eroded bank sediment. Herein we present a coupling procedure that comprises two discrete elements. First, immediately following bank failure, the slumped debris comes to rest at the bank toe as a planar surface inclined at an “angle of repose.” Second, we use a fractional transport model to simulate the subsequent erosion and deposition of the failed bank material debris. The method is demonstrated with an example in which an equilibrium bed topography model is combined with a river bank erosion model to predict the morphological response of a river bend to a large flow event.

The components of fractional transport rate

Fractional transport rates are defined as the product of spatial grain entrainment, displacement length, and displacement frequency. Grain entrainment is defined relative to the population of grains on the bed surface at an initial time in order to account for partial transport, a condition in which only a portion of the exposed grains of a single size Di are mobilized over the duration of a transport event. A relation for grain displacement frequency is calculated from an assumed relation for displacement length and observed values of fractional transport rate, bed surface size distribution, and spatial entrainment in flume experiments. Fractional transport rates are independent of Di for fully mobilized fractions and decrease rapidly with Di for partially mobile fractions; the latter results from a decrease in both the displacement frequency and the mobilized proportion of each fraction. These two terms, which together determine the fractional entrainment rate, have different relations with flow strength and Di and should be independently included in a general fractional transport model. Grain‐size similarity in fractional transport rates does not hold for conditions of partial transport. Partial transport is a grain‐scale representation of incipient motion and can be directly related to the reference shear stress τri a surrogate for the critical shear stress for incipient motion based on transport rate rather than grain mobility. Fractional mobilization at different flows may be predicted from τri, and partial transport appears to be the general condition at the reference transport rate. Using τri as input, the transport component relations are used to predict the bed mobilization and fractional transport rates for a field case.