Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid

In recent years, non-Newtonian fluids have received much attention due to their numerous applications, such as plastic manufacture and extrusion of polymer fluids. They are more complex than Newtonian fluids because the relationship between shear stress and shear rate is nonlinear. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluid, which is modelled using terms involving multi-term time fractional diffusion and reaction. In this paper, we consider the application of the finite difference method for this class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete $H^1$ norm and prove that their accuracy is of $O(\tau+h^2)$ and $O(\tau^{\min\{3-\gamma_s,2-\alpha_q,2-\beta\}}+h^2)$, respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.Comment: 19 pages, 8 figures, 3 table

Rheological synergistic thermal conductivity of CMC-based Fe3O4 and Al2O3 nanofluids in shear flow fields

In this paper, considering the variation of the viscous dissipative heat in the transfer direction, a new theoretical formula for thermal conductivity measurement was proposed based on the energy equation of the rotational Couette flow field. This theoretical formula shows that thermal conductivity and rheology have a synergistic effect. Based on this theoretical formula, the rheological synergistic thermal conductivity of CMC-based Fe3O4 nanofluids and Al2O3 nanofluids was experimentally investigated. The experimental results show that the thermal conductivity rises with the increase of shear rate and volume fraction, moreover, volume fraction and shear rate have mutually reinforcing effects on thermal conductivity enhancement. Non-Newtonian effects of rheology and heat transfer reduce with shear rate and increase with volume fraction, with consistent synergistic effects. According to the experimental data, the expressions of the thermal conductivity and dynamic viscosity of these two nanofluids as functions of shear rate and volume fraction were presented

Effects of fractional mass transfer and chemical reaction on MHD flow in a heterogeneous porous medium

This paper presents a study on space fractional anomalous convective-diffusion and chemical reaction in the magneto-hydrodynamic fluid over an unsteady stretching sheet. The fractional diffusion model is derived from decoupled continuous time random walks in a heterogeneous porous medium. A novel transformation which features time finite difference is introduced to reduce the governing equations into ordinary differential ones in each time level. Numerical solutions are established by an implicit finite difference scheme. The stability and convergence of the method are analyzed. Results show that increasing fractional derivative parameter enhances concentration near the surface while an opposite phenomenon occurs far away from the wall. There is a reduction of mass transfer rate on the sheet with an increase in the fractional derivative parameter. Moreover, the numerical solutions are compared with exact solutions and good agreement has been observed.</p

Anomalous diffusion in rotating Casson fluid through a porous medium

This paper investigates the space-fractional anomalous diffusion in unsteady Casson fluid through a porous medium, based on an uncoupled continuous time random walk. The influences of binary chemical reaction and activation energy between two horizontal rotating parallel plates are taken into account. The governing equations of motion are reduced to a set of nonlinear differential equations by time derivatives discretization and generalized transformation, which are solved by bvp4c and implicit finite difference method (IFDM). Stability and convergence of IFDM are proved and some numerical comparisons to the previous study are presented with excellent agreement. The effects of involved physical parameters such as fractional derivative parameter, rotation parameter and time parameter are presented and analyzed through graphs. Results indicate that the increase of fractional derivative parameter triggers concentration increase near the lower plate, while it causes a reduction near the upper plate. It is worth mentioning that the decrease of heat transfer rate on the plate is observed with the higher time parameter.</p

Fractal aggregation kinetics contributions to thermal conductivity of nano-suspensions in unsteady thermal convection

Nano-suspensions (NS) exhibit unusual thermophysical behaviors once interparticle aggregations and the shear flows are imposed, which occur ubiquitously in applications but remain poorly understood, because existing theories have not paid these attentions but focused mainly on stationary NS. Here we report the critical role of time-dependent fractal aggregation in the unsteady thermal convection of NS systematically. Interestingly, a time ratio λ = t(p)/t(m) (t(p) is the aggregate characteristic time, t(m) the mean convection time) is introduced to characterize the slow and fast aggregations, which affect distinctly the thermal convection process over time. The increase of fractal dimension reduces both momentum and thermal boundary layers, meanwhile extends the time duration for the full development of thermal convection. We find a nonlinear growth relation of the momentum layer, but a linear one of the thermal layer, with the increase of primary volume fraction of nanoparticles for different fractal dimensions. We present two global fractal scaling formulas to describe these two distinct relations properly, respectively. Our theories and methods in this study provide new evidence for understanding shear-flow and anomalous heat transfer of NS associated non-equilibrium aggregation processes by fractal laws, moreover, applications in modern micro-flow technology in nanodevices

A Novel Equivalent Agglomeration Model for Heat Conduction Enhancement in Nanofluids

We propose a multilevel equivalent agglomeration (MEA) model in which all particles in an irregular cluster are treated as a new particle with equivalent volume, the liquid molecules wrapping the cluster and in the gaps are considered to assemble on the surface of new particle as mixing nanolayer (MNL), the thermal conductivity in MNL is assumed to satisfy exponential distribution. Theoretical predictions for thermal conductivity enhancement are highly in agreement with the classical experimental data. Also, we first try to employ TEM information quantitatively to offer probable reference agglomeration ratio (not necessary a very precise value) to just test rational estimations range by present model. The comparison results indicate the satisfactory priori agglomeration ratio estimations range from renovated model