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In this paper, the magnetic field effects on natural convection of power-law non-Newtonian fluids in rectangular enclosures are numerically studied by the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). To maintain the locality of the LBM, a local computing scheme for shear rate is used. Thus, all simulations can be easily performed on the Graphics Processing Unit (GPU) using NVIDIA’s CUDA, and high computational efficiency can be achieved. The numerical simulations presented here span a wide range of thermal Rayleigh number ($10^4\le \mathrm{\text{Ra}}\le 10^6$), Hartmann number ($0\le \mathrm{\text{Ha}}\le 20$), power-law index ($0.5\le n\le 1.5$) and aspect ratio ($0.25\le \mathrm{\text{AR}}\le 4.0$) to identify the different flow patterns and temperature distributions. The results show that the heat transfer rate is increased with the increase of thermal Rayleigh number, while it is decreased with the increase of Hartmann number, and the average Nusselt number is found to decrease with an increase in the power-law index. Moreover, the effects of aspect ratio have also investigated in detail.

This research is focused on mathematical modeling of fluids with yield stress in the squeeze mode. The focus of the research is on magneto-rheological (MR) fluids but the results can be extended to any non-Newtonian fluid exhibiting a yielding behavior. There is no universally accepted mathematical model of squeeze flow of MR fluids to date because of the complexity of the flow and lack of theoretical and experimental studies. In this research, the squeeze flow problem of MR fluids is solved using perturbation techniques and squeeze force and flow field are determined. The model is verified and validated using experimental test data and it is shown that the model can be used as a design tool for design of MR devices that operate in the squeeze mode.

Resonance Acoustic MixingⓇ(RAM) technology applies an external low-frequency vertical harmonic vibration to mix ultrafine granular materials and subsequently non-Newtonian fluids. Although this system is used for various applications, its mechanism is yet not well understood, especially in the mixing of non-Newtonian fluids. To address this gap in knowledge, a phase model of the shear-thinning and shear-thickening non-Newtonian power-law fluid in a low-frequency vertical harmonic vibration container is established in this study, and the different power-law index is also considered. During the initial mixing process, there is Faraday instability at the gas–liquid interface, and Faraday waves are related to the power-law index. With the continuous input of external energy, the flow field is further destabilized, so that the uniform mixing is finally completed. In addition, the rheology of non-Newtonian fluids is consistent with the constitutive relation of power-law fluids. The dynamic viscosity of shear-thinning non-Newtonian fluid decreases rapidly with the increase of mixing time, while the shear-thickening non-Newtonian fluid decreases rapidly with the increase of mixing time. The variation of shear rate for Newtonian and non-Newtonian fluids is identical. Finally, a proper vibration parameter for the high mixing efficiency of RAM is proposed.

In this paper we consider a lattice dynamical system generated by a parabolic equation modeling suspension flows. We prove the existence of a global compact connected attractor for this system and the upper semicontinuity of this attractor with respect to finite-dimensional approximations. Also, we obtain a sequence of approximating discrete dynamical systems by the implementation of the implicit Euler method, proving the existence and the upper semicontinuous convergence of their global attractors.

The paper concerns the model of a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor. Contrary to standard power-law type rheology, we propose the formulation with the help of the spatially-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide the existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. Basing on the energy equality, we exploit the tools of Young measures.

We study mathematically a system of partial differential equations arising in the modeling of an aging fluid, a particular class of non-Newtonian fluids. We prove well-posedness of the equations in appropriate functional spaces and investigate the longtime behavior of the solutions.

We deal with flows of non-Newtonian fluids in three-dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index $p\ge 11/5$, we establish regularity properties of a solution with respect to time variable. Consequently, we can use this better information for showing the uniqueness of the solution provided that the initial data are good enough for all power–law indices $p\ge 11/5$. Such a result was available for $p\ge 12/5$ and therefore the paper fills the gap and extends the uniqueness result to the whole range of $p$’s for which the energy equality holds.

In this paper, we study the stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise in three dimensions with a hereditary viscous term which depends on the past history. We establish the local solvability of the Cauchy problem for such systems. The local monotonicity property of the nonlinear term of the cutoff problem and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. Finally, we show that the global solvability results hold under smallness condition on the initial data and suitable assumptions on the noise coefficients.

We consider the stochastic non-Newtonian fluids defined on a two-dimensional Poincaré unbounded domain, and prove that it generates an asymptotically compact random dynamical system. Then, we establish the existence of random attractor for the corresponding random dynamical system. Random attractor is invariant and attracts every pullback tempered random set.

The majority of numerical simulations assumes blood as a Newtonian fluid due to an underestimation of the effect of non-Newtonian blood behavior on hemodynamics in the cerebral arteries. In the present study, we evaluated the effect of non-Newtonian blood properties on hemodynamics in the idealized 90${}^{\circ }$-bifurcation model, using Newtonian and non-Newtonian fluids and different flow rate ratios between the parent artery and its branch. The proposed Local viscosity model was employed for high-precision representation of blood viscosity changes. The highest velocity differences were observed at zones with slow recirculating flow. During the systolic peak the average difference was 17–22%, whereas at the end of diastole the difference increased to 27–60% depending on the flow rate ratio. The main changes in the viscosity distribution were observed distal to the flow separation point, where the non-Newtonian fluid model produced 2.5 times higher viscosity. A presence of such high viscosity region substantially affected the size of the flow recirculation zone. The observed differences showed that non-Newtonian blood behavior had a significant effect on hemodynamic parameters and should be considered in the future studies of blood flow in cerebral arteries.

Shock waves are plane discontinuities, pulse-like in nature, and are sometimes more appropriately called shock fronts. Formation and characteristics of shock waves can be easily understood by studying the flow of a compressible fluid through a nozzle. One can see that in some cases, continuity considerations lead to the formation of plane shock waves across which there are discontinuities in pressure, density, temperature, etc. The principles of conservation of mass, momentum, and energy still apply across these plane discontinuities, but there is an entropy gain. In this chapter, we mainly discuss the recent developments of shock waves, shock-wave solutions, and traveling-wave solutions in viscoelastic generalization of Burgers’ equation. We also discuss the shock wave solutions of differential equations arising due to flow of some viscoelastic fluids of differential type.

A brief review of convection heat transfer in thermodynamically compatible fluids of grade three is presented. Some recent results for (i) the natural convection of such a fluid between two infinitely long cylinders, and (li) the effect of variable viscosity in the forced convection of a third grade fluid in a pipe are presented.

A finite-volume discretization over an unstructured grid is used to integrate the differential form of the Lattice Boltzmann equation with a variable viscosity using a cell-vertex finite-volume technique. The use of such technique for blood flow simulations combines the advantages of the LBM for Non-Newtonian fluids with the enhanced geometrical flexibility, allowing to suit the complex shapes typical of blood vessels with a limited number of nodes. Being the stress tensor expressed in terms of the distribution function, an explicit and accurate form for the shear stress is available at the wall surface. The methodology is applied to a selected number of test flow problems.