We find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from periodic and localized initial conditions as separate cases. The particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist) at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localized initial condition, are shown to approach the appropriate particular solution.

We generalize the known collision results for a solid in a 3D compressible Newtonian fluid to compressible non-Newtonian ones, and to Newtonian fluids with temperature-depending viscosities.

We show that the energy norm of weak solutions to Vlasov equation coupled with a shear thickening fluid on the whole space has a decay rate the energy norm $E(t) \leq {C}/{(1+t)^{\alpha }}, \forall t \geq 0$ for $\alpha \in (0,3/2)$.

This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

Suspended fibres significantly alter fluid rheology, as exhibited in for example solutions of DNA, RNA and synthetic biological nanofibres. It is of interest to determine how this altered rheology affects flow stability. Motivated by the fact thermal gradients may occur in biomolecular analytic devices, and recent stability results, we examine the problem of Rayleigh–Bénard convection of the transversely isotropic fluid of Ericksen. A transversely isotropic fluid treats these suspensions as a continuum with an evolving preferred direction, through a modified stress tensor incorporating four viscosity-like parameters. We consider the linear stability of a stationary, passive, transversely isotropic fluid contained between two parallel boundaries, with the lower boundary at a higher temperature than the upper. To determine the marginal stability curves the Chebyshev collocation method is applied, and we consider a range of initially uniform preferred directions, from horizontal to vertical, and three orders of magnitude in the viscosity-like anisotropic parameters. Determining the critical wave and Rayleigh numbers, we find that transversely isotropic effects delay the onset of instability; this effect is felt most strongly through the incorporation of the anisotropic shear viscosity, although the anisotropic extensional viscosity also contributes. Our analysis confirms the importance of anisotropic rheology in the setting of convection.

The Stokes axisymmetric flow of an incompressible viscous fluid past a micropolar fluid spheroid whose shape deviates slightly from that of a sphere is studied analytically. The boundary conditions used are the vanishing of the normal velocities, the continuity of the tangential velocities, continuity of shear stresses and spin-vorticity relation at the surface of the spheroid. The hydrodynamic drag force acting on the fluid spheroid is calculated. An exact solution of the problem is obtained to the first order in the small parameter characterizing the deformation. It is observed that due to increase spin parameter value, the drag coefficient decreases. Well known results are deduced and comparisons are made with classical viscous fluid and micropolar fluid.

A theoretical study of an unsteady two-layered blood flow through a stenosed artery is presented in this article. The geometry of a rigid stenosed artery is assumed to be $w$-shaped. The flow regime is assumed to be laminar, unsteady and uni-directional. The characteristics of blood are modelled by the generalized Oldroyd-B non-Newtonian fluid model in the core region and a Newtonian fluid model in the periphery region. The governing partial differential equations are derived for each region by using mass and momentum conservation equations. In order to facilitate numerical solutions, the derived differential equations are nondimensionalized. A well-tested explicit finite-difference method (FDM) which is forward in time and central in space is employed for the solution of a nonlinear initial boundary value problem corresponding to each region. Validation of the FDM computations is achieved with a variational finite element method algorithm. The influences of the emerging geometric and rheological parameters on axial velocity, resistance impedance and wall shear stress are displayed graphically. The instantaneous patterns of streamlines are also presented to illustrate the global behaviour of the blood flow. The simulations are relevant to haemodynamics of small blood vessels and capillary transport, wherein rheological effects are dominant.

This paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress are also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.

In oscillatory shear experiments, the values of the storage and loss moduli, $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$, respectively, are only measured and recorded for a number of values of the frequency $\unicode[STIX]{x1D714}$ in some well-defined finite range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$. In many practical situations, when the range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ is sufficiently large, information about the associated polymer dynamics can be assessed by simply comparing the interrelationship between the frequency dependence of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$. For other situations, the required rheological insight can only be obtained once explicit knowledge about the structure of the relaxation time spectrum $H(\unicode[STIX]{x1D70F})$ has been determined through the inversion of the measured storage and loss moduli $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$. For the recovery of an approximation to $H(\unicode[STIX]{x1D70F})$, in order to cope with the limited range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ of the measurements, some form of localization algorithm is required. A popular strategy for achieving this is to assume that $H(\unicode[STIX]{x1D70F})$ has a separated discrete point mass (Dirac delta function) structure. However, this expedient overlooks the potential information contained in the structure of a possibly continuous $H(\unicode[STIX]{x1D70F})$. In this paper, simple localization algorithms and, in particular, a joint inversion least squares procedure, are proposed for the rapid recovery of accurate approximations to continuous $H(\unicode[STIX]{x1D70F})$ from limited measurements of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$.

We show that in an infinite straight pipe of arbitrary (sufficiently smooth) cross-section, a generalized non-Newtonian liquid admits one and only one fully developed time-periodic flow (Womersley flow) when either the flow rate (problem 1) or the axial pressure gradient (problem 2) is prescribed in analogous time-periodic fashion. In addition, we show that the relevant solution depends continuously upon the data in appropriate norms. As is well known from the Newtonian counterpart of the problem, the latter is pivotal for the analysis of flow in a general unbounded pipe system with cylindrical outlets (Leray's problem). It is also worth remarking that problem 1 possesses an intrinsic interest from both mathematical and physical viewpoints, in that it constitutes a (nonlinear) inverse problem with a significant bearing on several applications, including blood flow modelling in large arteries.

The linear operator plays an important role in the computational process of Homotopy Analysis Method (HAM). In HAM frame any kind of linear operator can be chosen to develop a solution. Hence, it is easy to introduce the modified/physical parameter dependent linear operators. The effective use of these operators has been judged through solving fluid flow problems. Modification in linear operators affects the solution and improves the computational efficiency of HAM for larger values of parameters. The convergence rate of the solution is rapid and several times higher resulting in lesser computational time.

In this study, we compare different diffuse and sharp interface schemes of direct-forcing immersed boundary — thermal lattice Boltzmann method (IB-TLBM) for non-Newtonian flow over a heated circular cylinder. Both effects of the discrete lattice and the body force on the momentum and energy equations are considered, by applying the split-forcing Lattice Boltzmann equations. A new technique based on predetermined parameters of direct forcing IB-TLBM is presented for computing the Nusselt number. The study covers both steady and unsteady regimes (20<Re<80) in the power-law index range of 0.6<n <1.4, encompassing both shear-thinning and shear-thickening non-Newtonian fluids. The numerical scheme, hydrodynamic approach and thermal parameters of different interface schemes are compared in both steady and unsteady cases. It is found that the sharp interface scheme is a suitable and possibly competitive method for thermal-IBM in terms of accuracy and computational cost.

A biphase mixture continuum mechanics model is derived for neutral heat-shrinkable thermo-sensitive hydrogels in this paper. The mixing free energy of the special mixture is recalculated based on the partition function of Bose system, and it evaluates the contribution of the hydrophilic, hydrophobic interaction and hydrogen bonding to the volume phase transition behaviors. The ideas of the Flory lattice theory and the UNIFAC group contribution method are employed to get the expression of the mixing free energy. Then we deduce a particular model by combining this mixing free energy with the conservation laws equations and constitutive relations of both phases to predict the volume transition behaviors of these special hydrogels.

Stokes’ axisymmetrical translational motion of a slip sphere, located anywhere on the diameter of a virtual spherical fluid ‘cell’, is investigated. The fluid is micropolar and flows are parallel to the line connecting the two centres. An infinite-series solution is presented for the stream function, pressure field, vorticity, microrotation component, shear stress and couple stress of the flow. Basset-type slip boundary conditions on the sphere surface are used for velocity and microrotation. The Happel and Kuwabara boundary conditions are used on the fictitious surface of the cell model. Numerical results for the normalized drag force acting on the sphere are obtained with excellent convergence for various values of the volume fraction, the relative distance between the centre of the sphere and the virtual envelope, the vortex viscosity parameter and the slip coefficients of the sphere. In the special case when the spherical particle is in the concentric position with the cell surface, the numerical values of the normalized drag force agree with the available values in the literature.

Understanding blood flow in human body’s cerebral arterial system is of both fundamental and practical significance for prevention and treatment of vascular diseases. The mechanism and treatment for the growth of daughter aneurysm on its mother aneurysm are not yet fully understood. Themain purpose of the present paper is to elucidate the relationships between hemodynamics and the genesis, growth, subsequent rupture of the mother and daughter aneurysm on the cerebral vascular. The intensified stents with different porosities and structures are investigated to reduce the wall shear stress and pressure of mother and daughter aneurysm. The simulation is based on a lattice Boltzmann modeling of non-Newtonian blood flow. A novel stent structurewith “dense in front and sparse in rear” is proposed,which is verified to have good potential to reduce the wall shear stress of both mother and daughter aneurysm. The simulation is based on a lattice Boltzmann modeling of non-Newtonian blood flow. A novel stent structurewith “dense in front and sparse in rear” is proposed,which is verified to have good potential to reduce the wall shear stress of both mother and daughter aneurysm.

In this paper, the power-law fluid flows in a two-dimensional square cavity are investigated in detail with multi-relaxation-time lattice Boltzmann method (MRT-LBM). The influence of the Reynolds number (Re) and the power-law index (n) on the vortex strength, vortex position and velocity distribution are extensively studied. In our numerical simulations, Re is varied from 100 to 10000, and n is ranged from 0.25 to 1.75, covering both cases of shear-thinning and shear-thickening. Compared with the Newtonian fluid, numerical results show that the flow structure and number of vortex of power-law fluid are not only dependent on the Reynolds number, but also related to power-law index.

We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

The gas-kinetic scheme is applied to a depth-integrated continuum model for avalanche flows, namely the Savage-Hutter model. In this method, the continuum fluxes are calculated based on the pseudo particle motions which are relaxed from nonequilibrium to equilibrium states. The processes are described by the Bhatnagar-Gross-Krook (BGK) equation. The benefit of this scheme is its capability to resolve shock discontinuities sharply and to handle the vacuum state without special treatments. Because the Savage-Hutter equation bears an anisotropic stress on the tangential space of the topography, the equilibrium distribution function of the microscopic particles are shown to be bi-Maxwellian. These anisotropic stresses are the key to preserve the coordinate objectivity in the Savage-Hutter model. The effect of the anisotropic stress is illustrated by two examples: an axisymmetric dam break and a finite mass sliding on an inclined plane chute. It is found that the propagation of the flow fronts significantly depends on the orientation of the principal axes of the tangential stresses.

Stent placement is considered as a promising and minimally invasive technique to prevent rupture of aneurysm and favor coagulation mechanism inside the aneurysm. Many scholars study the effect of the stent on blood flow in cerebral aneurysm by numerical simulations, and usually regard blood as the Newtonian fluid, blood, however, is a kind of non-Newtonian fluid in practice. The main purpose of the present paper is to investigate the effect of non-Newtonian behavior on the hemodynamic characteristics of blood flow in stented cerebral aneurysm with lattice Boltzmann method. The Casson model is used to describe the blood non-Newtonian character, which is one of the most popular models in depicting blood fluid. In particular, hemodynamic characteristics derived with Newtonian and non-Newtonian models are studied, and compared in detail. The results show that the non-Newtonian effect gives a great influence on hemodynamic characteristics of blood flow in stented cerebral aneurysm, especially in small necked ones.

We study the biofilm-flow interaction resulting in biofilm growth and deformation in a water channel in a 3-D setting using the phase field model developed recently. In this biofilm model, the biofilm made up of the EPS, bacteria and solvent is tracked using a biofilm volume fraction which vanishes outside the biofilm region. The interface between the biofilm and the solvent is marked by the zero level surface of the volume fraction measured from the biofilm to the solvent. The growth of the biofilm and the solvent-biofilm interaction with the top nutrient feeding condition is simulated in the viscous regime (growth regime) of the biofilm-solvent mixture flow. In quiescent flows, the model predicts growth patterns consistent with experimental findings for single or multiple adjacent biofilm colonies, in which the known mushroom shape growth pattern is obtained. Shear induced deformation in biofilms is simulated in a shear cell, providing a viable numerical evidence for using simulation tool to study biofilm growth and interaction dynamics in aqueous environment.