GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

Waves on the surface of a falling power-law fluid film

Waves that occur at the surface of a falling film of thin power-law fluid on a vertical plane are investigated. Using the method of integral relations an evolution equation is derived for two types of waves equation which are possible under long wave approximation. This equation reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and the energy is acquired from the mean flow during the interaction of the waves, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. It is also found that this exchange of energy between kinematic and inertial waves strongly depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that the surface tension plays a double role: for a kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for a dynamic wave process it yields dispersion. Further, it is shown that the non-Newtonian character n plays a vital role in controlling the role of the term that contains surface tension in the above processes.     

MHD flow of power-law fluid on moving surface with power-law velocity and special injection/blowing

The problem of magnetohydrodynamic （MHD） flow on a moving surface with the power-law velocity and special injection/blowing is investigated. A scaling group transformation is used to reduce the governing equations to a system of ordinary differen- tial equations. The skin friction coefficients of the MHD boundary layer flow are derived, and the approximate solutions of the flow characteristics are obtained with the homotopy analysis method （HAM）. The approximate solutions are easily computed by use of a high order iterative procedure, and the effects of the power-law index, the magnetic parameter, and the special suction/blowing parameter on the dynamics are analyzed. The obtained results are compared with the numerical results published in the literature, verifying the reliability of the approximate solutions.     

Effects of shear thinning on migration of neutrally buoyant particles in pressure driven flow of Newtonian and viscoelastic fluids

The pattern of cross stream migration of neutrally buoyant particles in a pressure driven flow depends strongly on the properties of the suspending fluid. These migration effects have been studied by direct numerical simulation in planar flow. Shear thinning has a large effect when the inertia or elasticity is large, but only a small effect when they are small. At moderate Reynolds numbers, shear thinning causes particles to migrate away from the centerline, creating a particle-free zone in the core of the channel, which increases with the amount of shear thinning. In a viscoelastic fluid with shear thinning, particles migrate either toward the centerline or toward the walls, creating an annular particle-free zone at intermediate radii. The simulations also give rise to precise determination of slip velocity distributions in the various cases studied.     

Stability of high‐speed two‐layer film casting of Newtonian fluids

The steady‐state flow and its linear stability are investigated for the isothermal two‐layer film casting process. Newtonian fluids are considered in this study. The continuity of traction is ensured at the interface, and the axial velocity is assumed to be uniform across each film layer separately. The effects of inertia, gravity, fluid parameters and processing conditions on the steady‐state flow and its stability are studied. The results indicate that the fluid properties and the processing conditions have significant influence on the flow. The flow stability is strongly dependent on the layer layout with respect to the take‐up rolling process. The frequency of the (unstable) disturbance is insensitive to flow and processing parameters. Copyright © 2006 John Wiley & Sons, Ltd.     

A spectral method for free surface flows of inviscid fluids

In this paper, an efficient numerical method for unsteady free surface motions, with simple geometries, has been devised. Under the potential flow assumption, the governing equation of free surface flows becomes a Laplace equation, which is treated here by means of a series expansions of the velocity potential. The free surface is represented with a height function. The present method is applied to surface gravity waves to test the stability and accuracy of the method. To show the versatility of the method, a model for a dip formation is considered. © 1998 John Wiley & Sons, Ltd.     

Exact solution to peristaltic transport of power-law fluid in asymmetric channel with compliant walls

Effects of compliant wall properties on the peristaltic flow of a non-Newtonian fluid in an asymmetric channel are investigated.The rheological characteristics are characterized by the constitutive equations of a power-law fluid.Long wavelength and low Reynolds number approximations are adopted in the presentation of mathematical developments.Exact solutions are established for the stream function and velocity.The streamlines pattern and trapping are given due attention.Salient features of the key parameters entering into the present flow are displayed and important conclusions are pointed out.     

Stability of viscoelastic shear flows in the limit of high Weissenberg and Reynolds numbers

We consider a limit of the upper convected Maxwell model where both the Weissenberg and Reynolds numbers are large. The limiting equations have a status analogous to that of the Euler equations for the high Reynolds number limit. These equations admit parallel shear flows with an arbitrary profile of velocity and normal stress. We consider the stability of these flows. An extension of Howard’s semicircle theorem can be used to show that the flow is stabilized if elastic effects are sufficiently strong. We also show how to analyze the long wave limit in a fashion similar to the inviscid case.     

An iteration method for integral equations arising from gravity flows with free surface

Gravity potential flows with free surface still present considerable difficulties in non-linear mathematical problems. Previous researchers using analytic function theory could consider only simple geometrical boundaries. Analyzing curvilinear solid boundaries by means of analytic theory is a difficult problem that has not been solved. In this paper, using Muskhelishvili's singular integral equation theory, we turn the gravity flow problem into the Riemann-Hilbert problem. Taking the length of the streamline of the boundary as the independent variable and the velocity potential of the boundary as the function to be determined, we avoid the difficulty that the angle of the curved fixed part is unknown. Following the difference method and the finite element method, we develop a new numerical method that is suitable for complex solid boundaries and overcome the difficulties encountered in applying analytic function theory. Under known discharge, the convergence and stability of the method have been proved and an estimation of error has been obtained. The method has been successfully applied to the calculation of the flow past spillway buckets. The calculated values agree well with the measured results.     

内管自转且公转时环管幂律流的受力分析

数值模拟了环管中内管偏心自转且公转时由轴向压力所驱动的幂律流体充分发展层流,分析了内管上的流体作用力。结果表明,内管偏心自转时流体作用力具有推动内管作和自转同向公转的效果。当只有外力矩驱动内管自转时,由于流体的作用,随内管线密度的不同,内管能达到的受力平衡态也不同:线密度较小时内管仅能在同心自转时达到受力平衡;线密度较大时内管能在作具有不变角速度和偏心率公转时达到受力平衡,且内管线密度越大,对应的受力平衡的公转的偏心率也越大。     

Effect of thermal conductivity on heat transfer for a power-law non-Newtonian fluid over a continuous stretched surface with various injection parameters

The two-dimensional non-Newtonian steady flow on a power-law stretched surface with suction or injection is studied. Thermal conductivity is assumed to vary as a linear function of temperature. The transformed governing equations in the present study are solved numerically using the Runge-Kutta method. Through a comparison, results for a special case of the problem show excellent agreement with those in a previous work. Two cases are considered, one corresponding to a cooled surface temperature and the other to a uniform surface temperature. Numerical results show that the thermal conductivity variation parameter, the injection parameter, and the power-law index have significant influences on the temperature profiles and the Nusselt number.     

Squeeze flow of a power-law fluid between two rigid spheres with wall slip

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An SPH model for free surface flows with moving rigid objects

This paper presents a computational model for free surface flows interacting with moving rigid bodies. The model is based on the SPH method, which is a popular meshfree, Lagrangian particle method and can naturally treat large flow deformation and moving features without any interface/surface capture or tracking algorithm. Fluid particles are used to model the free surface flows which are governed by Navier–Stokes equations, and solid particles are used to model the dynamic movement (translation and rotation) of moving rigid objects. The interaction of the neighboring fluid and solid particles renders the fluid–solid interaction and the non‐slip solid boundary conditions. The SPH method is improved with corrections on the SPH kernel and kernel gradients, enhancement of solid boundary condition, and implementation of Reynolds‐averaged Navier–Stokes turbulence model. Three numerical examples including the water exit of a cylinder, the sinking of a submerged cylinder and the complicated motion of an elliptical cylinder near free surface are provided. The obtained numerical results show good agreement with results from other sources and clearly demonstrate the effectiveness of the presented meshfree particle model in modeling free surface flows with moving objects. Copyright © 2013 John Wiley & Sons, Ltd.     

Doppler effects of an oscillating line source in shear flow with a free surface

The linearised water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. There is a nonzero surface velocity. The depth is infinite and the vorticity is uniform. The amplitudes radiated from the source are calculated analytically. Due to Doppler effects, there may be up to four different emitted waves, and there is resonance with zero group velocity and infinite amplitude.     

Particle manipulation via integration of electroosmotic flow of power-law fluids with standing surface acoustic waves (SSAW)

Standing surface acoustic wave (SSAW) based microfluidic devices have shown great promise toward fluid and particle manipulation applications in medicine, chemistry, and biotechnology. In this article, we present an analytical model for investigating continuous manipulation of particles (both synthetic and biological) within electroosmotic flow of non-Newtonian bio-fluids in a microfluidic channel under the influence of standing surface acoustic waves (SSAW). The particles are injected along the center of channel into the electroosmotically driven flow of power-law fluids, wherein their transport through the SSAW region is dictated by the hydrodynamic, electrophoretic, and acoustic forces. We first present a mathematical model to analyze the characteristics of electroosmotic flow of non-Newtonian power-law fluids in a hydrophobic slit microchannel. Next, we investigate the trajectories of particles in the flow field due to the combined effect of electroosmotic, electrophoretic, and acoustophoretic forcing mechanisms. The effect of key parameters such as particle size, their physical properties, input power, flow rate, and flow behavior index on the particle trajectories is examined while including the effect of the channel walls. The presented model delineates the methodologies of improving SSAW-based particle separation technology by considering the fluid rheology as well as the surface properties of the channel walls. Therefore, we believe that this model can serve as an efficient tool for device design and quick optimizations to explore novel applications concerning the integration of electroosmotic flows with acoustofluidic technologies.     

Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt

In this paper, we prove in general that the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the homotopy analysis method (HAM) profound in 1992 when ħ = −1. Besides, by using the thin film flows of Sisko and Oldroyd 6-constant fluids on a moving belt as examples, we show that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.: Chaos Solitons and Fractals (2006) in press) are divergent, and thus useless. However, by choosing a proper value of the auxiliary parameter ħ, we give convergent series solution by means of the HAM. These two examples also show that, different from the HPM and other traditional analytic techniques, the HAM indeed provides us with a simple way to ensure the convergence of the solution.     

Thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined plane

This study is devoted to the investigation of thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined isothermal plane. It is assumed that the reaction is exothermic under Arrhenius kinetics, neglecting the consumption of the material. The governing non-linear equations for conservation of momentum and energy are obtained and solved by using a new computational approach based on a special type of Hermite-Padé approximation technique implemented in MAPLE. This semi-numerical scheme offers some advantages over solutions obtained with traditional methods such as finite differences, spectral method, and shooting method. It reveals the analytical structure of the solution function. Important properties of overall flow structure including velocity field, temperature field, thermal criticality, and bifurcations are discussed.     

The steady annular extrusion of a Newtonian liquid under gravity and surface tension

The steady extrusion of a Newtonian liquid through an annular die and its development outside and away from the die are studied under the influence of gravitational and surface tension forces. The finite element method (FEM) is used for the simulations. The positions of the inner and outer free surface profiles are calculated simultaneously with the other unknown fields, i.e. using the Newton–Raphson iterative scheme. The effects of three relevant parameters, i.e. the Reynolds, the Stokes and the capillary numbers, on the shape of the annular film are studied for two values of the inner to the outer diameter ratio, corresponding to a thick and a thin annular film respectively. A one‐dimensional model for the extrudate region, valid for thin annular films, is also presented, and its predictions are compared with the two‐dimensional finite element calculations. Despite the fact that it is valid away from the die exit, the one‐dimensional model predicts satisfactorily the effects of the Stokes and capillary numbers. Copyright © 2000 John Wiley & Sons, Ltd.     

Extension of the volume-of-fluid method for analysis of free surface viscous flow in an ideal gas

The volume-of-fluid (VOF) method is a simple and robust technique for simulating free surface flows with large deformations and intersecting free surfaces. Earlier implementations used Laplace's formula for the normal stress boundary condition at the interface between the liquid and vapour phases. We have expanded the interfacial boundary conditions to include the viscous component of the normal stress in the liquid phase and, in a limited manner, to allow the pressure in the vapour phase to vary. Included are sample computations that show the accuracy of added third-order-accurate differencing schemes for the convective terms in the Navier-Stokes equation (NSE), the viscous terms in the normal stress at the interface and the solution of potential flow in the vapour phase coupled with the solution of the NSE in the liquid phase. With these modifications we show that the VOF method can accurately predict the instability of a thin viscous sheet flowing through a stagnant vapour phase.     

FLOQUET STABILITY ANALYSIS OF TWO-LAYER FLOWS IN VERTICAL PIPE WITH PERIODIC FLUCTUATION

Based on linear stability theory, parametric resonance phenomenon of a liquid-gas cylindrical flow in a vertical pipe with periodic fluctuation was discussed with the help of Floquet theory and Chebyshev spectral collocation method. The effects of different physical parameters were investigated on the properties of parametric resonance and the stability characteristics of flow field.