Computational Modelling of Non-Newtonian Effects on Flow in Channels with Moving Wall Indentations

Non-Newtonian effects in a channel with moving wall indentations are assessed numerically by a finite volume method for solving the unsteady incompressible Navier-Stokes equations and using a power-law model exhibiting shear thinning viscosity and Casson's model as the constitutive equations for the non-Newtonian fluid. The computations show that for a non-Newtonian fluid, there are differences in the velocity profiles and in the structure and size of the reversed flow regions as compared with the corresponding Newtonian fluid. The comparison of non-Newtonian and Newtonian wall shear stress reveals a slight decrease in the magnitude on the average for the non-Newtonian case, eventually resulting in the strength of the “wave train” being slightly weaker than those corresponding to a Newtonian fluid.     

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. The flow near a wall suddenly set in motion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model. Exact solutions of velocity and stress are obtained by using the discrete inverse Laplace transform of the sequential fractional derivatives. It is found that the effect of the fractional orders in the constitutive relationship on the flow field is significant. The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate, for large times the viscoelastic effects become weak. The project supported by the National Natural Science Foundation of China (10002003), Foundation for University Key Teacher by the Ministry of Education, Research Fund for the Doctoral Program of Higher Education     

Numerical breakdown at high Weissenberg number in non-Newtonian contraction flows

 Planar contraction flows of non-Newtonian fluids with integral constitutive models are studied to investigate the problem of numerical breakdown at high Weissenberg or Debrorah numbers. Spurious shear stress extrema are found on the wall downstream of the re-entrant corner for both sharp and rounded corners. Moreover, a non-monotonic relation between shear stress and strain rate is found when the Deborah number limit is approached, which correlates with these shear extrema. This strongly suggests that non-monotonicity between shear stress and strain rate may be responsible for the Deborah number limit problem in contraction flow simulations. This non-monotonicity is caused by the inaccuracy of the quadrature, using constitutive equations that do not have shear stress maxima when exactly evaluated. This conclusion agrees with recent analytical findings by others that inaccuracy of the integration along the streamlines – either by numerical integration or asymptotic approximation – makes the problem ill-conditioned, with spurious growth occurring on the wall downstream of the re-entrant corner. Received: 5 March 1999/Accepted: 1 September 1999     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

FLOW OF A VISCOELASTIC FLUID BETWEEN TWO ROTATING CIRCULAR CYLINDERS SUBJECT TO SUCTION OR INJECTION

The steady flow of an Oldroyd-B fluid between two porous concentric circular cylinders is studied. The equation of motion and the constitutive equations form a system of non-linear ODEs that is solved numerically, and in a few cases the numerical results are compared with a known analytical solution. We consider the effect of the non-Newtonian nature of the fluid on the drag and on the boundary layer structure near the walls. Numerical computations show the effect of the non-Newtonian quantities on the velocity and on the shear stress as the dimensionless parameters are varied. © by 1997 John Wiley & Sons, Ltd.     

An investigation on internal thermal flow of non-Newtonian fluid

In the present paper an unsteady thermal flow of non-Newtonian fluid is investigated which is of the fiow into axisymmetric mould cavity. In the second part an unsteady thermal flow of upper-convected Maxwell fluid is studied, For the flow into mould cavity the constitutive equation of power-law fluid is used as a rheological model of polymer fluid. The apparent viscosity is considered as a function of shear rate and temperature. A characteristic viscosity is introduced in order to avoid the nonlinearity due to the temperature dependence of the apparent viscosity. As the viscosity of the fluid is relatively high the flow of the thermal fluid can be considered as a flow of fully developed velocity field. However, the temperature field of the fluid fiow is considered as an unsteady one. The governing equations are constitutive equation, momentum equation of steady flow and energy conservation equation of non-steady form. The present system of equations has been solved numerically by the splitting differen     

液晶高分子各向异性粘弹性流体本构方程理论

将液晶高分子－各向异性流体的本构方程,建立在Oldroyd随体导数观点基础上。推广上随机Oldroyd B流体模型,提出共转OldroydB流体模型,同时将微观结构的影响通过宏观参数表示出来,使在宏观理论中包含微观结构的贡献,即引入取向物质函数,非线性各向异性黏度函数和各向异性松弛时间及推迟时间等,表征取向运动对黏度和松弛及推迟现象的影响,在此基础上开展了一类新的液晶高分子－Oldroyd型本构方程理论,由该类型本构方程得出的物质函数,液晶高分子流体的第一、第二法向应力差与实验结果一致,解释了液晶高分子溶液的第一、第二法向应力差的特殊流变学行为。     

Non-Newtonian Behavior of Ballistic Gelatin at High Shear Rates

A coordinated modeling and experimental effort to investigate the shear stress-shear strain rate response of ballistic gelatin is presented. A power-law constitutive model that captures non-Newtonian shear-thickening behavior, the evolution of viscosity, and the momentum diffusion at high shear rates is adopted. A simple asymptotic relationship between the maximum wall shear stress and the maximum striking wall velocity is derived in the high diffusion rate regime for a shear flow between two parallel plates. Experimental investigation is conducted on double lap-shear test fixture with gelatin specimens of different thicknesses subjected to high strain rate input on the inner surface, generated by a polymer split Hopkinson pressure bar. This test fixture allows measurement of transmitted shear stress as well as visualization of momentum diffusion through gelatin when imaged by a high speed camera. Gelatin specimens of various thicknesses were used for extracting the power-law model parameters. It is found that ballistic gelatin behaves as a shear-thickening fluid at high shear rates with a power-law exponent of 2.22.     

Moving boundary in a non-Newtonian fluid

Moving boundary value problem in non-Newtonian fluid is considered. Exact analytical solution for the flow of second-grade fluid for a rigid moving plate oscillating in its own plane, is obtained. The Doppler effect has been observed due to the motion of the plate. The shearing stress on the plate is also calculated. It is concluded that the solutions for stationary porous boundaries can be obtained from the solutions of moving rigid boundaries.     

An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis

The problem of non-Newtonian and nonlinear blood flow through a stenosed artery is solved numerically where the non-Newtonian rheology of the flowing blood is characterised by the generalised Power-law model. An improved shape of the time-variant stenosis present in the tapered arterial lumen is given mathematically in order to update resemblance to the in vivo situation. The vascular wall deformability is taken to be elastic (moving wall), however a comparison has been made with nonlinear visco-elastic wall motion. Finite difference scheme has been used to solve the unsteady nonlinear Navier-Stokes equations in cylindrical coordinates system governing flow assuming axial symmetry under laminar flow condition so that the problem effectively becomes two-dimensional. The present analytical treatment bears the potential to calculate the rate of flow, the resistive impedance and the wall shear stress with minor significance of computational complexity by exploiting the appropriate physically realistic prescribed conditions. The model is also employed to study the effects of the taper angle, wall deformation, severity of the stenosis within its fixed length, steeper stenosis of the same severity, nonlinearity and non-Newtonian rheology of the flowing blood on the flow field. An extensive quantitative analysis is performed through numerical computations of the desired quantities having physiological relevance through their graphical representations so as to validate the applicability of the present model.     

On the Generation of Discontinuous Shearing Motions of a Non-Newtonian Fluid

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, starting from rest at , tends as to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit a large increase in mean flow rate when the driving pressure is raised above a critical value. (Accepted April 22, 1996)     

Viscoelastic blood flow through arterial stenosis—Effect of variable viscosity

Current theoretical investigation deals with mathematical model of unsteady non-Newtonian flow of blood through a stenosed artery. The flowing blood is considered as a viscoelastic fluid having shear-thinning rheology and characterized by generalised Oldroyd-B model. The arterial wall is considered to be rigid having cosine shaped stenosis in its lumen. The governing equations of motion accompanied by appropriate choice of the initial and boundary conditions are solved numerically by MAC (Marker and Cell) method and the results are checked for numerical stability with desired degree of accuracy. The quantitative analysis has been carried out finally which includes the respective profiles of the flow-field. The key factors like the wall shear stress and flow separation are also examined for further qualitative insight into the flow through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.     

填隙二阶流体下圆球平行于壁平移的粘性阻力

离散元法是分析散体力学行为的数值方法。存在填隙流体时，颗粒之间或颗粒与壁之间产生的法向挤压力和切向阻力、阻力矩，是湿颗粒离散元法的理论基础。二阶流体是以微小偏离牛顿流体本构而考虑时间影响的一种流体。它具有常粘度，并且第一和第二法向应力差正比于剪切率的平方。根据Reynolds润滑理论，采用小参数法，导出了存在填隙二阶流体时，圆球沿平行于平壁缓慢移动时流体的速度场和压力方程，进而求出切向阻力和阻力矩的解析解。有趣的是在推导时所得的速度场和压力方程形式比牛顿流体要复杂得多，但最终结果表明圆球沿平行于平壁移动时因填隙二阶流体引起的切向阻力和阻力矩与牛顿流体时的结果相同。     

Tresca-type plastic materials in the theory of hypoelasticity II. Optical constitutive equations and birefringence in simple shear

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.     

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.     

厚板的高阶剪切变形理论研究

已有多种厚板理论和高阶剪切变形模型,但仍需要进一步研究以更加完善.首先根据平均转角及上下表面剪应力自由这两个条件,提出了具有统一高阶剪切变形模型的中面位移模式,并将之表示为正交分解形式.根据正交特性,定义了板的广义应力;运用板问题应变能密度表示的等价性,提出了与广义应力功共轭的广义应变表示形式,建立了板的本构关系.证明了不同转角定义时虚功原理板理论表示的客观性,以及与三维弹性理论表示的等价性.运用虚功原理,建立了变分自洽的高阶厚板理论和变分渐近的低阶厚板理论,推导了相应的平衡方程及边界条件,分析了与已有板理论的异同.以广义应力形式建立了厚板理论的平衡方程,厘清了不同转角表示时板理论间的关系、低阶厚板理论与高阶厚板理论间的关系以及剪切系数计算等若干基本问题.对圣维南扭转问题的求解证明了该理论的正确性.      

基于分形导数对非牛顿流体层流的数值研究

非牛顿流体具有复杂的流变特性,揭示该流变特性可以更加合理地指导非牛顿流体在工农业生产中的应用.经典的非牛顿流体本构模型往往形式复杂,仅能应用于某些特定的情况.分数阶导数模型具有参数少和形式简单的特点,己成功地应用于描述非牛顿流体的运动.Hausdorff分形导数作为一个备选的建模方法,相比分数阶导数具有更简单的形式以及更高的计算效率.本文基于Hausdorff分形导数改进现有牛顿黏性模型,提出分形黏壶模型.通过研究分形黏壶在常应变率下表观黏度的变化情况,以及在加、卸载条件下的蠕变及恢复特性,发现分形黏壶模型适合于描述具有黏弹性的非牛顿流体(本文称之为分形流体).结合连续性方程及运动微分方程,推导出分形流体在平行板间层流的基本方程.按是否拖动上板和是否存在水平的压力梯度分为3种工况,分别用数值方法计算这3种工况下流速在板间的分布及其随时间变化的情况.通过分析不同工况下的流速分布,发现水平的压力梯度会改变流速随时间变化的形状,且会推迟流速到达稳定的时间.在水平压力梯度不存在的情况下,不同阶数的分形流体具有相同的流速分布或是演变过程.另外,在水平压力梯度存在的情况下,上板速度不影响不同阶数分形流体间稳定速度的差值.     

A generalized continuum theory for granular materials

A generalized continuum theory for granular media is formulated by allowing for the possibility of rotation of granules. The basic balance laws are presented and based on thermodynamical consideration a set of constitutive equations are derived. The theory naturally gives rise to the generation of antisymmetric stress tensor and existence of couple stresses. The basic equations of motion are derived and it is shown that the theory contains Mohr-Coulomb criterion of limiting equilibrium as a special case. The problem of coupled porosity and microrotational wave propagation is investigated and the rectilinear shear flow of granular materials is discussed.     

Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes

The results of direct numerical simulation of turbulent flows of non-Newtonian pseudoplastic fluids in a straight pipe are presented. The data on the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances are obtained for steady turbulent flows at the Reynolds numbers of 10⁴ and 2×10⁴. As distinct from Newtonian fluid flows, the viscous shear stresses turn out to be significant even far from the wall. In power-law fluid flows the mechanism of the energy transport from axial to transverse component fluctuations is suppressed. It is shown that with decrease in the fluid index the turbulent transfer of the momentum and the velocity fluctuations between the wall layer and the flow core reduces, while the turbulent energy flux toward the wall increases. The earlier-proposed models for the average viscosity and the non-Newtonian one-point correlations are in good agreement with the data of direct numerical simulation.     

The influence of flow-induced non-Newtonian fluid properties on turbulent drag reduction

Friction factors and velocity profiles in turbulent drag reduction can be compared to Newtonian fluid turbulence when the shear viscosity at the wall shear rate is used for the Reynolds number and the local shear viscosity is used for the non-dimensional wall distance. On this basis, an apparent maximum drag reduction asymptote is found which is independent of Reynolds number and type of drag reducing additive. However, no shear viscosity is able to account for the difference between the measured Reynolds stress and the Reynolds stress calculated from the mean velocity profile (the Reynolds stress deficit). If the appropriate local viscosity to use with the velocity fluctuation correlations includes an elongational component, the problem can be resolved. Taking the maximum drag reduction asymptote as a non-Newtonian flow, with this effective viscosity, leads to agreement with the concept of an asymptote only when the solvent viscosity is used in the non-dimensional wall distance.