The Effect of Double Dispersion on Natural Convection Heat and Mass Transfer in a Non-Newtonian Fluid Saturated Non-Darcy Porous Medium

The effect of power law index parameter of the non-Newtonian fluid on free convection heat and mass transfer from a vertical wall is analyzed by considering double dispersion in a non-Darcy porous medium with constant wall temperature and concentration conditions. The Ostwald–de Waele power law model is used to characterize the non-Newtonian fluid behavior. In this case a similarity solution is possible. The variation of heat and mass transfer coefficients with the governing parameters such as power law index, thermal and solutal dispersion parameters, inertia parameter, buoyancy ratio, and the Lewis number is discussed for a wide range of values of these parameters.     

Unsteady rotative flow of non-Newtonian fluid in an annular pipe

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Similitude considerations in laminar flow of modified power law fluids in circular ducts

A numerical solution is presented for the friction factorReynolds number relation for a fluid which is Newtonian at low shear rates, power law at high shear rates with a transition region at intermediate shear rates. The solution makes possible the conservation of similitude when designing duct systems for such fluids since both Reynolds numbers and shear rates are considered.     

Relaxation and retardation functions of the Maxwell model with fractional derivatives

A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type.The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles.This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value.In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.     

A non-Newtonian model of flow in a kneading disc region of a modular intermeshing corotating twin screw extruder

The flow of a non-Newtonian fluid in the kneading disc region of an intermeshing co-rotating twin screw extruder has been analyzed using hydrodynamic lubrication theory based on a power law shear viscosity model. The Tadmor et al. FAN method is used as a numerical solution method. Three different approximation methods for introducing power law behavior into the FAN method are described. These were subsequently used to calculate velocity and pressure fields in the kneading disc region. The effects of machine throughput and tip clearance as well as power law index on the flow field were determined. The computation results for the different methods are compared.     

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

Separating forces in blade coating of viscous and viscoelastic liquids

Coating of viscous and viscoelastic liquids is examined both theoretically and experimentally. A single simple geometry, a blade over a rotating roll, is considered. A perturbation solution to the Navier-Stokes equations yields a lubrication theory with first order corrections for curvature and inertia. A numerical solutions by the Finite Element Method (FEM) is compared to the analytical solutions. For Newtonian fluids, agreement between these mathematical models, and data on blade loading, is quite good.The effect of a non-Newtonian viscosity is explored by adopting a purely viscous power law model. The zeroth-order (lubrication) equations are solved by the method of Steidler and Horowitz, and predictions for coating thickness and blade loading agree quite well with those obtained from a FEM solution of the full equations of motion for a power law fluid. Data on blade loading, obtained using a strongly elastic polymer solution, are compared to these mathematical models, and discrepancies are noted.     

Long wavelength approximation to peristaltic motion of a power law fluid

Peristaltic motion of a power law fluid in a two-dimensional channel is studied. Assuming that the wavelength of the peristaltic wave is large in comparison to the mean half-width of the channel, a solution for the stream function is obtained as an asymptotic expansion in terms of slope parameter. Expressions for axial pressure gradient and shear stress are derived. The effect of flow behaviour indexn on the streamline pattern and shear stress is studied and the phenomenon of trapping is discussed.     

Numerical solution of the three‐dimensional fluid flow in a rotating heterogeneous porous channel

A numerical solution to the problem of the three‐dimensional fluid flow in a long rotating heterogeneous porous channel is presented. A co‐ordinate transformation technique is employed to obtain accurate solutions over a wide range of porous media Ekman number values and consequent boundary layer thicknesses. Comparisons with an approximate asymptotic solution (for large values of Ekman number) and with theoretical predictions on the validity of Taylor–Proudman theorem in porous media for small values of Ekman number show good qualitative agreement. An evaluation of the boundary layer thickness is presented and a power‐law correlation to Ekman number is shown to well‐represent the results for small values of Ekman number. The different three‐dimensional fluid flow regimes are presented graphically, demonstrating the distinct variation of the flow field over the wide range of Ekman numbers used. Copyright © 1999 John Wiley & Sons, Ltd.     

幂律型浆液扩散压力环向分布模型

为研究幂律型浆液注浆时注浆压力的变化情况,考虑盾尾断面新注入浆液与已注入浆液间阻碍作用,假设壁后注浆时盾尾形成三维环形空隙,提出了幂律型浆液扩散压力环向分布模型,并利用流体力学理论推导了幂律型浆液扩散压力环向分布式,分析了公式适用范围以及各参数的实际意义。与实际工程数据对比,验证了模型和计算式的正确性。结果表明,计算式可以反映注浆时环向分布各个位置压力值的大小;当公式中稠度系数n=1时,该式即为牛顿流体计算式,环向压力扩散模型同样适用,且幂律型流体环向扩散压力小于牛顿流体;受浆液自重影响,注浆孔注浆时向上表现为减压,向下表现为加压;压力环向分布断面呈现出上窄下宽的不规则环形;同一注浆孔幂律型浆液水灰比越大,浆液扩散压力越小。     

Viscous dissipation of a power law fluid in axial flow between isothermal eccentric cylinders

We study the temperature distribution of a power law fluid in a pressure-driven axial flow between isothermal eccentric cylinders in bipolar cylindrical coordinates. We begin our analysis by writing the equation of energy in bipolar cylindrical coordinates. We then obtain a dimensionless algebraic analytic solution for temperature profiles under a steady, laminar, incompressible and fully developed flow [Eq. (64)]. We find that the dimensionless temperature profile depends upon the radius ratio of the inner to outer cylinders, the eccentricity, the angular position, and the power law exponent n. The temperature is a strong function of the gap between the cylinders. The temperature profiles are flat in the middle of the gap and then, near the wall, suddenly drop to the wall temperature.     

Rotating disk heat transfer in a fluid swirling as a forced vortex

The paper represents results of an exact solution of a laminar heat transfer problem for a rotating disk in a fluid co-rotating with the disk as a solid body. The angular speed of the fluid is less than the angular speed of the disk. Disks surface temperature varies radially accordingly to a power law. Results for the laminar regime are compared with computations for turbulent heat transfer obtained using an integral method developed earlier. On the basis of the exact solution for laminar flow and basic ideas of the integral methods solution for turbulent flow, an integral method for laminar regime is designed and an approximate analytical solution of the considered problem is derived. Inaccuracies of the laminar approximate solution over the main range of variation of the influencing parameters and Prandtl numbers from 0.71 to 1 do not exceed 2.5%. It is shown that the dependence of the Nusselt number on the ratio of the angular speeds of disk and fluid varying from 0 to 0.3 is weak and has a point of maximum within this region for laminar flow. The obtained results are important in predictions of fluid flow and heat transfer in different types of rotating machinery.     

Natural convection of non-Newtonian power law fluids in a shallow horizontal rectangular cavity uniformly heated from below

Analytical and numerical study is conducted for two-dimensional steady-state buoyancy driven flow of a non-Newtonian power law fluid confined in a shallow rectangular horizontal cavity uniformly heated from below, while its short vertical rigid sides are considered adiabatic. The effect of the non-Newtonian behaviour on the onset of convection, fluid flow, temperature field, and heat transfer is examined. A closed approximate analytical solution is developed on the basis of the parallel flow assumption and the obtained results are validated numerically by solving the full governing equations.     

A computer model of the two-roll mill for generalised Newtonian flow

We investigate the flow of a generalised Newtonian liquid between two contra-rotating cylinders of equal size — the so called two-roll mill problem. A finite element method is used to obtain solutions for the case of a Carreau model and for a power law model. Consideration is given to the influence of shearthinning on the flow pattern around the cylinders. Results are presented for different speeds of rotation of the cylinders and for various values of fluid parameters. A comparison is made with the analytical solution of Jeffrey.     

Peristaltic motion of a non-Newtonian fluid

Summary To understand theoretically the flow properties of physiological fluids, we have considered as a model the peristaltic motion of a power law fluid in a tube, with a sinusoidal wave of small amplitude travelling down its wall. The solution for the stream function is obtained as a power series in terms of the amplitude of the wave. The stream function and the velocity components are evaluated by solving numerically two point boundary value problems with a singular point at the origin. The influence of the applied pressure gradient along with non-Newtonian parameters on the streamlines and velocity profiles are discussed in detail.     

Temporal analysis of breakup for a power law liquid jet in a swirling gas

The breakup mechanism and instability of a power law liquid jet are investigated in this study. The power law model is used to account for the non-Newtonian behavior of the liquid fluid. A new theoretical model is established to explain the breakup of a power law liquid jet with axisymmetric and asymmetric disturbances, which moves in a swirling gas. The corresponding dispersion relation is derived by a linear approximation, and it is applicable for both shear-thinning and shear-thickening liquid jets. Analysis results are calculated based on the temporal mode. The analysis includes the effects of the generalized Reynolds number, the Weber number, the power law exponent, and the air swirl strength on the breakup of the jet. Results show that the shear-thickening liquid jet is more unstable than its Newtonian and shear-thinning counterparts when the effect of the air swirl is taken into account. The axisymmetric mode can be the dominant mode on the power law jet breakup when the air swirl strength is strong enough, while the non-axisymmetric mode is the domination on the instability of the power liquid jet with a high We and a low Re _n . It is also found that the air swirl is a stabilizing factor on the breakup of the power law liquid jet. Furthermore, the instability characteristics are different for different power law exponents. The amplitude of the power law liquid jet surface on the temporal mode is also discussed under different air swirl strengths.     

Ideal incompressible flow around a wedge tip

A planar analog of conical flows is considered: an inviscid incompressible fluid flow around a wedge tip. A class of conical flows is found where vorticity is transported along streamlines by the potential component of velocity. Problems of a wave “locked” in the corner region and of a flow accelerating along the rib of a dihedral angle are considered. By analogy with an axisymmetric quasi-conical flow, a planar quasi-conical flow of the fluid is determined, namely, the flow inside and outside the region bounded by tangent curves described by a power law. Conditions are found where vorticity and swirl produce a significant effect. An approximate solution of the problem of the fluid flow inside a “zero” angle is obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 57–65, March–April, 2007.     

Wave regimes on a nonlinearly viscous fluid film flowing down a vertical plane

The flow of a thin film of a nonlinearly viscous fluid whose stress tensor is modeled by a power law, flowing down a vertical plane in the field of gravity, is considered. For the case of low flow rates, an equation that describes the evolution of surface disturbances is derived in the long-wave approximation. The domain of linear stability of the trivial solution is found, and weakly nonlinear, steady-state travelling solutions of this equation are obtained. The mechanism of branching of solution families at the singular point of the neutral curve is described. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 73–84, May–June, 2005.     

Quadratic programming algorithm for wall slip and free boundary pressure condition

Wall slip is often observed in a highly sheared fluid film in a solid gap. This makes a difficulty in mathematical analysis for the hydrodynamic effect because fluid velocity at the liquid–solid interfaces is not known a priori. If the gap has a convergent–divergent wedge, a free boundary pressure condition, i.e. Reynolds pressure boundary condition, is usually used in the outlet zone in numerical solution. This paper, based on finite element method and parametric quadratic programming technique, gives a numerical solution technique for a coupled boundary non‐linearity of wall slip and free boundary pressure condition. It is found that the numerical error decreases with the number of elements in a negative power law having an index larger than 2. Our method does not need an iterative process and can simultaneously gives rise to fluid film pressure distribution, wall slip velocity and surface shear stress. Wall slip always decreases the hydrodynamic pressure. Large wall slip even causes a null hydrodynamic pressure in a pure sliding solid gap. Copyright © 2005 John Wiley & Sons, Ltd.     

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

The linear stability analysis of vertical throughflow of power law fluid for double-diffusive convection with Soret effect in a porous channel is investigated in this study. The upper and lower boundaries are assumed to be permeable, isothermal and isosolutal. The linear stability of vertical through flow is influenced by the interactions among the non-Newtonian Rayleigh number (Ra), Buoyancy ratio (N), Lewis number (Le), Péclet number (Pe), Soret parameter (Sr) and power law index (n). The results indicate that the Soret parameter has a significant influence on convective instability of power law fluid. It has also been noticed that buoyancy ratio has a dual effect on the instability of fluid flow. Further, it is noticed that the basic temperature and concentration profiles have singularities at \(Pe = 0\) and \(Le = 1\), the convective instability is looked into for the limiting case of \(Pe\rightarrow 0\) and \(Le \rightarrow 1\). For the case of pure thermal convection with no vertical throughflow, the present numerical results coincide with the solution of standard Horton–Rogers–Lapwood problem. The present results for critical Rayleigh number obtained using bvp4c and two-term Galerkin approximation are compared with those available in the literature and are tabulated.