Drag reduction of a nonnewtonian fluid by fluid injection on a moving wall

Summary A boundary layer problem of a nonnewtonian fluid flow with fluid injection on a semi-infinite flat plate whose surface moves with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. Concluding similarity equations are solved numerically to show the dependence of the problem to the velocity ratio λ of the plate to uniform flow and to the injection velocity parameter C. The critical values of λ and C for each nonnewtonian power-law index n are obtained, and their significance in drag reduction is discussed. Received 26 August 1997; accepted for publication 21 October 1998     

Mass transfer effects on the non-Newtonian fluids past a vertical plate embedded in a porous medium with non-uniform surface heat flux

 The present study is devoted to investigate the influences of mass transfer on buoyancy induced flow over vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald–de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solution for the transformed governing equations is obtained with prescribed variable surface heat flux. Numerical results for the details of the velocity, temperature and concentration profiles are shown on graphs. Excess surface temperature as well as concentration gradient at the wall associated with heat flux distributions, which are entered in tables, have been presented for different values of the power-law index n, buoyancy ration B and the exponent λ as well as Lewis number Le. Received on 26 April 2000     

On thermal boundary layer of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection

 Heat transfer characteristics of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection were investigated. Similarity solutions of the laminar boundary layer equations describing heat transfer and fluid flow in a quiescent fluid were obtained and solved numerically. Velocity and temperature profiles as well as the Nusselt number, Nu, were studied for two thermal boundary conditions; uniform surface temperature and variable surface temperature, for different parameters; Prandtl number Pr, temperature exponent b, velocity exponent m, injection parameter d and power-law index n. It was found that decreasing injection parameter d, and power-law index n and increasing Prandtl number Pr and surface temperature exponent b enhance the heat transfer coefficient. Received on 27 April 2000     

The Falkner–Skan Wedge Flows of Power-Law Fluids Embedded in a Porous Medium

The non-Darcy flow characteristics of power-law non-Newtonian fluids past a wedge embedded in a porous medium have been studied. The governing equations are converted to a system of first-order ordinary differential equations by means of a local similarity transformation and have been solved numerically, for a number of parameter combinations of wedge angle parameter m, power-law index of the non-Newtonian fluids n, first-order resistance A and second-order resistance B, using a fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting method. Velocity and shear stress at the body surface are presented for a range of the above parameters. These results are also compared with the corresponding flow problems for a Newtonian fluid. Numerical results show that for the case of the constant wedge angle and material parameter A, the local skin friction coefficient is lower for a dilatant fluid as compared with the pseudo-plastic or Newtonian fluids.     

Non-Newtonian flow in microporous structures under the electroviscous effect

Single phase non-Newtonian microporous flow combined with the electroviscous effect is investigated in the pore-scale under conditions of various rheological properties and electrokinetic parameters. The lattice Boltzmann method is employed to solve both the electric potential field and flow velocity field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends on both the fluid rheological behavior and pore surface area ratio significantly. For the shear thinning fluid with power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electrovicous effect plays a more important role compared to the Newtonian fluid and shear thickening fluid. The high pore surface area ratio in the porous structure increases the electroviscous force and the induced flow resistance becomes important even to the flow of Newtonian and shear thickening fluids.     

Analytic homotopy solution of generalized three-dimensional channel flow due to uniform stretching of the plate

In this communication a generalized threedimensional steady flow of a viscous fluid between two infinite parallel plates is considered. The flow is generated due to uniform stretching of the lower plate in x- and y-directions. It is assumed that the upper plate is uniformly porous and is subjected to constant injection. The governing system is fully coupled and nonlinear in nature. A complete analytic solution which is uniformly valid for all values of the dimensionless parameters β, Re and λ is obtained by using a purely analytic technique, namely the homotopy analysis method. Also the effects of the parameters β, Re and λ on the velocity field are discussed through graphs.     

Generalized flow analysis of non-Newtonian visco-elastic fluid flow through fractal reservoir

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Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as x ^m and the velocity outside the boundary layer varies as x ^(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λ _c (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ _c is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 2: Numerical Simulations

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Generalized Solution for 1-D Non-Newtonian Flow in a Porous Domain due to an Instantaneous Mass Injection

Non-Newtonian fluid flow through porous media is of considerable interest in several fields, ranging from environmental sciences to chemical and petroleum engineering. In this article, we consider an infinite porous domain of uniform permeability k and porosity f{\phi} , saturated by a weakly compressible non-Newtonian fluid, and analyze the dynamics of the pressure variation generated within the domain by an instantaneous mass injection in its origin. The pressure is taken initially to be constant in the porous domain. The fluid is described by a rheological power-law model of given consistency index H and flow behavior index n; n, < 1 describes shear-thinning behavior, n > 1 shear-thickening behavior; for n = 1, the Newtonian case is recovered. The law of motion for the fluid is a modified Darcy’s law based on the effective viscosity μ _ef , in turn a function of f, H, n{\phi, H, n} . Coupling the flow law with the mass balance equation yields the nonlinear partial differential equation governing the pressure field; an analytical solution is then derived as a function of a self-similar variable η = rt ^β (the exponent β being a suitable function of n), combining spatial coordinate r and time t. We revisit and expand the work in previous papers by providing a dimensionless general formulation and solution to the problem depending on a geometrical parameter d, valid for plane (d = 1), cylindrical (d = 2), and semi-spherical (d = 3) geometry. When a shear-thinning fluid is considered, the analytical solution exhibits traveling wave characteristics, in variance with Newtonian fluids; the front velocity is proportional to t ^(n-2)/2 in plane geometry, t ^{(2n-3)/(3−n)} in cylindrical geometry, and t ^{(3n-4)/[2(2−n)]} in semi-spherical geometry. To reflect the uncertainty inherent in the value of the problem parameters, we consider selected properties of fluid and matrix as independent random variables with an associated probability distribution. The influence of the uncertain parameters on the front position and the pressure field is investigated via a global sensitivity analysis evaluating the associated Sobol’ indices. The analysis reveals that compressibility coefficient and flow behavior index are the most influential variables affecting the front position; when the excess pressure is considered, compressibility and permeability coefficients contribute most to the total response variance. For both output variables the influence of the uncertainty in the porosity is decidedly lower.     

幂律流体圆柱绕流的格子波尔兹曼模拟

基于插值补充格子波尔兹曼方法和幂律流体的本构方程,建立了贴体坐标系下适用于幂律流体的格子波尔兹曼模型,模拟了幂律流体的圆柱绕流问题,采用非平衡外推格式处理圆柱表面的速度无滑移边界,利用应力积分法确定曳力系数和升力系数,并与基于标准的格子波尔兹曼方法和有限容积法获得的数值数据进行对比,吻合良好. 进行了网格无关性验证之后,分析了稳态流动时,不同雷诺数下幂律指数对于尾迹长度、分离角、圆柱表面黏度分布、表面压力系数及曳力系数的影响,以及非定常流动中,幂律指数对于流场、曳力系数、升力系数和斯特劳哈尔数的影响. 获得的变化规律与基于其他数值模拟方法得到的结果相一致,充分验证了模型的有效性和正确性. 结果表明:插值补充格子波尔兹曼方法可以用来模拟幂律流体在具有复杂边界流场内的流动问题,通过引入不同的非牛顿流体本构方程,该方法还可以进一步应用于其他类型的非牛顿流体研究中.      

Electroviscous effect on non-Newtonian fluid flow in microchannels

Understanding non-Newtonian flow in microchannels is of both fundamental and practical significance for various microfluidic devices. A numerical study of non-Newtonian flow in microchannels combined with electroviscous effect has been conducted. The electric potential in the electroviscous force term is calculated by solving a lattice Boltzmann equation. And another lattice Boltzmann equation without derivations of the velocity when calculating the shear is employed to obtain flow field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends significantly on the fluid rheological behavior. For the shear thinning fluid of the power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electroviscous effect plays a more important role. And its effect on the flow increases as the ratio of the Debye length to the channel height increases and the exponent n decreases. While the shear thickening fluid of n > 1 is less affected by the electroviscous force, it can be neglected in practical applications.     

Effects of viscous dissipation on a power-law fluid over plate embedded in a porous medium

The present study is devoted to investigate the influences of viscous dissipation on buoyancy induced flow over a horizontal or a vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald-de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solutions for the transformed governing equations are obtained with prescribed variable surface temperature (PT) or with prescribed variable surface heat flux (PHF) for the horizontal plate case. While, the similarity solutions are obtained with prescribed variable surface heat flux for the vertical plate case. Different similar transformations, for each case, are used. Numerical results for the details of the velocity and temperature profiles are shown on graphs. Nusselt number associated with temperature distributions and excess surface temperature associated with heat flux distributions which are entered in tables have been presented for different values of the power-law index n and the exponent as well as Eckert number.     

A remark on similarity analysis of boundary layer equations of a class of non-Newtonian fluids

A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluid in which the stress, an arbitrary function of rates of strain, is studied. It is shown that under any group of transformation, for an arbitrary stress function, not all non-Newtonian fluids possess a similarity solution for the flow past a wedge inclined at arbitrary angle except Ostwald-de-Waele power-law fluid. Further it is observed, for non-Newtonian fluids of any model only $90°$ of wedge flow leads to similarity solutions. Our results contain a correction to some flaws in Pakdemirli׳s [14] (1994) paper on similarity analysis of boundary layer equations of a class of non-Newtonian fluids.     

On the unsteady rotational flow of a generalized Maxwell fluid through a circular cylinder

In this paper, the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized Maxwell fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. Initially, the fluid is at rest, and the motion is produced by the rotation of the cylinder about its axis with a unsteady angular velocity. The solutions that have been obtained are presented under series form in terms of the generalized G _a,b,c (, t)-functions. The similar solutions for the ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases, when β → 1, respectively β → 1 and λ → 0, from general solutions. Finally, the solutions that have been obtained are compared by graphical illustrations, and the influence of the pertinent parameters on the fluid motion is also underlined by graphical illustrations.     

幂律流体圆柱绕流的格子波尔兹曼模拟简

基于插值补充格子波尔兹曼方法和幂律流体的本构方程,建立了贴体坐标系下适用于幂律流体的格子波尔兹曼模型,模拟了幂律流体的圆柱绕流问题,采用非平衡外推格式处理圆柱表面的速度无滑移边界,利用应力积分法确定曳力系数和升力系数,并与基于标准的格子波尔兹曼方法和有限容积法获得的数值数据进行对比,吻合良好. 进行了网格无关性验证之后,分析了稳态流动时,不同雷诺数下幂律指数对于尾迹长度、分离角、圆柱表面黏度分布、表面压力系数及曳力系数的影响,以及非定常流动中,幂律指数对于流场、曳力系数、升力系数和斯特劳哈尔数的影响. 获得的变化规律与基于其他数值模拟方法得到的结果相一致,充分验证了模型的有效性和正确性. 结果表明：插值补充格子波尔兹曼方法可以用来模拟幂律流体在具有复杂边界流场内的流动问题,通过引入不同的非牛顿流体本构方程,该方法还可以进一步应用于其他类型的非牛顿流体研究中.     

Effects of drift angle on model ship flow

The effects of drift angle on model ship flow are investigated through towing tank tests for the Series 60 C_B=0.6 cargo/container model ship. Resistance, side force, drift moment, sinkage, trim, and heel data are procured for a range of drift angles β and Froude numbers (Fr) and the model free condition. Detailed free-surface and mean velocity and pressure flow maps are procured for high and low Fr=0.316 and 0.16 and β=5° and 10° (free surface) and β=10° (mean velocity and pressure) for the model fixed condition (i.e. fixed with zero sinkage, trim, and heel). Comparison of results at high and low Fr and previous data for β=0° enables identification of important free-surface and drift effects. Geometry, conditions, data, and uncertainty analysis are documented in sufficient detail so as to be useful as a benchmark for computational fluid dynamics (CFD) validation. The resistance increases linearly with β with same slope for all Fr, whereas the increases in the side force, drift moment, sinkage, trim, and heel with β are quadratic. The wave profile is only affected near the bow, i.e. the bow wave amplitude increases/decreases on the windward/leeward sides, whereas the wave elevations are affected throughout the entire wave field. However, the wave envelope angle on both sides is nearly the same as β=0°, i.e. the near-field wave pattern rotates with the hull and remains within a similar wave envelope as β=0°. The wave amplitudes are significantly increased/decreased on the windward/leeward sides. The wake region is also asymmetric with larger wedge angle on the leeward side. The boundary layer and wake are dominated by the hull vortex system consisting of fore body keel, bilge, and wave-breaking vortices and after body bilge and counter-rotating vortices. The occurrence of a wave-breaking vortex for breaking bow waves has not been previously documented in the literature. The trends for the maximum vorticity, circulation, minimum axial velocity, and trajectories are discussed for each vortex. Received: 16 September 1999/Accepted: 8 November 2001     

Modelling of Power-law Fluid Flow Through Porous Media Using Smoothed Particle Hydrodynamics

The flow of non-Newtonian fluids through two-dimensional porous media is analyzed at the pore scale using the smoothed particle hydrodynamics (SPH) method. A fully explicit projection method is used to simulate incompressible flow. This study focuses on a shear-thinning power-law model (n < 1), though the method is sufficiently general to include other stress-shear rate relationships. The capabilities of the proposed method are demonstrated by analyzing a Poiseuille problem at low Reynolds numbers. Two test cases are also solved to evaluate validity of Darcy’s law for power-law fluids and to investigate the effect of anisotropy at the pore scale. Results show that the proposed algorithm can accurately simulate non-Newtonian fluid flows in porous media.     

Influence of lateral mass flux on mixed convection heat and mass transfer over inclined surfaces in porous media

 The effect of lateral mass flux on mixed convection heat and mass transfer in a saturated porous medium adjacent to an inclined permeable surface is analyzed. A similarity solution is obtained when surface temperature and concentration, free stream velocity and injection/suction velocity of fluid are prescribed as power functions of distance from the leading edge. The cases when the flow and buoyancy forces are in the same and opposite directions are discussed both for aiding and opposing buoyancy effects. The governing parameters are the mixed convection parameter Gr, the Lewis number Le, the buoyancy ratio N, the lateral mass flux parameter f _w, representing the effects of injection or withdrawal of fluid at the wall, and λ which specifies three cases of the inclined plate. The interactive effect of these parameters on heat and mass transfer rates are presented. It is observed that the diffusion ratio (Le) has a more pronounced effect on concentration field than on flow and temperature fields. It is found that the rates of heat and mass transfer increase with suction and decrease with injection of the fluid. Received on 31 August 2000 / Published online: 29 November 2001     

Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation, (2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values of ξ ( = Gr _x /Re _x ², defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied. We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity gradient is higher than that of temperature. On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh.