An Investigation of Stokes' Second Problem for Non-Newtonian Fluids

ABSTRACT

Stokes flow produced by an oscillatory motion of a wall is analyzed in the presence of a non-Newtonian fluid. A total of eight non-Newtonian models are considered. A mass balance approach is introduced to solve the governing Equations. The velocity and temperature profiles for these models are obtained and compared to those of Newtonian fluids. For the power law model, correlations for the velocity distribution and the time required to reach the steady periodic flow are developed and discussed. Furthermore, the effects of the dimensionless parameters on the flow are studied. For the temperature distribution, an analytical solution for Newtonian fluid is developed as a comparative source. To simulate the rheological behavior of blood at unsteady state, three non-Newtonian constitutive relationships are used to study the wall shear stress. It is found that in the case of unsteady stokes flow, although the patterns of velocity and wall shear stress is consistent across all models, the magnitude is affected by the model utilized.     

LAMINAR IMPINGING JET HEAT TRANSFER WITH A PURELY VISCOUS INELASTIC FLUID

Laminar impinging flow heat transfer is considered with a purely viscous inelastic fluid. The rheology of the fluid is modeled using a strain rate dependent viscosity coupled with asymptotic Newtonian behavior in the zero shear limit. The velocity and temperature fields are computed numerically for a confined laminar axisymmetric impinging flow. Important features of the non-Newtonian developing flow field are described and contrasted with the Newtonian situation. It is demonstrated that very small departures from Newtonian rheology lead to qualitative changes in the Nusselt number distribution along the impinging surface. In particular, a mildly shear thinning fluid displays a pronounced off-stagnation point heat transfer maxima, a feature that is not observed with a Newtonian fluid. Hence, Newtonian fluid approximations cannot adequately describe experimental heat transfer measurements in such situations even though they may be deemed acceptable in terms of describing the velocity field in the incoming nozzle. Numerical results are presented to analyze the effect of the dimensionless nozzle-to-plate distance, the rheological parameters, and the Reynolds and Prandtl numbers on the magnitude of the off-stagnation point peak heat transfer rate. The influence of the rheology of the fluid is particularly significant at low nozzle-to-plate distances since the mean strain rate in the flow field increases as the nozzle-to-plate distance is reduced. The numerical heat transfer results are interpreted in the context of the developing flow field.     

Peristaltic Transport of a Herschel–Bulkley Fluid in an Elastic Tube

In this paper, we investigate the peristaltic transport of a non‐Newtonian viscous fluid in an elastic tube. The governing equations are solved using the assumptions of long wavelength and low Reynolds number approximations. The constitution of blood has a non‐Newtonian fluid model and it demands the yield stress fluid model: The blood transport in small blood vessels is done under peristalsis. Among the available yield stress fluid models for blood flow, the non‐Newtonian Herschel–Bulkley fluid is preferred (because Bingham, power‐law and Newtonian models can be obtained as its special cases). The Herschel–Bulkley model has two parameters namely the yield stress and the power‐law index. The expressions for velocity, plug flow velocity, wall shear stress, and the flow rate are derived. The flux is determined as a function of inlet, outlet, external pressures, yield stress, amplitude ratio, and the elastic property of the tube. Further when the power‐law index n = 1 and the yield stress and in the absence of peristalsis, our results agree with Rubinow and Keller [J. Theor. Biol. 35 , 299 (1972)]. Furthermore, it is observed that, the yield stress, peristaltic wave, and the elastic parameters have strong effects on the flux of the non‐Newtonian fluid flow. Effects of various wave forms (namely, sinusoidal, trapezoidal and square) on the flow are discussed. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study on the non‐Newtonian fluid phenomena, especially the shear‐thinning phenomena. Shear thinning reduces the wall shear stress.     

Exact solution of two phase stratified flow through the pipes for non-Newtonian Herschel–Bulkley fluids

Steady state, laminar and fully developed stratified two phase flow including two immiscible fluids through the pipe has been studied analytically. One of the phases is Newtonian and the other one is non-Newtonian which obeys the Herschel–Bulkley fluid model. The dimensionless velocity distribution, Martinelli correction factor and non-Newtonian liquid holdup have been reported. The effect of interface curvature and wide range of viscosity ratio of two phases on flow behavior has been investigated. The results illustrate that the non-Newtonian rheological properties have significant effects on dimensionless velocity and consequently on two phase flow pressure drop specially for larger viscosity ratio.     

Mass and heat transport impact on the peristaltic flow of a Ree–Eyring liquid through variable properties for hemodynamic flow

Variable properties play a prominent role in analyzing the blood flow in narrow arteries. Specifically, considering the variation of thermal conductivity and viscosity helps in the understanding of the rheological behavior of blood and other biological fluids, such as urine, spermatozoa, and eye drops. Inspired by these applications, the current study incorporates the impact of variable thermal conductivity and viscosity for modeling the peristaltic flow of a Ree–Eyring liquid through a uniform compliant channel. The governing equations are nondimensionalized with the assistance of similarity transformations. The long-wavelength and small Reynolds wide variety approximation are utilized for solving the governing differential equations. Furthermore, the series solution method (perturbation technique) is utilized for solving the nonlinear temperature equation. The obtained results show that the velocity is greater in the case of the Newtonian liquid than that of the non-Newtonian liquid.     

Modulating fluid rheology and confinement toward augmenting the performance of a double layered microchannel heat sink

The improvement of the cooling performance of liquid-cooled microchannel heat sinks used for densely packed electronic circuits is sorted via passive techniques like tuning substrate or coolant properties. We propose a design for enhancing heat sink performance by simulataneously modifying the channel geometry and tuning the fluid rheology. By modeling the coolant as a power law fluid, its rheological behavior is varied ranging from shear-thinning to shear-thickening, alongside Newtonian fluid. We introduced tapering to the middle wall that separates the bottom and top channels of a double layered microchannel heat sink (DL-MCHS), causing both channels to converge. This convergence not only increases the flow velocity within the downstream microchannel but also reduces the apparent viscosity of the shear-thinning fluid being subjected to shear, resulting in enhanced thermal and hydraulic performance. We analyze the results from both the first and the second law of thermodynamics context, demonstrating that a tapered DL-MCHS with shear-thinning fluid outperforms a straight partition wall DL-MCHS with Newtonian coolant. However, we also discovered that extreme tapering compromises thermodynamic viability, but by fine-tuning the extent of tapering, we inferred that a DL-MCHS with shear-thinning fluid can become viable with little compromise in the thermal performance.     

Transitional processes of flow and heat transfer in a circular pipe with short static mixer

The transitional processes of flow and heat transfer in a circular pipe fitted with a short static mixer were studied with Newtonian and pseudoplastic fluids. An experimental formula, which was derived from the same concept as a well-known transition model of boundary layer flow on a flat plate, coincided well with the experimental results of friction and heat transfer. The transitional Reynolds number for heat transfer was larger than that for flow for both fluids. In heat transfer experiments, the transitional Reynolds number for a pseudoplastic fluid was smaller than that for a Newtonian fluid, and heat transfer augmentation in the transitional region was larger in a pseudoplastic fluid than in a Newtonian fluid. © 1997 Scripta Technica, Inc. Heat Trans Jpn Res 25(4): 254–266, 1996     

Thermal and rheological characteristics of CuO–Base oil nanofluid flow inside a circular tube

In the present experimental investigation, stable CuO–Base oil nanofluids with different particle weight fractions of 0.2% to 2% are prepared. Then, these fluids are used for heat transfer measurements as well as rheological behavior investigation. Density, thermal conductivities, viscosities and specific heat capacities of base fluid and all nanofluids at different temperatures are measured and the effect of nanoparticles concentration on fluid properties is investigated. Also, heat transfer characteristics of CuO–Base oil nanofluids laminar flow in a smooth tube under constant heat flux are studied experimentally. Experimental results clearly indicate that addition of nanoparticles into the base fluid enhances the thermal conductivity of the fluid and the enhancement increases with increasing of particle concentration. For the particle concentrations tested, nanofluids exhibit Newtonian behavior. It is observed that the dynamic viscosity substantially increases with the increase in nanoparticle concentration and this increase is more pronounced at the lower temperatures of the nanofluid. The specific heat capacity of nanofluids is significantly less than that of base fluid and it is decreased with the increase in nanofluid concentration. The results show that for a specific nanoparticle concentration, there is an increase in heat transfer coefficient of nanofluid flow compared to pure oil flow. A maximum increase of 12.7% in Heat Transfer coefficient was observed for 2 wt.% nanofluid at the highest Reynolds number studied in this investigation. Furthermore, heat transfer coefficients obtained using experimental fluid properties are compared to those obtained using the existing theoretical models for fluid properties.     

Thermal effects of two immiscible fluids through a permeable stenosed artery having nanofluid in the core region

A theoretical investigation of two-layered fluid flow in a stenosed tube having permeable walls is studied. The fluid (blood with nanoparticles) within the core region behaves as a non-Newtonian fluid (nanofluid) and the fluid within the peripheral layer behaves as a Newtonian fluid. Flow equations are linearized considering mild stenoses. The closed form mathematical expressions for flow resistance and wall shear stress are computed. The problem is solved using HPM (homotopy perturbation method). The numerical calculations of flow parameters (like flow resistance, wall shear stress) are performed and are discussed graphically. A novel result is found that with increased permeability and viscosity, the resistance of the fluid flow and shear at the wall is found to decrease. Moreover, the velocity profiles are increasing in the radial direction with the enhancement of viscosity of the fluid in the peripheral layer but decrease with permeability. Streamlines are drawn to examine the flow pattern.     

Slip Flow of Casson Rheological Fluid Under Variable Thermal Conductivity with Radiation Effects

This paper investigates the radiation and chemical reaction effects on Casson non‐Newtonian fluid towards a porous stretching surface in the presence of thermal and hydrodynamic slip conditions. The governing boundary layer conservation equations are normalized into nonsimilar form using similarity transformations. A numerical approach is applied to the resultant equations. The behavior of the velocity, temperature, concentration, as well as the skin friction coefficient, Nusselt number, and Sherwood number for various governing physical are discussed. Increasing the radiation parameter decreases the temperature. An increase in the rheological parameter (Casson parameter) induces an elevation in the skin friction coefficient, the heat and mass transfer rates. The larger the β values the closer the fluid is in behavior to a Newtonian fluid and further departs from plastic flow. Temperature of the fluid was found to decrease with increasing values of the Casson rheological parameter. The most important non‐Newtonian fluid possessing a yield value is the rheological Casson fluid, which finds significant applications in polymer processing industries, biomechanics, and chocolate food processing.     

微泡沫钻井液的具体流变模型建立和流变特性研究

实验研究表明,由于微泡沫具有假塑性的流动特性,在静止状态时又存在随微泡质量的增加而增加的静切力的特殊流体,使用屈服值幂律模式及赫-巴流变模式能较真实地反映泡沫的流动特性。但是,由于幂律模式简单、使用方便,且拟合效果与赫-巴模式相当,故综合考虑认为,幂律模式是描述该钻井液的较佳模式。通过压力实验进一步证明微泡在压力释放后,体积可以恢复90%以上。温度和压力变化对微泡流变性的影响从本质上讲,主要是对泡沫流体黏度的影响。换句话说,是温度和压力变化改变了泡沫流体的黏度。当剪切速率在比较低的范围内,增加压力会增加泡沫流体的表观黏度;当剪切速率在比较高的范围内,这种影响明显减弱。泡沫流体的表观黏度不稳定地随温度的升高而降低,但趋势逐渐变缓。     

Unsteady Convective Heat Transfer of Power-Law Fluid with Variable Fluid Properties in a Concentric Annulus Originating from a Polymer Flooding Process

We study the unsteady convective heat transfer of power-law fluid with variable fluid properties in a concentric annulus with isothermal surface. The problem is originated from the polymer flooding process between a sucker rod and oil well. A new power-law rheological model is proposed, which takes the effects of temperature on fluid viscosity and thermal conductivity into account. Numerical solutions are presented for velocity and temperature fields using the Chebyshev spectral method coupled with the strong stability-preserving Runge–Kutta time discretization. The exponential convergence is verified by accuracy testing between a smooth exact solution of the Partial Differential Equations (PDEs) with source terms and the numerical approximation of manufactured solutions. It is found that heat transfer is enhanced in the variable power-law index model, and a decrease in power-law index of pseudoplastic fluids promotes heat transfer due to the increased Nusselt number. Moreover, the influences of other parameters on convective heat transfer behaviors are discussed in detail.     

含油污泥的流变特性试验研究

针对舟山三种含油污泥样品,研究了温度、剪切速率、含渣率等参数对含油污泥流变特性的影响。通过幂律本构方程对实验数据进行分析处理,发现含油污泥在剪切速率增加时,剪切应力增加,粘度减小,含油污泥逐渐由假塑性流体向宾汉流体转变;温度越低,含油污泥的粘度越大且维持假塑性流体能力增强;含油污泥的含渣率越高,样品的非牛顿性越强,粘度值也越高。     

幂律流体环膜射流破碎特征的试验

通过自行搭建的环膜射流试验系统,采用高速摄影技术对牛顿流体(水)与剪切变稀型幂律流体(卡波姆凝胶)环膜射流的破碎模式及破碎特征进行了试验.结果表明:牛顿流体与幂律流体都存在相同的3种射流破碎模式,且两种流体在射流破碎机理上没有本质性的差异.在3种射流破碎模式中,圣诞树状破碎模式下的射流破碎效果最佳,泡状破碎模式次之,波...     

Heat Transfer in a Casson Rheological Fluid from a Semi‐infinite Vertical Plate with Partial Slip

The laminar boundary layer flow and heat transfer of Casson non‐Newtonian fluid from a semi‐infinite vertical plate in the presence of thermal and hydrodynamic slip conditions is analyzed. The plate surface is maintained at a constant temperature. Increasing velocity slip induces acceleration in the flow near the plate surface and the reverse effect further from the surface. Increasing velocity slip consistently enhances temperatures throughout the boundary layer regime. An increase in thermal slip parameter strongly decelerates the flow and also reduces temperatures in the boundary layer regime. An increase in the Casson rheological parameter acts to elevate considerably the skin friction (non‐dimensional wall shear stress) and this effect is pronounced at higher values of tangential coordinate. Temperatures, however, are very slightly decreased with increasing values of Casson rheological parameter. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21115     

Effect of heat generation on low Reynolds number fluid and mass transport in a single lymphatic blood vessel with uniform magnetic field

The paper studies fluid and mass transfer flow of blood in a single blood vessel. Blood is modeled as a Newtonian fluid of constant properties while the blood vessel is modeled as a long tube of circular section of slowly varying radius. The coupled governing equations are solved using a series expansion about a small parameter ε, (ε ≪ 1) characterizing the radius variation, and expressions are obtained for the velocity, temperature, concentration, shear stress and species transfer at the wall of the blood vessel. Results obtained show that the concentration of species in blood vessels drifts from the axis towards the walls of the blood vessel in qualitative agreement with existing clinical observations.     

Rheological effects on peristaltic transport of Bingham fluid through an elastic tube with variable fluid properties and porous walls

The peristaltic pumping of non‐Newtonian material configured by a tube is quite interesting phenomenon to examine the behavior of physiological fluids. The present paper investigates the role of variable viscosity and thermal conductivity on the peristaltic mechanism of Bingham fluid in an elastic tube with porous and convective boundary conditions. The long wavelength and small Reynolds number approximation is used to obtain the analytical solutions for velocity, plug flow velocity, streamlines, flow rate, and temperature. The theoretical determination of flux is calculated using the equilibrium condition, and an application to flow through an artery is highlighted by using the tension relation in an elastic tube. It is observed that the rate of flow within the elastic tube declined with variation of variable viscosity. An enhanced temperature distribution near the tube axis has been observed with increment of variable viscosity. An increasing influence of the porous parameter in the bolus size can also be seen from the streamlines plotted.     

Free surface dynamics of MHD third-grade fluid model over a heated stretching sheet with variable fluid properties

A thorough investigation of MHD third-grade differential-type fluid flow over a heated stretching sheet is performed in this work. In particular, we analyze the film thinning process, when the thermal sensitive fluid parameters vary due to the effect of heat supplied to the stretching sheet. Starting with a two-dimensional (2D) free surface boundary value problem of non-Newtonian third-grade fluid, we present a systematic derivation of a 1D transient thin-film height equation using longwave analysis with respect to the small aspect ratio of the fluid domain. The derived model is used to study the impact of Newtonian and non-Newtonian parameters with variable fluid properties on the thin film height. The model is discretized using an upwind discretization in space and implicit time integration to guarantee first-order convergence. The model is analyzed thoroughly with the help of numeric computing software MATLAB. The existing findings for a Newtonian fluid are in excellent agreement with derived evidence. In comparison to Newtonian fluid, the study finds that the third-grade parameter causes thinning under different parametric restrictions. Simulations on the coupling effect explain that, the film thickness can be reduced with a high Marangoni number for highly viscous fluids. Also, since the effect of the conductivity parameter can be reduced at a low Prandtl number, the fluid shows a thinning effect. The film thinning rate, on the other hand, is reduced by the magnetic field.     

Analytical solution for Newtonian–Bingham plastic two-phase pressure driven stratified flow through the circular ducts

For steady state, stratified, laminar, fully developed two-phase flow which one of them is Newtonian and the other one is Bingham plastic, the motion equations in horizontal pipe with appropriate boundary conditions have been solved analytically. Pressure drop, velocity distribution and location of plug region related to Bingham plastic fluid have been reported. The results show that the non-Newtonian rheological properties have negligible effects on two-phase velocity profile and consequently on pressure gradient in small viscosity ratio of two fluids. With promotion of viscosity ratio, the influence of yield stress on two-phase velocity profile is more considerable.     

Study on forced convective heat transfer of non-newtonian nanofluids

This paper is concerned with the forced convective heat transfer of dilute liquid suspensions of nanoparticles (nanofluids) flowing through a straight pipe under laminar conditions. Stable nanofluids are formulated by using the high shear mixing and ultrasonication methods. They are then characterised for their size, surface charge, thermal and rheological properties and tested for their convective heat transfer behaviour. Mathematical modelling is performed to simulate the convective heat transfer of nanofluids using a single phase flow model and considering nanofluids as both Newtonian and non-Newtonian fluid. Both experiments and mathematical modelling show that nanofluids can substantially enhance the convective heat transfer. Analyses of the results suggest that the non-Newtonian character of nanofluids influences the overall enhancement, especially for nanofluids with an obvious non-Newtonian character.