Laminar Flow Of Power-law Fluids Research Articles

Influence of Permeability and Shear-Thinning Behavior on the Hydrodynamics Flow Features Around Porous Square Cylinders

Abstract This article investigates the laminar flow of power-law fluids through two porous square cylinders in a side-by-side configuration. The effects of power-law index (n), Darcy number (Da), and gap ratio (g/W) are examined within ranges of g/W = 0.5–5, n = 0.4–0.8, and Da = 10−6–10−2, respectively. Two flow conditions are considered: first, for a creeping flow (unseparated flow) at Re = 1 where Darcy's law is applicable; second, for a viscous dominant flow at Re = 100, where Darcy–Forchheimer-extended model is exercised. Flow patterns behind the porous cylinders are analyzed using streamlines, velocity profiles, pressure distribution curves, and vorticity structural parameters (Г). In low permeability levels, the flow exhibits an irregular nonperiodic vortex shedding characterized by a single large vortex street far off the downstream for g/W = 0.5. However, synchronized wake patterns were observed in either antiphase or in-phase modes for higher gap ratios. Leading-edge separation with two-side recirculation induces quasi-periodicity in the flow for all g/W. It was found that increasing the permeability can prevent the leading edge separation. Additionally, a transition from antiphase to in-phase mode occurs when the permeability is altered while maintaining constant flow-time. The presence of a jet-like flow between cylinders significantly impacts unsteady wake patterns. The impact of g/W, power-law index, and permeability on drag is also examined. A jump in some flow parameters was observed at higher Re for the midrange Darcy number, but no such increase was noted for the high shear-thinning behavior. These findings provide a potential approach for improving the design of fluidic systems involving porous cylinders.

A numerical study of two-dimensional laminar flow of power-law fluids around three circular cylinders in side-by-side arrangement

PurposeThis study aims at investigating two-dimensional laminar flow of power-law fluids around three unconfined side-by-side cylinders.Design/methodology/approachThe numerical study is performed by solving the governing (continuity and momentum) equations using a finite volume-based code ANSYS Fluent. The numerical results have been presented for different combinations of the governing dimensionless parameters (dimensionless spacing, 1.2 = L = 4; Reynolds number, 0.1 = Re = 100; power-law index, 0.2 = n = 1.8). The dependence of the kinematic and macroscopic characteristics of the flow such as streamline patterns, distribution of the surface pressure coefficient, total drag coefficient with its components (pressure and friction) and total lift coefficient on these dimensionless parameters has been discussed in detail.FindingsIt is found that the separation of the flow and the apparition of the wake region accelerate as the dimensionless spacing decreases, the number of the cylinder increases and/or the fluid behavior moves from shear-thinning to Newtonian then to shear-thickening behavior. In addition, the distribution of the pressure coefficient on the surface of the cylinders presents a complex dependence on the fluid behavior index and Reynolds number when the dimensionless spacing between two adjacent cylinders is varied. At low Reynolds numbers, the drag coefficient of shear-thinning fluids is stronger than that of Newtonian fluids; this tendency decreases progressively with increasing of Re until a critical value; beyond the critical Re, the opposite trend is observed. The lift coefficient of the middle cylinder is null, whereas, the exterior cylinders experience opposite lift coefficients, which show a complex dependence on the dimensionless spacing, the Reynolds number and the power-law index.Originality/valueThe flow over bluff bodies is a practical engineering problem. In the literature, it can be seen that the previous studies on non-Newtonian fluids are limited to the flow over one or two cylinders (effect of an odd number of cylinders on each other). Besides that, the available results concerning the flow of Newtonian fluids over three cylinders are limited to the high Reynolds numbers region only. However, this work treats the flow of non-Newtonian power-law fluids past three circular cylinders in side-by-side arrangements under a wide range of Re. The outcome of the present study demonstrates that the augmentation of the geometry complexity to three cylinders (effect of pair surrounding cylinders on the surrounded ones in what concerns Von Karman Street phenomenon) causes a drastic change in the flow patterns and in the macroscopic characteristics. The present results may be used to predict the flow behavior around multiple side-by-side cylinders.

Friction factor of shear thinning fluids in non-circular ducts—a simplified approach for rapid engineering calculation

This work reports on an investigation of the relationship between the Reynolds number and the friction factor of the fully developed laminar flow of power-law fluids through non-circular cross-sections (elliptical, concentric annuli, symmetrical L-shape, isosceles triangle, eccentric annuli, and square duct with a central cylindrical core), with the aim of providing a simple method for rapid and simplified engineering calculations. The friction factor—Reynolds number relation has been derived for non-circular ducts on the basis of the modified Reynolds number proposed by Metzner and Reed. It has been shown that the proposed correlation enables a prediction of the friction factor for the fully developed laminar flow of shear thinning fluids in non-circular cross-sections by using a calculation method similar to the method used for laminar flow of a Newtonian fluid. The correlation that is provided has been validated by analytical and numerical methods. It has been found that the proposed method can be used to determine the friction factor with high accuracy. This method yields a friction factor with a deviation of less than 5% in comparison with the conventional method.

Laminar Power-Law Fluid Flow in a T-Shaped Channel at Given Pressure Differences

The steady laminar flow of an incompressible power-law fluid in a plane T-shaped channel is studied at given pressure differences between the inflow and outflow sections. A non-Newtonian fluid, whose rheological behavior is governed by the Ostwald—de Waele law is called the power-law fluid. The mathematical formulation of the problem includes the equations of motion and continuity. The no-slip condition is imposed on solid walls. The problem solution is obtained using a finite-difference method invoking the SIMPLE procedure. A parametric investigation of the kinematic and dynamic flow parameters is carried out for different values of the relevant parameters of the problem. The flow regime map is constructed for different pressure differences between the inflow and outflow boundaries and different nonlinearity exponents of the rheological model.

A practical method for predicting the friction factor of power-law fluids in a rectangular duct

This article presents a simple and unique method for predicting the friction factor for the fully developed, laminar flow of power-law fluids in ducts with a rectangular cross-section by means of an engineering calculation for rapid equipment design. The relationship between friction factor f and the Reynolds number defined by Metzner–Reed (ReMR) for a rectangular cross-section was investigated and rearranged with very good engineering accuracy using the results of very simple correlations based on the application of the well-known and long-established coefficient for Newtonian fluid flow. The investigated value of the flow index was up to 2, and the aspect ratio ranged from 0 to 1, with calculation error not higher than 3.5%. The proposed method was compared with conventional methods from the literature, and was validated by means of numerical computations, and also with experimental data from the literature. The product of friction factor and ReMR is a linear function of geometrical parameter C and flow behavior index n and also the proposed method predicts the friction factor as accurately as conventional and traditional methods.

Non-Newtonian Power-Law Fluid Flow around a Heated Square Bluff Body in a Vertical Channel under Aiding Buoyancy

The present study focuses on momentum and heat transfer characteristics of the laminar flow of power-law fluids in a vertical channel with a built-in square cylinder under the influence of aiding buoyancy. The simulations are carried out at low Reynolds numbers (1–40) for the range of settings as power-law index = 0.4–1, Richardson number = 0–1, blockage ratio = 25%–50%, and Prandtl number = 50. It is found that the drag coefficient and the Nusselt number increase with the aiding buoyancy, while the shear-thinning behavior reduces the drag coefficient and increases the heat transfer rate.

Collocated Discrete Least Squares (CDLS) meshless method for the simulation of power-law fluid flows

Least squares is a robust and simple method in function approximation. Collocated Discrete Least Squares (CDLS) is a meshless method based on least squares technique. In this paper CDLS is used with a non-incremental projection method for the solution of incompressible generalized Newtonian fluid flow equations in the simulation of laminar flow of power-law fluids. The scheme is used to solve two benchmark problems named lid-driven cavity flow and flow past a circular cylinder for the power-law fluids.

Laminar flow of power-law fluids past a hemisphere: Momentum and forced convection heat transfer characteristics

The governing field equations describing the momentum and heat transfer characteristics of the laminar power-law fluid flow past a hemisphere have been solved numerically. The present results span wide ranges of Reynolds number (1 to 150), power-law index (0.1 to 1) and Prandtl number (0.72 to 100) in order to delineate their influence on the momentum and heat transfer characteristics of a hemisphere oriented with its curved surface towards the oncoming fluid stream. Over the range of Reynolds number, the flow is expected to be steady and axisymmetric. Owing to the sudden loss of rear surface, the flow detaches itself from the hemisphere at much lower Reynolds numbers than that for a sphere. Similarly, the effect of power-law index gradually weakens with the increasing Reynolds number due to the increasing significance of inertial forces. All else being equal, the drag on a hemisphere is increased in shear-thinning fluids than that in Newtonian fluids at low Reynolds numbers and it is the other way round at high Reynolds numbers. Similarly, irrespective of the type of thermal boundary condition prescribed on the surface of the hemisphere, heat transfer is facilitated in shear-thinning fluids with reference to that in Newtonian fluids otherwise under identical conditions.

Transient and steady-state analysis of heat, mass, and momentum transfer in developing and fully-developed regions of homogeneous tubular reactors with non-Newtonian fluid flow

In this article, the problem of simultaneous heat, mass, and momentum transfer in the developing region of tubular reactors with homogeneous chemical reaction and laminar power-law fluid flow under unsteady-state conditions has been solved. In this regard, the general governing equations were solved using finite-difference method and analyzed carefully. Moreover, the influences of various parameters and dimensionless numbers such as power-law index, heat of reaction, reaction order, and Damköhler number value on the numerical results were investigated. In addition, the numerical results obtained for the fully-developed velocity distribution of Newtonian fluid flow and Sherwood number value were validated against the well-known fully-developed velocity distribution and reported Sherwood number value in the literature and good agreement was obtained.

Measurement of shear rate on the surface of a cylinder submerged in laminar flow of power-law fluids

Micro-electrodes made of platinum have been used to measure the local shear rate on the surface of a cylinder (length-to-diameter ratio of 11–12) exposed to the fully developed laminar flow of power-law fluids in a vertical pipe. Two Newtonian and two shear-thinning solutions are used as model test fluids to ascertain the influence of shear-thinning viscosity on the distribution of shear rate on the surface of the cylinder. The results reported herein encompass a wide range of Reynolds numbers 0.16⩽Re⩽75 based on the cylinder diameter and centreline velocity. Over the range of conditions, it is observed that the shear rate is a maximum at about θ=130° and it tends to be higher in shear-thinning fluids than that in Newtonian fluids otherwise under identical conditions.

Flow of power-law fluids past an equilateral triangular cylinder: Momentum and heat transfer characteristics

Extensive numerical simulations for the 2-D laminar flow of power-law fluids over an equilateral triangular cylinder are performed to elucidate the role of power-law index (0.2 ≤ n ≤ 1.8) on the critical Reynolds number denoting the onset of flow separation and of vortex shedding. Results are presented for both orientations of the triangular cylinder, namely, its apex facing in the upstream and the downstream directions. For shear-thinning fluids, the onset of wake formation and vortex shedding both are seen to be delayed than that in Newtonian and shear-thickening fluids. After delineating the limits of the steady flow regime, the effects of the Reynolds and Prandtl numbers and the power-law index on drag phenomena and heat transfer characteristics of triangular cylinder for shear-thinning fluids have been studied in the steady flow regime. The results reported herein embrace the following ranges of conditions: 0.2 ≤ n ≤ 1; 1 ≤ Re ≤ 30 and 1 ≤ Pr ≤ 100. Detailed results on isotherm contours, local Nusselt number and its surface-averaged values are presented. Finally, the present numerical values of the critical Reynolds numbers and Nusselt number have been correlated using simple forms which are convenient for interpolating the present results for the intermediate values of the governing parameters in a new application.

Laminar flow of power-law fluids past a rotating cylinder

In this work, the continuity and momentum equations have been solved numerically to investigate the flow of power-law fluids over a rotating cylinder. In particular, consideration has been given to the prediction of drag and lift coefficients as functions of the pertinent governing dimensionless parameters, namely, power-law index (1 ≥ n ≥ 0.2), dimensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (0.1 ≤ Re ≤ 40). Over the range of Reynolds number, the flow is known to be steady. Detailed streamline and vorticity contours adjacent to the rotating cylinder and surface pressure profiles provide further insights into the nature of flow. Finally, the paper is concluded by comparing the present numerical results with the scant experimental data on velocity profiles in the vicinity of a rotating cylinder available in the literature. The correspondence is seen to be excellent for Newtonian and inelastic fluids.

Flow of Newtonian and Power-Law Fluids Past an Elliptical Cylinder: A Numerical Study

Extensive numerical simulations of the 2-D laminar flow of power-law fluids over elliptical cylinders with different aspect ratios have been carried out to establish the conditions for the onset of wake formation and the onset of vortex shedding. The continuity and momentum equations were solved numerically using FLUENT (version 6.3.26). The influence of the power-law index (0.3 ≤ n ≤ 1.8) and the aspect ratio (E = b/a; 0.2 ≤ E ≤ 5) of the cylinder on the critical values of the Reynolds number denoting the onsets of flow separation and vortex shedding are presented. For shear-thinning (n 1). Also, the values of the Strouhal number (Stc) and the time-average drag coefficient (CD) corresponding to the cessation of the steady-flow regime are presented. Velocity vector plots denoting the flow separation and vorticity profiles showi...

Two-dimensional laminar flow of a power-law fluid across a confined square cylinder

Two-dimensional laminar flow of power-law fluids past a long square cylinder confined in a planar channel is investigated numerically for the range of conditions as 60 ≤ R e ≤ 160 , 0.5 ≤ n ≤ 1.8 and β = 1 / 6 , 1 / 4 , and 1/2. A semi-explicit finite volume method is used on a non-uniform collocated grid arrangement. The third order QUICK scheme and the second-order central difference scheme are used to discretize the convective and diffusive terms respectively. Depending upon the value of blockage ratio, power-law index and Reynolds number, the nature of flow in the above range of conditions is either steady or unsteady (periodic in time). An increase in the blockage ratio delays the onset of vortex shedding to higher Reynolds number in both shear-thinning and shear-thickening fluids whereas it advances the occurrence of the quasi-periodicity in flow to lower Reynolds numbers in shear-thinning fluids. Extensive numerical results are presented to elucidate the effects of blockage, power-law index and Reynolds number on the drag coefficient, stream function, vorticity, Strouhal number and amplitudes of drag and lift coefficients in the unsteady flow regime.

Transient and steady-state forced convection to power-law fluids in the thermal entrance region of circular ducts: Effects of viscous dissipation, variable viscosity, and axial conduction

A numerical study was conducted on the transient behavior of a hydrodynamically fully developed, laminar flow of power-law fluids in the thermally developing entrance region of circular ducts with taking into account the effects of viscous dissipation, axial conduction, and variations of viscosity with temperature. In this regard, the unsteady-state thermal energy and momentum equations were solved numerically using a finite-difference method, whereas the steady-state thermal energy equation with constant wall heat flux as the boundary condition was solved analytically as the initial condition of the former. The numerical procedure used in the present work was validated with an analytical solution for the special case of Newtonian fluids. The effects of the power-law index, axial conduction, wall heat flux, and variations of fluid viscosity on the local Nusselt number and thermal entrance length were investigated. Moreover, the local Nusselt number values of steady-state conditions were correlated as a function of the power-law index and wall heat flux. Furthermore, a correlation was derived for the thermal entrance length as a function of the power-law index and wall heat flux.

Transient forced convection with viscous dissipation to power-law fluids in thermal entrance region of circular ducts with constant wall heat flux

A numerical investigation was conducted on the transient behavior of a hydrodynamically, fully developed, laminar flow of power-law fluids in the thermally developing entrance region of circular ducts taking into account the effect of viscous dissipation but neglecting the effect of axial conduction. In this regard, the unsteady state thermal energy equation was solved by using a finite difference method, whereas the steady state thermal energy equation without wall heat flux was solved analytically as the initial condition of the former. The effects of the power-law index and wall heat flux on the local Nusselt number and thermal entrance length were investigated. Moreover, the local Nusselt number of steady state conditions was correlated in terms of the power-law index and wall heat flux and compared with literature data, which were obtained by an analytic solution for Newtonian fluids. Furthermore, a relationship was proposed for the thermal entrance length.

Complete monotonicity of flow and cross-sectional impulse response functions for rigid right circular tubes

Indicator dilution theory is the conceptual base for many flow measurement methods within living organisms; a whole animal, an organ, or an individual vessel. In this manuscript, we provide a characterization of the system impulse response function for flow- and cross-sectionally sampled indicator curves within a rigid right circular tube. The curves are measured downstream from the position of an injector, which labels uniformly the streamlines of a steady, laminar power law fluid flow. The system impulse response function is shown to be the weighted sum of a Dirac delta function and a completely monotonic function. A numerical example employing data from a physical experiment involving a vertically orientated, perfused glass tube is presented, illustrating the characterization of the impulse response function between cross-sectionally sampled upstream and downstream measurement sites.

Numerical analysis of the transient conjugated heat transfer in a circular duct with a power-law fluid

We treat numerically in this paper, the tran- sient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluids is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the transverse and longitudinal heat conduction effects are taken into account. The numerical results for the temporal evolution of the average tem- perature of the tube wall, hav, is plotted for different nondimensional parameters such as conduction param- eter, a, the aspect ratios of the tube, e and e0 and the index of power-law fluid, n.

Transient analysis of the conjugated heat transfer in circular ducts with a power-law fluid

The transient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow, is studied. This conjugated heat transfer process is characterized by the simultaneous thermal interaction between the wall tube and the adjacent thermal boundary layer of the power-law fluid. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluid is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the axial heat conduction effects are taken into account. The governing equation is reduced to a integro-differential equation which is solved by muliple-scale analysis and a conventional numerical technique. The results for the temperature of the wall tube and the corresponding average temperature are plotted for differents parameters such as conduction parameter, α, the aspect ratios of the tube, ε and ε 0 and the index of power-law fluid, n.

Laminar flow of power-law fluids in the entrance region of a pipe

The study is concerned with developing laminar flow of a power-law fluid in a circular tube. The analysis extends the filled-region concept, used previously to study Newtonian fluid flow, to the more general class of power-law fluids. Flow is analyzed in both the inlet and filled regions using an integral boundary layer method. Results obtained provide the lengths of the entrance, inlet and filled regions as a function of the generalized Reynolds number and the power-law fluid index. In addition, the variations of the local friction factor, the pressure drop and the centerline velocity along the axial coordinate are also provided. The available models are compared with the present one on the basis of experimental data. The present results are found to reach asymptotically the fully developed values, and also to be in good agreement with all available experimental data.