We use a generic and general numerical method to obtain solutions for the flow of generalized Newtonian fluids through circular pipes and plane slits. The method, which is simple and robust can produce highly accurate solutions which virtually match any analytical solutions. The method is based on employing the stress, as a function of the pipe radius or slit thickness dimension, combined with the rate of strain function as represented by the fluid rheological constitutive relation that correlates the rate of strain to stress. Nine types of generalized Newtonian fluids are tested in this investigation and the solutions obtained from the generic method are compared to the analytical solutions which are obtained from the Weissenberg-Rabinowitsch-Mooney-Schofield method. Very good agreement was obtained in all the investigated cases. All the required quantities of the flow which include local viscosity, rate of strain, flow velocity profile and volumetric flow rate, as well as shear stress, can be obtained from the generic method. This is an advantage as compared to some traditional methods which only produce some of these quantities. The method is also superior to the numerical meshing techniques which may be used for resolving the flow in these systems. The method is particularly useful when analytical solutions are not available or when the available analytical solutions do not yield all the flow parameters.
In this paper, we use a generic and general variational method to obtain solutions to the flow of generalized Newtonian fluids through circular pipes and plane slits. The new method is not based on the use of the Euler-Lagrange variational principle and hence it is totally independent of our previous approach which is based on this principle. Instead, the method applies a very generic and general optimization approach which can be justified by the Dirichlet principle although this is not the only possible theoretical justification. The results that were obtained from the new method using nine …
In this paper, analytical expressions correlating the volumetric flow rate to the pressure drop are derived for the flow of Carreau and Cross fluids through straight rigid circular uniform pipes and long thin slits. The derivation is based on the application of Weissenberg-Rabinowitsch-Mooney-Schofield method to obtain flow solutions for generalized Newtonian fluids through pipes and our adaptation of this method to the flow through slits. The derived expressions are validated by comparing their solutions to the solutions obtained from direct numerical integration. They are also validated by c…
In this article we challenge the claim that the previously proposed variational method to obtain flow solutions for generalized Newtonian fluids in circular tubes and plane slits is exact only for power law fluids. We also defend the theoretical foundation and formalism of the method which is based on minimizing the total stress through the application of the Euler-Lagrange principle.
This paper presents a method for calculating the wall shear rate in pipe turbulent flow. It collapses adequately the data measured in laminar flow and turbulent flow into a single flow curve and gives the basis for the design of turbulent flow viscometers. Key words: non-Newtonian, wall shear rate, turbulent, rheometer
This study focuses on a parametric study of the laminar fast transient flow of non-Newtonian fluids through helical pipes. Classical simulations of fluid hammer do not deal with the pipeline helicity and non-Newtonian characteristics of the fluid, while the present work addresses those features. To this end, the power-law model is employed to accommodate the non-Newtonian behavior of the fluid. Effects of the pipe wall elasticity and compressibility of the working fluid are taken into account through a modified bulk modulus elasticity of the fluid. The results of the three-dimensional numerica…
This paper presents a new theory of turbulence in time-independent non-Newtonian fluids. The wall layer is modelled in terms of unsteady exchange of viscous momentum between the wall and the main stream, following the classic visualisation of inrush-sweep-ejection/burst. The thickness of the wall layer is found to be the same for Newtonian and purely viscous non-Newtonian fluids, when normalised with the instantaneous wall parameters at the onset of bursting. The results indicate that the mechanisms of turbulence in Newtonian and time-independent fluids are identical when structural similarity…
We continue our investigation to the use of the variational method to derive flow relations for generalized Newtonian fluids in confined geometries. While in the previous investigations we used the straight circular tube geometry with eight fluid rheological models to demonstrate and establish the variational method, the focus here is on the plane long thin slit geometry using those eight rheological models, namely: Newtonian, power law, Ree-Eyring, Carreau, Cross, Casson, Bingham and Herschel-Bulkley. We demonstrate how the variational principle based on minimizing the total stress in the flo…
We investigate the possibility that the spatial dependency of stress in generalized Newtonian flow systems is a function of the applied pressure field and the conduit geometry but not of the fluid rheology. This possibility is well established for the case of a one-dimensional flow through simply connected regions, specifically tubes of circular uniform cross sections and plane thin slits. If it can also be established for the more general case of generalized Newtonian flow through non-circular or multiply connected geometries, such as the two-dimensional flow through conduits of rectangular o…
A zonal similarity analysis of friction factors in pipe flow of power law fluids is presented. It uses the critical Reynolds number and friction factor at transition as estimates of the normalised velocity and distance between the wall and log-law layers and successfully collapses both Newtonian and non-Newtonian data for laminar, transition and turbulent flows. Key words: zonal similarity analysis, friction factor, Reynolds number, transition
This paper presents a technique that collapses existing experimental data of heat transfer in pipe flow of Newtonian and power law fluids into a single master curve. It also discusses the theoretical basis of heat, mass and momentum analogies and the implications of the present analysis to visualisations of turbulence.
Newtonian pipe flow is known to be linearly stable at all Reynolds numbers. We report, for the first time, a linear instability of pressure driven pipe flow of a viscoelastic fluid, obeying the Oldroyd-B constitutive equation commonly used to model dilute polymer solutions. The instability is shown to exist at Reynolds numbers significantly lower than those at which transition to turbulence is typically observed for Newtonian pipe flow. Our results qualitatively explain experimental observations of transition to turbulence in pipe flow of dilute polymer solutions at flow rates where Newtonian …
A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number, the elasticity number, and the ratio of solvent to solution viscosity. The unstable mode has a phase speed close to the base-state maximum over the entire unstable region in the relevant parameter space, implying that the unstable mode belongs to a class of viscoelastic center modes. Unlike t…
Non-Newtonian fluid flow might be driven by spatially nonlocal velocity, the dynamics of which can be described by promising fractional derivative models. This short communication proposes a space FrActional-order Constitutive Equation (FACE) that links viscous shear stress with the velocity gradient, and then interprets physical properties of non-Newtonian fluids for steady pipe flow. Results show that the generalized FACE model contains previous non-Newtonian fluid flow models as end-members by simply adjusting the order of the fractional index, and a preliminary test shows that the FACE mod…
The flow of Ree-Eyring and Casson non-Newtonian fluids is investigated using a variational principle to optimize the total stress. The variationally-obtained solutions are compared to the analytical solutions derived from the Weissenberg-Rabinowitsch-Mooney equation and the results are found to be identical within acceptable numerical errors and modeling approximations.
The Bhatnagar-Gross-Krook kinetic model of the Boltzmann equation is solved for the steady cylindrical Poiseuille flow fed by a constant gravity field. The solution is obtained as a perturbation expansion in powers of the field (through fourth order) and for a general class of repulsive potentials. The results, which are hardly sensitive to the interaction potential, suggest that the expansion is only asymptotic. A critical comparison with the profiles predicted by the Navier-Stokes equations shows that the latter fail over distances comparable to the mean free path. In particular, while the N…
We present the dynamic velocity profiles of a Newtonian fluid (glycerol) and a viscoelastic Maxwell fluid (CPyCl/NaSal in water) driven by an oscillating pressure gradient in a vertical cylindrical pipe. The frequency range explored has been chosen to include the first three resonance peaks of the dynamic permeability of the viscoelastic fluid / pipe system. Three different optical measurement techniques have been employed. Laser Doppler Anemometry has been used to measure the magnitude of the velocity at the centre of the liquid column. Particle Image Velocimetry and Optical Deflectometry are…
This paper uses the estimates of phase-locked parameters at the onset of bursting presented in a companion paper to derive logarithmic correlations for turbulent friction factor losses in time-independent power law fluids. Two different techniques for analysis were used. They gave logarithmic correlations with different coefficients. But both correlations predicted 264 data points of friction factors published in the literature with a standard error of 1.3%. The derivations show how modern understanding of coherent structures embedded in turbulent flow fields can be used to derive engineering …
A large eddy simulation (LES) with an extended Smagorinsky model has been carried to investigate numerically the fully developed turbulent flow of a shear thinning fluid (n=0.75) in a stationary pipe at a simulation's Reynolds number equals to 4000 with a grid resolution of 65 3 gridpoints in axial, radial and circumferential directions and a domain length of 20R. The present study set out to critically evaluate the influence of the Non-Newtonian rheological and hydrodynamic behaviour on the turbulence main features, as well as to ascertain the accuracy and reliability of the laboratory code p…
The velocity and friction properties of laminar pipe flow of a viscoelastic solution are bounded by the corresponding values for two Newtonian fluids, namely, the solvent and a fluid with a viscosity identical to the total viscosity of the solution. The lower friction factor for the flow of the solution when compared to the latter is tracked to an increased strain rate needed to enhance viscous dissipation. Lastly, we show analytically that the effective viscosity varies similarly to the radial diagonal component of the conformation tensor as observed numerically in turbulent flows, and give a…
Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common brain diseases. Mathematical models could highlight possible mechanisms of aneurismal development and suggest specialized biomarkers for aneurysms. Existing mathematical models of intracranial aneurysms focus on mechanical interactions between blood flow and arteries. However, these models ca…
We perform fully Eulerian numerical simulations of an initially spherical hyperelastic particle suspended in a Newtonian pressure-driven flow in a cylindrical straight pipe. We study the full particle migration and deformation for different Reynolds numbers and for various levels of particle elasticity, to disentangle the interplay of inertia and elasticity on the particle focusing. We observe that the particle deforms and undergoes a lateral displacement while traveling downstream through the pipe, finally focusing at the pipe centerline. We note that the migration dynamics and the final equi…
In the present study, the ionic transport and selectivity of electrokinetically-driven flow of power-law fluids in a long pH-regulated rectangular nanochannel are analyzed. The electrical potential and momentum equations are numerically solved through a finite difference procedure for a non-uniform grid. Non-linear Poisson-Boltzmann equation along with the association/dissociation reactions on the surface is considered. In addition, numerical simulations with the finite element method in 3D space are performed to compare the results with those obtained from 2D analysis. Moreover, an analytical…
Significant improvements in the computational performance of the lattice-Boltzmann (LB) model, coded in FORTRAN90, were achieved through application of enhancement techniques. Applied techniques include optimization of array memory layouts, data structure simplification, random number generation outside the simulation thread(s), code parallelization via OpenMP, and intra- and inter-timestep task pipelining. Effectiveness of these optimization techniques was measured on three benchmark problems: (i) transient flow of multiple particles in a Newtonian fluid in a heterogeneous fractured porous do…
Polymer additives can substantially reduce the drag of turbulent flows and the upper limit, the so called "maximum drag reduction" (MDR) asymptote is universal, i.e. independent of the type of polymer and solvent used. Until recently, the consensus was that, in this limit, flows are in a marginal state where only a minimal level of turbulence activity persists. Observations in direct numerical simulations using minimal sized channels appeared to support this view and reported long "hibernation" periods where turbulence is marginalized. In simulations of pipe flow we find that, indeed, with inc…
Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. Phys. Rev. Lett., 121.024502 (2018). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidences of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expans…
Modal stability analysis provides information about the long-time growth or decay of small-amplitude perturbations around a steady-state solution of a dynamical system. In fluid flows, exponentially growing perturbations can initiate departure from laminar flow and trigger transition to turbulence. Although flow of a Newtonian fluid through a pipe is linearly stable for very large values of the Reynolds number ($Re \sim 10^7$), a transition to turbulence often occurs for $Re$ as low as $1500$. When a dilute polymer solution is used in the place of a Newtonian fluid, the transitional value of t…
Suspended particles can alter the properties of fluids and in particular also affect the transition from laminar to turbulent flow. In the present experimental study, we investigate the impact of neutrally buoyant, spherical inertial particles on transition to turbulence in a pipe flow. At low particle concentrations, like in single phase Newtonian fluids, turbulence only sets in when triggered by sufficiently large perturbations and, as characteristic for this transition localized turbulent regions (puffs) co-exist with laminar flow. In agreement with earlier studies this transition point ini…
The perturbations of the laminar shear-thinning viscoelastic pipe flow under Finitely Extensible Nonlinear Elastic model with Peterlin approximation (FENE-P) are shown to exhibit leading-order power-law behaviours, and the expected odd-even parities with respect to the radial coordinate that depend on the azimuthal wave\-number, $n$. The analysis helps regularizing the governing system of equations at the centreline, and allows for a complete stability analysis of three-dimensional perturbations for a general integer value of $n$, which has hitherto remained a challenge for FENE-P models. It i…
Laminar flow velocity profiles depend heavily on fluid rheology. Developing methods of laminar flow characterization, based on low-field magnetic resonance (MR), contributes to the widespread industrial application of the MR technique in rheology. In this paper, we designed a low-cost, palm-sized permanent magnet with a 1H resonance frequency of 20.48 MHz to measure laminar flow. The magnet consists of two disk magnets, which were each tilted at an angle of 1° from a starting separation of 1.4 cm to generate a constant gradient, 65 gauss/cm, in the direction of flow. Subsequently, a series of …
The study of creeping motion of viscoelastic fluid around a rotating rigid torus is investigated. The analysis of the problem is performed using a second-order viscoelastic model. The study is carried out in terms of the bipolar toroidal system of coordinates where the toroid is rotating about its axis of symmetry (z-axis). The problem is solved within the frame of slow flow approximation. Therefore, all variables in the governing equations are expanded in a power series of angular velocity. A set of successive partial differential equations is obtained. The equations of motion governing the f…
The flow of viscoelastic fluids in channels and pipes remain poorly understood, particularly at low Reynolds numbers. Here, we investigate the flow of polymeric solutions in straight channels using pressure measurements and particle tracking. The law of flow resistance is established by measuring the flow friction factor $f_η$ versus flow rate. Two regimes are found: a transitional regime marked by rapid increase in drag, and a turbulent-like regime characterized by a sudden decrease in drag and a weak dependence on flow rate. Lagrangian trajectories show finite transverse modulations not seen…
The problem of oscillating flows inside pipes under periodic forcing of viscoelastic fluids is addressed here. Starting from the linear Oldroyd-B model, a generalized Darcy's law is obtained in frequency domain and an explicit expression for the dependence of the dynamic permeability on fluid parameters and forcing frequency is derived. Previous results in both viscoelastic and Newtonian fluids are here shown to be particular cases of our results. On the basis of our calculations, a possible explanation for the observed damping of local dynamic response as the forcing frequency increases is gi…
This paper presents a new theory of turbulent mixing in stirred reactors. The degree of homogeneity of a mixed fluid may be characterized by the Kolmogorov micro-scale. The smaller its value, the better homogeneity. The micro-scale scales inversely with the fourth root of the energy dissipation rate in the stirring process. The higher this rate, the smaller lambda, and the better the homogeneity in the reactor. This is true for Newtonian fluids. In non-Newtonian fluids the situation is different. For instance, in shear-thickening fluids it is plausible that high shear rates thicken the fluid a…
Recently, detailed experiments on visco-elastic channel flow have provided convincing evidence for a nonlinear instability scenario which we had argued for based on calculations for visco-elastic Couette flow. Motivated by these experiments we extend the previous calculations to the case of visco-elastic Poiseuille flow, using the Oldroyd-B constitutive model. Our results confirm that the subcritical instability scenario is similar for both types of flow, and that the nonlinear transition occurs for Weissenberg numbers somewhat larger than one. We provide detailed results for the convergence o…
In this paper, we discover the fundamental quantity that Nature minimizes in almost all flows encountered in everyday life: river, rain, flow in a pipe, blood flow, airflow over an airplane, etc. We show that the norm of the pressure gradient over the field is minimum at every instant of time! We call it the principle of minimum pressure gradient (PMPG). The principle is deeply rooted in classical mechanics via Gauss' principle of least constraint. Therefore, while we prove mathematically that Navier-Stokes' equation represents the necessary condition for minimization of the pressure gradient,…
Turbulence is one of the most fascinating phenomena in nature and one of the biggest challenges for modern physics. It is common knowledge that a flow of a simple, Newtonian fluid is likely to be turbulent, when velocity is high, viscosity is low and size of the tank is large\cite{Landau,Tritt}. Solutions of flexible long-chain polymers are known as visco-elastic fluids\cite{bird}. In our experiments we show, that flow of a polymer solution with large enough elasticity can become quite irregular even at low velocity, high viscosity and in a small tank. The fluid motion is excited in a broad ra…
We numerically investigate the effect of entrance condition on the spatial and temporal evolution of multiple three-dimensional vortex pairs and wall shear stress distribution in a curved artery model. We perform this study using a Newtonian blood-analog fluid subjected to a pulsatile flow with two inflow conditions. The first flow condition is fully developed while the second condition is undeveloped (i.e. uniform). We discuss the connection along the axial direction between regions of organized vorticity observed at various cross-sections of the model and compare results between the differen…
