Power-law Liquids Research Articles

Realizing infrared power-law liquids in the cuprates from unparticle interactions

Recent photoemission experiments [1] reveal that the excitations along the nodal region in the strange metal phase of the cuprates, rather than corresponding to poles in the single-particle Green function, exhibit power-law scaling as a function of frequency and temperature. Because such power-law scaling is indicative of a scale-invariant sector, as a first step, we perturbatively evaluate the electron self-energy due to interactions with scale-invariant unparticles. We focus on a ${G}_{0}W$-type diagram with an interaction $W$ mediated by a bosonic scalar unparticle. We find that in the high-temperature limit, the imaginary part of the self-energy $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ is linear in temperature. In the low-temperature limit, $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ exhibits the same power law in both temperature and frequency, with an exponent that depends on the scaling dimension of an unparticle operator. Such behavior is qualitatively consistent with the experimental observations. We then expand the unparticle propagator into coherent and incoherent contributions and study how the incoherent part violates the density of states (DOS) and density-density correlation function sum rules (f-sum rule). Such violations, in principle, can be observed experimentally. Our work indicates that the physical mechanism for the origin of the power-law scaling is the incoherent background, which is generated from the Mott-scale physics.

Heat transfer characteristics of thin power-law liquid films over horizontal stretching sheet with internal heating and variable thermal coefficient

The effect of internal heating source on the film momentum and thermal transport characteristic of thin finite power-law liquids over an accelerating unsteady horizontal stretched interface is studied. Unlike most classical works in this field, a general surface temperature distribution of the liquid film and the generalized Fourier’s law for varying thermal conductivity are taken into consideration. Appropriate similarity transformations are used to convert the strongly nonlinear governing partial differential equations (PDEs) into a boundary value problem with a group of two-point ordinary differential equations (ODEs). The correspondence between the liquid film thickness and the unsteadiness parameter is derived with the BVP4C program in MATLAB. Numerical solutions to the self-similarity ODEs are obtained using the shooting technique combined with a Runge-Kutta iteration program and Newton’s scheme. The effects of the involved physical parameters on the fluid’s horizontal velocity and temperature distribution are presented and discussed.

Flow of power-law liquids in a Hele-Shaw cell driven by non-uniform electro-osmotic slip in the case of strong depletion

We analyse flow of non-Newtonian fluids in a Hele-Shaw cell, subjected to spatially non-uniform electro-osmotic slip. Motivated by their potential use for increasing the characteristic pressure fields, we specifically focus on power-law fluids with wall depletion properties. We derive a $p$-Poisson equation governing the pressure field, as well as a set of linearized equations representing its asymptotic approximation for weakly non-Newtonian behaviour. To investigate the effect of non-Newtonian properties on the resulting fluidic pressure and velocity, we consider several configurations in one and two dimensions, and calculate both exact and approximate solutions. We show that the asymptotic approximation is in good agreement with exact solutions even for fluids with significant non-Newtonian behaviour, allowing its use in the analysis and design of microfluidic systems involving electrokinetic transport of such fluids.

Marangoni Effects on Forced Convection of Power Law Fluids in Thin Film over an Unsteady Horizontal Stretching Surface with Heat Source

Objectives: In this paper, the effect of surface tension gradient on the transport phenomenon and temperature differences within a power-law liquid film on an unsteady horizontal stretching sheet with heat source is been investigated. Method: The flow of a fluid film and the heat transfer from the subsequent stretching surface is been found by applying similarity transformation. The numerical computation of the problem is done with aid of Maple. The system of governing non-linear partial differential equation is converted to a set of nonlinear ordinary differential equation using local similarity transformations, then solved numerically applying the Runge Kutta Felhberg-45 method. The effect of power law index n, Prandtl number Pr, the unsteadiness parameter S, Space dependent heat source parameter α 1 , temperature dependent heat source parameter β 1 , and film thickness parameter β(=η) on heat transfer and flow are studied and graphically presented. And it was found that the surface tension gradient and nonuniform heating source have tremendous impact on controlling the rate of convective heat transfer near the boundary layer region.

Foams built up by non-Newtonian polymeric solutions: Free drainage

A mathematical model of free drainage of foam built up by a power-law non-Newtonian liquid is developed. The theory predictions are compared with the experimental data on the drainage of foams formed using commercially available Aculyn™22 and Aculyn™33 polymeric solutions. The rheological parameters of the polymeric solutions were independently measured and used in the calculations. The deduced dimensionless equations were solved using finite element method with appropriate boundary conditions. The numerical simulations show that the decrease in the foam height and liquid content is very fast in the very beginning of the drainage; however, it reaches a steady state at longer time. The predicted values of the time evolution of the foam height and liquid content are in good agreement with the measured experimental data.

Momentum and Mass Transfer Phenomena of Contaminated Bubble Swarms in Power-Law Liquids

Momentum and Mass Transfer Phenomena of Contaminated Bubble Swarms in Power-Law Liquids

Dynamic Behavior of Gas Nano-sized Bubbles in Liquid Phase of the Metal being Electrodeposited

The paper deals with the experimental results which confirm the occurrence of foam on the crests of waves of metals being electrodeposited in the course of their wavelike flow under action of the external force of insignificant value directed parallel to the crystallization front. The mechanism of foam formation on the crests of waves of metals being electrodeposited, conditioned by the dynamic behavior of gas nanosized bubbles in the liquid phase of the metal being electrodeposited, is offered. The paper also presents the mathematical models for analysis of the dynamic behavior of gas nano- and micro-sized bubbles in the Newtonian and power-law liquids. Results of calculations on obtained mathematical models with the use of MathCAD package are given.

Analytical determination of process windows for bilayer slot die coating

Slot die coating is a film casting process with a highly diverse variety of everyday applications. As a pre-metered process it not only guarantees excellent film uniformity, but is also suitable for simultaneously applied multilayer coatings. Characteristic singularities like the behavior of the liquid–liquid interface and the impact of the additional mid-lip on film uniformity were already investigated before. However, the effect of an altered gap pressure regime on commonly used coating windows has not yet been discussed. In this work, we therefore extended available single-layer coating windows for Newtonian and power-law liquids to the bilayer case. Here, the emphasis was laid on the air entrainment limit. Subsequently, the theoretical results were compared to experimental data. It was found that the onset of air entrainment strongly depends on the top to bottom film thickness ratio for bilayer coatings. A critical film thickness ratio which delivers similar coating limits as those for single-layer coatings was derived and confirmed by experimentally gained results.

Motion of partially contaminated bubbles in power-law liquids: Effect of wall retardation

Momentum transfer characteristics of partially contaminated confined bubbles in power-law non-Newtonian fluids are numerically investigated within the framework of the spherical stagnant cap model. The solver is thoroughly benchmarked by comparing the present results with existing literature counterparts. Further extensive new results are obtained in the range of conditions as: Reynolds number, Re: 0.1–200; power-law behavior index, n=0.2–1.6; stagnant cap angle, α: 0°–180° and confinement ratio, λ: 0.2–0.5. Briefly results indicate that the size of recirculation wake at the rear end of bubble decreases with the increasing confinement ratio and/or with the decreasing Reynolds number and/or with the decreasing contamination angle and/or with increasing power-law behavior index. The wall retardation suppresses sudden rise in the surface pressure coefficient at leading edge of the stagnant cap of contaminated bubbles in the case of highly shear-thinning fluids. The total drag coefficients of partially contaminated confined bubbles decrease with decreasing confinement ratio and/or with the decreasing power-law behavior index and/or with the decreasing cap angle. Furthermore, unlike in the case of motion of unconfined bubbles in power-law fluids, the crossover Reynolds number do not exist in Cd vs. Re curves with respect to the power-law behavior index because of the domination of wall retardation effects.

A numerical study on flow and drag phenomena of spheroid bubbles in Newtonian and shear-thinning power-law fluids

Effects of the Reynolds number, aspect ratio of spheroid bubbles, and power-law behavior index of shear-thinning liquids on the flow and drag behavior of spheroid bubbles are elucidated using a computational fluid dynamics-based numerical solver, ANSYS Fluent 14. The solution methodology is extensively benchmarked via detailed domain and grid independence study and by comparing present results of spherical bubbles in Newtonian and shear-thinning power-law fluids with their literature counterparts. Further extensive new results are reported over a wide range of pertinent conditions as follows: Reynolds number, Re: 1 – 200; aspect ratio of spheroid bubbles, e: 0.5 – 2.5, and power-law behavior index, n: 0.2 – 1. The size of the recirculation wake decreases with decreasing power-law index, and/or with decreasing bubble aspect ratio, and/or with decreasing Reynolds number. For bubbles of aspect ratio e > 1, a crossover Reynolds number is observed with respect to the power-law index, i.e. below the crossover Reynolds number, the drag coefficient of bubble increases with decreasing power-law index; whereas, above the crossover Reynolds number, a reverse trend is observed. Finally, based on the present numerical results, a simple predictive correlation is proposed for the total drag coefficients of spheroid bubbles rising in Newtonian and power law liquids, which can be used in new applications.

Experimental research on breakup of 2D power law liquid film

On account of limited knowledge of the breakup of power law liquid film, the process of its disintegration and atomization was studied by using a planar liquid film. A linear stability analysis was adopted to predict the breakup characteristics of the power law film. The predicting formulas of stripping breakup length and diameter of ligament were put forward presently. Through high-speed photography and laser light sheet illumination, different breakup characteristics of flat power law film under different conditions were derived. The characteristic dimension of breakup regimes were defined and extracted. The effects of several parameters (injection pressure, ambient pressure, nozzle structure and fluid property) on the stripping breakup length and spray angle were investigated. The results revealed that increasing both the velocity of liquid film and the ambient pressure facilitated the breakup of film, reduced the stripping breakup length and enlarged the spray angle in different extents. The comparison between theoretical and experimental results was conducted to validate the feasibility of the linear stability theory.

Slip in flows of power-law liquids past smooth spherical particles

The effect of slip in flows of power-law liquids past smooth spherical particles is numerically studied by using Navier’s linear slip model. For computational simplicity, a sphere-in-sphere type computational domain has been chosen. Thus, the governing conservation equations of mass and momentum are considered in spherical coordinates. These are solved by using a finite difference method-based simplified marker and cell algorithm implemented on a staggered grid arrangement. The non-Newtonian terms of the momentum equation are discretized by a second-order central differencing scheme, whereas the convective terms are discretized by using QUICK scheme. The reliability and accuracy of the solver is established by comparing the present numerical results with the existing literature counterparts. Furthermore, extensive new results are obtained in the range of conditions for Reynolds number, Re: 0.1–200; power-law behavior index, n: 0.5–1.6; and a dimensionless slip parameter, λ: 0.01–100. Finally, effects of these dimensionless parameters on near surface flow kinematics are thoroughly delineated.

Flow and Heat Transfer of Quiescent Non-Newtonian Power-Law Fluid Driven by a Moving Plate: An Integral Approach

The problem of boundary layer flow and heat transfer from a plate moving in a quiescent non-Newtonian power law fluid medium is solved numerically to obtain results showing the effects of power-law index and Reynolds number on friction and heat transfer coefficients. A decrease in friction coefficients and an increase in heat transfer coefficients is observed for power-law liquids in comparison with water. Further in the case of a power-law liquid, an increase in the velocity of the plate also produces a reduction in drag and an increase in heat transfer. Compared to a Newtonian liquid, the friction coefficient decreases by 57% for a non-Newtonian liquid of power-law index, n = 0.568 at Re=2000. By increasing the Reynolds number to 10000, the friction coefficient decreases to 66% in comparison with a Newtonian liquid.

Stability of an annular power-law liquid sheet

Because of the mathematical difficulties dealing with the nonlinear viscous stress term in momentum equation for power-law liquid, the study of stability analysis of a power-law liquid sheet has been lacking. In the present study, a temporal stability analysis has been carried out for an annular power-law liquid sheet exposed to both inner and outer-gas streams, by integrating the governing equations for the power-law liquid sheet. The dimensionless dispersion equation that governs the instability of liquid sheet is obtained by considering the velocity profile of liquid sheet. It is found that the instability of liquid sheet can be enhanced by independently increasing either the outer gas stream velocity, or the inner gas stream velocity. The liquid sheet is more unstable when both inner and outer gas streams are applied. To promote the instability of annular liquid sheet, a gas stream applied to the outer interface is more effective than when applied to the inner surface. The effects of rheological parameters on the instability of the liquid sheet are actually determined by the relative velocity across the gas–liquid interfaces. The surface tension, liquid sheet thickness, and outer surface radius of annular liquid sheet have been tested for their influence on the instability of annular power-law liquid sheet.

Temporal Instability of a Power-Law Planar Liquid Sheet

A linear stability analysis has been carried out to reveal the atomization mechanism of a power-law planar liquid sheet. The dispersion equation that governs the symmetric instability of a power-law liquid sheet is obtained by considering the gas boundary-layer thickness and the velocity profile of power-law liquid sheet. The dispersion equation is worked out by numerical solution to test the effects of the physical properties and flow parameters of liquid on the stability of a liquid sheet. Regarding the effects of rheological parameters, it is found that there is a critical value of consistency coefficient above which increasing will make the liquid sheet unstable, and below which the effect of is opposite. A larger flow index number makes the instability of the power-law liquid sheet damped. As the gas boundary-layer thickness, liquid sheet thickness, and surface tension increase, the disturbance wave growth rate will decrease and the instability will be damped. It is the gas-to-liquid relative velocity that governs the instability of the liquid sheet. A larger liquid density, viscosity, and density of ambient air can accelerate the growth of symmetric disturbance waves.

Natural convection from a vertical plate immersed in a power-law fluid saturated non-Darcy porous medium with viscous dissipation and Soret effects

In this work, we study the viscous dissipation and thermal-diffusion effects on natural convection from a vertical plate embedded in a fluid saturated non-Darcy porous medium. The non-Newtonian behaviour of fluid is characterized by the generalized power-law model. The governing partial differential equations are transformed into a system of ordinary differential equations using a local non-similarity solution and the resulting boundary value problem is solved using a novel successive linearisation method (SLM). The accuracy of the SLM has been established by comparing the results with the shooting technique. The effects of physical parameters on heat and mass transfer coefficients for the convective motion of the power-law liquid are presented both qualitatively and quantitatively. The results show that the Nusselt number is reduced by viscous dissipation and enhanced by the Soret number but the Sherwood number increases with viscous dissipation and decreases with the Soret number. An increasing viscosity enhances heat and mass transfer coefficients in both cases of aiding buoyancy and opposing buoyancy.

Non-Newtonian power-law gravity currents propagating in confining boundaries

The propagation of viscous, thin gravity currents of non-Newtonian liquids in horizontal and inclined channels with semicircular and triangular cross-sections is investigated theoretically and experimentally. The liquid rheology is described by a power-law model with flow behaviour index $$n$$ , and the volume released in the channel is taken to be proportional to $$t^{\alpha }$$ , where $$t$$ is time and $$\alpha $$ is a non-negative constant. Some results are generalised to power-law cross-sections. These conditions are representative of environmental flows, such as lava or mud discharges, in a variety of conditions. Theoretical solutions are obtained in self-similar form for horizontal channels, and with the method of characteristics for inclined channels. The position of the current front is found to be a function of the current volume, the liquid rheology, and the channel inclination and geometry. The triangular cross-section is associated with the fastest or slowest propagation rate depending on whether $$\alpha <\alpha _c$$ or $$\alpha >\alpha _c$$ . For horizontal channels, $$\alpha _c=n/(n+1)<1$$ , whereas for inclined channels, $$\alpha _c=1$$ , irrespective of the value of $$n$$ . Experiments were conducted with Newtonian and power-law liquids by independently measuring the rheological parameters and releasing currents with constant volume ( $$\alpha =0$$ ) or constant volume flux ( $$\alpha =1$$ ) in right triangular and semicircular channels. The experimental results validate the model for horizontal channels and inclined channels with $$\alpha =0$$ . For tests in inclined channels with $$\alpha =1$$ , the propagation rate of the current front tended to lower values than predicted, and different flow regimes were observed, i.e., uniform flow with normal depth or instabilities resembling roll waves at an early stage of development. The theoretical solution accurately describes current propagation with time before the transition to longer roll waves. An uncertainty analysis reveals that the rheological parameters are the main source of uncertainty in the experiments and that the model is most sensitive to their variation. This behaviour supports the use of carefully designed laboratory experiments as rheometric tests.

Breakup characteristics of power-law liquid sheets formed by two impinging jets

The breakup characteristics of the shear-thinning power-law liquid sheets formed by two impinging jets have been investigated with the shadowgraph technique. This paper focuses on the effects of spray parameters (jet velocity), physical parameters (viscosity) and geometry parameters (impinging angle and nozzle cross-sectional shape) on the breakup behaviors of liquid sheets. The breakup mode, sheet length and expansion angle of the sheet are extracted from the spray images obtained by a high speed camera. Impinging angle and Weber number play the similar roles in promoting the breakup of liquid sheets. With the increase of jet velocity, five different breakup modes are observed and the expansion angle increases consistently after the closed-rim mode while the sheet length first increases and then decreases. But there exists a concave consisting of a fierce drop and a second rising process on the sheet length curve for the fluid with smaller viscosity. Different nozzle cross-sectional shapes emphasize significant effects on the sheet length and expansion angle of liquid sheets. At a fixed Weber number, the liquid sheet with greater viscosity has a greater sheet length and a smaller expansion angle due to the damping effect of viscosity.

Analytical solution for two-phase flow between two rotating cylinders filled with power law liquid and a micro layer of gas

This paper deals with the effects that a thin gas layer exerts on the hydrodynamic aspects of power law liquid in a radial Couette flow between two cylinders. Analytical solution is made to determine the velocity profile in the two-phase flow system occupied by the power law liquid and the micro layer of a gas. It is shown that the thin (micro) gas layer contributes in reducing torque to set the fluid in motion in most cases. However, by considering generalized power law liquids, this paper limits the credibility for the positive role of the gas layer on reducing the torque for lubrication. For instance, when n < 0.5 (n is the behavior index of the liquid), slight increment in the torque (about 6%) is reported. Finally, energy gradient method is used for stability analysis. It is shown that the stability nature may be changed based on behavior index of the liquid.

On the insertion of a thin gas layer in micro cylindrical Couette flows involving power-law liquids

Consideration is given to the fluid flow and heat transfer repercussions for introducing a thin (micro) gas layer into a cylindrical Couette flow assembly dealing with a power law liquid (lubricant). The trio influence of the thin gas layer on (1) the torque required for activating the lubrication process, (2) on the maximum temperature of the shaft (inner cylinder) and (3) on Nusselt number of the two-phase flow configuration are studied analytically. The results demonstrate that the thin gas layer normally contributes to reduce in the torque to set the fluid in motion and to downscale the maximum temperature at the shaft, especially for the sub-category of shear thickening liquids. However, in the sub-category of shear thinning liquids, the above-mentioned positive roles attributed to the micro gas layer hold true only for a limited number of flow configurations depending on several factors such as the Knudsen number, the accommodation coefficients and the rheological parameters of the liquid. It is further corroborated that the thin gas layer stabilizes or destabilizes the flow, depending on the magnitude of the power index number characterizing the liquid.