Exponential Dependence Of Viscosity Research Articles

Numerical modelling of high velocity impact problem involving non-linear viscosity

The paper is aimed at developing numerical models for simulation of hypervelocity collisions of small fragments with honeycomb shields relevant to Space structures protection from impact of debris fragments. Numerical modelling of hypervelocity collision within hydrodynamic approach reveals effects that are inconsistent with physical sense. In order to overcome these inconsistencies and to achieve justified description of the process of penetration and fracturing numerical model was developed assuming the impactor and projectile being viscous materials. Despite constant viscosity we investigate two phenomenological relationships for viscosity: depending on temperature and depending on deformation rate. The exponential dependence of viscosity on material state (temperature and second invariant of strain rate deviator) makes the material manifest both properties of fluid and solid state of matter. Taking into account viscosity allows to suppress jets at symmetry axis and small-scaled ejecta. The usage of non-constant viscosity results in a more physically based scenarios for fracturing of the shell obtained in numerical simulations.

ANALYTICAL AND NUMERICAL SOLUTIONS FOR SOME

Oscillatory motions of incompressible viscous fluids with exponential dependence of viscosity on the pressure between infinite horizontal parallel plates are analytically and numerically studied. The fluid motion is generated by the lower plate that oscillates in its plane and exact expressions are established for the steady-state solutions. The convergence of starting solutions to the corresponding steady-state solutions is graphically proved. The steady solutions corresponding to the simple Couette flow of the same fluids are obtained as limiting cases of the previous solutions. As expected, the fluid velocity diminishes for increasing values of the pressure-viscosity coefficient and ordinary fluids flow faster. The time required to reach the steady-state is graphically approximated. The spatial profiles of the starting solutions are presented both for oscillatory motions and the simple Couette flow.

Analytical solutions of upper-convected Maxwell fluid flow with exponential dependence of viscosity on the pressure

Exact and simple expressions for the permanent solutions corresponding to two oscillatory motions of incompressible upper-convected Maxwell fluids with exponential dependence of viscosity on the pressure between parallel plates have been established using suitable changes of the spatial variable and the unknown function and the Laplace transform technique. The solutions that have been obtained satisfy the boundary conditions and governing equations but are independent of the initial conditions. They are important for those who want to eliminate the transients from their experiments. The similar solutions for the simple Couette flow of the same fluids as well as known results for the Newtonian fluids performing the same motions were obtained as limiting cases. The convergence of starting solutions to the corresponding permanent components that has been graphically proved could constitute an asset on the correctness of obtained results. The influence of pertinent parameters on the fluid motion and the spatial profiles of starting solutions have been graphically depicted and discussed. The oscillations’ amplitude is an increasing function with respect to the dimensionless pressure–viscosity coefficient and the Weissenberg number. It is lower for the shear stress as compared to the fluid velocity. The three-dimensional distribution of the starting velocity fields has been numerically visualized by means of the two-dimensional contour graphs.

Fractional Walden rule for aprotic ionic liquids: Experimental verification over a wide range of temperatures and pressures

The Walden rule, revealing a universal conductivity-viscosity relationship, is considered as a useful tool to investigate the transport mechanism in ionic liquids (ILs). In this work, we constructed the Walden plots covering 10 orders of magnitude for two ILs: 1-butyl-2,3-dimethylimidazolium bis(trifluoromethylsulfonyl)imide [BMMIm][NTf2] and 1-butyl-3-methylimidazolium bis(perfluoroethylsulfonyl)imide [BMIm][BETI] using experimental viscosity data at pressures up to 950 MPa supplemented by high-pressure conductivity and density data. Moreover, the high-pressure viscosity data of studied ILs revealed an inflection point denoting a transition from slower to faster than exponential dependence of viscosity on pressure. Finally, we found that the experimental viscosity data conform to the thermodynamic density scaling idea for each IL, with the scaling parameter equal to the ratio of scaling exponent obtained for conductivity data to the slope of the Walden plot. Consequently, our results show a promising method to generate high-pressure viscosity data especially for highly viscous aprotic ionic liquids, which are technically impossible to measure.

Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure

Some unsteady motions of incompressible upper-convected Maxwell (UCM) fluids with exponential dependence of viscosity on the pressure are analytically studied. The fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies time-dependent shear stresses to the fluid. Exact expressions, in terms of standard Bessel functions, are established both for the dimensionless velocity fields and the corresponding non-trivial shear stresses using the Laplace transform technique and suitable changes of the unknown function and the spatial variable in the transform domain. They represent the first exact solutions for unsteady motions of non-Newtonian fluids with pressure-dependent viscosity. The similar solutions corresponding to the flow of the same fluids due to an exponential shear stress on the boundary as well as the solutions of ordinary UCM fluids performing the same motions are obtained as limiting cases of present results. Furthermore, known solutions for unsteady motions of the incompressible Newtonian fluids with/without pressure-dependent viscosity induced by oscillatory or constant shear stresses on the boundary are also obtained as limiting cases. Finally, the influence of physical parameters on the fluid motion is graphically illustrated and discussed. It is found that fluids with pressure-dependent viscosity flow are slower when compared to ordinary fluids.

First exact solutions for mixed boundary value problems concerning the motions of fluids with exponential dependence of viscosity on pressure

This paper presents exact expressions for dimensionless starting solutions corresponding to some oscillatory motions of fluids with exponential dependence of viscosity on pressure. These expressions are established in terms of standard Bessel functions of order zero and one. Fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies oscillatory shear stresses to the fluid. The corresponding solutions, which are currently absent in the literature, are presented as sums of steady-state and transient components. These are useful for experimentalists who wish to eliminate transients from their experiments. For completeness, the dimensionless velocity field corresponding to the motion due to the lower plate applying a constant shear stress to the fluid is determined as a limiting case. Furthermore, to verify the results, it is shown that diagrams of the present steady-state solutions coincide with those of ordinary Newtonian fluids performing the same motions as the dimensionless pressure–viscosity coefficient tends to zero. The spatial distributions of starting solutions and some transversal sections are also presented and discussed.

Research of eigenfuctions perturbation of the transverse component velocity thermoviscous liquids flow

The viscous model fluid flow in a plane channel with a linear temperature profile is considered. The problem of the thermoviscous fluid flow stability is solved on the basis of the previously obtained generalized Orr–Sommerfeld equation by the spectral method of decomposition into Chebyshev polynomials. We study the effect of taking into account the linear and exponential dependences of the viscosity of a liquid on temperature on the eigenfunctions of the hydrodynamic stability equation and on perturbations of the transverse velocity of an incompressible fluid in a plane channel when various wall temperatures are specified. Eigenfunctions are found numerically for two eigenvalues of the linear and exponential dependence of viscosity on temperature. Presented pictures of their own functions. The eigenfunctions demonstrate the behavior of the transverse velocity perturbations, their possible growth or attenuation over time. For the given eigenfunctions, perturbations of the transverse flow velocity of a thermoviscous fluid are obtained. It is shown that taking the temperature dependence of viscosity into account affects the eigenfunctions of the equations of hydrodynamic stability and perturbations of the transverse flow velocity. Perturbations of the transverse velocity significantly affect the hydrodynamic instability of the fluid flow. The results show that when considering the unstable eigenvalue over time, the velocity perturbations begin to grow, which leads to turbulence of the flow. The maximum values of the eigenfunctions and perturbations of the transverse velocities are shifted to the hot wall. It is seen that for an unstable eigenvalue, the perturbations of the transverse flow velocity increase over time, and for a stable one, they decay.

Heat transfer in flow of nonlinear fluids with viscous dissipation

The rotational flow of viscoplastic fluids between concentric cylinders is examined while dissipation due to viscous effects through the energy balance. The viscosity of fluid is simultaneously dependent on shear rate and temperature. Exponential dependence of viscosity on temperature is modeled through Nahme law, and the shear dependency is modeled according to the Carreau equation. Hydrodynamically, stick boundary conditions are applied, and thermally, both constant temperature and constant heat flux on the exterior of cylinders are considered. The governing motion and energy balance equations are coupled adding complexity to the already highly correlated set of differential equations. Introduction of Nahme number has resulted in a nonlinear base flow between the cylinders. As well, the condition of constant heat flux has moved the point of maximum temperature toward the inner cylinder. Taking viscous heating into account, the effects of parameters such as Nahme and Brinkman numbers, material time and pseudoplasticity constant on the stability of the flow are investigated. Moreover, the study shows that the total entropy generation number decreases as the fluid elasticity increases. It, however, increases with increasing Nahme and Brinkman numbers.

Thermo-viscous fingering in quarter five-spot miscible displacements

A pseudo-spectral method is implemented to model thermo-viscous fingering in a quarter five-spot geometry. An exponential dependence of viscosity on temperature and concentration is represented by two parameters β T and β C , respectively. The effects of these two parameters as well as that of the thermal lag coefficient and the Lewis number on the flow development are analyzed. Time evolutions of nonlinear fingers are examined qualitatively by plotting concentration and temperature iso-surfaces. The qualitative observations are further substantiated through an examination of the relative contributions of solutal and thermal vorticity components in the development of the instability. This study reveals that the strong velocity gradients in this radial geometry and the coupling of heat and mass transfer result in some important differences with what is obtained in the case of the rectilinear geometry.

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 1: Linear Stability Analysis

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

Theoretical and numerical study of a thermal convection problem with temperature-dependent viscosity in an infinite layer

A convection problem with temperature-dependent viscosity in an infinite layer is presented. This problem has important applications in mantle convection. The existence of a stationary bifurcation is proven together with a condition to obtain the critical parameters at which the bifurcation takes place. A numerical strategy has been developed to calculate the critical bifurcation curves and the most unstable modes for a general dependence of viscosity on temperature. An exponential dependence of viscosity on temperature has been considered in the numerical calculations. Comparisons with the classic Rayleigh–Bénard problem with constant viscosity indicate that the critical temperature difference threshold decreases as the exponential rate parameter increases. The vertical velocity of the marginal mode exhibits motion concentrated in the region where viscosity is smaller.

Combined effect of variable fluid properties on nonisothermal flow of an inelastic fluid with viscous heating

This study investigates the effect of variable thermal conductivity and variable viscosity on the flow characteristics and flow rates of an inelastic fluid, including viscous heating terms in the analysis. The exponential dependence of viscosity on temperature is modeled through Arrhenius law and shear rate dependence is modeled through Carreau rheological equation while it is assumed that the thermal conductivity varies linearly with temperature. Flow governing motion and energy balance equations are coupled and the solution is found iteratively facilitating a pseudospectral method based on the Chebyshev polynomial expansions. Effect of variable thermal conductivity and viscosity as well as the effect of power law index, material time constant, Brinkman and Peclet numbers on the velocity and temperature profiles are presented. Influence of the above parameters on the flow rate is also calculated.

Initiation of localized shear zones in viscoelastoplastic rocks

Shear localization is a process of primary importance for the onset of subduction and the evolution of plate tectonics on Earth. In this paper we focus on a model in which shear localization is initiated through shear heating. The rheology employed is linear Maxwell viscoelastic with von Mises plasticity and an exponential dependence of viscosity on temperature. Dimensional analysis reveals that four nondimensional (0‐D) parameters control the initiation of shear zones. The onset of shear localization is systematically studied with 0‐D, 1‐D, and 2‐D numerical models, both under constant stress and under constant velocity boundary conditions. Mechanical phase diagrams demonstrate that six deformation modes exist under constant velocity boundary conditions. A constant stress boundary condition, on the other hand, exhibits only two deformation modes (localization or no localization). Scaling laws for the growth rate of temperature are computed for all deformation modes. Numerical and analytical solutions demonstrate that diffusion of heat may inhibit localization. Initial heterogeneities are required to initiate localization. The derived scaling laws are applied to Earth‐like parameters. For a given heterogeneity size, stable (nonseismic) localization only occurs for a certain range of effective viscosities. Localization is inhibited if viscosity is smaller then a minimum threshold, which is a function of the heterogeneity size. The simplified rheological model is compared with a more realistic and more complex model of olivine that takes diffusion, power law, and Peierls creep into account. Good agreement exists between the models. The simplified model proposed in this study thus reproduces the main physics of ductile faulting. Two‐dimensional late stage simulations of lithospheric‐scale shear localization are presented that confirm the findings of the initial stage analysis.

Reverse osmosis at elevated temperatures: influence of temperature on degree of concentration polarization and transmembrane flux

A model describing the influence of temperature on longitudinal distribution of the degree of concentration polarization (CP) and transmembrane flux in reverse osmosis systems was developed. The model is based on the following assumptions: (1) membrane morphology is temperature independent; (2) transport characteristics of membranes are invariant with their coordinate; (3) specific water permeability of membranes is based on exponential dependence of viscosity vs. temperature; (4) temperature dependence of membrane rejection is assumed to be linear. The proposed model gives quantitative correlations between the longitudinal mass flow and the CP degree at different operating temperatures. The model enables analyzing the influence of temperature, mass flow, bulk concentration, axial velocity, channel geometry, physical properties and membrane rejection on longitudinal distribution of the degree of CP. This model can be hybridized with sub-models describing driving force, gel resistance, etc., and segmented into a more advanced algorithm for modeling longitudinal variation of characteristics. The model does not contain digital integration and can be easily incorporated into optimization algorithms.

Viscous heating effects on the linear stability of Poiseuille flow of an inelastic fluid

In this paper, the effect of viscous heating on the stability of a non-Newtonian fluid flowing between two parallel plates under the effect of a constant pressure gradient is investigated. The viscosity of the fluid depends on both temperature and shear rate. Exponential dependence of viscosity on temperature is modeled through Arrhenius law. Non-Newtonian behavior of the fluid is modeled according to the Carreau rheological model. Motion and energy balance equations that govern the base flow and the stability of the flow are coupled and the solution to the problem is found iteratively using a pseudospectral method based on the Chebyshev polynomials. In the presence of viscous heating, the effect of activation energy parameter, Prandtl and Brinkman numbers, material time and power-law constants on the stability of the flow is presented in terms of neutral stability curves.

Modeling the influence of temperature on degree of concentration polarization in reverse osmosis systems

A model describing the influence of temperature on degree of concentration polarization is proposed. The model is based on the following assumptions: (1) membrane morphology is temperature-independent; (2) transport characteristics of membrane are invariant with coordinate; (3) specific water permeability of the membrane is based on exponential dependence of viscosity vs. temperature; (4) temperature-dependence of membrane rejection is assumed to be linear. Proposed models permit quantitative correlations of longitudinal mass flow and degree of concentration polarization at different operating temperatures. It enables analysis of the influence of temperature, mass flow, bulk concentration, axial velocity, channel geometry, physical properties and membrane rejection on longitudinal distribution of concentration polarization degree. The submitted model does not contain digital integration and can be segmented into optimization algorithms.

Modeling the Influence of Temperature or Resistance of Concentration Layer and Transmembrane Flux in Reverse Osmosis Systems

ABSTRACTModel permits analyzing the influence of temperature on resistance of concentration layer and transmembrane flux. Proposed model is based on the following assumptions: (1) membrane morphology doesn’t depend on temperature; (2) membrane rejection, and other transport characteristics of membrane are invariant with coordinate; (3) specific water permeability of membrane was based on exponential dependence of viscosity vs. temperature; (4) dependence of membrane rejection on temperature is assumed to be linear. Dependence of normalized permeability upon temperature is strongly influenced by the CP degree (and surface concentration), namely, (A) at low CP degree the growth of temperature causes growth of normalized permeability, while at (B) high values of CP degree the system is characterized by opposite behavior: the normalized permeability goes down with growth of temperature. The model allows analyzing the influence of temperature and degree of membrane rejection on concentration resistance and transmembrane flux. Calculated data are attached. The presented solution can be segmented and built into algorithm for calculation of longitudinal profile of CP degree and for optimization of regime parameters against temperature.

Correction for Exponential Dependence of Viscosity on Temperature for Natural Convection by an Integral Method

It is common to correct for the effect of exponential dependence on inverse absolute temperature of the viscosity on the Nusselt number for laminar natural convection by multiplying the correlation for constant properties by a factor μ∞/μwn to obtain Nu=C Ra1/4μ∞/μwn. That there is a sound basis for the functional form of this corrective term is newly shown by employing an integral method to obtain closed-form solutions to the laminar boundary-layer equations for the case of a vertical plate of specified temperature immersed in such a liquid. The basis for the alternative means of correction by using the correlation for constant properties with viscosity evaluated at a special reference temperature is also shown.

Influence of temperature and permeate recovery on energy consumption of a reverse osmosis system

A model permits analysis of the influence of temperature on permeate recovery and energy consumption. The proposed model is based on the following assumptions: (1) membrane morphology is temperature-independent; (2) membrane rejection and other transport characteristics of membranes are position-independent; (3) specific water permeability of membranes was based on exponential dependence of viscosity vs. temperature; (4) temperature-dependence depembrane rejection is assumed to be linear. This allows for analyzing the influence of channel geometry, feed concentration, flow rate and temperature on permeate recovery and energy consumption. Calculated data are included. The solutionpresented can be segmented andbuilt into systems for comprehensive techno-economic evaluation of the RO-based process where temperature-dependence of process characteristics has to be considered.

NONISOTHERMAL CHANNEL FLOW OF A NON-NEWTONIAN FLUID WITH VISCOUS HEATING

This study investigates the pressure gradient-flow rate relationship for steady-state nonisothermal pressure-driven flow of a non-Newtonian fluid in a channel including the effect of viscous heating. The viscosity of the fluid depends on both temperature and shear-rate. Exponential dependence of viscosity on temperature is modeled through Arrhenius law. Non-Newtonian behaviour of the fluid is modeled according to the Carreau rheological equation. Flow governing motion and energy balance equations are coupled and the solution of this non-linear boundary value problem is found iteratively using a pseudospectral method based on Chebyshev polynomials. The effect of activation energy parameter and Brinkman number, as well as the power-law index and material time constant on the flow is studied. It is found that while the pressure gradient-flow rate graph is monotonic for certain ranges of flow controlling parameters, there is a large jump in the graph under certain values of these parameters.