Values Of Magnetic Parameters Research Articles

Роль замещающих марганец ионов и кислородной нестехиометрии в формировании свойств манганитов

Structural and magnetic characteristics of La0.7Sr0.3Mn0.957Fe0.05Zn0.05O3+γ and La0.7Sr0.3Mn0.957Fe0.05Mg0.05O3+γ manganites, in which the iron is Mossbauer isotope 57Fe, are investigated and compared. Ceramic samples were sintered in air at 1473 K. They were then exposed to heat treatments at 1223 K and various partial pressure of oxygen in the gas phase, which ensured the production of manganites with following values of non-stoichiometry index: γ = −0.005; 0.000; 0.007; 0.008 for Zn-containing manganites (ZnM), and γ = −0.022; 0.000; 0.002; 0.005 for Mg-containing manganites (MgM). All synthesized manganites have rhombohedral crystal structure. Mössbauer spectroscopy data correspond to Fe3+ (3d5) ions. Curie point and magnetization depend on γ nonmonotonically. ZnM have essentially higher values of magnetic parameters and narrowest temperature interval of “ferromagnetic-paramagnetic” transition as compared to MgM. Their structure is considered as more homogeneous, which corresponds to lower value of quadrupole splitting of Mössbauer spectra. The results obtained indicate that different effect of Zn2+ and Mg2+ ions on electromagnetic characteristics of manganites is mainly determined by the configuration of their electron shells (3d10 and 2p6, correspondingly).

Second law and entropy generation analysis of magnetized viscous fluid flow over a permeable expandable sheet with nonlinear thermal radiation: Brownian and thermophoresis effect

The second law, thermal, magnetic field, and concentration of viscous fluid across a permeable stretching surface are the focus of this study. The transverse and longitudinal velocities, temperature, and concentration with boundary conditions are computed numerically by applying Runge-Kutta 4th-order method. For this determination the system of governing equations are first converted to the first order ordinary linear equations and then solve by RK4 built-in function in MATHLAB SOFTWARE by taking step size [Formula: see text]. The existing work is compared with the difference between existing and published work. The iteration procedure was stopped until all of the nodes in the η-direction met the convergence condition 10−5. For confirmation of the results, the BVPh2 package is also applied and excellent agreement is found. Both on longitudinal and transverse kinematics, the impact of ferromagnetic and viscoelastic factors are examined. The temperature is studied in relation to the Prandtl number, the magnetism factor, and the heat reference factor. The concentration is also shown, as well as how it varies well with Schmidt number and the magnetic factor. The entropy generation number is calculated using fluid velocity, energy, and volume fraction coefficient. Moreover, the current work is also equated with the available work for limiting cases. The longitudinal and transverse velocities are declined when the viscoelastic and magnetic parameters are increased. The temperature profile enhances as the magnetic parameters and heat source-sink parameters increase, but declines as Prandtl number increases. At the same time, concentration raises as the magnetic parameters rises. From this investigation it is also observed that the concentration falls, as the Sc enhances. The entropy generation number decreases with the increasing values of magnetic parameter. It is perceived that as Pr enhances, the Ns increases near the surface but with increasing η, the situation reverses. For higher values of [Formula: see text], the generation number [Formula: see text] declines at the surface, however, the situation reverses via η rises.

Entropy optimized MHD fluid flow over a vertical stretching sheet in presence of radiation

AbstractThe present article describes the influence of radiation on two‐dimensional laminar magnetohydrodynamic fluid flow passing over a convective surface. The behavior of the thermal equation is explored through Joule heating, heat generation/absorption, and viscous dissipation. The aim of this study is to examine the physical behavior of the entropy optimization rate. The Cartesian coordinates system is used to model the flow equations. Using similarity variables, a system of partial differential equations is converted into a system of ordinary differential equations. The problem is solved using HAM. The influence of various pertinent parameters on fluid characteristics is graphically explored. Velocity decreases for an increased amount of magnetic parameter, suction parameter, and velocity slip parameter, while behaves the opposite for Grashof number. Temperature increases for a large amount for Brinkman number, magnetic parameter, and radiation parameter, while decreases for Prandtl number. Entropy generation rate increases for Brinkman number, magnetic parameter, and temperature difference parameter. Bejan number decreases for Brinkman number while behaves the opposite for magnetic parameter and temperature difference parameter. Skin friction decreases for large values of magnetic parameters while behaving the opposite for a large amount of velocity slip parameter. Nusselt number decreases for a large amount of Brinkman number. For a better understanding of the study, comparison between numerical outcomes of entropy generation rate and Bejan number for different values of Prandtl number has been done through tables. Also, numerical outcomes of skin friction and Nusselt number are discussed using pertinent parameters through tables.

Entropy generation on flow and heat transfer of a reactive MHD Sisko fluid through inclined walls with porous medium

The present research focuses on flow, heat transfer, and entropy production properties in the context of their applications. The use of non-Newtonian materials in biomedical rheological models has garnered considerable attention. Having such practical and potential applications, the current study explores the incompressible, gravity-driven flow of a combustible Sisko liquid down inclined heated permeable walls embedded in a porous medium. The flow is said to be fully developed and modified Darcy’s law is employed to express the porous medium. Such a flow is critical to simulate free-surface flows in trapezoidal reservoirs or channels, and around curved or inclined hulls. The spectral collocation technique with expansions in Chebyshev polynomials was instrumental in achieving numerical solutions. Furthermore, a parametric study is done to explore the behavior of flow, thermal, entropy generation, and Bejan number profiles. The significant outcomes of the current study are that the rise in values of material parameter decreases the velocity, thermal, and entropy generation profiles but increases the Bejan number profile. The upsurge in values of magnetic parameter decreases the velocity and entropy generation profiles but increases the thermal and Bejan number profile. The rise in values of the Frank-Kameneskii parameter increases the Bejan number, thermal, and entropy generation profiles.

Numerical simulation of carbon nanotubes nanofluid flow over vertically moving disk with rotation

In this article, we explored the impact of magnetic field on nanofluid flow in the presence of SWCNT and MWCNT with water as a base fluid and flow is caused due to a vertically upward/downward moving rotating disk. The problem is constructed in such a way that the equilibrium of controlling the physical object is finally reduced for those conveyed in the traditional von Karman’s investigation of viscous pumping in a rotating disk. The framed partial differential equations (PDEs) are reduced into ordinary differential equations (ODES) by considering suitable similarity variables. The numerical solutions are obtained by using Runge–Kutta–Fehlberg’s fourth fifth order method (RKF-45) by adopting shooting technique. The computation of rate of heat transfer is analyzed through graphs. Further variation in velocity and thermal profiles for various dimensionless parameters are studied briefly and illustrated with the help of graphs. The outcomes reveal that, upsurge in expanding/contraction parameter increases the fluid velocity and it is more in MWCNT–water than SWCNT–water nanofluid flow. The escalation in values of magnetic parameter increases the thermal profile of both single and multi-wall CNTs with water based nanofluid but declines radial, azimuthal velocity profiles.

Numerical investigation on magnetohydrodynamics boundary layer flow of micropolar Carreau nanofluid with nonlinear thermal radiation

The current work examines the behaviour of micropolar Carreau nanofluid flow past an infinite vertical plate with nonlinear thermal radiation. Magnetic field, heat source/sink, viscous dissipation, internal and Joule heating effects are considered. The governing equations are reframed and expressed in the non-dimensional form with appropriate dimensionless quantities to analyse the model. The nonlinear non-dimensional governing equations are solved numerically using finite difference approximation to acquire the solutions. Impact of involving flow controlling parameters on fluid flow, heat and mass transport characteristics are discussed and presented via tables and graphical illustrations. Our outcomes figured out that the fluid motion and angular velocity declines with greater magnetic parameter values, and reverse trend is observed for micropolar constant. Further, computational results depict that thermal distribution is a rising function of radiation parameter and high thermal performance is observed; that is, 10.45% more temperature distribution is noticed in presence of nonlinear thermal radiation.

Entropy analysis on nonlinear radiative MHD flow of Diamond-Co3O4/ethylene glycol hybrid nanofluid with catalytic effects

In the present study, flow of Diamond-Co3O4/ethylene glycol (EG) hybrid nanofluid towards a stretching sheet with stagnation point has been analysed under the influence of magnetic field and nonlinear thermal radiation. The catalytic property of homogeneous–heterogeneous reactions has also been included in the system. Additionally, the entropy generated in the system due to irreversible processes has been analysed here. The governing equations of the flow are obtained according to the given system and are converted to non-dimensional forms by using their appropriate similarity transformations. BVP4C tool of matlab has been utilised to solve these equations. Nature of various properties such as velocity, temperature and concentration of the flow with respect to different parameters has been represented graphically. Also a graphical comparison has been made between the nature of skin-friction and Nusselt number for hybrid-nanofluid and nanofluid respectively. The major outcome of the study is that the velocity profile of the fluid decreases for higher values of magnetic parameter. The temperature profile of the fluid enhances in the presence of nonlinear radiation. Also, the hybrid nanofluid was more efficient in transferring heat than the corresponding nanofluid. This work can be utilised by various large scale industries to increase their efficiency.

Green synthesis and investigations of structural, cation distribution, morphological, and magnetic properties of nanoscale nickel ferrites: the effect of green fuel proportion

ABSTRACT Cubic spinel form nickel ferrite (NiFe2O4) nanoparticles were synthesized by the green synthesis approach using aqueous honey. The concentration of honey was 10%, 30%, 50%, 70% and 100%, and its effects on different properties of NiFe2O4 NPs were investigated. The prepared powder samples underwent thermal characterization (TGA) to know the ferritization temperature. The X-ray diffraction analysis showed that the ready samples have the mono-phase with cubic spinel symmetry. The surface morphology was investigated from SEM images and results revealed the slight agglomeration, which decrements with the increment in honey concentration. Elemental studies revealed the stoichiometric compositional preparation of nickel ferrite nanoparticles. The saturation magnetization, coercivity, magnetic parameters were measured using M-H plots. The values of magnetic parameters show increased values for 50% honey concentration compared to other concentrations of honey. All the results show the excellent impact of the green chemistry approach on the properties of nickel ferrite NPs.

Analysis on physical properties of micropolar nanofluid past a constantly moving porous plate

The computational analysis is presented for boundary layer heat and mass transfer flow of hydro magnetic micropolar nanofluid flow. In the flow model, viscosity of the fluid is taken as temperature-dependent and varies linearly and the other physical properties such as radiative heat flux, the magnetic field, the viscous dissipation, chemical reaction are additionally assumed in the energy equation and spices concentration equation respectively The PDEs representing the fluid flow have been changed into a framework of dimensionless ODEs and explained mathematically through the 4th order R-K and NS shooting technique. Temperature distribution, velocity distribution, micro rotation, and concentration distribution are explored graphically for a series of solid volume fraction (0<ϕ<2) of nano-solid particles. All the findings for various flow parameters agreed perfectly with physical situation of the flow. It is observed that for increasing value of magnetic parameter, the concentration and temperature of the micropolar nano fluid near the boundary layer declines and increasing value of the volume fraction of nano-solid particle ϕ leads to decrease in velocity and micro rotation of the fluid within the boundary layer decreases.

Nonlinear thermal radiation and temperature dependent viscosity effects on MHD heat and mass transfer in a thin liquid film over a stretching surface

This paper describes nonlinear thermal radiation effects on MHD heat and mass transfer in a thin liquid film over a permeable unsteady stretching surface taking temperature-dependent fluid viscosity with convective boundary condition. For the non-linearity of the momentum, energy and mass diffusion equations, the problem is solved numerically. At first, Similarity transformations is used to the governing equations to reduce the equations into a set of ordinary differential equations. Then the resulting nonlinear ordinary differential equations are solved using Runge-Kutta-Felberg method with shooting technique. Different physical parameters effects on heat and mass transfer in a thin liquid film are presented graphically. It is found that increase in the unsteadiness parameter leads to increase in the velocity distribution, temperature and concentration gradient. Further, increase in the value of magnetic parameter results in a decrease in the velocity profile and increase in the temperature and concentration gradient. For enhancement of thermal radiation decreases the temperature gradient of the thin film flow. Also, for increase in viscosity variation parameter is to decrease velocity distribution but reverse effects shown in case of temperature and concentration gradient.

Significance of Thermophoretic and Brownian Motion on MHD Nanofluids Flow towards a Circular Cylinder under the Inspiration of Multiple Slips: An Industrial Application

In this article, MHD flow of silver/water nanofluid past a stretched cylinder under the impact of thermal radiation with chemical reaction and slip condition is studied. The impact of Soret and Dufour effect is also analyzed during this flow. The uniqueness of the given problem is enlarged with the insertion of variable magnetic field, free stream velocity, thermal slip condition, and nonlinear thermal radiation. The PDEs are converted to ODEs by using suitable similarity transformation. The nonlinear system of ODEs is solved by applying convergent homotopy analysis method (HAM). The velocity, temperature, and concentration profiles for the free stream and at the plate are discussed through graphs and numerical tables. It is found that velocity field reduces, while the temperature profile rises for the increasing values of magnetic parameter. It is examined that effects of curvature on frication factor are increasing. Furthermore, temperature profile increases for greater Brownian motion and thermophoresis parameters. Transfer of heat enhances decreasing the radius of the cylinder also with heat generation parameter. The skin friction can be reduced by enhancing free stream and wall stretching velocities ratio. Velocity profile of the flow can be controlled by enhancing velocity slip and magnetic field.

RETRACTED: Analysis of radiative nonlinear heat transfer in a convective flow of dusty fluid by capitalizing a non-Fourier heat flux model

The study is concerned with the heat transfer in a slip flow of a dusty fluid with the impact of a magnetic field and nonlinear thermal radiation. Furthermore, for the heat transfer process the Cattaneo–Christov heat flux model is used. Suitable similarity transformations are used to transform the governing equations. Later, shooting method and the Runge-Kutta Fehlberg's fourth fifth order (RKF-45) process are utilized to solve these reduced system of nonlinear ordinary differential equations. Impact of numerous involved parameters on the flow, thermal fields of both dust and fluid phase, skin friction and rate of heat transfer are visually plotted through graphs and discussed quantitatively. The significant outcomes drawn from the current study are that, the rise in value of the velocity slip parameter decreases the velocity profile but improves the thermal profile of both the phases. The growing values of curvature parameter intensify the flow and the thermal fields of both phases. The cumulative values of magnetic parameter and dust particle mass concentration parameter declines the velocity and thermal gradients of both phases. The thermal relaxation time parameter decays the temperature profile. The heat transfer rate is strengthened with the growing values of the curvature parameter, the velocity slip parameter, and radiation parameter.

Multiple Solutions for Stagnation-Point Flow of Unsteady Carreau Fluid along a Permeable Stretching/Shrinking Sheet with Non-Uniform Heat Generation

The aim of the present study was to explore the effect of a non-uniform heat source/sink on the unsteady stagnation point flow of Carreau fluid past a permeable stretching/shrinking sheet. The novelty of the flow model was enhanced with additional effects of magnetohydrodynamics, joule heating, and viscous dissipation. The nonlinear partial differential equations were converted into ordinary differential equations with the assistance of appropriate similarity relations and were then tackled by employing the Runge-Kutta-Fehlberg technique with the shooting method. The impacts of pertinent parameters on the dimensionless velocity and temperature profiles along with the friction factor and local Nusselt number were extensively discussed by means of graphical depictions and tables. The current results were compared to the previous findings under certain conditions to determine the precision and validity of the present study. The fluid flow velocity of Carreau fluid increased with the value of the magnetic parameter in the case of the first solution, and the opposite behavior was noticed for the second solution. It was seen that temperature of the Carreau fluid expanded with the higher values of unsteadiness and magnetic parameters. It was visualized from multiple branches that the local Nusselt number declined with the Eckert number parameter for both the upper and lower branch.

Three Dimensional MHD Hybrid Nanofluid Flow with Rotating Stretching/Shrinking Sheet and Joule Heating

A three-dimensional hybrid nanofluid flow over a stretching/shrinking sheet is numerically studied. The hybrid nanofluid being considered in this study used water as the base fluid and mixed with two types of solid nanoparticles, namely alumina (Al2O3) and copper (Cu). The main focus of the current study is to examine the effect of magnetic field, Joule heating, and rotating sheet on the velocity, and temperature profiles. In addition, the impact of suction and stretching sheet on the variations of reduced skin friction, , and reduced heat transfer are studied as well. The fluid flow and heat transfer problem presented in this study is governed by a system of nonlinear partial differential equations (PDEs), which is then transformed into the corresponding system of high order nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting system of higher order nonlinear ODEs is solved numerically using a boundary value solver known as bvp4c, which operates on the MATLAB computational platform. Results revealed that dual solutions exist for shrinking sheet while unique solutions are observed for stretching sheet with various values of Cu nanoparticles volume fraction and magnetic parameter. Dual solutions also exist for the value of the suction parameter greater than its critical point with various values of Cu nanoparticles volume fraction. Velocity profile of the hybrid nanofluid increases alongside with the value of magnetic parameter but declination was observed in the profile of and temperature, for both solutions as the value of Cu nanoparticles volume fraction increases. When the value of rotational parameter increases, both velocity and profiles increase for both solutions. This indicates that the momentum boundary layer thickness increases with increasing values of for both solutions, but thermal boundary layer thickness decreases for the first solution and increases for the second solution. Finally, an increment in the value of Eckert number causes the temperature of the hybrid nanofluid to rise as well for both first and second solutions.

Entropy optimized dissipative flow of hybrid nanofluid in the presence of non-linear thermal radiation and Joule heating

Present article reads three dimensional flow analysis of incompressible viscous hybrid nanofluid in a rotating frame. Ethylene glycol is used as a base liquid while nanoparticles are of copper and silver. Fluid is bounded between two parallel surfaces in which the lower surface stretches linearly. Fluid is conducting hence uniform magnetic field is applied. Effects of non-linear thermal radiation, Joule heating and viscous dissipation are entertained. Interesting quantities namely surface drag force and Nusselt number are discussed. Rate of entropy generation is examined. Bvp4c numerical scheme is used for the solution of transformed O.D.Es. Results regarding various flow parameters are obtained via bvp4c technique in MATLAB Software version 2019, and displayed through different plots. Our obtained results presents that velocity field decreases with respect to higher values of magnetic parameter, Reynolds number and rotation parameter. It is also observed that the temperature field boots subject to radiation parameter. Results are compared with Ishak et al. (Nonlinear Anal R World Appl 10:2909–2913, 2009) and found very good agreement with them. This agreement shows that the results are 99.99% match with each other.

Analysis of MHD mixed convection in a Ag-TiO2 hybrid nanofluid flow past a slender cylinder

The concept of MHD mixed convection with Ag-TiO2 / H2O hybrid nanofluid flow over a slender cylinder find its several applications in the field of science and engineering. Especially, in designing and manufacturing of coating of wires, fibre sheets, optical fibres, photo electric devices, solar cells etc. A uniform magnetic field is applied to analyse its impact on transport characteristics of the flow. The different nanoparticles shape factor is considered in order to study the behaviour of thermal efficiency. The study of such flow problems involving physical aspects such as mixed convection, thermo-physical behaviours of hybrid nanofluid and uniform applied magnetic field is an innovative approach. The mathematical model describing the fluid flow has been formulated by nonlinear coupled partial differential equations with boundary constraints. Further, the governing equations have been transformed into dimensionless form by utilizing appropriate non-similar transformations and are attempted for the solution in combination with Quasilinearization technique and implicit finite difference approximation. The simulation comprises the various physical parameters, such as velocity ratio parameter, mixed convection parameter (Richardson number) Ri, magnetic field parameter M, Eckert number Ec, nanofluid volume fraction φ1, hybrid nanofluid volume fraction φ2 and nanoparticles shape factor s are demonstrated through graphs. The results unveil that inclusion of the hybrid nanoparticles in the base fluid result in high heat transfer than that of nanofluid and base fluid. Further, the velocity of the hybrid nanofluid is enhanced for the enhancing values of mixed convection parameter and velocity ratio parameter. The hybrid nanofluid temperature rises with raising values of magnetic parameter and volume fraction of Ag-TiO2, significantly.

Effect of Magnetohydrodynamics on Heat Transfer Behaviour of a Non-Newtonian Fluid Flow over a Stretching Sheet under Local Thermal Non-Equilibrium Condition

A mathematical model is proposed to describe the flow, heat, and mass transfer behaviour of a non-Newtonian (Jeffrey and Oldroyd-B) fluid over a stretching sheet. Moreover, a similarity solution is given for steady two-dimensional flow subjected to Buongiorno’s theory to investigate the nature of magnetohydrodynamics (MHD) in a porous medium, utilizing the local thermal non-equilibrium conditions (LTNE). The LTNE model is based on the energy equations and defines distinctive temperature profiles for both solid and fluid phases. Hence, distinctive temperature profiles for both the fluid and solid phases are employed in this study. Numerical solution for the nonlinear ordinary differential equations is obtained by employing fourth fifth order Runge–Kutta–Fehlberg numerical methodology with shooting technique. Results reveal that, the velocity of the Oldroyd-B fluid declines faster and high heat transfer is seen for lower values of magnetic parameter when compared to Jeffry fluid. However, for higher values of magnetic parameter velocity of the Jeffery fluid declines faster and shows high heat transfer when compared to Oldroyd-B fluid. The Jeffery liquid shows a higher fluid phase heat transfer than Oldroyd-B liquid for increasing values of Brownian motion and thermophoresis parameters. The increasing values of thermophoresis parameter decline the liquid and solid phase heat transfer rate of both liquids.

On bioconvection and mass transpiration of micropolar nanofluid dynamics due to an extending surface in existence of thermal radiations

This study examines the magnetic effects of heat and mass transmission on the flow of micropolar fluid over a permeable stretching geometry with dilute homogeneous dispersion of nano-particles and gyrotactic microorganisms. A system of coupled highly non-linear PDEs is renovated into corresponding ODEs by using similarity functions. These transmuted equations are resolved for a solution with shooting technique accompanied with Runge-Kutta fourth order. The variations of intricate physical quantities such as temperature, micro-motion, concentration, velocity, and motile micro-organism profiles are evaluated under the influence of the emerging parameters. The velocity profile decreases down with upsurge values of magnetic parameter M while micro-rotation is strengthened and its value becomes higher directly with increments in M. The microorganisms profile depict the diminishing behavior with the growing value of bioconvection Lewis number. These results are useful for obtaining better solutions for heat transfer devices and micropolar fuel cells. Additionally, the impact of the parameter of Brownian motion, Rayleigh number, and the parameter of thermophoresis, Peclet number, and buoyancy ratio parameter were discussed numerically and graphically. Moreover, the numerical results were validated by comparing them with previously obtained exact solution for special cases and acceptable compatibility between the two results is achieved. The findings from this work can be utilized for efficient heat exchangers and thermal balance in micro-electronics.

Magnetized flow of sutterby nanofluid through cattaneo-christov theory of heat diffusion and stefan blowing condition

Stefan blowing phenomenon in electrically conducting Sutterby material flow over stretchable rotating disk is demonstrated in this research. Cattaneo-Christov (CC) model of energy diffusion is adopted to analyze the heat transmission. Buongiorno model is carried out to evaluate the involvement of nanoparticles. The formulated system of partial differential expressions is re-structured by the enactment of similarity functions. Runge–Kutta-Fehlberg (RKF) fourth-fifth order process has been executed to communicate the solution of velocity, thermal and solutal fields. The velocity, concentration, thermal fields, skin friction, rate of mass and heat transportations are explored for the embedded non-dimensional parameters graphically. Result reveals that the rise in Stefan blowing factor leads to an enhancement in radial and tangential velocities gradients. The velocity of nanomaterial is reduced by the incrementing material parameter values. The augmenting magnetic parameter values reduced the liquid velocity but improves the temperature. The thermophoretic force and Brownian motion involvement resulted the higher thermal field.

Effect of rotational slip on the physical parameter in a micropolar fluid flow past a stretching sheet

Micropolar fluid flow studied in this paper is influenced by microstructural slip. The flow is directed by a scheme of Partial Differential Equalities. These Partial Differential Equalities are then converted to nonlinear set of Ordinary Differential Equalities via boundary layer conversions. MATLAB bvp4c built in code is taken into account to resolve the leading set of ODEs, along with the initial-boundary settings. Hydro dynamical and thermal boundary layer outlines are considered and deliberated and the results are confirmed by linking with available literature in the classical case. Microstructural slip effects are shown on the Nusselt number and skin-friction coefficient. This model can better predict the effects and characteristics of rotational slip. Specifically, it is predicted from the tabular and graphical results that the rotational slip affects the boundary layer thickness when second-order translational slip disappears. It is important to mention that augmenting the standards of Prandtl number and radiation constraint declines the fluid temperature in the nonappearance and existence of microstructural slip. Moreover, increasing the values of magnetic parameter enhances the fluid temperature in presence as well as in absence of microstructural slip.