Microrotation Fields Research Articles

Numerical investigation on transport of momenta and energy in micropolar fluid suspended with dusty, mono and hybrid nano-structures

Computational fluid dynamics (CFD) tools are applied to model transportation activity in micropolar fluid suspension with density particles and mono and hybrid metallic nano-sized structures. Highly complex computational models obtained by CFD simulations are executed via the finite element method. Galerkin residuals are approximated by Galerkin approximations. The assembled nonlinear system is linearized and solved iteratively under reasonable computational tolerance. A notable impact of vortex viscosity is observed during transportation of macro- and micro-momentum. The velocity of dust particles has been predicted via the variation of dust particle interaction parameter. Angular velocity for both mono and hybrid nanofluids is found to increase as a function of vortex viscosity. Furthermore, it is noted that the micro-rotation field associated with mono–nanofluid has a higher value than the micro-rotation field associated with hybrid nanofluids.

Application of CNT-based micropolar hybrid nanofluid flow in the presence of Newtonian heating

This research addresses the effects of Hall and ion slip in micropolar hybrid nanofluid in the presence of Newtonian heating. Furthermore, the influence of thermal radiation, Darcy–Forchheimer, viscous dissipation, and variable viscosity is discussed. Total entropy generation rate is calculated. Two distinct nanoparticles such as (SWCNT, MWCNT) used as a hybrid nanofluid. Built-in function bvp4c integrates the solution of simulated hydrodynamic boundary value problems. The effects on axial velocity, angular velocity, temperature field, concentration field, Bejan number, and entropy optimization of different flow field variables are displayed graphically. The nanoparticle fraction boosts the temperature field, Bejan number, and entropy generation, while the axial velocity and microrotation field reduces. Furthermore, the entropy generation enhances with higher the Brinkman number and variable viscosity parameter.

Numerical study on enhancement of heat transfer in hybrid nano-micropolar fluid

Hybrid nanoparticles correlations together with governing models for the flow of micropolar fluid are used to develop novel complex coupled nonlinear set of differential equations. The solutions are computed by using the finite element method (FEM) and results are validated after convergence and grid-free analysis. The macroscopic velocity and micro-rotation field are investigated versus parametric variation. The temperature field is examined for various emerging dimensionless parameters. A significant improvement in the thermal performance of micropolar fluid due to simultaneous dispersion of Cu and Al2O3 (hybrid nanoparticles) is noted. Therefore, simultaneous dispersion of Cu and Al2O3 for significant improvement in thermal performance of micropolar fluid is recommended rather the dispersion of copper nanoparticles. Surprisingly micro rotation field for nano micropolar mixture near the vicinity of the wall is less than the micro-rotation field for hybrid nano-micropolar fluid. However, away from the wall opposite dynamics for the micro-rotation field are observed.

Micromagnetorotation of MHD Micropolar Flows

The studies dealing with micropolar magnetohydrodynamic (MHD) flows usually ignore the micromagnetorotation (MMR) effect, by assuming that magnetization and magnetic field vectors are parallel. The main objective of the present investigation is to measure the effect of MMR and the possible differences encountered by ignoring it. The MHD planar Couette micropolar flow is solved analytically considering and by ignoring the MMR effect. Subsequently, the influence of MMR on the velocity and microrotation fields as well as skin friction coefficient, is evaluated for various micropolar size and electric effect parameters and Hartmann numbers. It is concluded that depending on the parameters’ combination, as MMR varies, the fluid flow may accelerate, decelerate, or even excite a mixed pattern along the channel height. Thus, the MMR term is a side mechanism, other than the Lorentz force, that transfers or dissipates magnetic energy in the flow direct through microrotation. Acceleration or deceleration of the velocity from 4% to even up to 45% and almost 15% deviation of the skin friction were measured when MMR was considered. The crucial effect of the micromagnetorotation term, which is usually ignored, should be considered for the future design of industrial and bioengineering applications.

Unsteady heat transfer in colloidal suspension containing hybrid nanostructures

Theoretical investigation about the role of hybrid nanoparticles to enhance the effective thermal conductivity of working micropolar fluid over a non-uniformly moving surface having non-uniform temperature and exposed to non-uniformly magnetic field is carried out. Additional conservation law of angular momentum is used to study the dynamics of microrotation field in the presence of hybrid nanoparticles. It is observed that rise in vortex viscosity causes a remarkable rise in microrotation velocity. However, opposite trend is noted for the case of macro-velocity. The vortex viscosity and spin gradient viscosity both have significant effects on temperature, and an increase in a vortex and spin gradient viscosities results in a significant decrease in temperature. It is also noted that impact of vortex and spin gradient viscosities on the temperature of a micropolar mixture containing only copper particles is less than the impact of the vortex and spin gradient viscosities on the temperature of a micropolar mixture containing Cu and Al2O3.

Extended Hill’s lemma for non-Cauchy continua based on a modified couple stress theory

Two extended versions of Hill’s lemma for non-Cauchy continua are provided using a modified couple stress theory, which includes a constraint relation between the displacement gradient and the micro-rotation, unlike the Cosserat (micro-polar) elasticity theory. The first version is used to obtain the classical elasticity tensor that relates the Cauchy (force) stress to the strain, while the second version is employed to determine the couple stress elasticity tensor that links the couple stress to the curvature. The kinematic (displacement- or strain-based) boundary conditions used in each version of the extended Hill’s lemma are then modified to accommodate composites with periodic microstructures by introducing periodic parts to the displacement and micro-rotation fields and reconstructing the Hill–Mandel condition. To illustrate the two newly proposed versions of the extended Hill’s lemma, homogenization analyses are performed for a two-phase composite using a meshfree radial point interpolation method incorporating the extended Hill’s lemma and the modified couple stress theory, which is newly developed in the current study.

Magnetohydrodynamics mixed convective flow driven through a static wedge including TiO2 nanomaterial with micropolar liquid: Similarity dual solutions via finite difference method

This research scrutinizes the influence of titanium oxide (TiO2) nanoparticle on electrically conducting micropolar liquid driven through wedge with mixed convection and thermal radiation. Buoyancy opposing flow and buoyancy assisting flow are taken into consideration. Similarity parameters are employed to transmute the governing partial differential equations into ordinary differential equations and then obtained the dual solutions through finite difference method. Impacts of ensuing parameters on liquid velocity, temperature distribution, and microrotation field are described and argued. Dual solutions are realized in buoyancy opposing flow, whereas in buoyancy assisting flow outcome is unique. Nanoliquid velocity tends to decline in the first solution and enhances in the second solution due to nanoparticle volume fraction, whereas microrotation profiles increases in both solutions. Temperature distribution increases in the first solution and decreases in the second solution due to φ. Due to micropolar parameter, the velocity and microrotation profiles decrease in both solutions, whilst the temperature of fluid behaves in opposite manner. Results also showed that separation of boundary layer can be controlled through micropolar parameter and nanoparticle volume fraction. In addition, support of present outcomes is arranged through benchmarking by previous well-known limiting conditions and pledged that a fabulous agreement with these results.

Analytical aspects in strain gradient theory for chiral Cosserat thermoelastic materials within three Green-Naghdi models

This work is motivated by the recent interest in using strain gradient theory to model the chiral behavior of elastic materials. In this paper, we derive a linear strain gradient theory for Cosserat thermoelastic materials according to the three models (types I, II and III) of Green-Naghdi theory. Models II and III permit propagation of thermal waves at finite speeds, while model I coincides with the classical Fourier’s law. The thermal field is influenced by the displacement and the microrotation fields and by some additional parameters that describe the chiral behavior. We prove the well-posedness for the three models and the asymptotic behavior for models I and III by the semigroup theory of linear operators.

G-space theory and weakened-weak form for micropolar media: Application to smoothed point interpolation methods

The concepts of G-space and weakened-weak form already available for the classic continuum model are extended to the case of the micropolar theory, in order to allow the use of smoothed point interpolation methods for the analysis of micropolar media. A new G-space, defined as a cartesian product of standard G-spaces, is introduced in order to represent both the displacement and the microrotation fields of a micropolar medium. Based on this G-space, a micropolar weakened-weak form is formulated, and the existence and uniqueness of its solution are proven.

A micropolar fluid model for hydrodynamics in the renal tubule

The creeping flow of an incompressible micropolar fluid in a small-diameter permeable tube is studied. The fluid absorption at the tube walls is taken as a function of wall permeability and the pressure gradient across the tube wall. Closed-form solutions of the governing equations for the pressure, velocity and micro-rotation fields are obtained. Expressions describing variations in the axial volume flow rate, mean pressure, wall leakage flux, wall shear stress and the fractional re-absorption are derived. Effects of the wall permeability coefficient K , the micropolar parameter m and the coupling number N on flow variables are graphically presented and implications of results are discussed briefly. The obtained solutions are applied to the flow problem in the proximal renal tubule and probable values of the wall permeability parameter and the mean pressure drop in the tubule are obtained. It is found that the mean pressure drop in the proximal tubule is higher for the micropolar fluid than for the Newtonian fluid. It is also found that the volume flow rate of the micropolar fluid decreases exponentially as a function of the axial distance. This is a biologically significant result which is well accepted and is reported in the literature by earlier researchers too. The derived solutions coincide well with the existing solutions for the Newtonian fluid flow in a permeable tube as a special case.

Unsteady micropolar fluid flow in a thin domain with Tresca fluid–solid interface law

We consider a micropolar fluid flow in a two-dimensional domain. We assume that the velocity field satisfies a non-linear slip boundary condition of friction type on a part of the boundary while the micro-rotation field satisfies non-homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of a solution. Then motivated by lubrication problems we assume that the thickness and the roughness of the domain are of order 0<ε<<1 and we study the asymptotic behaviour of the flow as ε tends to zero. By using the two-scale convergence technique we derive the limit problem which is totally decoupled for the limit velocity and pressure (v0,p0) on one hand and the limit micro-rotation Z0 on the other hand. Moreover we prove that v0, p0 and Z0 are uniquely determined via auxiliary well-posed problems.

The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments

The rotation ${\rm polar}(F) \in {\rm SO}(3)$ arises as the unique orthogonal factor of the right polar decomposition $F = {\rm polar}(F) \cdot U$ of a given invertible matrix $F \in {\rm GL}^+(3)$. In the context of nonlinear elasticity Grioli (1940) discovered a geometric variational characterization of ${\rm polar}(F)$ as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli's variational approach with weights (material parameters) $\mu > 0$ and $\mu_c \geq 0$ (Grioli: $\mu = \mu_c$). The energy subject to minimization coincides with the Cosserat shear-stretch contribution arising in any geometrically nonlinear, isotropic and quadratic Cosserat continuum model formulated in the deformation gradient field $F := \nabla\varphi: \Omega \to {\rm GL}^+(3)$ and the microrotation field $R: \Omega \to {\rm SO}(3)$. The corresponding set of non-classical energy-minimizing rotations $$ {\rm rpolar}^\pm_{\mu,\mu_c}(F) := \substack{{\rm argmin}\\ R\,\in\,{\rm SO(3)}} \Big\{ W_{\mu, \mu_c}(R\,;F) := \mu\, || {\rm sym}(R^TF - 1)||^2 + \mu_c\, ||{\rm skew}(R^TF - 1)||^2 \Big\} $$ represents a new relaxed-polar mechanism. Our goal is to motivate this mechanism by presenting it in a relevant setting. To this end, we explicitly construct a deformation mapping $\varphi_{\rm nano}$ which models an idealized nanoindentation and compare the corresponding optimal rotation patterns ${\rm rpolar}^\pm_{1,0}(F_{\rm nano})$ with experimentally obtained 3D-EBSD measurements of the disorientation angle of lattice rotations due to a nanoindentation in solid copper. We observe that the non-classical relaxed-polar mechanism can produce interesting counter-rotations. A possible link between Cosserat theory and finite multiplicative plasticity theory on small scales is also explored.

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part II: Non‐classical energy‐minimizing microrotations in 3D and their computational validation***

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field and the microrotation field , the shear–stretch energy is necessarily of the form urn:x-wiley:00442267:media:zamm201600030:zamm201600030-math-0003We aim at the derivation of closed form expressions for the minimizers of in SO(3), i.e., for the set of optimal Cosserat microrotations in dimension , as a function of . In a previous contribution (Part I), we have first shown that, for all , the full range of weights and can be reduced to either a classical or a non‐classical limit case. We have then derived the associated closed form expressions for the optimal planar rotations in SO(2) and proved their global optimality. In the present contribution (Part II), we characterize the non‐classical optimal rotations in dimension . After a lift of the minimization problem to the unit quaternions, the Euler–Lagrange equations can be symbolically solved by the computer algebra system Mathematica. Among the symbolic expressions for the critical points, we single out two candidates which we analyze and for which we can computationally validate their global optimality by Monte Carlo statistical sampling of SO(3). Geometrically, our proposed optimal Cosserat rotations act in the plane of maximal stretch. Our previously obtained explicit formulae for planar optimal Cosserat rotations in SO(2) reveal themselves as a simple special case. Further, we derive the associated reduced energy levels of the Cosserat shear–stretch energy and criteria for the existence of non‐classical optimal rotations.

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

In any geometrically nonlinear quadratic Cosserat‐micropolar extended continuum model formulated in the deformation gradient field and the microrotation field , the shear–stretch energy is necessarily of the form urn:x-wiley:00442267:media:zamm201500194:zamm201500194-math-0003where is the Lamé shear modulus and is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations as a function of and weights and . For , we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: and . In contrast to Grioli's theorem, we derive non‐classical minimizers for the parameter range in dimension . Currently, optimality results for are out of reach for us, but we contribute explicit representations for which we name and which arise for by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non‐classical optimal Cosserat rotations for simple planar shear.

A systematic approach to derive dynamic equations for homogeneous and functionally graded micropolar plates

This work considers a systematic derivation process to obtain hierarchies of dynamical equations for micropolar plates being either homogeneous or with a functionally graded (FG) material variation over the thickness. Based on the three dimensional micropolar continuum theory, a power series expansion technique of the displacement and micro-rotation fields in the thickness coordinate of the plate is adopted. The construction of the sets of plate equations is systematized by the introduction of recursion relations which relates higher order powers of displacement and micro-rotation terms with the lower order terms. This results in variationally consistent partial differential plate equations of motion and pertinent boundary conditions. Such plate equations can be constructed in a systematic fashion to any desired truncation order, where each equation order is hyperbolic and asymptotically correct. The resulting lowest order flexural plate equation is seen to be of a generalized Mindlin type. The numerical results illustrate that the present approach may render accurate solutions of benchmark type for both homogeneous and functionally graded micropolar plates provided higher order truncations are used. Moreover, low order truncations render new sets of plate equations that can act as engineering plate equations, e.g. of a generalized Mindlin type.

Chiral effects in piezoelectricity

This paper is concerned with the linear theory of piezoelectricity for Cosserat continua. The deformation of homogeneous and isotropic chiral materials is investigated. First, a counterpart of the Boussinesq–Somigliana–Galerkin solution in the classical elastostatics is presented. Then, the fundamental solutions in the stationary theory are established. In contrast with the case of achiral solids, in the theory of isotropic chiral materials the displacement and microrotation fields are coupled with the electric field.

Strain gradient theory of chiral Cosserat thermoelasticity without energy dissipation

In this paper, we use the Green–Naghdi theory of thermomechanics of continua to derive a linear strain gradient theory of Cosserat thermoelastic bodies. The theory is capable of predicting a finite speed of heat propagation and leads to a symmetric conductivity tensor. The constitutive equations for isotropic chiral thermoelastic materials are presented. In this case, in contrast with the classical Cosserat thermoelasticity, a thermal field produces a microrotation of the particles. The thermal field is influenced by the displacement and microrotation fields even in the equilibrium theory. Existence and uniqueness results are established. The theory is used to study the effects of a concentrated heat source in an unbounded homogeneous and isotropic chiral solid.

Second law violations, continuum mechanics, and permeability

The violations of the second law are relevant as the length and/or time scales become very small. The second law then needs to be replaced by the fluctuation theorem and mathematically, the irreversible entropy is a submartingale. First, we discuss the consequences of these results for the axioms of continuum mechanics, arguing in favor of a framework relying on stochastic functionals of energy and entropy. We next determine a Lyapunov function for diffusion-type problems governed by stochastic rather than deterministic functionals of internal energy and entropy, where the random field coefficients of diffusion are not required to satisfy the positive definiteness everywhere. Next, a formulation of micropolar fluid mechanics is developed, accounting for the lack of symmetry of stress tensor on molecular scales. This framework is then applied to employed to show that spontaneous random fluctuations of the microrotation field will arise in Couette—and Poiseuille-type flows in the absence of random (turbulence-like) fluctuations of the classical velocity field. Finally, while the permeability is classically modeled by the Darcy law or its modifications, besides considering the violations of the second law, one also needs to account for the spatial randomness of the channel network, implying a modification of the hierarchy of scale-dependent bounds on the macroscopic property of the network.

Radiation and Mass Transfer Effects on MHD Free Convection Flow of a Micropolar Fluid past a Stretching Surface Embedded in a Non-Darcian Porous Medium with Heat Generation

A comprehensive study of mass transfer and thermal radiation on a steady two-dimensional laminar flow of a viscous incompressible electrically conducting micropolar fluid past a stretching surface embedded in a non-Darcian porous medium in the presence of heat generation is analyzed numerically. The governing equations of momentum, angular momentum, energy, and species equations are solved numerically using Runge-Kutta fourth order method with the shooting technique. The effects of various parameters on the velocity, microrotation, temperature and concentration field as well as skin friction coefficient, Nusselt number and Sherwood number are shown graphically and tabulated. It is observed that the micropolar fluid helps in the reduction of drag forces and also acts as a cooling agent.

Prediction of microstructure in a Cosserat continuum using relaxed energies

AbstractWe develop a nonlinear incompressible multiphase material model in a Cosserat continuum with microstructure. The free energy of the material is enriched with an interaction potential taking into account the intergranular kinematics at the continuum scale. As a result the total energy becomes non‐convex, thus giving rise to the development of microstructural phases. To guarantee the existence of minimizers an exact quasi‐convex envelope of the corresponding energy functional is derived. As a result a three phase material energy appears, among them two of the phases are with microstructure in the translational motion (displacment field) and micromotion (microrotation field), whereas the third phase is without internal structure. The corresponding relaxed energy is then used for finding the minimizers of the two field minimization problem corresponding to a Cosserat continuum. Results from a numerical example predicting the development of microstructure in the material are presented. (© 2012 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)