Free convection boundary layer flow of a micropolar fluid past slender bodies

The transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past vertical cylinders are investigated using the theory of micropolar fluids formulated by Eringen. The governing equations for momentum, angular momentum and energy have been solved numerically. Missing values of the velocity, angular velocity and thermal functions are tabulated for a wide range of the material parameters, transverse curvature parameter and Prandtl number of the fluid. A comparison has been made with the corresponding results for Newtonian fluids. Micropolar fluids display drag reduction and reduced surface heat transfer rate as compared with Newtonian fluids.     

Numerical study for micropolar flow over a stretching sheet

The paper presents a numerical study of the steady flow of a micropolar fluid flow from a stretching sheet. Approximate analytical solution of high nonlinear momentum, angular momentum and confluent hypergeometric similarity solution of the heat transfer equation are obtained for a particular case when the vortex viscosity is neglected. Accuracy of the analytical solution is verified by numerical solutions obtained by employing finite element and Chebyshev finite difference methods. The good agreement between the numerical results of both methods, together with an excellent agreement with the analytical solutions for the special case, ensures the reliability of the obtained results. The velocity, microrotation and temperature functions are shown graphically and the effect of the permeability parameter is studied.     

Mixed convection boundary layer flow of a micropolar fluid along vertical cylinders and needles

An analysis is presented to investigate the effects of buoyancy and curvature on convection along vertical cylinders and needles placed in a micropolar fluid. The governing equations for momentum, angular momentum and energy are solved numerically by finite difference scheme. The heat transfer results are presented for a range of values of the buoyancy parameters, the curvature parameter and the material parameters of the fluid. The effect of the microrotation boundary conditions on heat transfer is discussed.     

The Lorentz reciprocal theorem for micropolar fluids

A generalization of the Lorentz reciprocal theorem is developed for the creeping flow of micropolar fluids in which the continuum equations involve both the velocity and the internal spin vector fields. In this case, the stress tensor is generally not symmetric and conservation laws for both linear and angular momentum are needed in order to describe the dynamics of the fluid continuum. This necessitates the introduction of constitutive equations for the antisymmetric part of the stress tensor and the so-called couple-stress in the medium as well. The reciprocal theorem, derived herein in the limit of negligible inertia and without external body forces and couples, provides a general integral relationship between the velocity, spin, stress and couple-stress fields of two otherwise unrelated micropolar flow fields occurring in the same fluid domain.     

Theory of micropolar gyroelastic continua

This paper is devoted to the dynamic modeling of micropolar gyroelastic continua and explores some of the modeling and analysis issues related to them. It can be considered as an extension of the previous studies on equivalent continuum modeling of truss structures with or without angular momentum devices. Assuming unrestricted or large attitude changes for the axes of the gyros and utilizing the micropolar theory of elasticity, the energy expressions and equations of motion for undamped micropolar gyroelastic continua are derived. Whereas the micropolar gyroelastic continuum model with extra coefficients and degrees of freedom is primarily developed to account for the asymmetric stress–strain analysis in the gyroelastic continua, it also proves to be beneficial for a more comprehensive representation of the actual gyroelastic structure. The dynamic equations of the general gyroelastic continua are reduced to the case of one-dimensional gyroelastic beams. Simplified micropolar beam torsion and bending theories are used to derive the governing dynamic equations of micropolar gyroelastic beams from Hamilton’s principle. A finite element model corresponding to the micropolar gyrobeams is built in MATLAB\({^{\circledR}}\) and is used in numerical examples to study the spectral and modal behavior of simply supported micropolar gyroelastic beams.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Flow and heat transfer in a micropolar fluid past a flat plate with suction and heat sources

The laminar boundary layer flow of a micropolar fluid past a flat plate subject to uniform suction has been examined. The heat transfer study has been made in the presence of both constant as well as temperature dependent heat sources. The governing equations of momentum, first stress momentum and energy have been solved using numerical integration and Gauss-Seidel iterative procedure. For a particular value of suction parameter λ, as compared to the Newtonian fluid, the velocity decreases on increasing the value of micropolar parameter R. The skin friction and the Nusselt number have been calculated and presented in tables.     

A spin-vorticity relation for unidirectional plane flows of micropolar fluids

The field equations of micropolar fluid theory are applied to consider transient unidirectional plane flows. The relative angular velocity is introduced and maximum and minimum principles of parabolic partial differential equations are utilized to establish a spin-vorticity relation which is valid for very general boundary and initial conditions on spin.     

Mixed convection of micropolar fluids along a vertical wavy surface

Summary This paper presents numerical results for the steady-state mixed convection in micropolar fluids along a vertical wavy surface. The problem has been formulated by a simple trnasposition theorem, and the spline alternating-direction implicit method has been applied to solve the governing momentum, angular momentum and energy equations. The influence of the micropolar parameters (R and ), the amplitude-wave length ratio and the Gr/Re² number on the skin-friction coefficient and Nusselt number have been studied. Results demonstrate that the skin friction coefficient and local Nusselt number consist of a mixture of two harmonics in micropolar fluids and in Newtonian fluids. As the vortex viscosity parameter (R) increases, the heat transfer rate decreases but the skin friction increases. In addition, when the spin gradient viscosity parameter () increases, the heat transfer rate and the skin friction decreases. However, the heat transfer rate of a micropolar fluid is smaller than a Newtonian fluid, but the skin friction of a micropolar fluid is larger than a Newtonian fluid under all circumstances.     

Flow and heat transfer of a micropolar fluid in an axisymmetric stagnation flow on a cylinder with variable properties and suction (numerical study)

Summary. In this paper, an analysis is presented to study the effects of variable properties, density, viscosity and thermal conductivity of a micropolar fluid flow and heat transfer in an axisymmetric stagnation flow on a horizontal cylinder with suction, numerically. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. However, the fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The similarity solution is used to transform the problem under consideration into a boundary value problem of nonlinear coupled ordinary differential equations which are solved numerically by using the Chebyshev finite difference method (ChFD). Numerical results are carried out for various values of the dimensionless parameters of the problem. The numerical results show variable density, variable viscosity, variable thermal conductivity and micropolar parameters, which have significant influences on the azimuthal and the angular velocities and temperature profiles, shear stress, couple stress and the Nusselt number. The numerical results have demonstrated that with increasing temperature ratio parameter the azimuthal velocity decreases. With increasing variable viscosity parameter the temperature increases, whereas the azimuthal and the angular velocities decrease. Also, the azimuthal and the angular velocities increase and the temperature decreases as the variable conductivity parameter increases. Finally, the pressure increases as the suction parameter increases.     

Relaxation of Anisotropic Systems of Quasi-particles in Superfluid Helium

We investigate the relaxation of boundless and anisotropic quasiparticle systems, in liquid ⁴He, with given initial values of momentum and energy densities. The evolution is governed by the dispersion curve which determines the interactions between quasi-particles. The first stage of the relaxation is the rapid creation of a quasistationary low-energy phonon system. This system transforms to another quasistationary system which contains high-energy phonons as well as low energy-phonons. Finally the system evolves to a stationary system of phonons and rotons. The temperature T and the drift velocity u are found for the closed quasi-particle system during its relaxation. This allows us to study the quasi-particle energy and the angular distribution of momentum, and the thermodynamic functions of the quasi-particles system, at each of stage of its evolution.      

Theory and FE-analysis for structures with large deformation under magnetic loading

We introduce and discuss a reduced micropolar continuum theory to simulate structures with large deformations under magnetic loading. Three numerical examples show the motivation of this model and its use in practical applications. The question of how to choose the micropolar material parameters is addressed. We use that a finite strain micropolar model would reduce to classical elasticity in the absence of curvature effects and body couples and for certain parameter ranges. This gives us information about a proper choice of material parameters. Thus, we introduce in fact a nearly classical model, but with the feature to cover large deformations and non-classical types of loading. As in shell theories, our continuum theory treats angular momentum as an explicit complementary principle. Thus, net couples—the typical loading of magnetized bodies in a magnetic field—can be modelled. Note that, in this case, the possibility for nonsymmetric Cauchy stresses is required for equilibrium, unlike classical shell theories. Micropolar theories are not commonly used, by comparison to the Boltzmann continuum. One reason may be that micropolar theories often require greater modelling effort without significant advantage. However, the simplicity of introducing physical effects like magnetic loading compensates those efforts.     

Finite element study of mixed convection micropolar flow in a vertical circular pipe with variable surface conditions

In the present paper the problem of fully developed mixed convection flow of a micropolar fluid with heat sources, in a vertical circular pipe has been studied. The governing non-linear coupled differential equations are solved by using the finite element technique. The effect of the micropolar parameter, heat source parameter, surface parameter and dissipation parameter on the velocity, micro-rotation and temperature functions have been discussed. The heat sources increase the velocity and temperature in the pipe while the heat sinks decrease them. The micropolar fluid thus behaves as a coolant and such a type of flow has useful applications in combustion chambers, exhaust nozzles of porous walled flow reactors, and the design of chemical processing equipment.     

Vortex configurations in a freely rotating superfluid drop

The equilibrium configurations of a single vortex in a freely rotating superfluid drop have been determined. The potential barrier to vortex nucleation has also been examined. A freely rotating drop is constrained to rotate at a fixed angular momentum, as opposed to fluid in a rotating vessel, where the vessel is usually constrained to rotate at fixed angular velocity. Vortex configurations that minimize the free energy have been found for a range of angular momenta. These configurations are not necessarily on the axis of rotation as is the case for a vessel rotating at a constant angular velocity. The nucleation of a vortex is shown to be opposed by a barrier in the free energy. This barrier narrows and decreases in height as the angular momentum of the drop is increased.     

Finite element analysis of combined heat and mass transfer in hydromagnetic micropolar flow along a stretching sheet

This paper presents a finite element solution of the problem of heat and mass transfer in a hydromagnetic flow of a micropolar fluid past a stretching sheet. The transformed equations for the flow regime are solved numerically by using finite element method. The effect of important parameters namely magnetic field parameter, material parameter, Eckert number and Schmidt number over velocity, microrotation, temperature and concentration functions has been studied. It has been observed that the magnetic field parameter has the effect of reducing the velocity and increasing the microrotation, temperature and concentration while the micropolar parameter has the opposite effect on these functions except temperature function. Temperature increases with the increase in Eckert number and concentration decreases with the increase in Schmidt number.     

Micropolar boundary layer flow at a stagnation point on a moving wall

The theory of micropolar fluids due to Eringen is used to formulate a set of boundary layer equations for 2-dimensional flow of an incompressible, constant density micropolar fluid at a stagnation point on a moving wall. The governing boundary layer equations are solved numerically. The development of the velocity of distribution has been illustrated for several positive and negative values of the wall velocity. A discussion is provided for the dependence of the important flow characteristics on the material parameters.     

Hybrid state‐space time integration in a rotating frame of reference

A time integration algorithm is developed for the equations of motion of a flexible body in a rotating frame of reference. The equations are formulated in a hybrid state‐space, formed by the local displacement components and the global velocity components. In the spatial discretization the local displacements and the global velocities are represented by the same shape functions. This leads to a simple generalization of the corresponding equations of motion in a stationary frame in which all inertial effects are represented via the classic global mass matrix. The formulation introduces two gyroscopic terms, while the centrifugal forces are represented implicitly via the hybrid state‐space format. An angular momentum and energy conserving algorithm is developed, in which the angular velocity of the frame is represented by its mean value. A consistent algorithmic damping scheme is identified by applying the conservative algorithm to a decaying response, which is rendered stationary by an increasing exponential factor that compensates the decay. The algorithmic damping is implemented by introducing forward weighting of the mean values appearing in the algorithm. Numerical examples illustrate the simplicity and accuracy of the algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet

This paper presents a finite element solution for the mixed convection micropolar flow driven by a porous stretching sheet with uniform suction. The governing partial differential equations are solved numerically by the using finite element method and the results have been compared with those obtained by using the quasi-linearization method. The effect of surface conditions on the velocity, microrotation as well as for temperature functions has been studied. It is noticed that the micropolar fluids help in the reduction of drag forces and also act as a cooling agent.     

Thermal radiation effects on magnetohydrodynamic mixed convection flow of a micropolar fluid past a continuously moving semi-infinite plate for high temperature differences

Summary An analysis is presented to study the effect of radiation on magnetohydrodynamic mixed convective steady laminar boundary layer flow of an optically thick electrically conducting viscous micropolar fluid past a moving semi-infinite vertical plate for high temperature differences. A uniform magnetic field is applied perpendicular to the moving plate. The density of the micropolar fluid is assumed to reduce exponentially with temperature. The usual Boussinesq approximation is neglected because of the high temperature differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting governing equations are transformed using a similarity transformation and then solved numerically by applying an efficient technique. The effects of radiation parameter R, magnetic parameter M, couple parameter Δ and density/temperature parameter n on the velocity, angular velocity and temperature profiles as well as the local skin friction coefficient, wall couple stress and the local Nusselt number are presented graphically and in tabular form.