Laminar flow of micropolar fluid in rectangular microchannels

Compared with the classic flow on macroscale, flows in microchannels have some new phenomena such as the friction increase and the flow rate reduction. Papautsky and co-workers explained these phenomena by using a micropolar fluid model where the effects of micro-rotation of fluid molecules were taken into account. But both the curl of velocity vector and the curl of micro-rotation gyration vector were given incorrectly in the Cartesian coordinates and then the micro-rotation gyration vector had only one component in the z-direction. Besides, the gradient term of the divergence of micro-rotation gyration vector was missed improperly in the angular moment equation. In this paper, the governing equations for laminar flows of micropolar fluid in rectangular microchannels are reconstructed. The numerical results of velocity profiles and micro-rotation gyrations are obtained by a procedure based on the Chebyshev collocation method. The micropolar effects on velocity and micro-rotation gyration are discussed in detail.The project was supported by the National Natural Science Foundation of China (10472054). The English text was polished by Boyi Wang.     

Mixed convection flow of a micropolar fluid over a stretching sheet

The mixed convective flow of a steady, incompressible micropolar fluid over a stretching sheet has been studied. This situation may arise in polymer technology involving the stretching of plastics sheets. The resulting system of non-linear ordinary coupled differential equations has been solved by the finite element method, using the variational Ritz model. Numerical results obtained for velocity, microrotation and temperature distributions are shown graphically. It was found that an increase in the micropolar parameter leads to a faster rate of cooling of the sheet. Also the velocity increases with an increase in micropolar effects. Microrotation effects are much smaller for the no-spin boundary condition as compared to the other boundary condition which assumes that the gyration vector is identical to the angular velocity of the fluid. Received on 9 February 1998     

Lubrication theory for micropolar fluids and its application to a journal bearing with finite length

In this paper, the field equation of micropolar fluid with general lubrication theory assumptions is simplified into two systems of coupled ordinary differential equation. The analytical solutions of velocity and microrotation velocity are obtained. Micropolar fluid lubrication Reynolds equation is deduced. By means of numerical method, the characteristics of a finitely long journal bearing under various dynamic parameters, geometrical parameters and micropolar parameters are shown in curve form. These characteristics are pressure distribution, load capacity, coefficient of flow flux and coefficient of friction. Practical value of micropolar effects is shown, so micropolar fluid theory further closes to engineering application.     

Dual solutions of time-dependent magnetohydrodynamic stagnation point boundary layer micropolar nanofluid flow over shrinking/stretching surface

Time-dependent, two-dimensional(2 D) magnetohydrodynamic(MHD)micropolar nanomaterial flow over a shrinking/stretching surface near the stagnant point is considered. Mass and heat transfer characteristics are incorporated in the problem. A model of the partial differential expressions is altered into the forms of the ordinary differential equations via similarity transformations. The obtained equations are numerically solved by a shooting scheme in the MAPLE software. Dual solutions are observed at different values of the specified physical parameters. The stability of first and second solutions is examined through the stability analysis process. This analysis interprets that the first solution is stabilized and physically feasible while the second one is un-stable and not feasible. Furthermore, the natures of various physical factors on the drag force, skin-friction factor, and rate of mass and heat transfer are determined and interpreted. The micropolar nanofluid velocity declines with a rise in the suction and magnetic parameters, whereas it increases by increasing the unsteadiness parameter.The temperature of the micropolar nanofluid rises with increase in the Brownian motion,radiation, thermophoresis, unsteady and magnetic parameters, but it decreases against an increment in the thermal slip constraint and Prandtl number. The concentration of nanoparticles reduces against the augmented Schmidt number and Brownian movement values but rises for incremented thermophoresis parameter values.     

Melting heat transfer effects on stagnation point flow of micropolar fluid saturated in porous medium with internal heat generation （absorption）

The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.     

Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.     

Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium

An analysis is presented for the problem of free convection with mass transfer flow for a micropolar fluid via a porous medium bounded by a semi-infinite vertical porous plate in the presence of a transverse magnetic field. The plate moves with constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which absorbs the micropolar fluid with a suction velocity varying with time. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient, the couple stress coefficient, the rate of the heat and mass transfers at the wall are prepared with various values of fluid properties and flow conditions.     

Slip effects on heat and mass transfer in MHD micropolar fluid flow over an inclined plate with thermal radiation and chemical reaction

The influence of partial slip, thermal radiation, chemical reaction and temperature‐dependent fluid properties on heat and mass transfer in hydro‐magnetic micropolar fluid flow over an inclined permeable plate with constant heat flux and non‐uniform heat source/sink is studied. The transverse magnetic field is assumed as a function of the distance from the origin. Also it is assumed that the fluid viscosity and the thermal conductivity vary as an inverse function and linear function of temperature, respectively. With the use of the similarity transformation, the governing system of non‐linear partial differential equations are transformed into non‐linear ordinary differential equations and are solved numerically using symbolic software MATHEMATICA 7.0 (Wolfram Research, Champaign, IL). The numerical values obtained for the velocity, microrotation, temperature, species concentration, skin friction coefficient and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of various physical parameters on the flow and heat transfer characteristics are discussed.Copyright © 2011 John Wiley & Sons, Ltd.     

化学反应对竖直平板边界磁流体动力学微极流体滑流的影响

Heat and mass transfer effects on the unsteady flow of a micropolar fluid through a porous medium bounded by a semi-infinite vertical plate in a slip-flow regime are studied taking into account a homogeneous chemical reaction of the first order. A uniform magnetic field acts perpendicular to the porous surface absorb micropolar fluid with a suction velocity varying with time. The free stream velocity follows an exponentially increasing or decreasing small perturbation law. Using the approximate method, the expressions for the velocity microrotation, temperature, and concentration are obtained. Futher, the results of the skin friction coefficient, the couple stress coefficient, and the rate of heat and mass transfer at the wall are presented with various values of fluid properties and flow conditions.     

MHD non-Newtonian micropolar fluid flow and heat transfer in channel with stretching walls

A study is presented for magnetohydrodynamics （MHD） flow and heat transfer characteristics of a viscous incompressible electrically conducting micropolar fluid in a channel with stretching walls. The micropolar model introduced by Eringen is used to describe the working fluid. The transformed self similar ordinary differential equations together with the associated boundary conditions are solved numerically by an algorithm based on quasi-linearization and multilevel discretization. The effects of some physical parameters on the flow and heat transfer are discussed and presented through tables and graphs. The present investigations may be beneficial in the flow and thermal control of polymeric processing.     

Thermal radiation effect on flow and heat transfer of unsteady MHD micropolar fluid over vertical heated nonisothermal stretching surface using group analysis

The aim of this paper is to study the thermal radiation effects on the flow and heat transfer of an unsteady magnetohydrodynamic (MHD) micropolar fluid over a vertical heated nonisothermal stretching surface in the presence of a strong nonuniform magnetic field. The symmetries of the governing partial differential equations are de- termined by the two-parameter group method. One of the resulting systems of reduced nonlinear ordinary differential equations are solved numerically by the Chebyshev spec- tral method. The effects of various parameters on the velocity, the angular velocity, and the temperature profiles as well as the skin-friction coefficient, the wall couple stress co- efficient, and the Nusselt number are studied.     

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

Using the theory of micropolar fluids developed by Eringen, the transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past slender vertical cones are investigated. The case of constant surface heat flux is considered in this paper. Using perturbation techniques, the governing equations for momentum, angular momentum and energy have been solved numerically. Graphical representations for the velocity, angular velocity and thermal functions are presented for various physical and fluid property parameters.     

Effect of melting on an MHD micropolar fluid flow toward a shrinking sheet with thermal radiation

The effect of melting on a steady boundary layer stagnation-point flow and heat transfer of an electrically conducting micropolar fluid toward a horizontal shrinking sheet in the presence of a uniform transverse magnetic field and thermal radiation is studied. A similarity transformation technique is adopted to obtain self-similar ordinary differential equations, which are solved numerically. The present results are found to be in good agreement with previously published data. Numerical results for the dimensionless velocity and temperature profiles, as well as for the skin friction and the rate of heat transfer are obtained.     

Axisymmetric magnetohydrodynamic flow of micropolar fluid between unsteady stretching surfaces

This investigation examines the time dependent magnetohydrodynamic (MHD) flow problem of a micropolar fluid between two radially stretching sheets. Both strong and weak concentrations of microelements are taken into account. Suitable transformations are employed for the conversion of partial differential equations into ordinary differential equations. Solutions to the resulting problems are developed with a homotopy analysis method (HAM). The angular velocity, skin friction coefficient, and wall couple stress coefficient are illustrated for various parameters.     

MHD free-convective flow of micropolar and Newtonian fluids through porous medium in a vertical channel

We have studied the fully-developed free-convective flow of an electrically conducting fluid in a vertical channel occupied by porous medium under the influence of transverse magnetic field. The internal prefecture of the channel is divided into two regions; one region filled with micropolar fluid and the other region with a Newtonian fluid or both the regions filled by Newtonian fluids. Analytical solutions of the governing equations of fluid flow are found to be in excellent agreement with analytical prediction. Analytical results for the details of the velocity, micro-rotation velocity and temperature fields are shown through graphs for various values of physical parameters. It is noticed that Newtonian fluids prop up the linear velocity of the fluid in contrast to micropolar fluid. Also the skin friction coefficient at both the walls is derived and its numerical values are offered through tables.     

MHD flow and heat transfer of micropolar fluid between two porous disks

A numerical study is carried out for the axisymmetric steady laminar incompressible flow of an electrically conducting micropolar fluid between two infinite parallel porous disks with the constant uniform injection through the surface of the disks. The fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form through von Karman’s similarity transformation. An algorithm based on a finite difference scheme is used to solve the reduced coupled ordinary differential equations under associated boundary conditions. The effects of the Reynolds number, the magnetic parameter, the micropolar parameter, and the Prandtl number on the flow velocity and temperature distributions are discussed. The results agree well with those of the previously published work for special cases. The investigation predicts that the heat transfer rate at the surfaces of the disks increases with the increases in the Reynolds number, the magnetic parameter, and the Prandtl number. The shear stresses decrease with the increase in the injection while increase with the increase in the applied magnetic field. The shear stress factor is lower for micropolar fluids than for Newtonian fluids, which may be beneficial in the flow and thermal control in the polymeric processing.     

A revised Cattaneo-Christov micropolar viscoelastic nanofluid model with combined porosity and magnetic effects

The dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications. The current predominately predictive modeling deals with the flow of the viscoelastic micropolar fluid in the presence of nanoparticles. A progressive amendment in the heat and concentration equations is made by exploiting the Cattaneo-Christov(C-C) heat and mass flux expressions. Besides, the thermal radiation effects are contributed in the energy equation and aspect of the radiation parameter, and the Prandtl number is specified by the one-parameter approach.The formulated expressions are converted to the dimensionless forms by relevant similarity functions. The analytical solutions to these expressions have been erected by the homotopy analysis method. The variations in physical quantities, including the velocity,the temperature, the effective local Nusselt number, the concentration of nanoparticles,and the local Sherwood number, have been observed under the influence of emerging parameters. The results have shown good accuracy compared with those of the existing literature.     

Rotational flow of Oldroyd-B nanofluid subject to Cattaneo-Christov double diffusion theory

A nanofluid is composed of a base fluid component and nanoparticles, in which the nanoparticles are dispersed in the base fluid. The addition of nanoparticles into a base fluid can remarkably improve the thermal conductivity of the nanofluid, and such an increment of thermal conductivity can play an important role in improving the heat transfer rate of the base fluid. Further, the dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications. The present predominately predictive modeling studies the flow of the viscoelastic Oldroyd-B fluid over a rotating disk in the presence of nanoparticles. A progressive amendment in the heat and concentration equations is made by exploiting the Cattaneo-Christov heat and mass flux expressions. The characteristic of the Lorentz force due to the magnetic field applied normal to the disk is studied. The Buongiorno model together with the Cattaneo-Christov theory is implemented in the Oldroyd-B nanofluid flow to investigate the heat and mass transport mechanism. This theory predicts the characteristics of the fluid thermal and solutal relaxation time on the boundary layer flow. The von K′arm′an similarity functions are utilized to convert the partial differential equations(PDEs) into ordinary differential equations(ODEs). A homotopic approach for obtaining the analytical solutions to the governing nonlinear problem is carried out. The graphical results are obtained for the velocity field, temperature, and concentration distributions. Comparisons are made for a limiting case between the numerical and analytical solutions, and the results are found in good agreement. The results reveal that the thermal and solutal relaxation time parameters diminish the temperature and concentration distributions, respectively. The axial flow decreases in the downward direction for higher values of the retardation time parameter. The impact of the thermophoresis parameter boosts the temperature distribution.     

A transient natural convection of micropolar fluids over a vertical cylinder

A transient free convective boundary layer flow of micropolar fluids past a semi-infinite cylinder is analysed in the present study. The transformed dimensionless governing equations for the flow, microrotation and heat transfer are solved by using the implicit scheme. For the validation of the current numerical method heat transfer results for a Newtonian fluid case where the vortex viscosity is zero are compared with those available in the existing literature, and an excellent agreement is obtained. The obtained results concerning velocity, microrotation and temperature across the boundary layer are illustrated graphically for different values of various parameters and the dependence of the flow and temperature fields on these parameters is discussed. An increase in the vortex viscosity tends to increase the magnitude of microrotation and thus decreases the peak velocity of fluid flow. An increase in the vortex viscosity in micropolar fluids is shown to decrease the heat transfer rate.     

Micropolar fluid flow over a shrinking sheet

An analysis is carried out for the steady two-dimensional flow of a micropolar fluid over a shrinking sheet in its own plane. The shrinking velocity is assumed to vary linearly with the distance from a fixed point on the sheet. The features of the flow and heat transfer characteristics are analyzed and discussed. It is found that the solution exists only if adequate suction through the permeable sheet is introduced. Moreover, stronger suction is necessary for the solution to exist for a micropolar fluid compared to a classical Newtonian fluid. Dual solutions are obtained for certain suction and material parameters.