Numerical Solution for Thermal Elastohydrodynamic Lubrication of Line Contact with Couple Stress Fluid as Lubricant

In this paper,the detailed analysis of the influence of thermal and non-Newtonian aspects of lubricant(couple stress fluid)on EHL line contact as a function of slide-roll ratio is presented.The novel low complexity FAS(full approximation scheme),of the multigrid scheme,with Jacobi dipole and Gauss Seidel relaxation is used for the solution of coupled equations viz.modified Reynolds equation,film thickness equation and energy equation satisfying appropriate boundary conditions.The analysis reveals the combined influence of non-Newtonian,thermal and slide-roll ratio(of bearing moving with different speeds)on pressure,film thickness and pressure spike covering a wide range of physical parameters of interest.Results show that pressure spike is strongly influenced by thermal,slide-roll ratio and non-Newtonian character of lubricant with negligible effect on the overall pressure distribution.Also,the minimum film thickness is slightly altered and it increases with the increase in the couple stress parameter.These findings confirm the importance of non-Newtonian and thermal effects in the study of EHL.

Stagnation-point flow of couple stress fluid with melting heat transfer

Melting heat transfer in the boundary layer flow of a couple stress fluid over a stretching surface is investigated. The developed differential equations are solved for homotopic solutions. It is observed that the velocity and the boundary layer thickness are decreasing functions of the couple stress fluid parameter. However, the temperature and surface heat transfer increase when the values of the couple stress fluid parameter increase. The velocity and temperature fields increase with an increase in the melting process of the stretching sheet.

Peristaltic flow of couple stress fluid through uniform porous medium

Investigation concerning peristaltic motion of couple stress fluid is made. An incompressible couple stress fluid occupies the porous medium. Mathematical anal- ysis is presented through large wavelength and low Reynolds number. Exact analytical expressions of axial velocity, volume flow rate, pressure gradient, and stream function are calculated as a function of couple stress parameter. The essential feature of the analysis is a full description of influence of couple stress parameter and permeability parameter on the pressure, frictional force, mechanical efficiency, and trapping.

Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder

The unsteady natural convective couple stress fluid flow over a semi-infinite vertical cylinder is analyzed for the homogeneous first-order chemical reaction effect. The couple stress fluid flow model introduces the length dependent effect based on the material constant and dynamic viscosity. Also, it introduces the biharmonic operator in the Navier-Stokes equations, which is absent in the case of Newtonian fluids. The solution to the time-dependent non-linear and coupled governing equations is carried out with an unconditionally stable Crank-Nicolson type of numerical schemes. Numerical results for the transient flow variables, the average wall shear stress, the Nusselt number, and the Sherwood number are shown graphically for both generative and destructive reactions. The time to reach the temporal maximum increases as the reaction constant K increases. The average values of the wall shear stress and the heat transfer rate decrease as K increases, while increase with the increase in the Sherwood number.

Natural convection flow of a couple stress fluid between two vertical parallel plates with Hall and ion-slip effects

The Hall and ion-slip effects on fully developed electrically conducting couple stress fluid flow between vertical parallel plates in the presence of a temperature dependent heat source are investigated. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations. The resulting equations are then solved using the homotopy analysis method （HAM）. The effects of the magnetic parameter, Hall parameter, ion-slip parameter and couple stress fluid parameter on velocity and temperature are discussed and shown graphically

Exact solutions for different vorticity functions of couple stress fluids

In this paper, inverse solutions are obtained for the class of 2D steady incompressible couple stress fluid flows. This class consists of flows for which the vorticity distribution is given by ▽2ψ=ψ+f(x,y). The solutions are obtained by applying the inverse method, which makes certain hypotheses regarding the form of the velocity field and pressure but without making any regarding the boundaries of the domain occupied by the fluid. Inverse solutions are derived for three different forms of f(x,y).

NONLINEAR NUMERICAL ANALYSIS OF A FLEXIBLE ROTOR EQUIPPED WITH SQUEEZE COUPLE STRESS FLUID FILM JOURNAL BEARINGS

This study performs a dynamic analysis of a rotor supported by two squeeze couple stress fluid film journal bearings with nonlinear suspension. The numerical results show that the stability of the system varies with the non-dimensional speed ratios and the dimensionless parameter l＊. It is found that the system is more stable with higher dimensionless parameter l＊. Thus it can conclude that the rotor-bearing system lubricated with the couple stress fluid is more stable than that with the conventional Newtonian fluid. The modeling results thus obtained by using the method proposed in this paper can be used to predict the stability of the rotor-bearing system and the undesirable behavior of the rotor and bearing center can be avoided.

Flow of couple stress fluid with variable thermal conductivity

The steady flow and heat transfer of a couple stress fluid due to an inclined stretching cylinder are analyzed. The thermal conductivity is assumed to be temperature dependent. The governing equations for the flow and heat transfer are transformed into ordinary differential equations. Series solutions of the resulting problem are computed. The effects of various interested parameters, e.g., the couple stress parameter, the angle of inclination, the mixed convection parameter, the Prandtl number, the Reynolds number, the radiation parameter, and the variable thermal conductivity parameter, are illustrated. The skin friction coefficient and the local Nusselt number are computed and analyzed. It is observed that the heat transfer rate at the surface increases while the velocity and the shear stress decrease when the couple stress parameter and the Reynolds number increase. The temperature increases when the Reynolds number increases.

The Effects of Post-Stenotic Dilatations on the Flow of Couple Stress Fluid through Stenosed Arteries

The flow of incompressible couple stress fluid in a circular tube with stenosis and dilatations has been investigated. The stenosis was assumed to be axially symmetric and mild. The flow equations have been linearized and the expressions for the resistance to the flow, velocity, pressure drop, wall shear stress have been derived. The effects of various parameters on these flow variables have been investigated. It is found that the resistance to the flow and pressure drop increase with height of the stenosis and decrease with post stenotic dilatation. Pressure drop decreases with couple stress fluid parameter for both stenosis and post stenotic dilatation. Further, the wall shear stress increases with height of the stenosis and couple stress parameter but decreases with post stenotic dilatation and couple stress fluid parameter.

Couple stress and Darcy Forchheimer hybrid nanofluid flow on a vertical plate by means of double diffusion Cattaneo-Christov analysis

A three-dimensional Darcy Forchheimer mixed convective flow of a couple stress hybrid nanofluid flow through a vertical plate by means of the double diffusion Cattaneo-Christov model is presented in this study.The influence of highorder velocity slip flow,as well as a passive and active control,is also considered.The motive of the research is to develop a computational model,using cobalt ferrite(Co Fe_(2)O_(4))and copper(Cu)nanoparticles(NPs)in the carrier fluid water,to magnify the energy and mass communication rate and boost the efficiency and performance of thermal energy conduction for a variety of commercial and biological purposes.The proposed model becomes more significant,with an additional effect of non-Fick's mass flux and Fourier's heat model to report the energy and mass passage rate.The results are obtained through the computational strategy parametric continuation method.The figures are plotted to reveal the physical sketch of the obtained solution,while the statistical assessment has been evaluated through tables.It has been observed that the dispersion of Cu and Co Fe_(2)O_(4)NPs to the base fluid significantly enhances the velocity and thermal conductivity of water,which is the most remarkable property of these NPs from the industrial point of view.

Peristaltic Flow of Couple Stress Fluid in a Non-Uniform Rectangular Duct Having Compliant Walls

The present study investigates the peristaltic flow of couple stress fluid in a non-uniform rectangular duct with compliant walls.Mathematical modeling is based upon the laws of mass and linear momentum.Analytic solutions are carried out by the eigen function expansion method under long-wavelength and low-Reynolds number approximations.The features of the flow characteristics are analyzed by plotting the graphs of various values of physical parameters of interest.Trapping bolus scheme is also presented through streamlines.

Hall effects on peristaltic flow of couple stress fluid in an inclined asymmetric channel

Effect of Hall current on peristaltic transport of couple stress fluid in an inclined asymmetric channel is investigated. An incompressible fluid is conducting in the pres- ence of inclined magnetic field. Problem formulation is developed through heat and mass transfer aspects. Soret and Dufour effects are present. Long wavelength and low Reynolds number approaches are adopted. Closed form expression for longitudinal velocity, pres- sure gradient, temperature and concentration are presented. Important results reflecting the influence of embedded parameters in the problems have been pointed out. Pumping and trapping phenomena are especially focused.

Mathematical modelling of couple stresses on fluid flow in constricted tapered artery in presence of slip velocity-effects of catheter

This paper explores the mathematical model for couple stress fluid flow through an annular region. The above model is used for studying the blood flow be-tween the clogged （stenotic） artery and the catheter. The asymmetric nature of the stenosis is considered. The closed form expressions for the physiological parameters such as impedance and shear stress at the wall are obtained. The effects of various geomet-ric parameters and the parameters arising out of the fluid considered are discussed by considering the slip velocity and tapering angle. The study of the above model is very significant as it has direct applications in the treatment of cardiovascular diseases.

Couple stress fluid flow in a rotating channel with peristalsis

This article describes a new model for obtaining closed-form semi-analytical solutions of peristaltic flow induced by sinusoidal wave trains propagating with constant speed on the walls of a two-dimensional rotating infinite channel. The channel rotates with a constant angular speed about the z-axis and is filled with couple stress fluid. The governing equations of the channel deformation and the flow rate inside the channel are derived using the lubrication theory approach. The resulting equations are solved, using the homotopy perturbation method（HPM）, for exact solutions to the longitudinal velocity distribution, pressure gradient, flow rate due to secondary velocity, and pressure rise per wavelength. The effect of various values of physical parameters, such as, Taylor＇s number and couple stress parameter, together with some interesting features of peristaltic flow are discussed through graphs. The trapping phenomenon is investigated for different values of parameters under consideration. It is shown that Taylor＇s number and the couple stress parameter have an increasing effect on the longitudinal velocity distribution till half of the channel, on the flow rate due to secondary velocity, and on the number of closed streamlines circulating the bolus.

Cilia-assisted hydromagnetic pumping of biorheological couple stress fluids

A theoretical study is conducted for magnetohydrodynamic pumping of electroconductive couple stress physiological liquids(e.g.blood)through a two-dimensional ciliated channel.A geometric model is employed for the cilia which are distributed at equal intervals and produce a whip-like motion under fluid interaction which obeys an elliptic trajectory.A metachronal wave is mobilized by the synchronous beating of cilia and the direction of wave propagation is parallel to the direction of fluid flow.A transverse static magnetic field is imposed transverse to the channel length.The Stokes’couple stress(polar)rheological model is utilized to characterize the liquid.The normalized two-dimensional conservation equations for mass,longitudinal and transverse momentum are reduced with lubrication approximations(long wavelength and low Reynolds number assumptions)and feature a fourth order linear derivative in axial velocity representing couple stress contribution.A coordinate transformation is employed to map the unsteady problem from the wave laboratory frame to a steady problem in the wave frame.No slip conditions are imposed at the channel walls.The emerging linearized boundary value problem is solved analytically and expressions presented for axial(longitudinal)velocity,volumetric flow rate,shear stress function and pressure rise.The flow is effectively controlled by three geometric parameters,viz cilia eccentricity parameter,wave number and cilia length and two physical parameters,namely magnetohydrodynamic(MHD)body force parameter and couple stress non-Newtonian parameter.Analytical solutions are numerically evaluated with MATLAB software.Axial velocity is observed to be enhanced in the core region with greater wave number whereas it is suppressed markedly with increasing cilia length,couple stress and magnetic parameters,with significant flattening of profiles with the latter two parameters.Axial pressure gradient is decreased with eccentricity parameter whereas it is elevated with cilia length,in the channel core region.Increasing couple stress and magnetic field parameter respectively enhance and suppress pressure gradient across the entire channel width.The pressure-flow rate relationship is confirmed to be inversely linear and pumping,free pumping and augmented pumping zones are all examined.Bolus trapping is also analyzed.The study is relevant to MHD biomimetic blood pumps.

Approximate Solution of Oil Film Load-carrying Capacity of Turbulent Journal Bearing with Couple Stress Flow

The instability of the rotor dynamic system supported by oil journal bearing is encountered frequently, such as the half speed whirl of the rotor, which is caused by oil film lubricant with nonlinearity. Currently, more attention is paid to the physical characteristics of oil film due to an oil-lubricated journal bearing being the important supporting component of the bearing-rotor systems and its nonlinear nature. In order to analyze the lubrication characteristics of journal bearings efficiently and save computational c[~brts, an approximate solution of nonlinear oil film forces of a finite length turbulent journal bearing with couple stress flow is proposed based on Sommerfeld and Ocvirk numbers. Reynolds equation in lubrication of a finite length turbulent .journal bearing is solved based on multi-parametric principle. Load-carrying capacity of nonlinear oil film is obtained, and the results obtained by different methods are compared. The validation of the proposed method is verified, meanwhile, the relationships of load-carrying capacity versus eccentricity ratio and width-to-diameter ratio under turbulent and couple stress working conditions are analyzed. The numerical results show that both couple stress flow and eccentricity ratio have obvious influence on oil film pressure distribution, and the proposed method approximates the load-carrying capacity of turbulent journal bearings efficiently with various width-to-diameter ratios. This research proposes an approximate solution of oil film load-carrying capacity of turbulent journal bearings with different width-to-diameter ratios, whicb are suitable for high eccentricity ratios and heavy loads.

The efect of the couple stress parameter and Prandtl number on the transient natural convection flow over a vertical cylinder

An analysis is performed to study transient free convective boundary layer flow of a couple stress fluid over a vertical cylinder, in the absence of body couples. The solution of the time-dependent non-linear and coupled governing equations is carried out with the aid of an unconditionally stable Crank-Nicolson type of numerical scheme. Numerical results for the steady-state velocity, temperature as well as the time histories of the skin-friction coefficient and Nus- selt number are presented graphically and discussed. It is seen that for all flow variables as the couple stress control parameter, Co, is amplified, the time required for reaching the temporal maximum increases but the steady-state decreases.

Heat transfer analysis of MHD and electroosmotic flow of non-Newtonian fluid in a rotating microfluidic channel:an exact solution

The heat transfer of the combined magnetohydrodynamic(MHD)and electroosmotic flow(EOF)of non-Newtonian fluid in a rotating microchannel is analyzed.A couple stress fluid model is scrutinized to simulate the rheological characteristics of the fluid.The exact solution for the energy transport equation is achieved.Subsequently,this solution is utilized to obtain the flow velocity and volume flow rates within the flow domain under appropriate boundary conditions.The obtained analytical solution results are compared with the previous data in the literature,and good agreement is obtained.A detailed parametric study of the effects of several factors,e.g.,the rotational Reynolds number,the Joule heating parameter,the couple stress parameter,the Hartmann number,and the buoyancy parameter,on the flow velocities and temperature is explored.It is unveiled that the elevation in a couple stress parameter enhances the EOF velocity in the axial direction.

Electro-magnetohydrodynamic Flow of Biofluid Induced by Peristaltic Wave： A Non-newtonian Model

This article aims to develop a mathematical model for peristaltic transport of magnetohydrodynamic flow of biofluids through a micro-channel with rhythmically contracting and expanding walls under the influence of an applied electric field. The couple stress fluid model is considered to represent the non-Newtonian characteristics ofbiofluids. The velocity slip condition at the channel walls is taken into account because of the hyclrophilic/hydrophobic interaction with negatively charged walls. The essential features of the electromagnetohydrodynamic flow of biofluid through micro-channels are clearly highlighted in the variations of the non-dimensional parameters of the physical quantities of interest such as the velocity, wall shear stress, pressure gradient, pressure rise per wave length, frictional force at the channel walls and the distribution of stream function. It reveals that the flow ofhiofluid is appreciably influenced by the sufficient strength of externally applied magnetic field and electro-osmotic parameter. The velocity slip condition reduces the frictional force at the channel wall. Moreover, the formation of the trapping bolus strongly depends on electro-osmotic parameter and magnetic field strength.

Gravity flow of pulsatile blood through a porous medium under periodic body acceleration and magnetic field in an inclined tube

Mathematical model for the pulsatile blood flow through a porous medium under the influence of periodic body acceleration for gravity flow along an inclined tube by considering blood as a couple stress, incompressible and electrically conducting fluid in the presence of magnetic field has been investigated. Analytical expressions for axial velocity, flow rate, fluid acceleration and shear stress are obtained by applying the Laplace and finite Hankel＇s transforms. The velocity profiles for various values of Hartmann number, couple stress parameters and the angle of inclination are shown graphically. Also the effects of body acceleration, Womerseley parameters and permeability parameters have been discussed. The results obtained in the present mathematical model for different values of the parameters involved in the problem show that the flow of blood is influenced by the effect of magnetic field, the porous medium and the inclination angle. The present model is compared with the other existing models. Through this theoretical investigation, the applications of magnetic field have also been indicated in the field of biological, biomedical and engineering sciences.