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In this modern age of fast growing technology, it is required to attain the need of compact size and high speed of rotation in rotating machines. The lubrication is an important aspect along with the bearing geometry, which significantly influence the bearing conduct. The arc length studied here is 100 degrees for partial bearing with a medium for lubrication as fluid with couple stresses (i.e. CSF). The influence of elastic deformation along with the fluid on performance of bearing in static and dynamic manner is obtained by solving the modified Reynold’s equation and an elasticity equation. The set of system equations are solved by using FEM approach. The results show a significant improvement in static and dynamic characteristics of partial bearing and the stability of the bearing is also improved while operating with couple stress fluid compared to Newtonian.

The paper deals with a theoretical study on blood flow in a stenosed segment of an artery, when blood is mixed with nano-particles. Blood is treated here as a couple stress fluid. Stenosis is known to impede blood flow and to be the cause of different cardiac diseases. Since the arterial wall is weakened due to arterial stenosis, it may lead to dilatation /aneurysm. The homotopy perturbation technique is employed to determine the solution to the problem for the case of mild stenosis. Analytical expressions for velocity, shear stress at the wall, pressure drop, and flow resistance are derived. The impact of different physical constants on the wall shear stress and impedance of the fluid is examined by numerical simulation. Streamline patterns of the nanofluid are investigated for different situations.

Performance of bearings used in modern machinery is significantly affected by the elasticity of the bearing liner and viscosity of the lubricant. This paper considers couple stress fluid (CSF) model of the lubricant along with elasticity of bearing liner. The three dimensional (3-D) model of elasticity is used to consider the effect of elasticity. Finite element method (FEM) has been utilized to comprehend the governing equation of fluid flow in the clearance space between the journal and the bearing. Various performance characteristics such as load capacity, dynamic coefficients and stability parameters have been evaluated for a range of CSF parameter and elasticity parameter. Results show that elasticity and CSF have significant effect on the performance of the journal bearing. Stability of the journal bearing increases as the value of elastic coefficient increases. Further the stability of the elastic journal bearing system is found to be increased by increase in couple stress parameter.

The main objective of the present investigation is to examine the couple stress fluid flow occurring as a result of rotation of a disk. On implementing a suitable transformation, the governing system of partial differential equations (PDEs) is converted into nonlinear differential equations of a single independent variable. These equations are solved analytically by virtue of the Homotopy Analysis Method (HAM) which gives solutions in the form of a series. The solution of most of the governing problems is determined in terms of the absolute exponential and decaying functions by means of this powerful technique. To support analytic results, some graphs are plotted for determining the convergence of the solution. Also the graphical interpretation of velocity profiles corresponding to the effects of pertinent parameters are shown and discussed in detail. The numerical results are calculated for evaluation of the influence of fluid parameter. It can also be anticipated that the radial and axial velocity components show decreasing behavior due to rise in the values of couple stress parameter which conflict the behavior of the tangential component of velocity.

A recent challenging task in the field of nanotechnology is nanofluids, which are potential heat transfer fluids. Numerous researchers worked on nanofluid with different physical conditions. In this research work, we presented the three-dimensional flow of couple stress nanofluid with Hall current, viscous dissipation and Joule heating impacts past an exponentially stretching sheet. The Cattaneo–Christov heat flux model is implemented to examine the thermal relaxation properties. The modeled equations have been transformed to nonlinear ordinary differential equations with the help of correspondence transformations. The homotopy analysis method is used to solve the proposed model. The effect of dimensionless parameters, which are couple stress, Hartmann number, the ratio of rates, and Hall on velocity fields in $x$ - and $y$ -directions has been scrutinized. The rise in Hall parameter, Hartmann number, the ratio of rates parameter, and couple stress parameter are reducing the velocity function in the $x$ -direction. The rise in Hall parameter, Hartmann number, and the ratio of rates parameter are improving the velocity function in the $y$ -direction. The influence of Prandtl number, thermal relaxation time, and temperature exponent on temperature field are presented in this paper. The rise in thermal relaxation parameter, Prandtl number, and temperature exponent are reducing the temperature function. The influence of thermophoresis, the Schmidt number, and Brownian motion on concentration field are presented. The rise in thermophoresis parameter is increasing the concentration function while the rise in Brownian motion parameter and Schmidt number are reducing the concentration function. The impacts of implanted factors on skin friction, Nusselt number, and Sherwood number are accessible through tables. The determined result of skin friction is compared with the previous study.

Hybrid nanofluid is an emerging field due to the rapid enhancement of heat transfer and stable nanoparticles in base fluid properties. A three dimensional hybrid nanofluid flow model is constructed over biaxial porous stretching/shrinking sheet with heat transfer, Radiative heat and mass flux (3D-HNF-RHF). Bayesian Regularization technique based on Backpropagated neural networks (BRT-BNNs) is employed to estimate the solution of proposed model. It is the mathematical process which converts nonlinear regression fitting into a well-posed statistical process in the form of ridge regression. This method diminishes length cross validation need and more robust then Backpropagation technique. The proposed flow system 3D-HNF-RHF is transferred to ordinary differential equations (ODEs) possessing physical variations through self-similar transformations. The effect of derived variations such as thermal relaxation parameter, mass flux parameter, Stretching/shrinking parameter, Prandtl number, Skin friction and Nusselt number observed over the velocity and temperature fields. Numerical results of these physical parameters have been presented in tabulated form obtained from a dataset constructed through Homotopy Analysis Method imposed on 3D-HNF-RHF model. Statistical tests through mean square error, histogram curve and regression fitting curves are employed to check the accuracy and convergences of the solution obtained through BRT-BNNs. It is observed that Stretching/shrinking quantity and mass flux parameter slow down the flow rate whereas, increase radiative heat flux upsurges the temperature gradient. The impacts of surface drag force and heat transfer are illustrated through the different graphical illustrations.

The effects of an overlapping stenosis on blood flow characteristics in an artery have been studied. Blood has been represented by a couple stress fluid. The flow equations have been linearised and the expressions for pressure drop, resistance to the flow and wall shear stress have been derived. The results are shown graphically. It is observed that the resistance to the flow, pressure drop and wall shear stress increases with height and length of the stenosis. And it is noticed that the resistance to the flow and pressure drop decreases with couple stress fluid parameters. But wall shear stress increases with couple stress fluid parameters.

An attempt has been made to examine the effects of magnetohydrodynamic couple stress fluid in peristaltic flow with porous medium under the impact of slip, heat transfer and wall properties. The expressions are obtained for temperature, coefficient of heat transfer and velocity. Influences of different parameters, the Hartmann number, Brinkman number and adaptability parameters on the temperature and warmth trade coefficient are discussed through outlines.

A theoretical study of steady laminar two-dimensional flow of a non-Newtonian fluid in a parallel porous channel with variable permeable walls is carried out. Solution by Differential Transform Method (DTM) is obtained and the flow behavior is studied. The non-Newtonian fluid considered for the study is couple stress fluid. Thus, in addition to the effects of inertia and permeabilities on the flow, the couple stress effects are also analyzed. Results are presented and comparisons are made between the behaviour of Newtonian and non-Newtonian fluids.

The credibility of a computational analytical solution of time dependent mean velocity for magneto hydrodynamic (MHD) peristaltic flow of an electrically conducting couple stress fluid in an uniform elastic porous channel is analyzed and assessed. The effect of slip condition has been studied. Mathematical model is formulated assuming small Reynolds number and long wavelength approximations. Various aspects such as Magnetic parameter, Darcy number, slip parameter and couple stress parameter are discussed through graphs under the existence and non existence of both stiffness and viscous damping force. It is noticed that the time dependent mean velocity increases with increase in the Magnetic parameter in presence of elastic parameters and flow reversal occurs at the center line of the channel.

The aim of the present attempt was to investigate the hydromagnetic peristaltic motion of a reacting and radiating couple stress fluid in an inclined asymmetric channel filled with a porous medium. The governing equations for the flow with heat and mass transfer are presented. The Mathematical modeling is investigated by utilizing long wavelength and low Reynolds number assumptions. The exact solution has been evaluated for the stream function, velocity, temperature, concentration and pressure gradient. The pressure rise solution has been simplified using numerical integration. The effects of Hartmann number, Darcy number, thermal Grashof number, couple stress parameter, thermal radiation parameter, species Grashof number, heat generation parameter, inclination angle and chemical reaction parameter on the velocity characteristics, heat and mass transfer characteristics and pumping characteristics are presented through graphically and discussed in detail.

The creeping motion of a rigid slip sphere in an unbounded couple stress fluid is investigated. The linear slip boundary condition and the vanishing couple stress condition are applied on the surface of the sphere. A simple formula for the drag force acting on a slip sphere translating in an unbounded couple stress fluid is obtained. Special cases of the deduced drag formula are concluded and compared with analogous results in the literature. The normalized drag force experienced by the fluid on the slip sphere is represented graphically and the effects of slip parameter and viscosity coefficients are discussed.

Exact analysis of miscible dispersion of solute with interphase mass transfer in a poorly conducting couple stress fluid flowing through a rectangular channel bounded by porous layers is considered because of its application in many practical situations. The generalized dispersion model of Sankarasubramanian and Gill is used, which brings into focus the exchange coefficient, the convective coefficient and the dispersion coefficient. The exchange coefficient comes into picture due to the interphase mass transfer and independent of solvent fluid viscosity. It is observed that the convective coefficient increases with an increase in the porous parameter while it decreases with an increase in the couple stress parameter. The dispersion coefficient is plotted against wall reaction parameter for different values of porous parameter and couple stress parameter. It is noted that the dispersion coefficient decreases with an increase in the value of couple stress parameter but increases with porous parameter.

In this paper, we considered the pulsatile flow of blood through catheterized tapered artery in the presence of an ω -shaped stenosis. Blood flow is modelled as homogeneous incompressible couple stress fluid. Further the effects of velocity slip at the arterial wall are also examined. The analysis is carried out analytically and closed form solutions are obtained with the assumption of mild stenosis. In the present study, we analyze the effects of various fluid and geometric parameters on the physiological parameters such as resistance to flow and shear stress at the wall. The variation in the resistance to the flow and wall shear stress with respect to stenosis size ( ∊ , ψ ), radius of the catheter ( R c ), couple stress fluid parameters ( β , ω ), Reynolds number ( Re ) and pulsatile parameter ( σ ) has been studied. In particular shear stress at the wall is reckoned at both the locations corresponding to the maximum height of the stenosis. It has been observed that this physiological parameter is independent of the location of the maximum height in case of nontapered artery while these locations significantly impact the shear stress at the wall in case of tapered artery. The locations of the critical and maximum heights with corresponding annular radii are summarized in the form of Table 1 . We also focussed our attention on the analysis of the wall shear stress over the entire stenosis region for various values of the geometric and fluid parameters. It is observed that the impedance and wall shear stress are increasing with increase in the radius of catheter and stenosis size while they are decreasing as the tapered parameter and the couple stress fluid parameters are increasing. It is observed that slip velocity and diverging tapered artery facilitate the fluid flow.

This communication reports the peristaltic transport of a fluid in a uniform channel having compliant wall in porous medium. Taking into account the equations of the deformable boundaries and the fluid, the problem is inspected under low Reynolds number and long wave length approximation. Obtaining solution for the velocity, the behaviour of various parameters including the Saffman slip parameter (β), couple stress fluid parameter (α), elastic parameters (E 1 , E 2 , E 3 ) are examined through the graphical results for time average velocity. In case of non zero elasticity parameters, we observe that the time average velocity increases as the slip parameter as well as the Darcy number.

In this paper, we have investigated the effect of channel inclination on the peristaltic transport of a couple stress fluid in the presence of externally applied magnetic field. The slip velocity at the channel wall has been taken into account. Under the long wave length and low-Reynolds number assumptions, the analytical solutions for axial velocity, stream function, pressure gradient and pressure rise are obtained. The computed results are presented graphically by taking valid numerical data for non-dimensional physical parameters available in the existing scientific literatures. The results revealed that the trapping fluid can be eliminated and the central line axial velocity can be reduced with a considerable extent by the application of magnetic field. The flow phenomena for the pumping characteristics, trapping and reflux are furthermore investigated. The study shows that the slip parameter and Froude number play an important role in controlling axial pressure gradient.

In the present article, the exact solutions for fundamental flows namely Couette, Poiseuille and generalized Couette flows of an incompressible couple stress fluid between parallel plates are obtained using slip boundary conditions. The effect of various parameters on velocity for each problem is discussed. It is found that, for each of the problems, the solution in the limiting case as couple stresses approaches to zero is similar to that of classical viscous Newtonian fluid. The results indicate that, the presence of couple stresses decreases the velocity of the fluid.

The Couette&ndash, Poiseuille flow of couple stress fluid with magnetic field between two parallel plates was investigated. The flow was driven due to axial pressure gradient and uniform motion of the upper plate. The influence of heating at the wall in the presence of spherical and homogeneous Hafnium particles was taken into account. The temperature dependent viscosity model, namely, Reynolds&rsquo, model was utilized. The Runge&ndash, Kutta scheme with shooting was used to tackle a non-linear system of equations. It was observed that the velocity decreased by increasing the values of the Hartman number, as heating of the wall reduced the effects of viscous forces, therefore, resistance of magnetic force reduced the velocity of fluid. However, due to shear thinning effects, the velocity was increased by increasing the values of the viscosity parameter, and as a result the temperature profile also declined. The suspension of inertial particles in an incompressible turbulent flow with Newtonian and non-Newtonian base fluids can be used to analyze the biphase flows through diverse geometries that could possibly be future perspectives of proposed model.

The present study is about the pressure-driven heated bi-phase flow in two slippery walls. The non-Newtonian couple stress fluid is suspended with spherically homogenous metallic particles. The magnetic susceptibility of Hafnium allures is taken into account. The rough surface of the wall is tackled by lubrication effects. The nonlinear coupled partial differential equations along with the associated boundary conditions are first reduced into a set of ordinary differential equations by using appropriate transformations and then numerical results were obtained by engaging the blend of Runge&ndash, Kutta and shooting techniques. The sway of physical quantities are examined graphically. An excellent agreement within graphical illustration and numerical results is achieved.

The purpose of this paper is to study the non-isothermal Poiseuille flow between two heated parallel inclined plates using incompressible couple stress fluids. Reynold’s model is used for temperature dependent viscosity. We have developed highly non-linear coupled ordinary differential equations from momentum and energy equations. The Perturbation technique is used to obtain the approximate analytical expressions for velocity and temperature distributions. Expressions for velocity field, temperature distribution, dynamic pressure, volume flow rate, average velocity and shear stress on the plates are obtained. The influence of various emerging parameters on the flow problem is discussed and presented graphically.

The effect of rotation on the onset of double diffusive convection in a horizontal couple stress fluid-saturated porous layer, which is heated and salted from below, is studied analytically using both linear and weak nonlinear stability analyses. The extended Darcy model, which includes the time derivative and Coriolis terms, has been employed in the momentum equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. The effect of Taylor number, couple stress parameter, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the rotation, couple stress parameter and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The Lewis number has a stabilizing effect in the case of stationary and finite amplitude modes, with a destabilizing effect in the case of oscillatory convection. The Darcy–Prandtl number and normalized porosity advances the onset of oscillatory convection. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer. The transient behavior of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge–Kutta method.

The Soret and Dufour effects in the presence of chemical reaction on natural convection heat and mass transfer of a couple stress fluid in a vertical channel formed by two vertical parallel plates is presented. 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 Homotopy Analysis Method (HAM). Profiles of dimensionless velocity, temperature and concentration are shown graphically for various values of Dufour number, Soret number, Couple stress parameter and chemical reaction parameter.

This paper presents the Hall and Ion-slip effects on electrically conducting couple stress fluid flow between two circular cylinders in the presence of a temperature dependent heat source. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations and then solved using 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.

Consider the flow of an incompressible fluid between two parallel plates, initially induced by a constant pressure gradient. After steady state is attained, the pressure gradient is suddenly with drawn while the plates are impulsively started simultaneously. The arising flow is referred to as run up flow and the present paper aims at studying this flow in the context of a couple stress fluid. Using Laplace transform technique, the expression for velocity is obtained in Laplace transform domain which is later inverted to the space time domain using a numerical approach. The variation of velocity with respect to various flow parameters is presented through graphs.

Stokes’ first and second problems for an incompressible couple stress fluid are considered under isothermal conditions. The problems are solved through the use of Laplace transform technique. Inversion of the Laplace transform of the velocity component in each case is carried out using a standard numerical approach. Velocity profiles are plotted and studied for different times and different values of couple stress Reynolds number. The results are presented through graphs in each case.

The standard decoupled method with Newton-generalized minimum residual procedure performs poorly or may break down when used to solve certain elastohydrodynamic lubrication problems. The authors Zargari et al., 2007 presented decoupled and coupled methods in which the limitations of the decoupled method for some set of physical parameters and slight variation in these values (physical parameters) give a non-convergence solution. To overcome this aspect the wavelet-based preconditioners technique is used in this paper to solve the elastohydrodynamic lubrication problem. The effect of coupled stress fluids on elastohydrodynamic lubrication behavior in smooth, as well as rough, contact is investigated using the proposed method, numerically, in a low-speed-high load condition. The elastohydrodynamic lubrication characteristics computed for couple stress fluids are found to have strong dependence on the couple stress parameter, which shows the versatility and applicability of the proposed method.

The unsteady three-dimensional flow of couple stress fluid over a stretched surface is investigated. Analysis has been performed in the presence of mass transfer and chemical reaction. Nonlinear flow analysis is computed by a homotopic approach. Plots are presented and analyzed for the various parameters of interest. A comparative study with existing solutions in a limiting sense is made.

The radiation and thermal diffusion effects on mixed convection flow of couple stress fluid through a channel 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 Spectral Quasi-linearization Method (QLM). The results, which are discussed with the aid of the dimensionless parameters entering the problem, are seen to depend sensitively on the parameters.