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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.

Contemporary investigation studies the silent features of the dissipative free convection couple stress fluid flow over a cylinder under the action of magnetic field, thermal radiation and porous medium with chemical reaction effect. Present two-dimensional viscous incompressible physical model is designed based on the considered flow geometry. Present physical problem gives the highly complicated nonlinear coupled partial differential equations (PDE's) which are not amenable to any of the known techniques. Thus, unconditionally stable, most accurate and speed converging with flexible finite difference implicit technique is utilized to simplify the dimensionless flow field equations. It is apparent from the current results that; the velocity profiles are diminished with enhancing values of magnetic field. Temperature profile increases with enhancing values of thermal radiation parameter. Velocity contours deviates away from the wall with enhancing magnetic parameter. Also, the effects of magnetic field, porous medium, thermal radiation, chemical reaction, buoyancy ratio parameter and Eckert number on couple stress flow velocity, temperature, and concentration profiles are studied. However, the present study has good number of applications in the various fields of engineering such as; polymer processing, solidification of liquid crystals, colloidal solutions, synovial joints, geophysics, chemical engineering, astrophysics and nuclear reactors etc. Finally, the current solutions are validated with the available results in the literature review and found to be in good agreement.

In this paper, we have explored the steady Poiseuille flow of couple stress fluid between two parallel plates under ‎the influence of non-isothermal effects of Reynold viscosity model, using Optimal Homotopy Asymptotic Method ‎‎(OHAM) and New Iterative Method (NIM). We obtained expressions for velocity profile, temperature distribution, ‎average velocity, volume flux and shear stress. The solutions obtained using these methods are in the form of ‎infinite series; therefore, they can be easily computed. Comparative results of solutions obtained by both methods ‎are given using different tables and graphs.

In this article, the generalized plane Couette flow of Vogel’s model of incompressible, non-isothermal, couple stress fluid flowing steadily between two parallel walls is investigated. The governing equations are reduced to ordinary differential equations. To investigate the non-linear coupled system of differential equations, the optimal homotopy asymptotic method with DJ polynomial and asymptotic homotopy perturbation method have been used. Important flow properties are presented and discussed. We have obtained expressions for velocity, average velocity, shear stress, volume flux and temperature. The results gained employing these techniques are in the form of infinite series; thus, the results can be easily calculated. Comparison of various results, obtained through the suggested approaches, is carried out and an excellent agreement is achieved.

In this paper, the mixed convective flow of an electrically conducting, viscous incompressible couple stress fluid through a vertical channel filled with a saturated porous medium has been investigated. The fluid is assumed to be driven by both buoyancy force and oscillatory pressure gradient parallel to the channel plates. A uniform magnetic field of strength B0 is imposed transverse to the channel boundaries. The temperature of the right channel plate is assumed to vary periodically, and the temperature difference between the plates is high enough to induce radiative heat transfer. Under these assumptions, the equations governing the two-dimensional couple stress fluid flow are formulated and exact solutions of the velocity and the temperature fields are obtained. The effects of radiation, Hall current, porous medium permeability and other various flow parameters on the flow and heat transfer are presented graphically and discussed extensively.

The non-isothermal couple stress fluid inside a reverse roll coating geometry is considered. The slip condition is considered at the surfaces of the rolls. To develop the flow equations, the mathematical modelling is performed using conservation of momentum, mass, and energy. The LAT (lubrication approximation theory) is employed to simplify the equations. The closed form solution for velocity, temperature, and pressure gradient is obtained. While the pressure and flow rate are obtained numerically. The impact of involved parameters on important physical quantities such as temperature, pressure, and pressure gradient are elaborated through graphs and in tabular form. The pressure and pressure gradient decreases for variation of the couple stress parameter and velocity ratio parameter K. While the variation of the slip parameter increases the pressure and pressure gradient inside the flow geometry. Additionally, flow rate decreases for the variation of the slip parameter as fluid starts moving rapidly along the roller surface. The most important physical quantity which is responsible for maintaining the quality of the coating and thickness is flow rate. For variation of both the couple stress parameter and the slip parameter, the flow rate decreases compared to the Newtonian case, consequently the coating thickness decreases for the variation of the discussed parameter.

The present article deals with the effects of nanoparticles and magnetic field on the peristaltic flow of a couple stress fluid in an asymmetric channel with different wave forms. Mathematical modelling for 2-D and two directional flows of a couple stress fluid along with nanofluid are first given and then simplified under the assumptions of long wavelength and low Reynolds number approximation. After invoking these approximations we get coupled non-linear differential equations. The exact solutions of temperature distribution, the nanoparticle concentration, velocity, stream function and pressure gradient are calculated. Finally graphical results of various physical parameters of interest are discussed to examine the behavior of flow quantities.

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.

This study presents the theoretical investigation of a roll-over thin layer formation under the lubrication approximation theory. The set of differential equations derived by lubrication approximation is solved by the optimal homotopy asymptotic method (OHAM) to obtain precise expressions for pressure and velocity gradients. Critical quantities such as velocity, pressure gradient, and coating layer depth are numerically estimated. The impact of parameters affecting the coating and layer formation is revealed in detail. Results indicate that the transport properties of the higher-grade fluid play an essential role in regulating velocity, pressure, and the final coated region. Moreover, couple stress effects on the properties of fluid particles to be coated on roller-surface have also been studied.

This investigation addresses the influence of a buoyancy force on the flow of a couple stress hydromagnetic heat generating fluid across a porous channel with isothermal boundaries. The analytical formulations for the momentum and energy equations are derived to seek the solutions for the rate of fluid momentum, heat transfer and the rate of entropy generation with the use of a well known and efficient series solution of Adomian decomposition method (ADM). The findings are compared with earlier acquired findings for validation and hereby showed the speedy convergence of the series solution. The results showed the substantial influence of inward warmth inside the stream and buoyancy force on the motion and thermal energy of the flow system. Also, the activities of entropy generation generally occur maximally at the centreline of the flow stream with significant reduction with respect to buoyancy force and magnetic field strength., Applied mathematics; Computational mathematics; Mathematics; Mechanical engineering; Thermodynamics; Computational fluid dynamics; Computational heat transfer; Couple stress fluid; Porous medium; Entropy generation; Internal heat generation; Buoyancy force; Adomian decomposition method

Different from a Newtonian fluid, couple stress fluid (CSF) includes a new material constant, which is responsible for couple stress and the lubricant viscosity. This material constant comes with the fourth-order spatial derivative term, and due to this higher-order derivative term in the momentum equation, this fluid (CSF) is comparatively less investigated even for the classical fluid problems. This paper aims to study the fractional model of CSF, based on the Atangana-Baleanu (AB) fractional derivatives definition. Since this AB definition is new, therefore, for the sake of comparison and correctness, this problem is also solved using the Caputo-Fabrizio (CF) fractional derivative definition. The CSF is considered to flow between two parallel plates, one of which is at rest and the other is moving with constant velocity. The external pressure gradient is also applied. This type of flow situations is usually called generalized Couette flow. The problem is first written in dimensionless form and then solved for the exact solution using the Laplace transform and the finite Fourier sine transform. The CSF velocity obtained via an AB fractional derivative is compared with the CSF velocity obtained via a CF fractional derivative approach, and the results obtained are shown graphically. The CSF results for interesting fluid parameters are displayed in various graphs for both the AB and CF fractional derivatives. It is observed that the CSF velocities obtained with the AB and CF fractional derivatives are the same for unit time. For time less than one and greater than one, variation in CSF velocities is observed. In limiting sense, the present CSF solutions are reduced to a similar Newtonian fluid problem solution in the absence of external pressure gradient.

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.

In this investigation, we have studied the problem of peristaltic flow with heat transfer through the gap between coaxial inclined tubes where the inner tube is rigid and the outer tube has sinusoidal wave travelling down its wall. The problem has been formulated in cylindrical coordinate system. The equations governing the flow have been simplified under the long wavelength and low Reynolds number assumptions. The exact solution is obtained for the temperature profile. The perturbation solutions for the velocity and pressure gradient are obtained for small couple stress parameter. Pressure difference per wavelength and frictional forces on the tube walls have been computed numerically. Results are demonstrated for various flow parameters. The better pumping results occur in vertical tube, while less pumping is seen in horizontal tube. The size of trapped bolus is small in triangular wave as compared to other waves. The present study has a wide range of applications in bio-medical engineering like the transport phenomenon in peristaltic micro pumps.

In this work, the time dependent flow of an incompressible couple stress fluid passing two infinite parallel plates is investigated. The fluid motion is generated by applying a constant pressure gradient together with a sudden motion of one of the two plates. The linear slip velocity and the vanishing couple stresses on the boundary are applied on both plates. A generalized solution is obtained analytically by utilizing the integral Laplace transform together with the state space technique in the Laplace domain. The inversion of Laplace transform is obtained numerically by using a standard numerical method adopted by Honig and Hirdes. The effects of couple stress viscosity coefficient and slip parameters on both plates are studied numerically. The results show that the increase in the couple stress and slip coefficients results in a decrease in the fluid velocity in the case of stationary plates. However the velocity of the fluid is amplified by the increase of these parameters when the upper plate is set in motion.

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.

In the last decade, nanoparticles have provided numerous challenges in the field of science. The nanoparticles suspended in various base fluids can transform the flow of fluids and heat transfer characteristics. In this research work, the mathematical model is offered to present the 3D magnetohydrodynamics Darcy&ndash, Forchheimer couple stress nanofluid flow over an exponentially stretching sheet. Joule heating and viscous dissipation impacts are also discussed in this mathematical model. To examine the relaxation properties, the proposed model of Cattaneo&ndash, Christov is supposed. For the first time, the influence of temperature exponent is scrutinized via this research article. The designed system of partial differential equations (PDE&rsquo, s) is transformed to set of ordinary differential equations (ODE&rsquo, s) by using similarity transformations. The problem is solved analytically via homotopy analysis technique. Effects of dimensionless couple stress, magnetic field, ratio of rates, porosity, and coefficient of inertia parameters on the fluid flow in x- and y-directions have been examined in this work. The augmented ratio of rates parameter upsurges the velocity profile in the x-direction. The augmented magnetic field, porosity parameter, coefficient of inertia, and couple stress parameter diminishes the velocity field along the x-direction. The augmented magnetic field, porosity parameter, coefficient of inertia, ratio of rates parameter, and couple stress parameter reduces the velocity field along the y-axis. The influences of time relaxation, Prandtl number, and temperature exponent on temperature profile are also discussed. Additionally, the influences of thermophoresis parameter, Schmidt number, Brownian motion parameter, and temperature exponent on fluid concentration are explained in this work. For engineering interests, the impacts of parameters on skin friction and Nusselt number are accessible through tables.

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.

In this paper, an incompressible couple stress fluid flow with magnetic field in a porous channel with expanding or contracting walls and slip boundary condition is considered. Lie group analysis and group invariant solutions are obtained, the governing equations are reduced to nonlinear ordinary differential equations. The resulting equations are solved analytically, also this equations are solved by using Adomian method. The graphs for the axial and the normal velocity components and the pressure distribution for different values of the physical and geometric parameters are plotted and discussed. Finally, the comparison between the analytic and Adomian methods is discussed. It is found that for no-slip case ϕ = 0 the fluid adheres to the walls and axial velocity is maximum at the center of the channel, also by increasing the slip parameter the velocity at the channel walls increases. However, it decreases at center of the channel by increasing slip parameter.

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.

In this work, an analytical study of the effects of Hall current and Joule heating on the entropy generation rate of couple stress fluid is performed. It is assumed that the applied pressure gradient induces fluid motion. At constant velocity, hot fluid is injected at the lower wall and sucked off at the upper wall. The obtained equations governing the flow are transformed to dimensionless form and the resulting nonlinear coupled boundary value problems for velocity and temperature profiles are solved by Adomian decomposition method. Analytical expressions for fluid velocity and temperature are used to obtain the entropy generation and the irreversibility ratio. The effects of Hall current, Joule heating, suction/injection and magnetic field parameters are presented and discussed through graphs. It is found that Hall current enhances both primary and secondary velocities and entropy generation. It is also interesting that Joule heating raises fluid temperature and encourages entropy production. On the other hand Hartman number inhibited fluid motion while increase in suction/injection parameter resulted into a shift in flow symmetry.

In this work, an analytical study of the effects of Hall current and Joule heating on the entropy generation rate of couple stress fluid is performed. It is assumed that the applied pressure gradient induces fluid motion. At constant velocity, hot fluid is injected at the lower wall and sucked off at the upper wall. The obtained equations governing the flow are transformed to dimensionless form and the resulting nonlinear coupled boundary value problems for velocity and temperature profiles are solved by Adomian decomposition method. Analytical expressions for fluid velocity and temperature are used to obtain the entropy generation and the irreversibility ratio. The effects of Hall current, Joule heating, suction/injection and magnetic field parameters are presented and discussed through graphs. It is found that Hall current enhances both primary and secondary velocities and entropy generation. It is also interesting that Joule heating raises fluid temperature and encourages entropy production. On the other hand Hartman number inhibited fluid motion while increase in suction/injection parameter resulted into a shift in flow symmetry.

In this paper, we have investigated the onset of double diffusive convection (DDC) in a couple stress fluid saturated rotating anisotropic porous layer in the presence of Soret and Dufour effects using linear stability analyses which is based on the usual normal mode technique. The onset criteria for both stationary and oscillatory modes obtained analytically. The effects of the Taylor number, mechanical anisotropy parameter, Darcy Prandtl number, solute Rayleigh number, normalized porosity parameter, Soret and Dufour parameters on the stationary and oscillatory convections shown graphically. The effects of couple stresses are quite significant for large values of the non-dimensional parameter and delay the onset of convection. Taylor number has stabilizing effect on double diffusive convection, Dufour number has stabilizing effect in stationary mode while destabilizing in oscillatory mode. The negative Soret parameter stabilizes the system and positive Soret parameter destabilizes the system in the stationary convection, while in the oscillatory convection the negative Soret coefficient destabilize the system and positive Soret coefficient stabilizes the system.

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.

The paper deals with the flow of two immiscible couple stress fluids between two homogeneous permeable beds. The flow is considered in two zones: zone I and II contain free flow of two immiscible couple stress fluids between two permeable porous beds at the bottom and top. The flow in the free channel bounded by two permeable beds is assumed to be governed by Stokes’s couple stress fluid flow equations and that in the permeable beds by Darcy’s law. The continuity of velocity, vorticity, shear stress and couple stress are imposed at the fluid-fluid interface and Beavers-Joseph (BJ) slip boundary conditions are employed at the fluid-porous interface. The equations are solved analytically and the expressions for velocity, skin friction and volumetric flow rate are obtained. The effects of the physical governing parameters on velocity are investigated.

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.

This paper describes the effects of induced magnetic field and partial slip on the peristaltic flow of a couple stress fluids in an asymmetric channel. The two dimensional equation of couple stress fluid are simplified by making the assumptions of long wave length and low Reynolds number. The exact solutions of reduced momentum equation and magnetic force function have been computed in wave frame. The expressions for stream function, magnetic force function and pressure rise per wave length have been also computed. Finally, the pressure rise, pressure gradient, velocity, magnetic force function and stream lines for different physical parameters of interest are plotted.

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.

In this paper, we investigate the generalized Stokes’ problems for an incompressible couple stress fluid. Analytical solution of the governing equations is obtained in Laplace transform domain for each problem. A standard numerical inversion technique is used to invert the Laplace transform of the velocity in each case. The effect of various material parameters on velocity is discussed and the results are presented through graphs. It is observed that, the results are in tune with the observation of V.K.Stokes in connection with the variation of velocity in the flow between two parallel plates when the top one is moving with constant velocity and the bottom one is at rest., {"references":["V.K. Stokes, Couple stresses in fluids, 9(9) (1966), pp.1710-1715.","V.K. Stokes, Theory of fluids with microstructure-An introduction, Springer Verlag, (1984).","N.B. Naduvinamani, P.S. Hiremath, G. Gurubasavaraj, Squeeze film lubrication of a short porous journal bearing with Couple stress fluids, Tribology International, 34(11) (2001), pp. 739-747.","N.B. Naduvinamani, P.S. Hiremath, G. Gurubasavaraj, Surface roughness effects in a short porous journal bearing with a Couple stress fluid, Fluid Dynamics Research, 31(5-6) (2002), pp. 333-354.","N.B. Naduvinamani, P.S. Hiremath, G. Gurubasavaraj, Effects of surface roughness on the Couple stress squeeze film between a sphere and a flat Plate, Tribology International, 38(5) (2005), pp. 451-458.","N.B. Naduvinamani, Syeda Tasneem Fathima, P.S. Hiremath, Hydrodynamic lubrication of rough slider bearings with Couple stress fluids, Tribology International, 36(12) (2003), pp. 949-959.","N.B. Naduvinamani, Syeda Tasneem Fathima, P.S. Hiremath, Effect of surface roughness on characteristics of couplestress squeeze film between anisotropic porous rectangular plates, Fluid Dynamics Research, 32(5) (2003), pp. 217-231.","Jaw-Ren Lin, Chi-Ren Hung, Combined effects of non-Newtonian couple stresses and fluid inertia on the squeeze film characteristics between a long cylinder and an infinite plate, Fluid Dynamics Research, 39(8) (2007), pp. 616-639.","H. Schlichting, K. Gersten, Boundary Layer Theory, Springer, (2002).\n[10] P.R. Gupta, K.L. Arora, Hydromagnetic flow between two parallel planes, one is oscillating and the other fixed, Pure. Appl. Geophy., 112(2) (1974), pp.498-505.\n[11] I.A. Hassanien, M.A. Mansour, Unsteady magnetohydrodynamic flow through a porous medium between two infinite parallel plates, Astrophy. Spac. Scie., 163(2) (1990), pp.241-246.\n[12] I.A. Hassanien, Unsteady hydromagnetic flow through a porous medium between two infinite parallel porous plates with time varying suction, Astrophy. Spac. Scie., 175(1) (1991), pp.135-147.\n[13] T. Hayat, S. Asghar, A.M. Siddiqui, Some unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Engg. Sci., 38(3) (2000), pp 337-346.\n[14] M.E. Erdogan, On the unsteady unidirectional flows generated by impulsive motion of a boundary or sudden application of a pressure gradient, Int. J. Non-Linear Mech., 37(6) (2002), pp.1091-1106.\n[15] T. Hayat, Masood Khan, A.M. Siddiqui, S. Asghar, Transient flows of a second grade fluid\", Int. J. Non-Linear Mech., 39(10) (2004), pp.1621-1633.\n[16] M.E. Erdogan, C.E. Imrak, On unsteady unidirectional flows of a second grade fluid, Int. J. Non-Linear Mech., 40(10) (2005), pp.1238-1251.\n[17] M.E. Erdogan, C.E. Imrak, On some unsteady flows of a non-Newtonian fluid, Appl. Math. Model., 31(2) (2007), pp.170-180.\n[18] C. Fetecau, C. Fetecau, Starting solutions for some unsteady unidirectional flows of second grade fluid, Int. J. Engg. Sci., 43(10) (2005), pp.781-789.\n[19] G. Honig, U. Hirdes, A method for the numerical inversion of Laplace Transforms, J. Comp. Appl. Math., 10(1) (1984)., pp.113-132.\n[20] M. Devakar, T.K.V. Iyengar, Stokes' problems for an incompressible couple stress fluid\". Non-Linear Anal. Mode. Contr., 13(2) (2008), pp.181-190."]}

Steady incompressible couple stress fluid flow through two dimensional symmetric channel with stenosis is investigated. The flow consisting of a core region to be a couple stress fluid and a peripheral layer of plasma (Newtonian fluid). Assuming the stenosis to be mild, the equations governing the flow of the proposed model are solved using the slip boundary condition and closed form expressions for the flow characteristics (the dimensionless resistance to flow and wall shear stress at the maximum height of stenosis) are derived. The effects of various parameters on these flow variables have been studied. It is observed that the resistance to flow as well as the wall shear stress increase with the height of stenosis, viscosity ratio and Darcy number. However, the trend is reversed as the slip and the couple stress parameter increase., {"references":["","D. F. Young, Effects of a time-dependent stenosis on flow through a tube, J. Engrg. Ind.Trans.ASME.,vol.90, pp. 248-254, 1968.","G.R.Zendehbudi and M.S.Moayeri, Comparison of physiological and simple plusatile flows through stenosed arteries, J. Biomech., vol.32, pp. 959-965,1999.","G. Radhakrishnamacharya and P. Srinivasa Rao, Flow of a magnetic fluid through a non-uniform wavy tube, Proc.Nat.Acad.Sci.India, vol.76 , pp. 241-245, 2007.","R.L.Whitmore, Rheology of the Circulation, Pergamon Press, New York ,1968.","J.H. Forrester and D.F. Young, Flow through a converging-diverging tube and its implications in occlusive vascular disease, J. Biomech., vol.3, pp. 307-316, 1970.","J. B. Shukla, R. S. Parihar and B. R. P. Rao, Effects of stenosis on non-Newtonian flow of the blood in an artery,Bull. Math. Biol., vol.42, pp. 283-294, 1980.","J.C.Misra and S.K. Ghosh S K, A mathematical model for the study of blood flow through a channel with permeable walls, Acta Mechanica, vol.12, pp. 137-153, 1997.","N.Jain , S.P.Singh and M. Gupta, Steady flow of blood through an atherosclerotic artery: A non-Newtonian model, International Journal of Applied Mathematics and Mechanics,vol.8, pp. 52-63, 2012.","S. Gupta, M. Gupta and S.P. Singh SP, Effect of radial viscosity variation on non-Newtonian flow of blood in a stenosed artery.International Journal of Applied Mathematics and Mechanics, vol.8, pp. 51-61, 2012.\n[10] V. K. Stokes, Couple Stresses in Fluids, Phys. Fluids., vol.9, pp. 1709-1715,1966.\n[11] S.C.Cowin, The theory of polar fluids, Advances in Applied Mechanics, Academic Press, New York, pp. 279-347, 1974.\n[12] G. C. Sankad and G. Radhakrishnamacharya, Effect of Magnetic field on the peristaltic transport of couple stress fluid in a channel with wall properties, Int. J. Biomath., vol.4, pp. 365-378,2011.\n[13] D. Srinivasacharya and D. Srikanth, Effect of couple stresses on the pulsatile flow through a constricted annulus,Comptus Rendus Mecanique., vol.336, pp. 820-827,2008.\n[14] R.K. Naeem, S.Younus and Dania, Inverse solutions for unsteady incompressible couple stress fluid flows, International Journal of Applied Mathematics and Mechanics, vol.6, pp.1-17, 2010.\n[15] G. Bugliarello and J.W. Hyden, Detailed characteristics of the flow of blood in vitro, Trans.Soc.Rheol., vol.7, pp. 209-230, 1963.\n[16] G.Bugliarello and J.Sevilla, Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes, Biorheology, vol.7, pp. 85-107,1970.\n[17] J.B.Shukla, R.S.Parihar and B.R.P.Rao BRP, Effect of peripheral layer viscosity on blood flow through the artery with mild stenosis, Bull.Math.Biol., vol.42, pp. 797-805, 1980.\n[18] J.B. Shukla, R.S. Parihar and B.R.P.Rao, Biorheological aspects of blood flow through artery with mild stenosis: Effect of peripherial layer, Biorheology, vol.17, pp. 403-410, 1980.\n[19] P.Chaturani and P.N. Kaloni, Two-layered poiseuille flow model for blood flow through arteries of small diameter and arterioles, Biorheology, vol.13, pp. 243-250, 1976.\n[20] P. Chaturani and R.Ponalagusamy, A two-layered model for blood flow through stenosed arteries, Proc. Of 11th National Conf. on fluid mechanics and fluid power,B.H.E.L.(R and D),Hydrabad, India, pp. 6-22, 1982.\n[21] R.Ponalagusamy and R.Tamil Selvi, A study on two layered model (Casson-Newtonian) for blood through an arterial stenosis: Axially variable slip velocity at the wall, J.Franklin Inst., vol.348, pp. 2308-2321, 2011.\n[22] H. Alemayehu and G. Radhakrishnamacharya, Dispersion of solute in peristaltic motion of couple stress fluid in the presence of magnetic field, Int. J of Engineering and Appl. Sciences., vol.7, pp. 156-160, 2011.\n[23] B. S. Bhatt and N. C. Sacheti, On the analogy in slip flows, Indian Journal of Pure and Applied Mathematics., vol.10, pp. 303-306, 1979.\n[24] K. Maruthi Prasad and G.Radhakrishnamacharya, Flow of Herschel-Bulkley fluid through an inclined tube of non-uniform cross-section with multiple stenosis, Arch. Mech., vol.60, pp. 161-172, 2008."]}

The paper deals with the pulsating flow of an incompressible couple stress fluid between permeable beds. The couple stress fluid is injected into the channel from the lower permeable bed with a certain velocity and is sucked into the upper permeable bed with the same velocity. The flow between the permeable beds is assumed to be governed by couple stress fluid flow equations of V. K. Stokes and that in the permeable regions by Darcy-s law. The equations are solved analytically and the expressions for velocity and volume flux are obtained. The effects of the material parameters are studied numerically and the results are presented through graphs., {"references":["D.N.Ku, D.P.Giddens, C.K. Zairns, S. Glagov, Pulsatile flow and\natherosclerosis in human carotid bifurcation: positive correlation between\nplaque location and low and oscillating shear stress, Arteriosclerosis 5\n(1985) 293-302.","R.M.Nerem, M.J.Levesque, Hemodynamics and the arterial wall, Vasc.\nDisc. (1987) 295-317.","F.Fedele, D.Hitt, R.D.Prabhu, Revisiting the stability of pulsatile pipe\nflow, European J. of Mech. ÔÇö B/Fluids 24 (2005) 237-254.","Y.C. Wang, Pulsatile flow in a porous channel, J. Appl. Mech. 38 (1971)\n553-555.","K. Vajravelu, K. Ramesh, S. Sreenadh, P.V. Arunachalam, Pulsatile flow\nbetween permeable beds, Int. J. Non-Linear Mech. 38 (2003) 999-1005","G.S. Beavers, D.D.Joseph : Boundary conditions at a naturally permeable\nwall. J. Fluid Mech., Vol 30, (1967) 197-207.","V.K.Stokes, Couple Stresses in Fluids, Phys. Fluids., Vol 9,(1966) 1709-\n1715.","V.K.Stokes, Theories of Fluids with Microstructure, Springer-Verlag,\nBerlin 1984.","S.K.Lakshmana Rao and T.K.V.Iyengar, Analytical and computational\nstudies in couple stress fluid flows, U.G.C. Research project C-8-4/82\nSR III, (1985)\n[10] L.M.Srivastava,Flow of couple stresss fluid through stenotic blood\nvessels, J.Bio Mech., Vol 18, 479-485 (1985)\n[11] L.M.Srivastava, Perstaltic transport of a couple stresss fluid, Rheo.Acta,\nVol 25, 638-641 (1986).\n[12] N.B.Naduvinamani, P.S.Hiremath and G.Gurubasavaraj : Squeeze film\nlubrication of a short porous journal bearing with couple stress fluids,\nTribo.Inter.,Vol 34, 739-747 (2001).\n[13] N.B.Naduvinamani, P.S.Hiremath and G.Gurubasavaraj : Surface roughness\neffects in a short porous journal bearing with couple stress fluid,\nFluid Dyn. Res.,Vol 31, 333-354 (2002).\n[14] N.B.Naduvinamani,Syeda Taseem Fathima and P.S.Hiremath: Effects of\nsurface roughness on characteristics of couple stress squeeze film between\nanisotropic porous rectangular plates, Fluid Dyn. Res., Vol 32, 217-231\n(2003).\n[15] M.Devakar and T.K.V.Iyengar, Stoke-s problems for an incompressible\ncouple stress fluid, Nonlinear Analysis: Modeling and Control, Vol 1,\n181-190 (2008).\n[16] M.Devakar and T.K.V.Iyengar, Runup flow a couple stress fluid between\nparallel plates , Nonlinear Analysis: Modeling and Control, Vol 15, 29-37\n(2010).\n[17] T.S.L.Radhika and T.K.V.Iyengar, Stokes flow of an incompressible couple\nstress fluid past a porous spherical shell, Proceedings of International\nMulti conference of Engineers and Computer Scientists, Vol 3, 1634-1639\n(2010)"]}

This present investigation carried out the effects of Joule heating and Hall current on free convection flow of electrically conducting couple stress fluid in a vertical channel. The final system of ordinary differential equations is obtained from the governing non- linear partial differential equations by using the similarity transformations. Homotopy Analysis Method has been used to solve the non-linear system. The influence of the emerging parameters (Hall, magnetic, Joule heating and the couple stress parameters) on velocity and temperature profiles are presented through plots.

This paper aims at presenting exact solutions for MHD flow of couple stress fluid with heat transfer. The governing partial differential equations (PDEs) for an incompressible MHD flow of couple stress fluid are reduced to ordinary differential equations by employing wave parameter. The methodology is implemented for linearizing the flow equations without extra transformation and restrictive assumptions. Comparison is made with the result obtained previously.

In this paper, we have studied the flow of incompressible fluids in a straight square duct through the porous medium. The couple stress fluid model and Jeffrey fluid model are considered separately to study the flow properties. The governing partial differential equations have been solved numerically using finite difference method in each case. In both the cases, the variation of different flow parameters on the fluid velocity is illustrated graphically and the numerical results for the volume flow rate have been presented through tables. It is observed that, the velocity and volume flow rate decrease with an increase in couple stress parameter and porosity parameter, while the velocity and volume flow rate increase with an increase in Jeffrey parameter and pressure gradient.

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.

In the present paper, the pulsatile flow of blood through a bifurcated artery with mild stenosis in parent artery is investigated by taking blood as couple stress fluid. The arteries pattern of bifurcation is treated to be symmetric about the axis of the parent artery and straight circular cylinders of limited length. The governing equations are made dimensionless and suitable coordinate transformation is used to convert the irregular boundary to a well designed boundary. The resulting system of equations is solved numerically using the finite difference method. The influence of physical parameters on the velocity, shear stress, flow rate and impedance near the apex is studied graphically. Further, the oscillatory nature of impedance, flow rate and shear stress against time near the apex in both parent and daughter arteries is presented. Keywords: Blood flow, Pulsatile flow, Couple stress fluid, Bifurcated artery, Impedance