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Here, unsteady boundary layer flow under the action of magnetic effect over a moving stagnation surface has been investigated numerically. This study examines non-Newtonian fluid flow with respect to combined effect of magnetic field, movement of surface and time. Governing equations are dimensionless by applying nondimensional quantities. Then system of coupled ordinary differential equations are obtained by appropriate similarly transformations, in terms of the governing parameter including unsteady parameter ([Formula: see text], magnetic number ([Formula: see text], power-law index ([Formula: see text] and three-dimensional parameter ([Formula: see text]. Thus, the obtained modelled equations are solved numerically by the shooting technique. The problem’s parameters are thoroughly discussed and verified physically and graphically. The obtained results validate to literature result. The two relevant flows’ velocity profile’s numerical results have been examined.

The characteristics of magnetohydrodynamics hybrid nanofluid flow in an irregular wall are explored theoretically. Since the movement of nanoparticles significantly impacts the flow, the Brownian motion and thermophoresis effect are incorporated in this investigation. The flow model is formulated through the non-dimensionalized nonlinear partial differential equations which are solved by means of the finite difference method. Using a judicial combination of quasilinearization technique and Thomas algorithm, the system of equations is solved for the velocity variation, heat transfer, and concentration of hybrid nanofluid. The range of parameters considered in the study is varied as required to obtain the physically meaningful flow situations. The present study is restricted only to positive values of the physical parameters. The wall shear stress, heat, and mass transfer gradients on the surface are also computed. The various results show that the effect of the magnetic field makes the boundary layer thinner and enhances the heat transfer rate. It is also found that the thermophoresis effect and Brownian motion are capable to reduce friction near the wavy wall. In addition, deduction in the values of the magnetic parameter, the volume fraction of nanoparticles, and amplitude of wavy wall result into enhancement in the heat transfer rate. The physical dynamics behind these interesting result is discussed in detail.

This paper investigates the linear stability of the flow in the two-dimensional boundary-layer flow of the Carreau fluid over a wedge. The corresponding rheology is analysed using the non-Newtonian Carreau fluid. Both mainstream and wedge velocities are approximated in terms of the power of distance from the leading edge of the boundary layer. These forms exhibit a class of similarity flows for the Carreau fluid. The governing equations are derived from the theory of a non-Newtonian fluid which are converted into an ordinary differential equation. We use the Chebyshev collocation and shooting techniques for the solution of governing equations. Numerical results show that the viscosity modification due to Carreau fluid makes the boundary layer thickness thinner. Numerical results predict an additional solution for the same set of parameters. Thus, a further aim was to assess the stability of dual solutions as to which of the solutions can be realized. This leads to an eigenvalue problem in which the positive eigenvalues are important and intriguing. The results from eigenvalues form tongue-like structures which are rather new. The presence of the tongue means that flow becomes unstable beyond the critical value when the velocity ratio is increased from the first solution.

The two-dimensional laminar boundary-layer flow of a viscous fluid and heat transfer is analyzed at arbitrary values of the exponent wall temperature and Prandtl number. This study is known to occur in several important classes of boundary-layer flow. The fluid flow outside the boundary-layer is approximated by a power of the distance from a leading edge and the flow is on an impermeable wedge. The steady non-dimensional partial differential equations with the physical boundary conditions are converted to a system of ordinary differential equations. Two methods are employed: the full nonlinear equations are solved numerically using the Keller-box method, and in order to support these results, we solve the linear system in an asymptotic limit. Both approaches give good agreement with each other in predicting the wall shear stress, the temperature gradient, velocity and temperature profiles. Our results show that there is a mutual interaction between wedge surface and fluid and the coupling between momentum and thermal boundary layers. It is observed further that, both thermal and momentum boundary layer thicknesses are found to be thinner for accelerated flows and smaller values of the Prandtl number. The results on wall shear stress and temperature gradients are found to decrease initially and become flatter as the Prandtl number increases. The present work shows that the results have various technological applications involving these kinds of flow.

We study the instability of plane Poiseuille flow of the viscoelastic second-order fluid in a homogeneous porous medium. The viscoelastic fluid between two parallel plates is driven by the pressure gradient. The effects of elasticity number E (depends on fluid properties, geometry; E is defined below) and Darcy number Da (gives the permeability of porous medium; Da is defined below) on flow stability are analyzed through the energy method that provides qualitative behavior of flow stability, and the numerical solution of generalized eigenvalue problem that gives the precise upper bound for stability. The plane Poiseuille flow of second-order fluid becomes unstable for increasing elasticity number while preserving Newtonian eigenspectrum up to a certain range of E. For large elasticity number, instability appears as a part of both wall and center modes for all Darcy numbers. We also noticed that along each neutral stability curve, the eigenfunctions are all antisymmetric with a single extremum near the channel walls. When E = 0.0011, we found an additional new elastic mode, which is unstable and also antisymmetric. For E c is found to be decreasing (increasing) for higher (lower) values of fluid elasticity (Darcy number). Physical mechanisms are discussed in detail.

This work examines the steady two-dimensional mixed convection boundary layer flow of non-Newtonian Carreau fluid embedded in a porous medium. The impermeable wedge is at rest over which the momentum and thermal boundary layers form due to the motion of Carreau fluid with a large Reynolds number. We consider local thermal nonequilibrium for which the temperature of the solid porous medium is different from that of the fluid phase, and hence, a single heat-transport equation is replaced by a two-temperature model. The governed equations for flow and heat transfer are converted into a system of ordinary differential equations using a similarity approach. It is observed that local thermal nonequilibrium effects are dominant for small interphase heat transfer rate and porosity scaled conductivity parameters. It is shown that the temperature at any location of the solid porous medium is always higher than that of the fluid phase. When these parameters are increased gradually, the local thermal equilibrium phase is recovered at which the temperatures of the fluid and solid are identical at each pore. A similar trend is noticed for both shear-thinning and shear-thickening fluids. The results further show that heat exchange between the fluid and solid porous medium is similar to both assisted and opposed flows and Carreau fluid. The velocity and temperature fields for the various increasing fluid index, Grashof number, and permeability show that the thickness of the momentum and thermal boundary layer is thinner.

This paper studies the local thermal nonequilibrium (LTNE) model for two-dimensional mixed convection boundary-layer flow over a wedge, which is embedded in a porous medium in the presence of radiation and viscous dissipation. It is considered that the temperature of the fluid and solid phases is not identical; hence, we require two energy equations: one for each phase. The motion of the mainstream and wedge is approximated by the power of distance from the leading boundary layer. The flow and heat transfer in the LTNE phase is governed by the coupled partial differential equations, which are then reduced to nonlinear ordinary differential equations via suitable similarity transformations. Numerical simulations show that when the interphase rate of heat transfer is large, the system attains the local thermal equilibrium (LTE) state and so is for porosity scaled conductivity. When LTNE is strong, the fluid phase reacts faster to the mainstream temperature than the corresponding solid phase. The state of LTE rather depends on radiation and viscous dissipation of the model. Further, numerical solutions successfully predicted the upper and lower branch solutions when the velocity ratio is varied. To assess which of these solutions is practically realizable, an asymptotic analysis on unsteady perturbations for a large time leading to linear stability needs to be performed. This shows that the upper branch solutions are always stable and practically realizable. The physical dynamics behind these results are discussed in detail.

This paper studies the linear stability of the unsteady boundary-layer flow and heat transfer over a moving wedge. Both mainstream flow outside the boundary layer and the wedge velocities are approximated by the power of the distance along the wedge wall. In a similar manner, the temperature of the wedge is approximated by the power of the distance that leads to a wall exponent temperature parameter. The governing boundary layer equations admit a class of self-similar solutions under these approximations. The Chebyshev collocation and shooting methods are utilized to predict the upper and lower branch solutions for various parameters. For these two solutions, the velocity, temperature profiles, wall shear-stress, and temperature gradient are entirely different and need to be assessed for their stability as to which of these solutions is practically realizable. It is shown that algebraically growing steady solutions do exist and their effects are significant in the unsteady context. The resulting eigenvalue problem determines whether or not the steady solutions are stable. There are interesting results that are linked to bypass an important class of boundary layer flow and heat transfer. The hydrodynamics behind these results are discussed in some detail.

Purpose This paper aims to present a detailed analysis to explore the various properties of non-Newtonian couple stress lubricants between parallel porous plates. Design/methodology/approach With reference to the theories based on micro-continuum analysis, a non-linear, non-Newtonian Reynolds type equation is arrived. The closed form solutions obtained clearly indicate the changes in pressure, load bearing capacity and response time because of variation in viscosity of couple stress fluid. Findings It is observed that the viscosity variation factor greatly influences the change in pressure, load carrying capacity and squeezing time. Originality/value It is observed that the nature of lubricants with suitable additives greatly helps in overcoming the adverse effect because of porous surface. Reynolds type equation is analysed using appropriate boundary conditions. The expression for pressure distribution arrived at in turn leads to the analysis of load bearing capacity and response time.

Both numerical and asymptotic analyses are performed to study the similarity solutions of three-dimensional boundary-layer viscous stagnation point flow in the presence of a uniform magnetic field. The three-dimensional boundary-layer is analyzed in a non-axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller-box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far-field behavior and in the limit of large shear-to-strain-rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near-field (due to viscous forces) and far-field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary-layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.

This paper concerns the LTNE model for two-dimensional mixed convection boundary layer flow of a viscous fluid over a wedge embedded in the porous medium. The temperature for the fluid and solid field is governed by the two separate energy equations: one each for fluid and solid phase due to LTNE condition. Since the flow velocity is sufficiently large, the momentum and thermal boundary layers form near the wedge surface. The resulting mathematical equations in terms of partial differential equations are converted into equivalent ordinary differential equations using appropriate similarity transformations and are solved numerically. Numerical results on various physical quantities reveal that the LTNE effects are more predominant for small inter-phase heat transfer rate and porosity scaled conductivity. For mixed convection thermal boundary layer flow, LTNE effects are not as strong as for forced convection heat transfer. However, when the inter-phase interaction and porosity-scaled conductivity are increased, the LTE is recovered for which both solid and fluid phases become almost equal. Simulations on velocity profiles show that the pressure gradient and permeability have less effects on the boundary layer but cannot be neglected. The physical dynamics behind these effects are discussed in some detail.

In this paper, we propose an analytic solution of a boundary value problem which models a steady, laminar, two dimensional, boundary layer flow of an incompressible and viscous micropolar fluid over a moving wedge. The governing non-linear partial differential equations are converted into highly non-linear ordinary differential equations using similarity transformations. An analytical exact solution obtained for particular values of parameters are then extended to obtain an exact solution for more general values of the parameters involved. We also propose asymptotic solution of the Micropolar boundary layer flow. The results thus obtained are compared with those of direct numerical solutions, which show a good agreement. The results are discussed in terms of velocity profiles and wall shear stresses for various physical parameters.

The present theoretical study investigates the effects of surface roughness and couple-stress fluid between two rectangular plates, of which an upper rough plate has a roughness structure and the lower plate has a porous material in the presence of transverse magnetic field. The lubricant in the gap is taken to be a viscous, incompressible, and electrically conducting couple-stress fluid. This gap is separated by a film thickness H which is made up of nominal smooth part and rough part. The modified Reynolds equation in the film region is derived for one-dimensional longitudinal roughness structure and solved numerically using multigrid method. The numerical results for various physical parameters are discussed in terms of pressure distribution, load capacity, and squeeze film time of the bearing surfaces. Our results show that, the pressure distribution, load capacity and squeeze film time are predominant for larger values of Hartman number and roughness parameter, and for smaller values of couple-stress parameters when compared to their corresponding classical cases.

In this paper, we study squeeze film lubrication characteristics between two rectangular plates, of which the lower plate is porous and the upper one has a roughness structure. The fluid in the film region is represented by a viscous, incompressible couple-stress fluid. The governing Reynolds equation, which incorporates the couple-stress fluid, roughness, and porosity of the material, is solved numerically using the multigrid method. Results show that the effects of couplestress fluid and roughness are more pronounced compared to the Newtonian fluid and smooth case, respectively, whereas the effect of permeability is to decrease the pressure distribution and load capacity. © 2013 by Begell House, Inc.

In this paper, we give an exact analytical solution of the Falkner–Skan equation for all values of β . Generalized similarity transformations are used to convert the Prandtl's boundary layer equations into a non-linear ordinary differential equation which accounts two important flow parameters: the pressure gradient parameter β and velocity ratio parameter ϵ . Our exact solution method embeds a known closed-form solution for β = − 1 as a special case. We also give the Dirichlet's series solution to the problem for ϵ = 0 , which is particularly useful when the derivative boundary condition at infinity is zero. We compare the results of both methods with that of direct numerical solution, and found that there is a good agreement between both the results. The results are presented in the form of velocity profiles and skin friction coefficient. Finally, the physical significance of the flow parameters is discussed in detail.

We study the linear stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension ${T}^{\ensuremath{\ast} } $. Far upstream the flow is parallel Poiseuille flow at Reynolds number $\mathit{Re}$; the width of the channel is $a$ and the length of the membrane is $\lambda a$, where $1\ll {\mathit{Re}}^{1/ 7} \lesssim \lambda \ll \mathit{Re}$. Steady flow was studied using interactive boundary-layer theory by Guneratne & Pedley (J. Fluid Mech., vol. 569, 2006, pp. 151–184) for various values of the pressure difference ${P}_{e} $ across the membrane at its upstream end. Here unsteady interactive boundary-layer theory is used to investigate the stability of the trivial steady solution for ${P}_{e} = 0$. An unexpected finding is that the flow is always unstable, with a growth rate that increases with ${T}^{\ensuremath{\ast} } $. In other words, the stability problem is ill-posed. However, when the pressure difference is held fixed (${= }0$) at the downstream end of the membrane, or a little further downstream, the problem is well-posed and all solutions are stable. The physical mechanisms underlying these findings are explored using a simple inviscid model; the crucial factor in the fluid dynamics is the vorticity gradient across the incoming Poiseuille flow.

For weakly hyperbolic heat equation a numerical scheme based on multiple scale technique is derived. The advantages over the conventional finite difference method are highlighted.

The Falkner–Skan equation, subject to appropriate physical boundary conditions arising from boundary layer theory, is exactly solved. The results obtained from this solution are compared with the numerical solution. The Blasius equation, subject to the same boundary conditions, is also solved exactly; the solution is compared with the earlier work on this equation. The analytic solution presented here agrees closely with the corresponding numerical results.

A simplified mathematical model has been developed for understanding the combined effects of surface roughness and couple stresses on lubrication aspects of synovial joints. The modified Reynolds equation which incorporates the elastic as well as randomized surface roughness structure of cartilage with couple-stress fluid as lubricant is derived. The mean pressure, load carrying capacity and time of approach as functions of film thickness during normal articulation of joints are obtained by using Christensen stochastic theory with the assumption that the roughness asperity heights are to be small compared to the film thickness. The effects of surface roughness and elasticity are considerably pronounced for the poroelastic bearings with couple-stress fluid as lubricant compared with classical case.

The combined study of effects of poroelasticity and couple stresses on the performance of lubrication aspects of porelastic bearings in general and that of synovial joints in particular are analyzed. The modified form of Reynolds equation which incorporates the elastic nature of cartilage and Stokes couple-stress fluid as lubricant is derived and solved using a recently developed wavelet multigrid method. This method has greatest advantage of minimizing the errors using wavelet transforms in obtaining accurate solution as grid size tends to zero. It is found that, 6–7 cycles are required to obtain a reasonably accurate solution in the multigrid scheme, whereas, only one cycle is required to obtain the solution in the wavelet-multigrid method. Also, matrix of discrete wavelet-transform acts as a natural preconditioner producing rapid convergence. It is observed that, the poroelastic bearings with couple-stress fluid as lubricant provide enhancement in pressure and ensure the increased load carrying capacity compared with viscous fluids. This may be one of the reasons in the efficient lubrication and proper functioning of synovial joints.

The effects of surface roughness on the squeeze film behaviour between two rectangular plates with an electrically conducting fluid in the presence of a transverse magnetic field are analysed. The modified Reynolds equation which incorporates randomized roughness structure with magnetic field is derived by assuming the roughness asperity heights to be small compared to the film thickness. The finite difference based multigrid method is used for the solution of the modified Reynolds equation. The method has the advantage of minimizing the errors using correction schemes in obtaining an accurate solution as grid size goes to zero. Lubrication characteristics which are functions of roughness, Hartmann number and aspect ratio parameter are obtained. It is observed that the effect of roughness and magnetic field is to provide a significant load carrying capacity and ensure a delayed squeezing time compared to a classical case.