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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.

This paper presents our recent work on investigating velocity slip boundary conditions’ effects on supersonic flat plate boundary layer flow stability. The velocity-slip boundary conditions are adopted and the flow properties are obtained by solving boundary layer equations. Stability analysis of two such boundary layer flows is performed by using the Linear stability theory. A global method is first utilized to obtain approximate discrete mode values. A local method is then utilized to refine these mode values. All the modes in these two scenarios have been tracked upstream-wisely towards the leading edge and also downstream-wisely. The mode values for the no-slip flows agree well with the corresponding past results in the literature. For flows with slip boundary conditions, a stable and an unstable modes are detected. Mode tracking work is performed and the results illustrate that the resonance phenomenon between the stable and unstable modes is delayed with slip boundary conditions. The enforcement of the slip boundary conditions also shortens the unstable mode region. As to the conventional second mode, flows with slip boundary conditions can be more stable streamwisely when compared with the results for corresponding nonslip flows.

We consider nonlinear Boussinesq-type equations that model the heat transfer and steady viscous flows of weakly concentrated water solutions of polymers in a bounded three-dimensional domain with a heat source. On the boundary of the flow domain, the impermeability condition and a slip condition are provided. For the temperature field, we use a Robin boundary condition corresponding to the classical Newton law of cooling. By using the Galerkin method with special total sequences in suitable function spaces, we prove the existence of a weak solution to this boundary-value problem, assuming that the heat source intensity is bounded. Moreover, some estimates are established for weak solutions.

In this paper, we develop an efficient lattice Boltzmann model for the two-phase moving contact line problem with variable density. The Navier–Stokes and Cahn–Hilliard equations are recovered from the lattice Boltzmann model proposed by Fakhari and Rahimian [5] . To improve numerical stability, we present a semi-implicit lattice Boltzmann method together with a mixed finite difference scheme. In order to describe the behavior of the contact line motion on the boundary, we incorporate the generalized Navier boundary condition [25] by the nonequilibrium extrapolation method [8] . The proposed method is easy to implement and retains the advantage of the standard lattice Boltzmann method. Numerical tests are carried out to verify the proposed method. Our numerical results show that the present approach is able to model two-phase flows with variable density and moving contact line. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem for a model of motion of aqueous polymer solutions in a bounded three-dimensional domain subject to the Navier slip boundary condition. We construct a global (in time) weak solution to this problem. Moreover, we establish some uniqueness results, assuming additional regularity for weak solutions. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Ballistic–diffusive heat conduction, which is predominantly affected by boundaries and interfaces, will occur in nanostructures whose characteristic lengths are comparable to the phonon mean free path (MFP). Here, we demonstrated that interactions between phonons and boundaries (or interfaces) could lead to two kinds of slip boundary conditions in the ballistic–diffusive regime: boundary temperature jump and boundary heat flux slip. The phonon Boltzmann transport equation (BTE) with relaxation time approximation and the phonon tracing Monte Carlo (MC) method were used to investigate these two slip boundary conditions for the ballistic–diffusive heat conduction in nanofilms on a substrate. For cross-plane heat conduction where the boundary temperature jump is the dominant non-Fourier phenomenon, ballistic transport causes the temperature jumps and thus introduces a ballistic thermal resistance. Importantly, when considering the interface effect, the corresponding model was derived based on the phonon BTE and verified by comparing with the MC simulations. In addition, an interface–ballistic coupling effect was identified, which indicates inapplicability of the standard thermal resistance analysis. In contrast, for the in-plane case that is controlled by boundary heat flux slip, both phonon boundary scattering and perturbation of the phonon distribution function induced by the interface can cause heat flux slip, leading to a variation in in-plane thermal resistance. In addition, a model beyond the Fuchs-Sondheimer formula, which can address both the boundary scattering and the interface effects, was derived based on the phonon BTE. The good agreements with the MC simulations indicate its validity. [ABSTRACT FROM AUTHOR]

A lattice Boltzmann bounce-back scheme that can simulate the first-order slip boundary condition at a fluid-solid interface is derived and verified with analytical and asymptotic solutions for shear and pressure driven flows in channels, through square arrays of cylinders, and in a Couette cell. We then simulated Stokes flows with surface slip through simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) arrays of spheres and established the slip (apparent) permeability of these arrays as a function of the no-slip (absolute) permeability, the slip length and the porosity. It is found that k s / k ns , the ratio between the slip permeability and the no-slip permeability, increases with increasing slip length and increases with decreasing porosity owning to enhancement of flow through the gaps between particles. Interestingly, k s / k ns is nearly independent of sphere configuration when the dimensionless slip length is small; with increasing dimensionless slip length, however, different sphere configurations start to have different k s / k ns . The dependence of k s / k ns on the slip length is slightly nonlinear owning to the curvature of the surface and this dependence can be correlated by quadratic equations. [ABSTRACT FROM AUTHOR]

The solvability and vanishing viscosity limit for three dimensional incompressible Navier-Stokes equations with a slip boundary condition were obtained. The proof of these results is based on standard energy estimates. [ABSTRACT FROM AUTHOR]

In this paper, using mixture theory we study the flow of a dense suspension, composed of solid particles and a fluid; the emphasis is on the influence of the slip boundary condition and the effect of normal stress differences. Very little work has been done considering both the slip at the walls and the normal stress effects in the frame of a two-component flow. In this paper, the stress tensor for the solid component is modeled as a nonlinear fluid which not only includes the viscous effects but also the normal stress effects; the fluid constituent is modeled as a viscous fluid. We look at the flow between two flat plates. [ABSTRACT FROM AUTHOR]

An attempt is made to study a steady two-dimensional flow of a viscous incompressible fluid incident at some angle onto a plate lubricated with a thin layer of a power-law fluid. Similar and nonsimilar solutions of the governing partial differential equations are obtained numerically by imposing the continuity of velocity and shear stress at the interface layer between the fluid and the lubricant. The Keller box method is applied to obtain the solutions. The limiting cases for full and no-slip conditions are compared. [ABSTRACT FROM AUTHOR]

Effects of slip at the boundary on the mixed convective boundary layer flow over a flat plate are investigated. Similarity transformations are employed to transform the governing partial differential equations into ordinary ones, which are then solved numerically by shooting method. The results are presented for different values of the governing parameters. Comparison with available results for certain cases is excellent. For increasing slip parameter, velocity increases at first but decreases after a point and thermal boundary layer becomes thinner in this case. Velocity overshoot and temperature overshoot are noted near the plate for increasing mixed convection parameter as well as both velocity and thermal slip parameters and also for Prandtl number.

Gas flows in micro-channels are, in general, theoretically treated with the Maxwell slip velocity as a boundary condition for the convection velocity at the wall. It is pointed out that wall slip is conventionally introduced in theoretical/numerical treatments of gas flows through micro-channels to obtain agreement with experimental results. In the present paper, we provide an alternative by solving the extended Navier-Stokes equations for compressible gas flows in micro-channels using the conventional no-slip velocity boundary condition for the convection velocity. Results obtained with this approach are presented and compared with experiments. It is also shown that the theoretical treatment of micro-channel gas flows using the “extended Navier-Stokes equations” also permits the phenomena such as the Knudsen paradox, to be treated in an analytical manner. Comparison with experimental data suggests that the derived analytical solution has excellent agreement up to Knudsen number of approximately 1, which shows the validity of extended Navier-Stokes equations with the conventional no-slip velocity boundary condition up to the early transition regime of micro-channel gas flows.

• The effects of adding TiO 2 and Al 2 O 3 nanoparticles to water on heat transfer and fluid flow characteristics are investigated. • Semi-analytical method was used. • Two nanofluids flow and heat transfer in a microtube are studied in presence of slip boundary conditions and magnetic fields. • The Nusselt number maximum augmentation of 33% is reported by applying magnetic field. According to the increasing desire towards heat transfer rate enhancement with various methods in different industries, this has been an interesting topic of many researchers. The effects of adding TiO 2 and Al 2 O 3 nanoparticles to water on heat transfer and fluid flow characteristics are investigated with a semi-analytical method in this study. Two nanofluids flow and heat transfer in a microtube are studied in presence of slip boundary conditions and magnetic fields. The nanoparticles volume fraction, magnetic fields strength, slip parameter and N BT coefficient (the ratio of Brownian to thermophoretic diffusions) effects on the nanoparticle's distribution, velocity profile, temperature distribution and heat transfer coefficient are studied. Furthermore, using the thermophysical characteristics of two nanofluids containing TiO2 and Al2O3 nanoparticles, their behavior under different conditions has been investigated. Results are compared with those of other researchers and acceptable similarity is obtained. It was observed that the heat transfer coefficient is increased by the NBT coefficient, magnetic field strength, and slip parameter, and it decrease by the nanoparticles volume fraction. The Nusselt number maximum augmentation of 33% is reported by applying magnetic field. The heat transfer coefficient average for Al 2 O 3 /water nanofluid is approximately 20% more than the average of TiO 2 /water nanofluid. [ABSTRACT FROM AUTHOR]

Electrical power generation via squeezing liquid motion is theoretically analyzed. Instead of squeezing discrete liquid droplets as extensively investigated in recent literature, we consider extracting the electrokinetic energy resulting from the streaming potential/current phenomenon by squeezing continuous liquid flows in an arbitrary number of slit channels arranged in parallel with an external load resistance and subjected to wall boundary slip. Analytical solutions to the instantaneous energy conversion efficiency, load current, and squeezing force and speed are derived assuming quasi-static, unidirectional flow conditions for the two squeezing modes of constant squeezing speed and force. Electrokinetic energy conversion efficiencies predicted by our present linear analysis for the two squeezing modes are found to be mathematically the same despite the drastic differences found in the load current and squeezing force-speed responses between the two modes. Parametric studies reveal that energy conversion efficiencies above 30% can be attained by introducing boundary slip, and that the value of the external load resistance at which the energy conversion efficiency approaches to maximum varies with respect to variations in the slip length, number of slit channels, and the slit channel geometry. Depending on the parametric conditions given, the electrokinetic energy conversion efficiency can either be increased or decreased as the separation distance between the slit channel parallel plates is reduced over the duration of the squeezing motion. Conversion efficiencies are also discussed in terms of the Debye parameter, liquid conductivity, and surface conductance for the electrokinetic power generator considered herein. [ABSTRACT FROM AUTHOR]

We investigate the system of nonlinear partial differential equations governing the unsteady motion of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain under Navier's slip boundary condition. We prove the existence of global weak solutions for the corresponding initial-boundary value problem without assuming that the model constants, body force or the initial values of the velocity and the stress tensor are small. [ABSTRACT FROM AUTHOR]

The pulsatile electroosmotic flow (PEOF) of a Newtonian fluid in a circular microchannel with slippage at the surface is theoretically analyzed. It is assumed that the electroosmotic flow is started from rest by the sudden imposition of a time-dependent external electric field. The PEOF is controlled by the following dimensionless parameters: the normalized slip length, δ , defined as the ratio of the Navier length to the microcapillary radius; the angular Reynolds number, R ω , which quantifies the competition between the time scale for the diffusion of momentum across the microcapillary and the externally imposed time scale due to the oscillatory electric field; the parameter ɛ , which characterizes the amplitude of the oscillating electric field; the electrokinetic parameter κ ¯ , defined as the ratio of the characteristic length scale to the Debye length; and the ionic energy parameter, α , which compares the electric potential at the surface to the thermal potential. The analysis is conducted for low and high zeta potentials; in the former case, an analytical solution is obtained, whereas in the latter, a numerical solution is found. The results reveal that the volumetric flow rate is notably increased in a microchannel with slippage at the wall surface. [ABSTRACT FROM AUTHOR]

This paper is concerned with large time behavior of solutions to the compressible Navier-Stokes equations in an infinite layer of under slip boundary condition. It is shown that if the initial data is sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one-dimensional diffusion waves. [ABSTRACT FROM AUTHOR]

The two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling (for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling () is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes. [ABSTRACT FROM PUBLISHER]

We address the effect of wall slip on the dynamics of a spherical particle suspended in an inertialess Newtonian or viscoelastic shear-thinning fluid under shear or Poiseuille flow. The study is performed through 3D direct finite element simulations by employing an Arbitrary Lagrangian-Eulerian method for the particle motion. In both shear and Poiseuille flows, wall slip reduces the difference between the particle translational velocity along the flow direction and the velocity of the unperturbed fluid, and slows down the particle rotational velocity. Remarkably, in a viscoelastic fluid, the presence of wall slip reverses the migration direction as compared to the no-slip case. Hence, for sufficiently high slip coefficients, all the particles migrate toward the channel midplane in shear flow and toward the channel centerline in Poiseuille flow, regardless of their initial position through the channel. [ABSTRACT FROM AUTHOR]

Direct numerical simulation (DNS) of a turbulent channel flow with the slip boundary condition was performed to investigate the spatial characteristics of large-scale motions (LSMs) in a drag-reducing flow and the influence of LSMs on drag reduction. The slip boundary condition was applied in the streamwise direction. DNS of a channel flow with the no-slip boundary condition was also conducted for comparison. The overall mean skin-friction coefficient of the flow with the slip boundary condition was reduced by 35% relative to the flow with the no-slip condition at the same bulk Reynolds number of 10,333. The streamwise and spanwise length scales of the LSMs in the outer region are larger with the slip boundary condition than with the no-slip condition due to a large population of long LSMs. These LSMs influence the near-wall region via roll-cell motions and footprints. Although the number of total streaks decreases, a portion of the footprints increases due to the influence of the outer LSMs. The streamwise vorticities around the low-speed footprints are attenuated more than those around the high-speed footprints. The LSMs and the near-wall footprints contribute 45% of the total reduction in the skin friction coefficient, with 25% of this contribution due to low-speed LSMs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this study, the combined effects of velocity slip and convective heating, Brownian motion and thermophoresis on the magnetohydrodynamic stagnation point flow, and heat transfer of a power-law nanofluid over a stretching sheet is investigated. Using appropriate similarity transformation and the Runge–Kutta–Fehlberg fourth fifth order with shooting technique, the model nonlinear governing differential equations are obtained and solved numerically. The effect of variations of parameters on nanofluid velocity, temperature, concentration, skin friction coefficient, local Nusselt number, and local Sherwood number are presented and discussed. It is found that the skin friction coefficient is an increasing function of a power-law index n and a decreasing function of the slip parameter δ. Both local Nusselt number and local Sherwood number are increasing function of a power-law index. Increase in Biot number increases local Nusselt number but decreases the local Sherwood number. The results of the study are compared with the previously published result and are found to be in excellent agreement with it. [ABSTRACT FROM AUTHOR]

The main emphasis of the present study is to discuss the effect of chemical reaction on flow, heat, and mass transfer of Williamson nanofluid over a stretching sheet. The flow is considered under the influence of nonlinear thermal radiation. To obtain the numerical results, the governing equations have been reduced to a set of nonlinear ordinary differential equations using appropriate similarity transformations. An efficient Runge-Kutta-Felhberg 45 order method along with a shooting technique is used to get the required solutions. The accuracy of the method used is verified with the existing results, and they are found to be in good agreement. Obtained numerical solutions are presented in the form of graphs and tables for a various range of slip boundary condition and for different values of flow pertinent parameters, such as permeability parameter, radiation parameter, Lewis number, Heat capacities ratio, diffusivity ratio, Schmidt number, chemical reaction parameter, volumetric volume expansion coefficient, and Prandtl number. Finally, the outcome of the problem is written in the form of a conclusion based on the tables and plotted graphs. [ABSTRACT FROM AUTHOR]

We consider the penalty method for the stationary Navier-Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate $$O(\epsilon )$$ in $$H^k$$ -norm, where $$\epsilon $$ is the penalty parameter. We are concerned with the finite element approximation with the P1 b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate $$O(h+\sqrt{\epsilon }+h/\sqrt{\epsilon })$$ for the non-reduced-integration scheme with $$d=2,3$$ , and the reduced-integration scheme with $$d=3$$ , where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with $$d=2$$ , we prove the convergence order $$O(h+\sqrt{\epsilon }+h^2/\sqrt{\epsilon })$$ . The theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

A comprehensive investigation of granular discharge through the eccentric opening of rectangular silos has been presented. Slip flow of the grains over the vertical wall closer to the orifice was observed experimentally. This has not been emphasized by the earlier researchers. Considering slip flow at the wall, the kinematic model of granular discharge has been extended to simulate the hydrodynamics during drainage through any eccentric opening. The simulation is applicable over the entire range of eccentricity possible and for both the uniform and the triangular velocity profile through the orifice. Results from the experiments using sand grains (300-500 μm) and glass beads (400-600 μm) have been used for validating the model. The excellent agreement not only justifies the application of the kinematic model for eccentric discharge, it also extends the model for irregular and much smaller grain size. Finally, some limitations of the kinematic model have been examined critically. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1439-1453, 2016 [ABSTRACT FROM AUTHOR]

Extrusion by Poiseuille flow and simple shear of hot lower crust has been deciphered from large hot orogens, and partial-slip boundary condition has been encountered in analogue models. Shear heat and velocity profiles are deduced from a simplified form of Navier-Stokes equation for simple shear together with extrusive Poiseuille flow and slip boundary condition for Newtonian viscous rheology. A higher velocity at the upper boundary of the shear zone promotes higher slip velocity at the lower boundary. The other parameters that affect the slip are viscosity and thickness of the shear zone and the resultant pressure gradient that drives extrusion. In the partial-slip case, depending on flow parameters (resultant pressure gradient, density and viscosity) and thickness of the shear zone, the velocity profiles can curve and indicate opposite shear senses. The corresponding shear heat profiles can indicate temperature maximum inside shear zones near either boundaries of the shear zone, or equidistant from them. [ABSTRACT FROM AUTHOR]

The effect of yield stress is numerically investigated on the viscous fingering phenomenon in a rectangular Hele-Shaw cell. It is assumed that the displacing fluid is Newtonian, while the displaced fluid is assumed to obey the bi-viscous Bingham model. The lubrication approximation together with the creeping-flow assumption is used to simplify the governing equations. The equations so obtained are made two-dimensional using the gap-averaged variables. The initially flat interface between the two (immiscible) fluids is perturbed by a waveform perturbation of arbitrary amplitude/wavelength to see how it grows in the course of time. Having treated the interfacial tension like a body force, the governing equations are solved using the finite-volume method to obtain the pressure and velocity fields. The volume-of-fluid method is then used for interface tracking. Separate effects of the Bingham number, the aspect ratio, the perturbation parameters (amplitude/wavelength), and the inlet velocity are examined on the steady finger width and the morphology of the fingers (i.e., tip-splitting and/or side-branching). It is shown that the shape of the fingers is dramatically affected by the fluid's yield stress. It is also shown that a partial slip has a stabilizing effect on the viscous fingering phenomenon for yield-stress fluids. [ABSTRACT FROM AUTHOR]

The stagnation-point flow of a second-grade fluid past a power law lubricated surface is considered in this paper. It is assumed that the fluid impinges on the wall obliquely. A suitable choice of similarity transformations reduces the governing partial differential equations into ordinary differential equations. The thin lubrication layer suggests that the interface conditions between the fluid and the lubricant can be imposed on the boundary. An implicit finite difference scheme known as the Keller-Box method is employed to obtain the numerical solutions. The effects of slip parameter and Weissenberg number on the fluid velocity and streamlines is discussed in the graphs. The limiting cases of partial-slip and no-slip can be deduced from the present solutions. [ABSTRACT FROM AUTHOR]