Dimensionless Slip Coefficient Research Articles

Newtonian flow with slip and pressure-drop predictions in hyperbolic confined geometries

We study theoretically the steady Newtonian flow in confined and hyperbolic long tubes (symmetric channels and axisymmetric pipes) considering slip along the walls. Using a stream function formulation, and the extended (or high-order) lubrication method in terms of the square of the aspect ratio of the tube, ε, the solution for the stream function is found analytically up to twentieth order in ε. At the classic lubrication limit, i.e. i.e. for a vanishing small aspect ratio, and for perfect slip conditions, the analysis predicts a plug-like velocity profile and a constant strain-rate on the midplane/axis of symmetry of the tube. A constant strain-rate is also predicted for the non-slip case. Furthermore, the high order asymptotic results for the stream function and fluid velocity are post-processed with an acceleration technique to investigate the convergence and accuracy of the solution. The results reveal the existence of a boundary layer at the inlet of the tube, the influence of which diminishes in a very short distance from the entrance. We discuss the effect of the contraction ratio of the tube and the dimensionless slip coefficient on the midplane/centerline and wall (slip) velocities, as well as on the average pressure-drop, required to maintain a constant flow-rate. The acceleration of converge technique on the solution for the pressure-drop revealed a remarkable convergence at a value slightly larger (∼1 %) than the value predicted by the classic lubrication theory. Finally, we comment on the common practice in the literature for approaching the velocity profile with the velocity profile at the classic lubrication limit, and we compare the high-order results for the strain rate at the midplane/centerline with the effective strain rate previously derived in the literature by Housiadas & Beris, J. Rheology, 68(3), 327–339, 2024.

Channel flow with variable geometry and Navier slip at the walls using high-order lubrication theory

We investigate the effect of Navier type slip for the steady, laminar flow of a highly viscous and incompressible Newtonian fluid in a microfluidic channel with variable geometry for one of the two walls of the channel. A formal perturbation expansion in terms of the square of the aspect ratio of the channel is used and an extended, high-order, lubrication theory is applied. The analysis generalizes the works of Tavakol et al. (2017) and Housiadas and Tsangaris (2022) both of which were studied under the classic no-slip and no-penetration conditions along the solid walls. Analytical expressions in series form for the pressure drop, required to maintain the constant flowrate through the channel, are derived, where the formulas are provided in terms of the function that describes the shape of the wall and the dimensionless slip coefficient that appears in the slip law. Furthermore, the formulas are processed with analytical non-linear techniques that increase the accuracy and extend the domain of convergence of series. It is shown that the slip at the walls decreases substantially the required pressure drop even for small values of the dimensionless slip coefficient, revealing the sensitivity of the results on the boundary conditions. It is also revealed that significant part of the kinetic energy of the flow is transferred along the walls due to slip decreasing the viscous dissipation energy and facilitating the fluid transport through the channel. Finally, it is demonstrated that the higher-order terms in the asymptotic solution provide important information for the field variables and the major quantities of interest (pressure drop, kinetic energy transferred along the walls, and viscous dissipation) for this type of internal and confined flows.

Mixed electroosmotic/pressure-driven flow for a generalized Phan–Thien–Tanner fluid in a microchannel with nonlinear Navier slip at the wall

This work derives an approximate solution of the laminar viscoelastic fluid flow through a parallel flat plate microchannel, driven by electroosmotic and external pressure forces. In contrast to other works published in the past, in the present analysis, the fluid’s rheology obeys the generalized Phan–Thien–Tanner model that has been proposed recently, which uses the Mittag-Leffler function instead of linear or exponential functions of the trace for the stress tensor. The hydrodynamic analysis considers a non-linear Navier slip law at the wall that follows a power-law behavior on the shear stress. For the electric potential in the electric double layer, the Debye–Hückel approximation was used, and it is assumed that the wall’s zeta potentials are symmetric. The solution obtained in this work depends on several dimensionless parameters that allow controlling the flow: ɛWi2 characterizes the fluid’s viscoelasticity, the electrokinetic parameter κ̄, the dimensionless slip coefficient L̄, the slip-power exponent m, and the ratio between pressure and electroosmotic forces, G. Besides, the influence of two parameters of the Mittag-Leffler function, α and β, notably affects the flow’s characteristics. Our results evidence the important consequences of considering the generalized Phan–Thien–Tanner model and the slippage effect on the flow field compared to that described by the Phan–Thien–Tanner model.

The singularity of the UCM/Oldroyd-B models at a finite Weissenberg number, for the steady sphere translation with Navier slip on the sphere

• The steady sphere translation with Navier slip on the sphere is studied. • A singular Weissenberg number is revealed for the UCM/Oldroyd-B models. • The singularity depends on the viscosity ratio and the slip coefficient. • Consistent and convergent approximations of the singularity are derived. For the steady sphere translation in an isothermal viscoelastic fluid which is modelled with the Upper Convected Maxwell and Oldroyd-B models, evidence of a singularity at a finite Weissenberg number is provided, when Navier type linear slip is applied at the surface of the sphere. The singular Weissenberg number depends on the dimensionless slip coefficient which appears in the slip-law, as well as on the polymer viscosity ratio. The analysis is performed following Housiadas et al. (2021) for similar flows of spherical micro-swimmers with prescribed slip velocity at the surface of the swimmer. Although here the exact singularity could not be found, the perturbation solution by Gkormpatsis et al. (2020) is utilized to evaluate consistent approximations of it is magnitude.

Numerical investigation of magnetic field on forced convection heat transfer and entropy generation in a microchannel with trapezoidal ribs

In this study, the effects of adding trapezoidal ribs to microchannel on functionalized multi-walled nano-tubes/water nanofluid heat transfer are examined. The dimensionless slip coefficient (0–0.1), Reynolds number (50–400) and Hartmann number (0–20) are considered as independent variables and the heat transfer along with the entropy generation are considered as the output parameters. The simulation outcomes confirm that the addition of trapezoidal ribs, on the one hand, increases the heat transfer area and, on the other hand, intensifies the possibility of vortex formation. The presence of a vortex decreases the heat transfer potential and thus reduces the performance of the trapezoidal-wall microchannel compared to the base one. With increasing Reynolds number (Re), the probability of vortex formation intensifies, which in turn diminishes the positive effects of using trapezoidal ribs. However, it is found that, with increasing Hartmann number (Ha) and dimensionless slip coefficient , the vortex strength is weakened, and consequently heat transfer is improved. Based on numerical computations, it is found that at Re = 400, Ha = 0 and = 0 and adding trapezoidal ribs to the base microchannel increases heat transfer by 11.12%.

The hydraulic conductivity of a shaped fracture with permeable walls

We investigate the flow-wise variation of the hydraulic conductivity inside a non-uniformly shaped fracture with permeable walls. Using lubrication theory for viscous flows, in conjunction with the Beavers–Joseph–Saffman boundary condition at the permeable walls, we obtain an analytical expression for the velocity profile, conductivity, and wall permeation velocity. These predictions highlight the effects of geometric variation (through the local slope of the aperture’s flow-wise variation), the permeability of the walls (through a dimensionless slip coefficient), and the effect of flow inertia (through a Reynolds number). The theory is validated against an OpenFOAMⓇ solver for the Navier–Stokes equations subject to a tensorial slip boundary condition, showing good agreement. The mathematical results have implications on system-level (multiscale) modeling of hydraulically fractured reservoirs, in which the Darcy conductivity of each non-uniform passage must be accurately accounted for, throughout the fractured porous rock.

The analytical solution of the Brinkman model for non-Brownian suspensions with Navier slip on the particles

The analytical solution of the Stokes-Darcy equations is derived for a Newtonian, statistically homogenous, suspension with non-Brownian (non-colloidal) rigid spherical particles. Navier-type linear slip over the surface of the particles is applied. The mathematical model is based on Brinkman's (1949) original idea for the viscous force exerted by a flowing fluid on a dense swarm of spherical particles. First, the equations are solved analytically for the pressure-driven and uniform flow of the suspension. Self-consistency of the model allows for the evaluation of the resistance parameter as function of the volume fraction of the solid phase and the dimensionless slip coefficient. The results show that for fixed solid concentration, the drag force on each particle decreases due to the slippage of the fluid over its surface. Consequently, the resistance parameter decreases compared to the classic no-slip case. However, in all cases, divergence of the resistance parameter occurs as the solid volume fraction approaches the value 2/3 which is very close to the maximum volume fraction 0.637( ≈ 2/π)for random close packing of spherical particles. The analysis is also performed for steady shear and steady uniaxial elongational flows and exact analytical solutions for the velocity and the pressure are derived. Then, a volume average of the total stress tensor in the suspension allows for the analytical evaluation of the shear and elongational viscosities of the complex flow system. Limiting expressions for the no-slip and perfect slip cases are also found and discussed.

Thermo-hydraulic characteristics investigation of nanofluid heat transfer in a microchannel with super hydrophobic surfaces under non-uniform magnetic field using Incompressible Preconditioned Lattice Boltzmann Method (IPLBM)

Present study concerns the heat transfer and fluid flow investigation of thermo-hydraulic characteristics of a nanofluid in a microchannel with super hydrophobic surfaces. In this regard, the walls of microchannel are kept in constant temperature. The incompressible version of lattice Boltzmann method with precondition factor (IPLBM) is employed to achieve a true prediction of friction factor and Nusselt number under the effect of ascending magnetic field. Simulations are performed for volume fraction of nanoparticles, Ha number (Ha) and dimensionless slip coefficient of respectively 0% to 2%, 0 to 40 and 0 to 0.1. Results show that volume fraction and Hartman numbers cause increase in Nu number and friction factor, whereas dimensionless slip coefficient has various effects on Nu number; unlike friction coefficient that causes it to reduce. The results indicated that with tuning the hydrophobicity level, one can yield a specific behavior in microchannel, so that upon using super hydrophobic surface having a dimensionless slip coefficient 0.1, at Ha number of 40 concerning a nanofluid that is of 2% volume fraction, the shear stress reduces to approximately 70%. Also, in this condition Nu number only reduces 1.7%. Numerical procedure has been validated by comparing with experimental results as well as analytical and numerical ones.

Newtonian and viscoelastic fluid flows through an abrupt 1:4 expansion with slip boundary conditions

In this work, we present a systematic numerical investigation of the 1:4 planar expansion creeping flow under the influence of slip boundary conditions for Newtonian and viscoelastic fluids, the latter modeled by the simplified Phan–Thien–Tanner constitutive model. The linear and nonlinear Navier slip laws were considered with the dimensionless slip coefficient kl* varying in the range 0, 4500 and the slip exponents m = 0.5, 1, and 2. The simulations were carried out for a low Reynolds number, Re = 0.001, and for Deborah numbers (De) between 0 and 100. Convergence could not be achieved for higher values of the Deborah number and large values of the slip coefficient due to the large stress gradients near the singularity point (reentrant corner). The results obtained allow us to conclude that for all De, the increase in slip velocity leads to vortex suppression. The flow characteristics are described in detail for low values of the Deborah number, De ≤ 5, while for higher De the main features are only shown for specific values of the slip coefficient. These results find application in polymer processing, where the use of lubricants that migrate to the wall is common, which promotes slip.

Lattice Boltzmann simulation of nanofluid conjugate heat transfer in a wide microchannel: effect of temperature jump, axial conduction and viscous dissipation

In the present study, conjugate heat transfer of nanofluid in a wide microchannel with thick wall, by considering the velocity slip and temperature jump on the fluid–solid interface and also the effect of viscous dissipation is investigated. For numerical solution of velocity field, preconditioned lattice Boltzmann method (PLBM) based on standard LBM, and for temperature field, standard LBM are used. Upper wall of the microchannel is insulated and uniform heat flux is imposed on the lower wall of the solid region. For applying the temperature jump boundary condition on the fluid–solid interface, a new algorithm reported here, is used. The problem is solved for dimensionless slip coefficient 0–0.1, volume fraction 0, 0.02 and 0.04, nanoparticles diameters (10–50) nm, and also Reynolds numbers 10–150. The results of the presented algorithm for conjugate heat transfer with temperature jump at the fluid–solid interface, show good agreement with analytical and other numerical solutions. Also, it is shown that in conjugate heat transfer of nanofluid, using super hydrophobic surfaces not only has no considerable negative effect on the average Nusselt number, but also it can increase it, especially in higher Reynolds numbers. As well as, in conjugate heat transfer, unlike the conditions of ignoring the wall thickness (at constant heat flux boundary condition), temperature jump on the wall is not constant and depends on the Reynolds number. On the other hands, using super hydrophobic surfaces (considering velocity slip and temperature jump on the wall), decreases the effect of viscous dissipation, specially at higher volume fraction of nanoparticles.

Investigation of permeability effect on slip velocity and temperature jump boundary conditions for FMWNT/Water nanofluid flow and heat transfer inside a microchannel filled by a porous media

The fluid flow and heat transfer of a nanofluid is numerically examined in a two dimensional microchannel filled by a porous media. Present nanofluid consists of the functionalized multi-walled carbon nanotubes suspended in water which are enough stable through the base fluid. The homogenous mixture is in the thermal equilibrium which means provide a single phase substance. The porous media is considered as a Darcy- Forchheimer model. Moreover the slip velocity and temperature jump boundary conditions are assumed on the microchannel horizontal sides which mean the influences of permeability and porosity values on theses boundary conditions are presented for the first time at present work. To do this, the wide range of thermo physical parameters are examined as like Da = 0.1 to 0.001, Re = 10,100, dimensionless slip coefficient from 0.001 to 0.1 at different mass fraction of nanoparticles. It is observed that less Darcy number leads to more local Nusselt number and also applying the porous medium corresponds to higher slip velocity.

Electrokinetic effects on pressure driven flow of viscoelastic fluids in nanofluidic channels with Navier slip condition

The flow of viscoelastic fluids through micro/nanofluidic systems is an important issue in the biological applications. The electrokinetic effects on pressure driven flow of a viscoelastic fluid through a nanochannel with slip boundary condition are investigated in this study. The rheological behavior of the viscoelastic fluids is described using constitutive equations and the linear Navier's law is employed to consider the slip condition on the channel walls. The Debye–Hückel linearization is employed to obtain a closed form analytical solution for the velocity distribution and induced electric field as function of slip parameter, Reynolds number, viscoelastic parameter, dimensionless Debye–Hückel parameter and dimensionless zeta potential. It is found that the induced electric field increases with increasing both dimensionless zeta potential and dimensionless slip coefficient while it decreases with increasing dimensionless Debye–Hückel parameter.

Slip flows of Newtonian and viscoelastic fluids in a 4:1 contraction

This work presents a numerical study of the 4:1 planar contraction flow of a viscoelastic fluid described by the simplified Phan-Thien–Tanner model under the influence of slip boundary conditions at the channel walls. The linear Navier slip law was considered with the dimensionless slip coefficient varying in the range [0;4500]. The simulations were carried out for a small constant Reynolds number of 0.04 and Deborah numbers (De) varying between 0 and 5. Convergence could not be achieved for higher values of the Deborah number, especially for large values of the slip coefficient, due to the large stress gradients near the singularity of the reentrant corner.Increasing the slip coefficient leads to the formation of two vortices, a corner and a lip vortex. The lip vortex grows with increasing slip until it absorbs the corner vortex, creating a single large vortex that continues to increase in size and intensity. In the range De=3–5 no lip vortex was formed. The flow is characterized in detail for De=1 as function of the slip coefficient, while for the remaining De only the main features are shown for specific values of the slip coefficient.

Viscous Dissipative Microchannel Flow of a Nanofluid Under Magneto-Hydrodynamic (MHD) Effect and With Boundary Slip

The present study deals with an analytical solution in forced convection of laminar fully developed flow and heat transfer of Al2O3–water nanofluid in a slit microchannel with constant wall heat flux in the presence of a viscous dissipation effect. The novelty in the solution is to include the magneto-hydrodynamic (MHD) effect, consider a slip condition for velocity at the wall, and use a temperature-dependent model for thermal conductivity that has not been applied to such a situation before. Closed-form solutions are obtained based on the Brinkman number, Hartman number, dimensionless slip coefficient, and nanoparticle volume fraction. The results are discussed in terms of dimensionless velocity and temperature distributions and the Nusselt number. It is observed that the Nusselt number diminishes with an increase in the dimensionless slip coefficient and Brinkman number and it increases with an increase in the Hartman number and nanoparticle volume fraction. In addition, it is shown that the classical Maxwell model for thermal conductivity overestimates the Nusselt number compared to the present model.

Some experiences with the slip boundary condition in viscous and viscoelastic flows

A slip boundary condition (SBC) has been known to occur at solid boundaries in a variety of materials processing including polymer processing. In the present work, implementation of the SBC has been tested for viscous and viscoelastic fluids obeying integral constitutive equations of the K-BKZ type. The Finite Element Method (FEM) is used to provide numerical results for tapered dies where sometimes slip is neglected in the reservoir and the converging entry section and applied only in the die. The present results show that this is valid for small values of a dimensionless slip coefficient Bsl (Bsl<1). However, slip must be applied to all the walls when Bsl>1, or the contraction angle or the contraction ratio is small; otherwise the pressure drops in the system are severely overpredicted. Viscoelasticity enhances those trends.

Viscoelastic Poiseuille flows with total normal stress dependent, nonlinear Navier slip at the wall

The effect of slip at the wall in steady, isothermal, incompressible Poiseuille flows in channel/slits and circular tubes of viscoelastic fluids is investigated analytically. The nonlinear Navier law at the wall, for the dependence on the shear stress, along with an exponential dependence of the slip coefficient on the total normal stress is assumed. The viscoelasticity of the fluid is taken into account by employing the Oldroyd-B constitutive model. The flow problems are solved using a regular perturbation scheme in terms of the dimensionless exponential decay parameter of the slip coefficient, ɛ. The sequence of partial differential equations resulting from the perturbation procedure is solved analytically up to third order. As a consequence of the nonlinearity of the slip model, a two-dimensional, continuously developing, flow field arises. Spectral analysis on the solution shows that the velocity and pressure profiles are fully resolved even for high values of ɛ, which indicates that the perturbation series up to third order approximates the full solution very well. The effects of the dimensionless slip coefficient, isotropic pressure, and deviatoric part of the total normal stress in the slip model, as well as the other parameters and dimensionless numbers in the flow are presented and discussed. Average quantities, in the cross section of the channel/slit or tube, with emphasis given on the pressure drop and the skin friction factor, are also offered.

Development Length in Planar Channel Flows of Newtonian Fluids Under the Influence of Wall Slip

This technical brief presents a numerical study regarding the required development length (L=Lfd/H) to reach fully developed flow conditions at the entrance of a planar channel for Newtonian fluids under the influence of slip boundary conditions. The linear Navier slip law is used with the dimensionless slip coefficient k¯l=kl(μ/H), varying in the range 0&lt;k¯l≤1. The simulations were carried out for low Reynolds number flows in the range 0&lt;Re≤100, making use of a rigorous mesh refinement with an accuracy error below 1%. The development length is found to be a nonmonotonic function of the slip velocity coefficient, increasing up to k¯l≈0.1-0.4 (depending on Re) and decreasing for higher k¯l. We present a new nonlinear relationship between L, Re, and k¯l that can accurately predict the development length for Newtonian fluid flows with slip velocity at the wall for Re of up to 100 and k¯l up to 1.

The effect of interfacial slip on the dynamics of a drop in flow: Part I. Stretching, relaxation, and breakup

Using a numerical method based on the boundary-integral technique, we assess the impact of interfacial slip on the dynamics of deformation and breakup of a single drop subjected to a uniaxial extensional flow under creeping-flow conditions. Interfacial slip is incorporated in our continuum development as a jump in the tangential velocity across the interface. This velocity jump is shown to reduce to the Navier-slip boundary condition to leading order and is characterized by a dimensionless slip coefficient α=(dI/μI)(μ/R), where dI is thickness of the diffuse interface between the liquids, μI is the viscosity of the interfacial region, μ is the viscosity of the suspending fluid, and R is the drop radius. A key contribution of this paper is the development of a stable, boundary-integral formulation to incorporate interfacial slip into existing, no-slip boundary-integral frameworks for drop deformation. Slip has a fourfold impact on the drop stretching, relaxation, and breakup phenomena. First, when the capillary number is small, the steady deformation of the drop with slip is smaller than the no-slip result, and the difference increases with the viscosity ratio and the capillary number. Slip thus leads to larger critical capillary numbers beyond which the drop stretches continuously in the extensional flow. Second, for capillary numbers greater than the critical value, we find that the shape of the deformed drop for the same drop elongation is relatively insensitive to the slip coefficient, but the time required to reach this deformation is a strong function of the slip coefficient—slip slows down the deformation process. Third, the end-pinch mechanism of drop breakup leads to a different number and sizes of droplets with the inclusion of slip. Finally, slip causes the capillary-instability mechanism of drop breakup to produce larger drops at faster rates relative to the no-slip case. In addition to the above results, we also show that slip moderates the viscosity and normal stress differences in a sheared, dilute emulsion. Our study has important implications in the area of blending of immiscible polymers and indicates that the drop size distribution, which ultimately governs the material properties of the blend and composites prepared from it, is influenced strongly by interfacial slip.

Axisymmetric stagnation-point flow over a lubricated surface

Axisymmetric stagnation-point flow is considered. A Newtonian fluid impinges orthogonally on a plane surface lubricated by a thin non-Newtonian liquid film of variable thickness. A slip-flow boundary condition is deduced, which allows for partial slip at the surface. The amount of slip, from full slip to no-slip, is controlled by a dimensionless slip coefficient. Similarity solutions are generally prohibited by the slip-flow boundary condition, except for one particular value of the power-law index of the lubricant. Solutions are presented for this case in order to demonstrate the influence of partial slip on the stagnation point flow. With increasing slip and reduced surface stress, a thinning of the viscous boundary layer is observed. The classical Homann flow is recovered in the no-slip limit.

Annular Extrudate Swell of Newtonian Fluids: Effects of Compressibility and Slip at the Wall

Numerical simulations have been undertaken for the benchmark problem of annular extrudate swell present in pipe extrusion and parison formation in blow molding. The effects of weak compressibility and slip at the wall are studied through simple linear laws. The finite element method is used to provide numerical results for different inner/outer diameter ratios κ under steady-state conditions for Newtonian fluids. The present results provide the shape of the extrudate, and, in particular, the thickness and diameter swells, as a function of the dimensionless compressibility and slip coefficients, B and Bsl, respectively. The pressures from the simulations have been used to compute the excess pressure losses in the flow field (exit correction). Weak compressibility slightly affects the thickness swell (about 1% in the range of simulations 0⩽B⩽0.02) mainly by a swell reduction, while slip drastically reduces the swelling to 1–2% for obvious slip (Bsl≈1) and to 0 for perfect slip (Bsl&gt;10). The exit correction increases with increasing compressibility levels and is highest for the tube (κ=0) and lowest for the slit (κ=1). It decreases monotonically to 0 as the dimensionless slip coefficient reaches its asymptotic limit of perfect slip. All results are ordered with the diameter ratio κ, between the limits of tube (κ=0) and slit (κ=1).