Linear Navier Slip Research Articles

Magnetohydrodynamic Flow of Immiscible Jeffrey Fluids Through a Horizontal Porous Cylinder

This paper investigates the steady and fully developed magnetohydrodynamic (MHD) flow of two nonmiscible Jeffrey fluids in a horizontal cylinder filled with a homogenous porous medium. A constant pressure gradient drives the flow in both regions, and a uniform magnetic field is applied transversely to the flow direction. Instead of the usual no‐slip condition, the linear Navier slip is used as a boundary condition on the cylinder’s surface, whereas the usual continuity conditions of velocity and shear stress are presumed at the fluid–fluid interface. The equations describing the problem under consideration are mathematically formulated and transformed into nondimensional forms with the proper choice of nondimensional variables. Analytical solutions have been obtained by solving the transformed equations of motion. The effects of different nondimensional parameters involved in the flow problem, including the magnetic number, Jeffrey parameter, Darcy number, ratio of viscosities, Reynolds number, slip parameter, and pressure gradient, on the velocity in each region are studied, and the results are exhibited graphically. In addition, numerical values for stress at the wall and volume flow rates are calculated for various fluid parameters and shown in tables. It is noticed that an increase in the values of the magnetic number and viscosity ratio reduces the velocity of the fluid, whereas increasing the Jeffrey parameters, Darcy number, Reynolds number, slip parameter, and pressure gradient enhances fluid velocity. Furthermore, the most favorable agreement was observed between the results of this study and the results of the previous studies.

Slip Backflow of Polymers in Elastic Fractures for Subsurface Heat Recovery

This research delves into the complexities of backflow phenomena in finite-length and flat-walled fractures with elastic walls, specifically focusing on power-law fluids, whose shear-thinning behavior distinguishes them from Newtonian fluids. We model the backflow process under the lubrication approximation and by incorporating the linear Navier slip law. We numerically examine the influence of parameters such as slip length, fluid rheology, and external pressure on the backflow propagation of the carrier fluid. Our findings underscore the significant role played by the rheological index in determining the fracture closure rate. Additionally, our investigations highlight the marked effect of external pressure variations on pressure distribution within the fracture. Notably, the friction coefficient at the fracture walls, as denoted by a dimensionless slip number, exhibits limited influence on the fundamental dynamics of the problem. These insights advance our understanding of power-law fluid backflow and have wide-ranging applications across various engineering disciplines.

The linear stability of slip channel flows

A linear stability investigation is undertaken on the two-dimensional flow that develops in a channel whose walls are coated with a superhydrophobic material. The surfaces are modeled as classical slip surfaces, exploiting a linear Navier slip condition imposed on the channel walls. Both symmetric and asymmetric slip walls are considered, whereby the linearized stability of the flow is determined via an Orr–Sommerfeld normal-mode approach. In both instances, the application of slip establishes a significant stabilizing effect and increases the critical Reynolds number associated with the onset of linearly unstable behavior. Indeed, for sufficiently large slip lengths, the upper and lower branches of the neutral stability curve coalesce. Consequently, the flow becomes linearly stable to all disturbances for all wavelengths and Reynolds numbers.

Effect of radiation and Navier slip boundary of Walters’ liquid B flow over a stretching sheet in a porous media

This paper investigates the steady-state momentum and radiation heat transfer flow of a viscoelastic fluid in a porous media in the presence of a linear Navier slip boundary condition. The velocity of the fluid over the linear stretching sheet is varied linearly with the axial distance while a Walters’ liquid-B model is assumed for the viscosity. A similarity transformation reduces the Navier–Stokes equations to a set of partial differential equations that are converted into ordinary differential equations and solved analytically for the velocity. Moreover, heat is balanced between a temperature dependent heat source and radiation and leads to a differential equation with variable coefficients. The temperature equation is transformed to a confluent hypergeometric differential equation using the Rosseland approximation for the radiation and solved analytically. Results are discussed for two boundary conditions of the sheet, the prescribed surface temperature and the wall heat flux. Parameters like the Reynolds number, the viscoelastic parameter and the boundary slip parameter are found to determine the flow field. In addition, the Prandtl number, the radiation number, the wall temperature and the heat source/sink parameters are found to control the temperature distribution inside the stretching sheet.

Transient electroosmotic slip flow of fractional Oldroyd-B fluids

The electroosmotic flow of a fractional Oldroyd-B fluid in a circular microchannel is studied. The linear Navier slip velocity model is used as the chosen slip boundary condition. Exact solutions for the electric potential and transient velocity are established by means of Laplace and finite Hankel transforms. And the velocity was presented as a sum of the steady part and the unsteady one. The corresponding solutions for the fractional Maxwell fluid, fractional second grade fluid, and Newtonian fluid can also be obtained from our results. Finally, numerical results for the fluid flow are obtained and some useful conclusions are drawn. Our results may be useful for the prediction of the flow behavior of viscoelastic fluids in microchannels and can benefit the design of microfluidic devices.

Heat transfer of nanofluids considering nanoparticle migration and second-order slip velocity

The heat transfer of a magnetohydrodynamics nanofluid inside an annu-lus considering the second-order slip condition and nanoparticle migration is theoret-ically investigated. A second-order slip condition, which appropriately represents the non-equilibrium region near the interface, is prescribed rather than the no-slip condi-tion and the linear Navier slip condition. To impose different temperature gradients, the outer wall is subjected to q2, the inner wall is subjected to q1, and q1 > q2. A modified two-component four-equation non-homogeneous equilibrium model is employed for the nanofluid, which have been reduced to two-point ordinary boundary value differential equations in the consideration of the thermally and hydrodynamically fully developed flow. The homotopy analysis method (HAM) is employed to solve the equations, and the h-curves are plotted to verify the accuracy and efficiency of the solutions. Moreover, the effects of the physical factors on the flow and heat transfer are discussed in detail, and the semi-analytical relation between NuB and NBT is obtained.

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.

Annular flow of viscoelastic fluids: Analytical and numerical solutions

This work provides analytical and numerical solutions for the linear, quadratic and exponential Phan–Thien–Tanner (PTT) viscoelastic models, for axial and helical annular fully-developed flows under no slip and slip boundary conditions, the latter given by the linear and nonlinear Navier slip laws. The rheology of the three PTT model functions is discussed together with the influence of the slip velocity upon the flow velocity and stress fields. For the linear PTT model, full analytical solutions for the inverse problem (unknown velocity) are devised for the linear Navier slip law and two different slip exponents. For the linear PTT model with other values of the slip exponent and for the quadratic PTT model, the polynomial equation for the radial location (β) of the null shear stress must be solved numerically. For both models, the solution of the direct problem is given by an iterative procedure involving three nonlinear equations, one for β, other for the pressure gradient and another for the torque per unit length. For the exponential PTT model we devise a numerical procedure that can easily compute the numerical solution of the pure axial flow problem.

Analytical and numerical study of the electro-osmotic annular flow of viscoelastic fluids

In this work we present semi-analytical solutions for the electro-osmotic annular flow of viscoelastic fluids modeled by the Linear and Exponential PTT models. The viscoelastic fluid flows in the axial direction between two concentric cylinders under the combined influences of electrokinetic and pressure forcings. The analysis invokes the Debye-Hückel approximation and includes the limit case of pure electro-osmotic flow. The solution is valid for both no slip and slip velocity at the walls and the chosen slip boundary condition is the linear Navier slip velocity model. The combined effects of fluid rheology, electro-osmotic and pressure gradient forcings on the fluid velocity distribution are also discussed.

Implementation of slip boundary conditions in the finite volume method: new techniques

SUMMARYTwo different techniques for the implementation of the linear and nonlinear slip boundary conditions into a finite volume method based numerical code are presented. For the linear Navier slip boundary condition, an implicit implementation in the system of equations is carried out for which there is no need for any relaxation, especially when handling high slip coefficients. For three different nonlinear slip boundary conditions, two different methods are devised, one based on solving a transcendental equation for the boundary and the other on the linearization of the slip law. For assessment purposes, comparison is made between these new methods and the usual iterative process. With these new methods, the convergence difficulties, typical of the iterative procedure, are eliminated, and for some of the test cases, the convergence rate even increased with the slip velocity. The details of these implementations are given first for a simple geometry using orthogonal meshes and Cartesian coordinates followed by their generalization to non‐Cartesian coordinates and nonorthogonal meshes. The developed code was tested in the benchmark slip‐stick and 4:1 contraction flows, evidencing the robustness of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.

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<k¯l≤1. The simulations were carried out for low Reynolds number flows in the range 0<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.

Flow through a circular tube with a permeable Navier slip boundary

For Newtonian fluid flow in a right circular tube, with a linear Navier slip boundary, we show that a second flow field arises which is different to conventional Poiseuille flow in the sense that the corresponding pressure is quadratic in its dependence on the length along the tube, rather than a linear dependence which applies for conventional Poiseuille flow. However, assuming that the quadratic pressure is determined, say from known experimental data, then the new solution only exists for a precisely prescribed permeability along the boundary. While this cannot occur for conventional pipe flow, for fluid flow through carbon nanotubes embedded in a porous matrix, it may well be an entirely realistic possibility, and could well explain some of the high flow rates which have been reported in the literature. Alternatively, if the radial boundary flow is prescribed, then the new flow field exists only for a given quadratic pressure. Our primary purpose here is to demonstrate the existence of a new pipe flow field for a permeable Navier slip boundary and to present a numerical solution and two approximate analytical solutions. The maximum flow rate possible for the new solution is precisely twice that for the conventional Poiseuille flow, which occurs for constant inward directed flow across the boundary.