Navier Slip Boundary Research Articles

Scaling laws for Rayleigh–Bénard convection between Navier-slip boundaries

We consider the two-dimensional Rayleigh–Bénard convection problem between Navier-slip fixed-temperature boundary conditions, and present a new upper bound for the Nusselt number ( ${\textit {Nu}}$ ). The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result of Whitehead & Doering (Phys. Rev. Lett., vol. 106, 2011, 244501) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (Phil. Trans. R. Soc. A, vol. 380, issue 2225, 2022, 20210025) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number ( ${\textit {Pr}}$ ) and slip length, our upper bounds indicate five possible scaling laws (where ${\textit {Ra}}$ is the Rayleigh number): ${\textit {Nu}}\sim (L_s^{-1}\,{\textit {Ra}})^{{1}/{3}}$ , ${\textit {Nu}}\sim (L_s^{-2/5}\,{\textit {Ra}})^{{5}/{13}}$ , ${\textit {Nu}}\sim {\textit {Ra}}^{{5}/{12}}$ , ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-4/3}\,{\textit {Ra}})^{{1}/{2}}$ and ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-1/3}\,{\textit {Ra}})^{{1}/{2}}$ .

Stokes flow past an array of circular cylinders through slip-patterned microchannel using boundary element method

Two-dimensional viscous incompressible, pressure-driven, creeping flow at low Reynold's number (Re ≪ 1) around a series of circular cylinders in a slip-patterned rectangular microchannel is investigated numerically by using the boundary element method (BEM) based on a non-primitive variables approach. The non-primitive variables approach refers to the combination of stream function and vorticity variables. The Stokes equations are used to govern the flow of creeping fluid through a microchannel. We consider the alteration of the slip on both the upper and lower surfaces of the microchannel maintain the same phase (i.e., in-phase configuration). Here, the slip boundary condition refers to Navier's slip boundary condition. We considered both small as well as large patterned slip on both surfaces of the microchannel. Moreover, we have assumed that a number of cylinders of equal diameter are present in the in-line configuration in the path of flow. We studied streamlines, velocity profiles, pressure gradients, and the shear stresses with varied slip-length, and the radius of the cylinder, to get a complete comprehension of flow dynamics. We observed that the velocity and shear stress profiles exhibit significant variability in the case of fine slip patterning. Additionally, the proposed investigation holds several potential applications, such as drug capsule delivery systems, hemodynamics, bio-MEMS technology, and so forth.

On determining Navier's slip parameter at a solid boundary in flows of a Navier–Stokes fluid

While the assumption of the “no-slip” condition at a solid boundary is unquestioningly applied to study the flow characteristics of a Navier–Stokes fluid, there was considerable debate among the early pioneers of fluid mechanics, Du Buat, Girard, Navier, Coulomb, Poisson, Prony, Stokes, and others, as to the proper condition that has to be met at a solid boundary due to a fluid, such as water flowing adjacent to the same. Contemporary usage of the no-slip boundary condition notwithstanding, in our previous study [Málek and Rajagopal, “On a methodology to determine whether the fluid slips adjacent to a solid surface,” Int. J. Non-Linear Mech. 157, 104512 (2023)], we outlined a methodology to test the validity of the assumption. In this study, we continue the investigation further by providing a scheme for determining the slip parameter that characterizes the extent of slip, if one presumes that Navier's slip boundary condition is satisfied. We find that depending on whether the volumetric flow rate is greater or less than the volumetric flow rate corresponding to the no-slip case, different scenarios present themselves regarding what transpires at the boundary.

A new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in a bounded domain

This paper is to derive a new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in terms of the density ρ and the pressure P. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm ||ρ||L∞(0,t;L∞)+||P||Lp0(0,t;L∞)<∞ for some constant p0 satisfying 1<p0≤2. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as ||ρ||L∞(0,t;L∞)+||P||L∞(0,t;L∞)<∞.

Mathematical modeling of graphene–PDMS Maxwell nanofluid flow over a stretching surface: A study on the efficient energy management†

AbstractThis study investigates the unsteady Maxwell nanofluid flow past a stretching surface, embedded in a porous medium, in the presence of magnetic field and thermal radiation effects. Polydimethylsiloxane (PDMS) is considered as a Maxwell base fluid and graphene is taken as nanoparticle to form the nanofluid. It is a well‐reported fact in the literature (Boland et al. Science, 354(2016)1257) that viscous graphene polymers can be used to make sensitive electromechanical sensors, thereby setting the direction and incentive for the study of graphene–PDMS nanofluid. In this paper, we have implemented the Navier's slip boundary condition at the nanofluid‐surface interface. Entropy analysis for the energy efficiency of the system in terms of the Bejan number is carried out extensively. By employing suitable similarity transformations, the set of partial differential equations has been transformed into a set of ordinary differential equations. The transformed equations are solved by the homotopy analysis method (HAM). The nanofluid velocity, temperature, skin‐friction coefficient, and heat transfer rate are analyzed in this paper under several regulatory physical parameters. The findings show that the rise in time relaxation parameter has adverse effects on nanofluid velocity but enhances the entropy generation in the system. The heat transfer rate decreases with an enhancement of the Deborah number but increases with a rise in the nanoparticle volume fraction. The Bejan number shows increasing trend with an enhancement of unsteady parameter, nanoparticle volume fraction, and porous permeability parameter. This study bears the potential application in powder technology, plastic extrusion process, as well as in bio‐engineering.

Heat Transfer of Magnetohydrodynamic Stratified Dusty Fluid Flow through an Inclined Irregular Porous Channel.

The primary objective of the study is to explore the phenomena of dusty fluid flow through an inclined irregular channel under the impact of the transversely applied magnetic field of fixed strength. The density and viscosity of the working fluid are assumed to vary along with the height of the channel as it behaves as a replica of many real world mechanisms. Hence, a stratified dusty fluid through a channel that tilts to an angle is the main objective of the present study. The prescribed flow is mathematically modeled and it is approached numerically under two distinct boundary conditions. The finite difference technique is employed to discretize the system of equations and solved using the Thomas algorithm. The velocity and temperature fields are discussed for different pertinent parameters which influence the flow. The friction factor and heat transfer rate are discussed as it has been a subject of interest in recent decades. The results show that the stratification decay parameter leads to enhancement in the momentum of the fluid flow. The temperature field is found to be higher in the convective boundary than the Navier slip boundary.

Electroosmotic thrusters in soft nanochannels for space propulsion

Space propulsion of electroosmotic thrusters (EOTs) with a soft charged nanochannel is investigated considering the Navier slip boundary and constant surface charge density on the walls of slit channels. The soft nanochannel is characterized by a wall-grafted ion-penetrable charged polyelectrolyte layer (PEL). The Poisson–Boltzmann equation is solved to give the electric potential distribution based on the assumption of the Debye–Hückel linearization for the low electric potential. An analytical solution of the electroosmotic velocity through the soft channel is obtained. The thrust, specific impulse, and total input power of EOTs produced by the electroosmotic flow are presented, and then, two significant physical quantities, thruster efficiency and thrust-to-power ratio, are described. It is found that these performance curves strongly depend on the slip length, surface charge density on the walls, drag coefficient, equivalent electric double layer thickness, PEL thickness, and density ratio of the PEL to the electrolyte solution layer. By analyzing and optimizing these design parameters, the simulated EOTs can deliver the thrust from 0 μN to 10 µN as well as the specific impulse from 40 s to 100 s, and the thruster efficiency up to 87.22% is realized. If more thrust control and kinetic energy are needed for different space missions, an array composed of thousands of single EOT emitters is constructed and maintains high thruster efficiency. Moreover, during mission operation, the total potential can be simply varied to optimize the performances of thrusters at any moment.

Navier–Stokes Equations in a Curved Thin Domain, Part II: Global Existence of a Strong Solution

We consider the Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier's slip boundary conditions. When the thickness of the thin domain is sufficiently small, we establish the global existence of a strong solution for large data. We also show several estimates for the strong solution with constants explicitly depending on the thickness of the thin domain. The proofs of these results are based on a standard energy method and a good product estimate for the convection and viscous terms following from a detailed study of average operators in the thin direction. We use the average operators to decompose a three-dimensional vector field on the thin domain into the almost two-dimensional average part and the residual part, and derive good estimates for them which play an important role in the proof of the product estimate.

Liquid slippage on rough hydrophobic surfaces with and without entrapped bubbles

The process of liquid slip on rough-walled hydrophobic surfaces with and without entrapped gas bubbles is modeled. Here, starting with the Navier–Stokes equations, a set of partial differential equations (PDE) and boundary conditions for the general effective slip tensor of a rough hydrophobic surface are constructed by an asymptotic analysis. The intrinsic slip and surface roughness are considered as the characteristics of the surface. The solution is based on a weak variation form that fully recovers the set of PDE and Navier slip boundary. For the surface with entrapped bubbles, a semi-analytical model based on eigenfunction expansion is developed. In addition to the surface characteristics, the size and contact angle of the bubbles are considered in the semi-analytical solution. Both models are validated with the published data as well as direct numerical simulation. Based on the model results, we present correlations of effective slip length with surface characteristics and entrapped bubbles. We found that surface roughness reduces liquid slippage on a surface. However, if the asperities on a surface are filled with gas bubbles, the effective slip length can significantly increase as long as the bubble contact angle is small. Interestingly, bubbles with a larger contact angle could act inversely and change a hydrophobic surface with a large intrinsic slip to a no-slip or even a sticky surface. These results shed light on the controversy over the order of magnitude of the actual slip length of water flow in carbon-based nanotubes and nanochannels. The model results also help understand the anomalies of high water production and high amounts of hydraulic fracturing fluid leak-off observed in tight oil and shale gas reservoirs.

A Pressure Associated with a Weak Solution to the Navier–Stokes Equations with Navier’s Boundary Conditions

We show that if u is a weak solution to the Navier-Stokes initial-boundary value problem with Navier's slip boundary conditions in $Q_T:=\Omega\times(0,T)$, where $\Omega$ is a domain in $R^3$, then an associated pressure $p$ exists as a distribution with a certain structure. Furthermore, we also show that if $\Omega$ is a "smooth" domain in $R^3$ then the pressure is represented by a function in $Q_T$ with a certain rate of integrability. Finally, we study the regularity of the pressure in sub-domains of $Q_T$, where $u$ satisfies Serrin's integrability conditions.

Incompressible limit of the compressible nematic liquid crystal flows in a bounded domain with perfectly conducting boundary

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

Novel discretization methods for improved simulation precision in injection molding

AbstractMold filling is an important stage of injection molding, which is one of the most commonly used manufacturing processes for the production of thermoplastic components in high volumes. As a consequence, the numerical simulation of this process based on computational fluid dynamics (CFD) is of great significance for production engineering [1, 2]. However, modeling of the mold filling is a tremendously demanding process, when considering interfacial physical phenomena, such as two‐phase flows, building a sharp interface between the molten plastics and the present air in the cavity or the dynamic wetting contact line at the cavity surface. A method for dealing with these phenomena is a local mesh refinement both in space and time. In this paper, the numerical solution of a mold filling problem using simplex‐type space‐time finite elements is presented and compared with experiments. These elements can be suited to increase efficiency, when used for the aforementioned refinement. In addition, a Navier's slip boundary condition is applied to the solid boundaries of the mold allowing the contact line to evolve along the boundary, while enabling a better prediction of the pressure distribution. The presented work was performed in collaboration of the subprojects B3 and B5 of the collaborative research center 1120 “Precision Melt Engineering”.

Electro-osmotic flow of Eyring fluids in a circular microtube with Navier's slip boundary condition

Electro-osmotic flows of non-Newtonian fluids play an important role in microdevices and microfluidic systems. A theoretical and numerical analyses are conducted to explore the characteristics of electro-osmotic flow of non-Newtonian Eyring fluids in a circular microtube under the Navier's slip boundary condition. The exact solution of velocity distribution as well as corresponding approximate solution is obtained. The effects of slip length, electrokinetic parameter and non-Newtonian characteristics on the electro-osmotic flows are discussed and plotted graphically.

Nanoparticle analysis of non-Newtonian fluid with slip and multiple convective boundary conditions

Slip flow of a Carreau nanofluid over a stretching sheet has been investigated. Thermophoresis and Brownian motion effects are taken into account. The Navier slip, thermal, and mass convective boundary conditions are taken into account. The prevailing partial differential equations are presented and transformed into a set of nonlinear ordinary differentials using scaling transformation. Effects of the physical parameters on velocity, temperature, and concentration profiles are computed numerically using the fourth-order Runge–Kutta–Fehlberg scheme. Numerical values of local Nusselt and Sherwood numbers are calculated and discussed. The obtained results revealed that enhancing the slip parameter consequently drops the temperature and concentration profiles. Heat and mass flux at the surface decreases with increasing thermophoresis parameter. Heat transfer rate enhances while the mass transfer rate decreases with the thermal Biot number.

Flows in slip-patterned micro-channels using boundary element methods

In this study we investigate steady, pressure-driven, two-dimensional flow of Newtonian fluid through slip-patterned, rectangular channels in the low Reynolds number limit. The slip flow regime is modeled using the Navier's slip boundary condition. In this work, we present only in-phase patterned slip. Subsequently, based on the characteristic length of the patterning, we have considered two subcases, namely large and fine patterned slip. Boundary element method (BEM) is used to numerically solve Stokes equation and obtain the streamline profiles. Streamlines, velocity profiles, pressure gradients, and shear stresses are analyzed to gain a proper understanding of the flow mechanics.

Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes

In the present study, a new soft computing framework is developed for solving nanofluidic problems based on fluid flow and heat transfer of multi-walled carbon nanotube (MWCNT) along a flat plate with Navier slip boundary with the help of artificial neural networks (ANNs), Genetic Algorithms (GAs), Interior-Point Algorithm (IPA), and hybridized approach GA-IPA. Original PDEs associated with the problem are transformed into system of nonlinear ODEs using similarity transformation. Mathematical model of transformed system is constructed by exploiting the strength of universal function approximation ability of ANNs and an unsupervised error function is formulated for the system in a least mean square sense. Learning of the design variable of the networks is carried out with GAs supported with IPA for rapid local convergence. The design scheme is applied to solve number of variants by taking water, engine oil, and kerosene oil as a base fluids mixed with different concentrations of MWCNTs. The reliability and effectiveness of the design scheme is measured with the help of results of statistical analysis based on sufficient large number of independent runs of the algorithms rather than single successful run. The comparative studies of the proposed solution are made with standard numerical results in order to establish the correctness of the given scheme.

Electro-osmotic slip flow of Eyring fluid in a slit microchannel

The electro-osmotic flow of a non-Newtonian fluid in a slit micro-channel under the Navier's slip boundary condition is investigated. The Eyring constitutive relationship model is adopted to describe the non-Newtonian characteristics of the flow driven by the applied electric field force and pressure. In consideration of the micro-scale effects, electric field, non-Newtonian behavior and slip boundary condition, a mechanical model is built and the effects of these factors on the flow are studied. Analytical expressions are derived for the electric potential and velocity profile by solving the linearized Poisson-Boltzmann equation and the modified Cauchy equation. Approximate expressions of the velocity distribution are also given and discussed. Furthermore, by comparing the effects of electric force with that of pressure on the velocity distribution, some meaningful conclusions are drawn from the obtained graphics.

Fully developed forced convection of alumina/water nanofluid inside microchannels with asymmetric heating

The effects of nanoparticle migration and asymmetric heating on the forced convective heat transfer of alumina/water nanofluid in microchannels have been investigated theoretically. Walls are subjected to different heat fluxes; qt″ for the top wall and qb″ for the bottom wall to form the asymmetric heating. Because of the microscopic roughness in microchannels, Navier's slip boundary condition is considered at the fluid–solid interface. A two-component mixture model is used for nanofluids with the hypothesis that Brownian motion and thermophoretic diffusivities are the only significant slip mechanisms between solid and liquid phases. Assuming a fully developed flow and heat transfer, the basic partial differential equations (including continuity, momentum, energy, and nanoparticle distribution equations) have been reduced to two-point ordinary boundary value differential equations and solved numerically. It is revealed that nanoparticles eject themselves from the heated walls, construct a depleted region, and accumulate in the core region, but they are more likely to accumulate toward the wall with the lower heat flux. In addition, the non-uniform nanoparticle distribution makes the velocities move toward the wall with the higher heat flux and enhances the heat transfer rate there. Moreover, the advantage of nanofluids is increased in the presence of a slip velocity at the walls.

Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field

The current study is a theoretical investigation of the laminar flow and convective heat transfer of alumina/water nanofluid inside a circular microchannel in the presence of a uniform magnetic field. A modified two-component four-equation nonhomogeneous equilibrium model was employed for nanofluids, which fully accounted for the effect of the nanoparticle volume fraction distribution. Because of the microscopic roughness in circular microchannels and also the non-adherence of the fluid–solid interface in the presence of nanoparticle migration, known as slip condition, the Navier's slip boundary condition is considered at the walls. The results indicated that nanoparticles migrate from the heated walls (nanoparticles depletion) towards the core region of the microchannel (nanoparticles accumulation) and construct a non-uniform nanoparticles distribution. The ratio of the Brownian to thermophoretic diffusivities (NBT) has relatively significant effects both on the distribution of the nanoparticles and the convective heat transfer coefficient of nanofluids. It was further observed that for smaller nanoparticles, the nanoparticle volume fraction is more uniform and abnormal variations in the heat transfer rate vanish. Moreover, in the presence of the magnetic field, the near wall velocity gradients increase, enhancing the slip velocity and thus the heat transfer rate and pressure drop increase.

Incompressible limit of global strong solutions to 3-D barotropic Navier–Stokes equations with well-prepared initial data and Navier's slip boundary conditions

This paper studies the incompressible limit of global strong solutions to the three-dimensional barotropic Navier–Stokes equations associated with Navier's slip boundary conditions, provided that the initial data are sufficiently small, and “well-prepared” in the sense that the time derivatives, up to first order, of solutions are bounded initially. The main idea is to derive a differential inequality with decay, so that the estimates are bounded uniformly both in the Mach number ϵ∈(0,ϵ0] (for some ϵ0>0) and the time t∈[0,+∞).