Navier Slip Research Articles

A multiscale volume of fluid method with self-consistent boundary conditions derived from molecular dynamics

Molecular dynamics (MD) and volume of fluid (VOF) are powerful methods for the simulation of dynamic wetting at the nanoscale and macroscale, respectively, but the massive computational cost of MD and the sensitivity and uncertainty of boundary conditions in VOF limit their applications to other scales. In this work, we propose a multiscale simulation strategy by enhancing VOF simulations using self-consistent boundary conditions derived from MD. Specifically, the boundary conditions include a particular slip model based on the molecular kinetic theory for the three-phase contact line to account for the interfacial molecular physics, the classical Navier slip model for the remaining part of the liquid–solid interface, and a new source term supplemented to the momentum equation in VOF to replace the convectional dynamic contact angle model. Each slip model has been calibrated by the MD simulations. The simulation results demonstrate that with these new boundary conditions, the enhanced VOF simulations can provide consistent predictions with full MD simulations for the dynamic wetting of nanodroplets on both smooth and pillared surfaces, and its performance is better than those with other VOF models, especially for the pinning–depinning phenomenon. This multiscale simulation strategy is also proved to be capable of simulating dynamic wetting above the nanoscale, where the pure MD simulations are inaccessible due to the computational cost.

Influence of the odd viscosity on a falling film down a slippery inclined plane

The stability of a thin viscous Newtonian fluid with broken time-reversal-symmetry draining down a slippery inclined plane is examined. The presence of the odd part of the Cauchy stress tensor with an odd viscosity coefficient brings new characteristics in fluid flow as it gives rise to new terms in the pressure gradient of the flow. By odd viscosity, it is meant that apart from the well-known coefficient of shear viscosity, a classical liquid with broken time-reversal symmetry is endowed with a second viscosity coefficient. The model implements a Navier slip condition at the solid–liquid interface with the slip length being the parameter that measures the deviation from the no-slip condition. The classical long-wave expansion technique is performed and a nonlinear evolution equation of Benney-type is derived in terms of film thickness h(x, t), which is significantly modified due to the presence of odd viscosity in the liquid. The parameters governing the film flow system and the slippery substrate strongly influence the waveforms and their amplitudes and hence the stability of the fluid. The linear stability analysis is performed using the normal mode approach and a critical Reynolds number is obtained. The results of the linear stability analysis reveal that larger odd viscosity leads to the higher critical Reynolds number while the higher slip length makes the critical Reynolds number lower. In other words, odd viscosity has a stabilizing effect while the slip length promotes instability. Based on the method of multiple scales, a weakly nonlinear stability analysis is carried out, which shows that there is a range of wave numbers with a supercritical bifurcation and a range of larger wave numbers with a subcritical bifurcation. Different instability zones are also demarcated. The weakly nonlinear study shows that with an increase in the odd viscosity, the supercritical stable region and the explosion area shrink, whereas the unconditional stable and the subcritical unstable regions increase. It has also been shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region. The spatiotemporal evolution of the model has been analyzed numerically by employing the Crank–Nicolson method in a periodic domain for different values of the odd viscosity and slip length. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The authors of the article agree to the retraction of the article effective 29 June 2022.

Hydrodynamic instability and wave formation of a viscous film flowing down a slippery inclined substrate: Effect of odd-viscosity

Hydrodynamic instability and formation of waves for thin viscous film flowing down a slippery inclined substrate with broken time-reversal-symmetry have been discussed in this present study. The effect of slip on the substrate is modeled using the Navier slip boundary condition. We have derived a nonlinear evolution model in the framework of the long-wave expansion technique. The one equation model can track the free surface evolution and involve the viscosity, gravity, surface tension, and the effect of the slip of the wall. The odd part of the Cauchy stress tensor with an odd-viscosity coefficient significantly modifies the characteristic of the film flow. We performed the linear stability analysis with the Orr Sommerfeld technique and classical temporal analysis from the model. Comparing the results shows a satisfactory agreement between Orr Sommerfeld and the model when the velocity scale is chosen as twice the free surface velocity. Analyzing the traveling wave solutions and studying different bifurcation analyzes and phase diagrams, we focused on and discussed two significant wave families, namely γ1 (low-wave-speed) and γ2 (high-wave-speed). The spatio-temporal study of the model has been performed numerically for different odd-viscosity and slip parameters. Results show that the odd-viscosity plays a stabilizing role while the slip on the substrate gives a destabilizing effect within the parameter range of our consideration.

Correlation in thermal fluctuations induced by phase-locked hydrodynamic modes.

Thermal fluctuations constitute a fundamental equilibrium phenomenon whose spatial and temporal correlations are governed by the relevant scales of molecular collisions. From the continuum point of view, thermal fluctuations in a fluid can be regarded as comprising a multitude of hydrodynamic modes (HMs) with random phases, each one having one degree of freedom. We show that in a two-dimensional fluid channel with the Navier slip boundary condition, in which the HMs are represented by periodic arrays of vortex and antivortex pairs, periodic modulation of the slip boundary condition can selectively suppress noncommensurate HMs while phase lock the remaining eigenmodes. As a result, thermal fluctuations would exhibit mesoscopic-scale spatial correlations, manifest as a spatially varying diffusion constant when evaluated from the fluctuation-dissipation theorem. Good agreement is shown with the molecular dynamics results. Such manifestation of equilibrium collective motion implies that instead of just being an alternative mathematical basis for expressing thermal fluctuations, in mesoscopic systems the HMs may be manipulated to have physical consequences very different from those expected in bulk fluid.

A cut finite element method for non-Newtonian free surface flows in 2D - application to glacier modelling

In ice sheet and glacier modelling, the Finite Element Method is rapidly gaining popularity. However, constructing and updating meshes for ice sheets and glaciers is a non-trivial and computationally demanding task due to their thin, irregular, and time dependent geometry. In this paper we introduce a novel approach to ice dynamics computations based on the unfitted Finite Element Method CutFEM, which lets the domain boundary cut through elements. By employing CutFEM, complex meshing and remeshing is avoided as the glacier can be immersed in a simple background mesh without loss of accuracy. The ice is modelled as a non-Newtonian, shear-thinning fluid obeying the p-Stokes (full Stokes) equations with the ice atmosphere interface as a moving free surface. A Navier slip boundary condition applies at the glacier base allowing both bedrock and subglacial lakes to be represented. Within the CutFEM framework we develop a strategy for handling non-linear viscosities and thin domains and show how glacier deformation can be modelled using a level set function. In numerical experiments we show that the expected order of accuracy is achieved and that the method is robust with respect to penalty parameters. As an application we compute the velocity field of the Swiss mountain glacier Haut Glacier d'Arolla in 2D with and without an underlying subglacial lake, and simulate the glacier deformation from year 1930 to 1932, with and without surface accumulation and basal melt.

Effect of magnetic field on electroosmotic flow of viscoelastic fluids in a microchannel.

Electroosmotic flow is an efficient transportation technology driven by applying an external electric field across the microchannel, which has a great potential for futureapplication. This work is presented to study the unsteady electroosmotic flow of viscoelastic fluids combined with a constant pressure gradient and a vertical magnetic field through a parallel plate microchannel. For the reason that the upper and bottom walls of the parallel plate microchannel in microfluidic devices can be made of different materials, this leads to different hydrophobic properties, asymmetric zeta wall potentials, and different slip boundary conditions. The Navier slip model with different slip coefficients at walls is considered. The generalized Maxwell fluid with fractional derivative is adopted for the constitutive equation of the fluid. The analytical and numerical solutions of velocity are derived by employing the integral transform method and finite difference method, respectively. Excellent agreement is found between the numerical solutions and analytical solutions. Finally, the effects of fractional parameter , relaxation time , slip coefficients and , the ratio of wall zeta potentials , Hartmann number , and electrical field strength parameter on velocity profiles are interpreted graphically in detail.

Thermocapillary instability on a film falling down a non-uniformly heated slippery incline

A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number Mn as well as slip length δ. This depicts that δ and Mn both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of Mn and δ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of Mn, when the slip length δ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude (ζa) decreases (increases) with the increment of Mn and δ. Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum hmax and minimum hmin thickness amplifies, for small change of Mn and δ. Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of δ. The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as δ increases but decreases as Mn increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length δ>max16Mn−13,12Mn−23−Mn.

Well-posedness of stochastic third grade fluid equation

In this paper, we establish the well-posedness for the third grade fluid equation perturbed by a multiplicative white noise. This equation describes the motion of a non-Newtonian fluid of differential type with relevant viscoelastic properties. We are faced with a strongly nonlinear stochastic partial differential equation supplemented with a Navier slip boundary condition. Taking the initial condition in the Sobolev space H2, we show the existence and the uniqueness of the strong (in the probability sense) solution in a two dimensional and non axisymmetric bounded domain.

Effects of gas-assisted extrusion on slip in the cable coating process

Abstract The isothermal viscoelastic finite element method is used to simulate and analyze the process of cable coating extrusion, in which the Navier slip model is adopted. The Phan–Thien–Tanner differential viscoelastic constitutive equation is used to describe the flow characteristics of the polymer melt. The polymer material used for simulation is polypropylene. The extrudate swell, velocity field, pressure field and shear stress field are calculated by finite element method. The influences of the gas-assisted extrusion and traditional extrusion on wall slip of cable coating extrusion are compared. The results indicate that the extrudate swell ratio is the largest under the condition of the complete slip between core wire and melt during traditional extrusion process. The increase of core wire dragging velocity can lead to the increase of slip velocity, the decrease of pressure and the increase of shear stress of melt. Gas-assisted extrusion can eliminate the negative effects caused by the slip of core wire or the increase of core wire dragging velocity. Therefore, gas-assisted extrusion can reduce the energy consumption, improve the cable coating layer quality and increase the production efficiency during extrusion process.

Layered Marangoni convection with the Navier slip condition

Layered Marangoni convection with the Navier slip condition

An atomistic model for the Navier slip condition

Abstract

Approximations for Steady Unidirectional Slip Flows in Elliptic Microchannels

Abstract Consider steady flows in a channel whose cross section Ω is an ellipse, flows with the Navier slip boundary condition. Denote the volume flow rate by Q. We apply to elliptic cross section a recent simple approximation, a rigorous lower bound R on Q, requiring, along with the channel's area and perimeter, the calculation of just the torsional rigidity and two other domain functionals. This avoids the need for solving the partial differential equation repeatedly for differing values of the slip parameter.

A partitioned numerical scheme for fluid–structure interaction with slip

We present a loosely coupled, partitioned scheme for solving fluid–structure interaction (FSI) problems with the Navier slip boundary condition. The fluid flow is modeled by the Navier–Stokes equations for an incompressible, viscous fluid, interacting with a thin elastic structure modeled by the membrane or Koiter shell type equations. The fluid and structure are coupled via two sets of coupling conditions: a dynamic coupling condition describing balance of forces, and a kinematic coupling condition describing fluid slipping tangentially to the moving fluid–structure interface, with no penetration in the normal direction. Problems of this type arise in, e.g., FSI with hydrophobic structures or surfaces treated with a no-stick coating, and in biologic FSI involving rough surfaces of elastic tissues or tissue scaffolds. We propose a novel, efficient partitioned scheme where the fluid sub-problem is solved separately from the structure sub-problem, and there is no need for sub-iterations at every time step to achieve stability, convergence, and its first-order accuracy. We derive energy estimates, which prove that the proposed scheme is unconditionally stable for the corresponding linear problem. Moreover, we present convergence analysis and show that under a time-step condition, the method is first-order accurate in time and optimally convergent in space for a Finite Element Method-based spatial discretization. The theoretical rates of convergence in time are confirmed numerically on an example with an explicit solution using the method of manufactured solutions, and on a benchmark problem describing propagation of a pressure pulse in a two-dimensional channel. The effects of the slip rate and fluid viscosity on the FSI solution are numerically investigated in two additional examples: a 2D cylindrical FSI example for which an exact Navier slip Poiseuille-type solution is found and used for comparison, and a squeezed ketchup bottle example with gravity enhanced flow. We show that the Navier-slip boundary condition increases the outflow mass flow rate by 21% for a bottle angled at 45 degrees pointing downward, in the direction of gravity.

Navier slip and suction/injection effects on oscillatory flow of nanofluid with fluid-particle interaction in a channel

In this paper, the slip and porosity effects on the channel walls are studied for the oscillatory flow of heat-absorbing/generating dusty nanofluid flowing through a porous medium. The channel is vertical and exposed to the transverse magnetic field and thermal radiation. The base fluid is water with silver as nanoparticles. The basic equations of the flow problem, which appeared to be dimensional, are remodelled in the dimension-less form with the help of non-dimensional variables. The obtained equations are solved analytically using the variable separable method. The graphs are presented to show the impact of these parameters on the flow fields, skin friction, heat transmission rate and discussed in detail. Results reveal that the flow velocities of fluid and particles for suction are greater than in the case of injection. The fluid velocity upsurges with the improved values of the velocity slip parameter.

Analysis of Chemically Reactive Hydromagnetic Maxwell Fluid Conveying Tiny Particles Due to Navier Partial Slip

The analysis of nanofluids in the solar thermal system is very fascinating owing to its important engineering applications (i.e., solar collectors). Aside from these the non-Newtonian boundary layer fluid flow has experienced considerable attention due to uprising engineering applications in the solar thermal field. This work investigates the analysis of chemically reactive hydromagnetic Maxwell fluid conveying tiny particles due to Navier partial slip. The governing equations that model the transport phenomena were transformed using suitable similarity variables. The boundary value problem of the corresponding coupled nonlinear ordinary differential equations was solved numerically using the shooting technique together with the fourth-order Runge-Kutta integration scheme and in-built bvp4c package of MATLAB. The effects of various controlling parameters on velocity, temperature, and concentration distributions were presented graphically and studied theoretically. Furthermore, the study reveals that the Navier slip parameter (δ) increases as the velocity distribution decreases, while it enhances both the temperature and concentration distributions, increase in the radiation parameter (Nr) enhances the temperature distribution, and the chemical reaction (γ) increment leads to decrease in concentration distribution.

Local null controllability of a fluid–rigid body interaction problem with Navier slip boundary conditions

The aim of this work is to show the local null controllability of a fluid–solid interaction system by using a distributed control located in the fluid. The fluid is modeled by the incompressible Navier–Stokes system with Navier slip boundary conditions and the rigid body is governed by the Newton laws. Our main result yields that we can drive the velocities of the fluid and of the structure to 0 and we can control exactly the position of the rigid body. One important ingredient consists in a new Carleman estimate for a linear fluid–rigid body system with Navier boundary conditions. This work is done without imposing any geometrical conditions on the rigid body.

Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

Investigation of Time-Dependent Microscale Close Contact Melting

Close-contact melting process occurs due to direct heating of phase change material (PCM) by a sliding heater plate. Due to the shearing motion, a squeeze film flow is developed between them, which generates the pressure needed to support the PCM. Such close-contact melting phenomena have numerous applications in engineering and natural settings. When a micro flow is developed in the squeeze film, for example in microelectromechanical systems (MEMS), slip effects in velocity and/or temperature can become significant. This study investigates the influence of slip velocity and/or temperature on the contact melting of an electrically conducting PCM in the presence of a magnetic field. An analytical solution for the thin film flow and energy transport coupled with unsteady phase change heat transfer under Navier slip conditions and their interactions with the electromagnetic fields specified via the Maxwell's equations is developed. Numerical solutions of the resulting model in non-dimensional form revealed the effect of various characteristic parameters on the transient variation of the melting rate and the liquid film thickness. In particular, it is found that increasing the slip velocity in the absence of temperature slip increases the melting rate, with an earlier onset of the steady state. More specifically, the melting rate increases by 24% and the film thickness decreases by 28 % when the dimensionless slip length for velocity λ˜ equals to 1 × 10−3 compared to the corresponding no slip case. On the other hand, when temperature slip is included, an increase in slip velocity leads to slower melting rate as well as taking longer to attain the steady state. The melting rate decreases 36 % and the film thickness increases by 39%, when the dimensionless slip length for temperature λ˜Hequals to 1 × 10−3 compared to corresponding no slip case.

Slow rotation of a spherical particle in an eccentric spherical cavity with slip surfaces

A semi-analytical study of the steady flow around a spherical particle rotating in an incompressible Newtonian fluid inside an eccentric spherical cavity with slip surfaces about their common diameter is presented at low Reynolds numbers. To solve the Stokes equation, a solution consisting of the general solutions in two systems of spherical coordinates is employed and the boundary conditions are fulfilled by a collocation method. Accurate results of the torque exerted by the fluid on the particle are obtained as a function of the dimensionless parameters a∕b, d∕(b−a), βa∕η, and βwb∕η, where a and b are the radii and β−1 and βw−1 are the Navier slip coefficients of the particle and cavity, respectively, d is the distance between the particle and cavity centers, and η is the fluid viscosity. The boundary effect of the cavity with a slip wall on the rotation of the slip particle is quite significant and interesting. The torque normalized by that on the particle in an unbounded identical fluid vanishes as βwb∕η=0 (the cavity wall is fully slip), equals unity for a value of βwb∕η very close to 3, and in general increases with an increase in βwb∕η (or the stickiness of the cavity wall). When βwb∕η>3, the normalized torque is in general greater than unity, an increasing/decreasing function of the eccentricity parameter d∕(b−a) if the value of βa∕η (or the stickiness of the particle surface) is large/small, and an increasing function of βa∕η and a∕b. When βwb∕η<3, conversely, the normalized torque is in general less than unity, a decreasing/increasing function of d∕(b−a) if the value of βa∕η is large/small, and a decreasing function of βa∕η and a∕b. The cavity wall exerts less torque on the particle when it rotates about their common diameter than about an axis normal to it.

A magneto-bioconvective and thermal conductivity enhancement in nanofluid flow containing gyrotactic microorganism

The current article scrutinizes the magneto-bioconvective of nanofluid flow through a rotating cone using Cattaneo-Christov heat flux model. Additionally, we analyzed about cross diffusion (thermo-diffusion, diffusion-thermo), porous medium, Stefan blowing and Navier slip effects respectively. The governing equations are converting into system of self-similar ordinary differential equations by using suitable transformation. A parametric study focusing the impact of involved parameters on different fields such as motile microorganisms density, nanoparticle concentration, temperature and (swirl, tangential) velocities are plotted via graphs along with the motile microorganisms density numbers, Sherwood number and local Nusselt number are analyzed through table. As a matter of fact is temperature field declined for increasing thermal relaxation parameter γ. An increase in magnetic parameter M resembles to diminish in energy field. The suction of wall is noted for different number of when flow of swirl is reduce meanwhile injection blowing reveals conversely. The swirl velocity field diminishes to free stream from wall (cone surface) with every numbers of Nr including for stationary cone (Nr=0). Further, Dufour (Du) and Soret (Sr) number have entirely reverse behavior on temperature and nanoparticle concentration fields.