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In this paper, we study the incompressible limit of all-time strong solutions to the isentropic compressible Navier-Stokes equations with ill-prepared initial data and slip boundary condition in three-dimensional bounded domains. The uniform estimates with respect to both the Mach number ϵ ∈ (0 , 1 ] and all time t ∈ [ 0 , + ∞) are derived by establishing a nonlinear integral inequality. In contrast to previous results for well-prepared initial data, the time derivatives of the velocity are unbounded which leads to the loss of strong convergence of the velocity. The novelties of this paper are to establish weighted energy estimates of new-type and to carefully combine the estimates for the fast variables and the slow variables, especially for the highest-order spatial derivatives of the fast variables. The convergence to the global strong solution of incompressible Navier-Stokes equations is shown by applying the Helmoltz decomposition and the strong convergence of the incompressible part of the velocity. [ABSTRACT FROM AUTHOR]

In this work, we examine the oblique flow and heat transfer properties of a micropolar ternary hybrid nanofluid flowing over a lubricated surface while accounting for a convective boundary condition. Ternary hybrid nanofluids are a latest concept in the field of research that offer higher rates of heat transfer than both hybrid and conventional nanofluids. The study also considers the effects of the slip condition and heat radiation. When set to 0.5, the power-law index yields similar results. Through a similarity transformation, the system of partial differential equations is transformed into ordinary differential equations, and the BVP4C technique is used to achieve the numerical solution. Outcomes are shown for ordinary Al2O3-water and trihybrid SiC-MWCNT∖water nanofluids. The primary aim of this study is to investigate how lubrication influences the movement of the stagnation point and the reduction of shear stress, particularly in comparison to viscous fluids. These findings may be beneficial for polymeric processing applications. The study results presented in this publication demonstrate good agreement and consistency with other findings and are closely related to the previously published research. [ABSTRACT FROM AUTHOR]

This study examines the influence of power-law nanofluid flow and velocity slip on the heat transfer performance of tetrahybrid nanofluids over a stretching sheet, with the objective of enhancing thermal efficiency in industrial applications. The tetrahybrid nanofluids, consisting of A l 2 O 3 , T i O 2 , Cu, and F e 3 O 4 nanoparticles suspended in a sodium alginate base fluid, represent a novel combination of materials whose collective thermal properties have not been extensively studied. The performance of these tetrahybrid nanofluids is compared to that of hybrid and tri-hybrid nanofluids to evaluate their relative heat transfer capabilities. The boundary layer equations are solved using MATLAB’s bvp4c solver, applying similarity transformations to investigate temperature and velocity profiles under varying parameter conditions. The results indicate that tetrahybrid nanofluids significantly enhance heat transfer rates compared to standard nanofluids, with improvements of 7.43%, 6.27%, and 5.35% over hybrid and tri-hybrid nanofluids, respectively. Furthermore, the analysis reveals that velocity profiles intersect at a specific point, and as the velocity slip parameter increases, the profiles move further away from the stretching sheet. Key factors, such as the Prandtl number, power-law index, slip parameter, magnetic field, and Reynolds number, are also examined for their effects on flow and thermal behavior. The key innovation of this research lies in the introduction of tetrahybrid nanofluids as an advanced medium for heat transfer, providing superior thermal efficiency compared to previously studied nanofluids. This study not only builds upon existing research but also provides new computational insights into the potential applications of tetrahybrid nanofluids in areas like food processing and other industries requiring effective thermal management.Article Highlights: A greater value for the slip parameter indicates a higher temperature and a thinner layer of film. Slip conditions alter the velocity profile, reducing the shear stress at the surface and modifying the overall flow behavior. Increasing the value of the solid volume fraction causes the tetrahybrid nanofluid’s velocity to decrease and its temperature to rise. The study demonstrated that the inclusion of velocity slip at the boundary significantly affects the flow and thermal characteristics of the nanofluid film. Slip conditions alter the velocity profile, reducing the shear stress at the surface and modifying the overall flow behavior. The non-Newtonian behavior of the power-law tetra hybrid nanofluid (where the viscosity depends on the shear rate) was analyzed. The study likely showed that fluids with different power-law indices exhibit distinct flow behaviors, with shear-thinning or shear-thickening properties affecting the temperature and flow distribution over the stretching sheet. [ABSTRACT FROM AUTHOR]

Abstract This study examines the influence of power-law nanofluid flow and velocity slip on the heat transfer performance of tetrahybrid nanofluids over a stretching sheet, with the objective of enhancing thermal efficiency in industrial applications. The tetrahybrid nanofluids, consisting of $$Al_2O_3$$ A l 2 O 3 , $$TiO_2$$ T i O 2 , Cu, and $$Fe_3O_4$$ F e 3 O 4 nanoparticles suspended in a sodium alginate base fluid, represent a novel combination of materials whose collective thermal properties have not been extensively studied. The performance of these tetrahybrid nanofluids is compared to that of hybrid and tri-hybrid nanofluids to evaluate their relative heat transfer capabilities. The boundary layer equations are solved using MATLAB’s bvp4c solver, applying similarity transformations to investigate temperature and velocity profiles under varying parameter conditions. The results indicate that tetrahybrid nanofluids significantly enhance heat transfer rates compared to standard nanofluids, with improvements of 7.43%, 6.27%, and 5.35% over hybrid and tri-hybrid nanofluids, respectively. Furthermore, the analysis reveals that velocity profiles intersect at a specific point, and as the velocity slip parameter increases, the profiles move further away from the stretching sheet. Key factors, such as the Prandtl number, power-law index, slip parameter, magnetic field, and Reynolds number, are also examined for their effects on flow and thermal behavior. The key innovation of this research lies in the introduction of tetrahybrid nanofluids as an advanced medium for heat transfer, providing superior thermal efficiency compared to previously studied nanofluids. This study not only builds upon existing research but also provides new computational insights into the potential applications of tetrahybrid nanofluids in areas like food processing and other industries requiring effective thermal management.

Two-layer thermal convection exists widely in nature. In the present work, an experiment was conducted to investigate the heat transport and flow structure in two-layer thermal convection. In a rectangular convection cell, two immiscible fluids, glycerol and 2 cs silicone oil, were used as the working fluids. In the lower-thin glycerol layer, the bottom boundary was subjected to a no-slip boundary condition (BC), and the interface was subjected to slip BC. The aspect ratio of glycerol layer (lower) was Γ1=10.4. The Rayleigh number and Prandtl number of the glycerol layer covered the ranges of 260≤Ra1≤6 000 and 3 708 < Pr1 < 7 000, respectively. In the upper-thick silicone oil layer, the boundary at the top was subjected to no-slip BC. The aspect ratio of silicone oil (upper) was Γ2=0.53. The Rayleigh number and Prandtl number of the silicone oil layer covered the ranges of 1.5×109≤Ra2≤2.0×1010 and 28 < Pr2 < 33. It is found that the two-layer thermal convection has different heat transfer efficiencies and flow structures in two regions. For region 1 where the heat flux is smaller than a certain value, the glycerol layer (lower) is in a stable stratified state. For region 2 where the heat flux is greater than the certain value, a cellular pattern was formed in glycerol layer and the global heat transport was sharply increased through a subcritical bifurcation. The heat transport of glycerol layer exhibits oscillatory instability at the critical Rayleigh number Ra1c=1 523, which is smaller than the theoretic value 1 708 of critical value Ra for the 2D infinite Rayleigh-Bénard convection (RBC) with both rigid BCs. It reveals that the slip BC makes the fluid become unstable easier and enhances the heat transport. A measurement with shadowgraph method was further conducted. The cellular pattern of glycerol layer, the interface and hot plumes were also studied.

In this paper, we study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier-Stokes equations with Coulomb force [S. Liu, X. Xu and J. Zhang, Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier-Stokes-Poisson equations subject to large oscillations and non-flat doping profile, J. Differ. Equ. 269 (2020) 8468-8508] and compressible MHD equations [H. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal. 45 (2013) 1356-1387] to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity. [ABSTRACT FROM AUTHOR]

In this paper we examine a class of elliptic quasi-variational inequalities, which involve a constraint set and a set-valued map. First, we establish the existence of a solution and the compactness of the solution set. The approach is based on results for an elliptic variational inequality and the Kakutani-Ky Fan fixed point theorem. Next, we prove an existence and compactness result for a quasi-variational-hemivariational inequality. The latter involves a locally Lipschitz continuous functional and a convex potential. Finally, we present an application to the stationary incompressible Navier-Stokes equation with mixed boundary conditions which model a generalized Newtonian fluid of Bingham type. [ABSTRACT FROM AUTHOR]

We investigate compressible nematic liquid crystal flows in three-dimensional (3D) bounded domains with slip boundary condition for velocity and Neumann boundary condition for orientation field. By applying piecewise-estimate method and delicate analysis based on the effective viscous flux and vorticity, we derive the global existence and uniqueness of strong solutions provided that the initial total energy is suitably small. Our result is an extension of the works of Huang–Wang–Wen (J. Differential Equations 252: 2222–2265, 2012) and Li–Xu–Zhang (J. Math. Fluid Mech. 20: 2105–2145, 2018), where the local strong solutions in three dimensions and the global strong solutions for 3D Cauchy problem were established, respectively. Moreover, it also shows that blow up mechanism for local strong solutions obtained by Huang–Wang–Wen (Arch. Ration. Mech. Anal. 204: 285–311, 2012) cannot occur if the initial total energy is sufficiently small. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary-value problem of the isentropic compressible Navier–Stokes–Poisson equations subject to large and non-flat doping profile in 3D bounded domain with slip boundary condition and vacuum. The global well-posedness of classical solution is established with small initial energy but possibly large oscillations and vacuum. The steady state (except velocity) and the doping profile are allowed to be of large variation. [ABSTRACT FROM AUTHOR]

Hybrid nanoparticles, which are nanoparticles composed of multiple materials or phases, find applications in various fields due to their unique properties resulting from the combination of different materials. In such fluids, viscosity changes significantly in response to alterations in temperature or pressure. For example, these fluids become less viscous as they are heated and more viscous as they are cooled. Similarly, in certain situations, the application of a temperature variation can cause a change in viscosity. This property is commonly observed in materials like polymers and some oils. The article aims to investigate the heat transfer and flow behavior of fluid from an elastic sheet of hybrid nanoparticles, considering the temperature-dependent variable viscosity and thermal conductivity. The obtained equations for the mathematical model are partial differential equations converted to ODEs by applying a suitable similarity transformation. The findings show that the magnetic field opposes fluid motion. Our main findings are that adding nanoparticles to the base fluid significantly increased its heat conductivity. This improvement has great potential for uses requiring effective heat transmission, especially in engineering systems where heat dissipation plays a crucial role. The study also reveals the complex relationships that influence thermal conductivity, including slip boundary effects, viscosity fluctuations, magnetohydrodynamic effects, and porous media dynamics. Understanding these interactions is critical for optimizing heat transport processes in porous media applications. Comprehending these interplays is essential for refining heat transport mechanisms in applications involving porous media. The graphical representations are used to explain the physical behavior of various model parameters. Previous outcomes are also contrasted with the current ones. The findings show that the magnetic field opposes fluid motion.We range the subsequent list of values for parameters in every graph unless indicated accordingly.ϵ=0.1,Pr=2,M=0.5,R=0.2,K=5,fw=0.1,s=0.1 ,λ=1,φ1=0.02,φ2=0.04.

In this work, a lattice Boltzmann method (LBM) for studying microchannel gas flow is developed in the multi-flow regime. In the LBM, by comparing previous studies' results on effective viscosity in multi-flow regimes, the values of the rarefaction factor applicable to multi-flow regions were determined, and the relationship between relaxation time and Kn number with the rarefaction factor is given. The Kn number is introduced into the second-order slip boundary condition together with the combined bounce-back/specular-reflection (CBBSR) scheme to capture the gas flow in the multi-flow regime. Sensitivity analysis of the dimensionless flow rate to adjustable parameters using the Taguchi method was carried out, and the values of adjustable parameters were determined based on the results of the sensitivity analysis. The results show that the dimensionless flow rate is more sensitive to j than h. Numerical simulations of Poiseuille flow and pulsating flow in a microchannel with second-order slip boundary conditions are carried out to validate the method. The results show that the velocity profile and dimensionless flow rate simulated by the present numerical simulation method in this work are found in the multi-flow regime, and the phenomenon of annular velocity profile in the microchannel is reflected in the phases. [ABSTRACT FROM AUTHOR]

Abstract We consider the two-phase flow model in 3D exterior domains with slip boundary conditions. We establish the global existence of classical solutions of this system, provided that the initial energy is suitably small. Furthermore, we prove that the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions.

We deal with the barotropic compressible magnetohydrodynamic equations in three-dimensional (3D) bounded domain with slip boundary condition and vacuum. By a series of a priori estimates, especially the boundary estimates, we prove the global well-posedness of classical solution and the exponential decay rate to the initial-boundary-value problem of this system for the regular initial data with small energy but possibly large oscillations. The initial density of such a classical solution is allowed to contain vacuum states. Moreover, it is also shown that the oscillation of the density will grow unboundedly with an exponential rate when the initial state contains vacuum. [ABSTRACT FROM AUTHOR]

We consider the two-phase flow model in 3D exterior domains with slip boundary conditions. We establish the global existence of classical solutions of this system, provided that the initial energy is suitably small. Furthermore, we prove that the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions. [ABSTRACT FROM AUTHOR]

The lattice Boltzmann (LB) method has been extensively applied to the simulation of various fluid flows. Nonetheless, the standard LB model in rectangular coordinates faces some challenges when attempts are made to use it to simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length. Therefore, in this paper, boundary conditions for an axisymmetric LB model are proposed for the simulation of liquid slip at both convex and concave cylindrical surfaces. Based on the Navier slip boundary conditions, an equilibrium–nonequilibrium boundary scheme is developed to obtain the liquid slip in the azimuthal direction, with a combined bounce-back and specular-reflection boundary condition being developed to obtain the liquid slip in the axial direction. These boundary schemes are verified by simulating several established cases, namely, cylindrical Couette flow, harmonic oscillatory cylindrical Couette flow, Hagen–Poiseuille flow, and annular Poiseuille flow. The results are in excellent agreement with analytical results. Furthermore, some more complex flows involving super-hydrophobic cylindrical surfaces, such as stepwise oscillatory cylindrical Couette flow, are also simulated and studied using the proposed LB model. The axisymmetric LB model presented here overcomes the inability of standard LB models to accurately and efficiently simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length. • Slip boundary scheme of axisymmetric LB model is proposed for cylindrical surfaces. • Liquid slip in both the azimuthal and axial directions can be well captured. • The LB model is applied to fluid flows with super-hydrophobic cylindrical surfaces. • Numerical tests validate the reliability and generality of the proposed scheme. [ABSTRACT FROM AUTHOR]

The dynamic coefficients of a dry gas seal affect the dynamic characteristics of rotor-seal systems. Fluid films in a dry gas seal can be laminar, turbulent or with slip conditions, according to various operating conditions and design parameters. They can be defined as laminar or turbulent, depending on the Reynolds number, and as slip or non-slip, depending on the Knudsen number. However, previous research did not consider the effect of laminar, turbulent and slip conditions on the dynamic coefficients of a dry gas seal. We proposed a mathematical perturbation method to calculate the dynamic coefficients of the dry gas seal according to laminar, turbulent, and slip effects. We derived the perturbed equations of the modified Reynolds equation, which includes the effects of laminar, turbulent and slip conditions. The pressure of the modified Reynolds equation was solved using the finite element method and the Newton–Raphson method, and the perturbed pressures with respect to three degrees of freedom were calculated by substituting the calculated pressure into the perturbed equations. We verified the proposed method by comparing the simulated results with prior studies. The dynamic coefficients of a T-grooved dry gas seal were investigated according to laminar, turbulent, and slip conditions in a fluid film with different clearances. [ABSTRACT FROM AUTHOR]

Energetic Variational Approach is used to derive a novel thermodynamically consistent three-phase model of a mixture of Newtonian and visco-elastic fluids. The model which automatically satisfies the energy dissipation law and is Galilean invariant, consists of coupled Navier-Stokes and Cahn-Hilliard equations. Modified General Navier Boundary Condition with fluid elasticity taken into account is also introduced for using the model to study moving contact line problems. Energy stable numerical scheme is developed to solve system of model equations efficiently. Convergence of the numerical scheme is verified by simulating a droplet sliding on an inclined plane under gravity. The model can be applied for studying various biological or biophysical problems. Predictive abilities of the model are demonstrated by simulating deformation of venous blood clots with different visco-elastic properties and experimentally observed internal structures under different biologically relevant shear blood flow conditions.

This article deals with the solvability of the boundary-value problem for the Navier-Stokes equations with a direction-dependent Navier type slip boundary condition in a bounded domain. Such problems arise when steady flows of fluids in domains with rough boundaries are approximated as flows in domains with smooth boundaries. It is proved by means of the Galerkin method that the boundary-value problem has a unique weak solution when the body force and the variability of the surface friction are sufficiently small compared to the viscosity and the surface friction. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the combined effects of induced magnetic field and chemical reaction on MHD nonlinear mixed convective flow of Casson fluid over an inclined vertical porous plate embedded in a porous medium. The influence of viscous dissipation, heat source/ sink, and slip phenomena is taken into consideration. The effect of thermal radiation is also considered in the energy equation. The Casson fluid model is used to characterize the non-Newtonian fluid behavior. The main objective here is to analyze the induced magnetic field in a nonlinear mixed convective flow. At first, the appropriate similarity transformation is used to transform the governing nonlinear coupled partial differential equations into nonlinear coupled ordinary differential equations. The nonlinear ordinary differential equations are solved by a shooting technique with the help of bvp4c Matlab package. For the validation of the obtained results through the bvp4c Matlab solver, we have also solved this problem via R-K fourth-order method in Matlab and a good agreement is noted in both the results. The results of different physical parameters involved in the problem on the velocity, temperature, induced magnetic field and concentration are discussed by using graphs. It is noticed that the increasing values of the inclination angle cause rising of the induced magnetic field while induced magnetic field has opposite nature with magnetic parameter and magnetic Prandtl number. With increasing values of the thermal radiation parameter, the temperature profile diminishes. Apart from this, the numerical values of skin friction coefficient, Nusselt number and Sherwood number for the various values of parameters are displayed in tabular form. [ABSTRACT FROM AUTHOR]

We consider in an infinite horizontal layer the stationary convective flow of a viscous compressible and heat-conductive fluid subject to gravitational force, where the slip boundary condition for the velocity and the boundary condition of the temperature near the hydrostatic distribution are assumed. The existence of stationary solution close to hydrostatic state is obtained in Sobolev spaces as limit of fixed points of some suitable operators. [ABSTRACT FROM AUTHOR]

The micro-scale equipment has many advantages, including high thermal performance, high surface-to-volume ratio in heat transfer, small size, low weight, low required fluid and high design flexibility. In this study, fluid flow inside a microchannel is modeled under the assumption of laminar, incompressible, and two-dimensional flow under symmetric boundary conditions. The slip boundary condition is applied to the walls and the flow in the channel output is assumed to be fully developed. The effect of sinusoidal wall with the domain of 0.1 on the hydrodynamic and thermal behavior of the fluid is investigated and the results are compared with the results of smooth wall. The results show that for a constant Reynolds number, the maximum velocity decreases in the microchannel center by increasing the slip coefficient. Also, the comparison between the results of the wavy-wall microchannel and the microchannel with a smooth wall indicates that the heat transfer in the smooth microchannel is less than that in wavy-wall one. Considering the boundary conditions, the thermal behavior of the fluid is approximately the same for two cases in which both walls are sinusoidal and the only upper wall is sinusoidal.

Nanoparticles are useful in improving the efficiency of convective heat transfer. The current study addresses this gap by making use of an analogy between Al 2 O 3 and γ -Al 2 O 3 nanoparticles in various base fluids across a stretched sheet conjunction with f. Base fluids include ethylene glycol and water. We address, for the first time, the stagnation-point flow of a boundary layer of γ -Al 2 O 3 nanofluid over a stretched sheet with slip boundary condition. Al 2 O 3 nanofluids employ Brinkman viscosity and Maxwell's thermal conductivity models with thermal radiations, whereas γ -Al 2 O 3 nanofluids use viscosity and thermal conductivity models generated from experimental data. For the boundary layer, the motion equation was solved numerically using the fourth-order Runge–Kutta method and the shooting approach. Plots of the velocity profile, temperature profile, skin friction coefficient and reduced Nusselt number are shown. Simultaneous exposure of the identical nanoparticles to water and ethylene glycol, it is projected, would result in markedly different behaviors with respect to the temperature profile. Therefore, this kind of research instills confidence in us to conduct an analysis of the various nanoparticle decompositions and profile structures with regard to various base fluids. [ABSTRACT FROM AUTHOR]

A dry gas seal is a mechanical seal that prevents leakage of gas from rotating machines utilizing gas as a medium. Fluid film in a dry gas seal can exhibit laminar, turbulent, and slip behavior due to operating conditions and design parameters. A modified Reynolds equation that considers the effects of laminar, turbulent, and slip behavior of a fluid film was proposed and solved using the finite-element and Newton–Raphson methods to calculate the pressure, opening force, and leakage rate. The accuracy of the developed program was verified by comparing the simulated pressure with that of prior research, and the measured leakage with simulated leakage. The characteristics of a T-groove dry gas seal were investigated according to laminar, turbulent, and slip behavior in the fluid film. The results show that the effects of laminar, turbulent, and slip behavior in a fluid film on a dry gas seal should be considered to accurately predict the characteristics of a dry gas seal. [ABSTRACT FROM AUTHOR]

In the present article, it is aimed to discuss a new boundary condition for a slip in peristaltic transport for a micropolar fluid within an asymmetric medium. Power-law fluid with a thin film of coating will be used for lubrication purposes. Flow analysis is carried out for a two-dimensional asymmetric medium. Exact solutions are obtained for micro-rotation component, velocity, and stream function, for low Reynolds number and long wavelength assumptions. Velocity increase near the lower wall of the channel, while opposite behavior is observed at the upper wall. For the case of pumping, pressure increases with lubricant parameter and coupling number in pumping region, while decrease with micropolar parameter, phase difference, and channel width. On enhancing the impacts of lubrication, boluses start to disappear and become streamlines, which are parallel to the channel walls giving the laminar flow. The reflux criteria in the presence of lubrication is also discussed. The effects of lubricant parameter on wall shear stress are also debated. The magnitude of shear stress decreases on enhancing the amount of lubrication. Our findings align with the available literature for a particular scenario.

In this paper, we consider the stationary Stokes equations in an exterior domain three-dimensional under a slip boundary condition without friction. We set the problem in weighted Sobolev spaces in order to control the behavior at infinity of the solutions. In this work, we try to investigate the existence and uniqueness of the weak solutions related to this problem in L p - theory when p > 3 . Our proof is based on obtaining Inf-Sup conditions that play a fundamental role. [ABSTRACT FROM AUTHOR]

In this paper, we study the incompressible limit of strong solutions to the isentropic compressible Navier—Stokes equations with ill-prepared initial data and slip boundary condition in a three-dimensional bounded domain. Previous results only deal with the cases of the weak solutions or the cases without solid boundary, where the uniform estimates are much easier to be shown. We propose a new weighted energy functional to establish the uniform estimates, in particular for the time derivatives and the high-order spatial derivatives. The estimates of highest-order spatial derivatives of fast variables are subtle and crucial for the uniform bounds of solutions. The incompressible limit is shown by applying the Helmholtz decomposition, the weak convergence of the velocity, and the strong convergence of its divergence-free component. [ABSTRACT FROM AUTHOR]

In this article, Cattaneo Christov heat transfer analysis in nanofluid (Ferro Fe3O4 and Aggregation) flow between two parallel rotating disks with different velocities determined. The relaxation time, velocity slip, heat convective boundary condition, and heat generation are also presented. The governing partial differential equation (PDEs) model is converted into a set of nonlinear ordinary differential equations (ODEs) system by similarity variables. The solution is computed of the resulting ODEs system by using the Runge Kutta (Rk) method. Here a decline is noticed in the tangential velocity for nanoparticles (Fe3O4 and Aggregation nanoparticle) for higher values of the porosity parameter (λ1), slip parameter γ1, magnetic parameter (M) and Reynolds number ( R e r ), while tangential velocity arises for higher values of rotation parameters (ß1). This reduces the temperature field for nanoparticles by higher values of Eckert number (Ec), Prandtl number (Pr), Reynolds number ( R e r ), porosity parameter (λ1), while increases for arising the values of thermal relaxation parameter λ 2 , and for both Biot numbers ( B 1 , B 2 ) nanoparticles (Fe 3 O 4 and Aggregation nanoparticle). Further we compute the characteristics of physical quantities, namely skin friction and Nusselt number are presented. [ABSTRACT FROM AUTHOR]

The micro-scale equipment has many advantages, including high thermal performance, high surface-to-volume ratio in heat transfer, small size, low weight, low required fluid and high design flexibility. In this study, fluid flow inside a microchannel is modeled under the assumption of laminar, incompressible, and two-dimensional flow under symmetric boundary conditions. The slip boundary condition is applied to the walls and the flow in the channel output is assumed to be fully developed. The effect of sinusoidal wall with the domain of 0.1 on the hydrodynamic and thermal behavior of the fluid is investigated and the results are compared with the results of smooth wall. The results show that for a constant Reynolds number, the maximum velocity decreases in the microchannel center by increasing the slip coefficient. Also, the comparison between the results of the wavy-wall microchannel and the microchannel with a smooth wall indicates that the heat transfer in the smooth microchannel is less than that in wavy-wall one. Considering the boundary conditions, the thermal behavior of the fluid is approximately the same for two cases in which both walls are sinusoidal and the only upper wall is sinusoidal. [ABSTRACT FROM AUTHOR]

The dynamic coefficients of a dry gas seal affect the dynamic characteristics of rotor-seal systems. Fluid films in a dry gas seal can be laminar, turbulent or with slip conditions, according to various operating conditions and design parameters. They can be defined as laminar or turbulent, depending on the Reynolds number, and as slip or non-slip, depending on the Knudsen number. However, previous research did not consider the effect of laminar, turbulent and slip conditions on the dynamic coefficients of a dry gas seal. We proposed a mathematical perturbation method to calculate the dynamic coefficients of the dry gas seal according to laminar, turbulent, and slip effects. We derived the perturbed equations of the modified Reynolds equation, which includes the effects of laminar, turbulent and slip conditions. The pressure of the modified Reynolds equation was solved using the finite element method and the Newton–Raphson method, and the perturbed pressures with respect to three degrees of freedom were calculated by substituting the calculated pressure into the perturbed equations. We verified the proposed method by comparing the simulated results with prior studies. The dynamic coefficients of a T-grooved dry gas seal were investigated according to laminar, turbulent, and slip conditions in a fluid film with different clearances.

Lattice Boltzmann (LB) method, which is a mesoscopic numerical method, has been considered as a powerful tool to study microscale gas and liquid flows. Boundary slip phenomenon, which is a significant feature of both microscale gas and liquid flows, has been extensively studied by the LB method in the past decade. However, most of the previous works have focused on the microchannel flows and studies on the microtube flows are very rare. In this paper, we investigate the widely used slip boundary conditions, i.e., combined bounce-back and specular-reflection (BS) scheme, combined Maxwell-diffusion and specular-reflection (MS) scheme, and combined Maxwell-diffusion and bounce-back (MB) scheme, for the axisymmetric LB model with multi-relaxation-time (MRT) in detail. In order to realize the Navier slip boundary condition for liquid flows, we put forward to a reasonable strategy for determining the combination coefficients and the relaxation time. The proposed boundary schemes are validated by some numerical tests including the Hagen–Poiseuille flow, axisymmetric Womersley flow, Poiseuille flow in a circular annulus, and Womersley flow in a circular annulus. Numerical results are consistent with the analytical solutions, which demonstrate that the proposed boundary schemes are suitable for microscale liquid flows in microtube. • Axisymmetric MRT LB model with slip boundary conditions for liquid flow is proposed. • BS, MS, and MB schemes are essentially equivalent for liquid flow in microtube. • Combination coefficients and relaxation time τ q are dependent on the slip length. • Numerical tests validate the reliability and generality of the proposed schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop the lower order stabilized finite element methods for the incompressible flow with the slip boundary conditions of friction type whose weak solution satisfies a variational inequality. The H1 $H^{1}$-norm for the velocity and the L2 $L^{2}$-norm for the pressure decrease with optimal convergence order. The reliable and efficient a posteriori error estimates are also derived. Finally, numerical experiments are presented to validate the theoretical results.

In this article, the electro-osmotic flow of Oldroyd-B fluid in a circular micro-channel with slip boundary condition is considered. The corresponding fractional system is represented by using a newly defined time-fractional Caputo-Fabrizio derivative without singular kernel. Closed form solutions for the velocity field are acquired by means of Laplace and finite Hankel transforms. Additionally, Stehfest’s algorithm is used for inverse Laplace transform. The solutions for fractional Maxwell, ordinary Maxwell and ordinary Newtonian fluids are obtained as limiting cases of the obtained solution. Finally, the influence of fractional and some important physical parameters on the fluid flow are spotlighted graphically.

A dry gas seal is a mechanical seal that prevents leakage of gas from rotating machines utilizing gas as a medium. Fluid film in a dry gas seal can exhibit laminar, turbulent, and slip behavior due to operating conditions and design parameters. A modified Reynolds equation that considers the effects of laminar, turbulent, and slip behavior of a fluid film was proposed and solved using the finite-element and Newton–Raphson methods to calculate the pressure, opening force, and leakage rate. The accuracy of the developed program was verified by comparing the simulated pressure with that of prior research, and the measured leakage with simulated leakage. The characteristics of a T-groove dry gas seal were investigated according to laminar, turbulent, and slip behavior in the fluid film. The results show that the effects of laminar, turbulent, and slip behavior in a fluid film on a dry gas seal should be considered to accurately predict the characteristics of a dry gas seal.

The heat transfer phenomenon is beneficial and applicable in engineering, industries, and technological processes. The production of energy with the help of some cheap resources plays a pivotal and renewable role in the industrial development of the countries. Owing to such a significant performance of heat transfer, the steady slip flow and heat transfer of tangent hyperbolic fluid over a lubricating surface of the stretchable rotatory disk is investigated. The effects of MHD, nonlinear radiation, and non-uniform heat source/sink subject to nonlinear boundary conditions are included in motion and energy equations. Because of the lubrication, a thin layer of the power-law fluid is produced at the surface of the disk. Since the lubricating layer is thin, the interfacial conditions are applied at the surface between the fluid and lubricant. The governing nonlinear partial differential equations (PDEs) have been converted into ordinary differential equations (ODEs), which are solved numerically using the Keller-box method. The upshots of pertinent parameters upon the dimensionless distributions of velocity and temperature are deliberated. The surface drag forces and heat transfer rates are computed, and the effects of governing parameters on them are examined. With the enhancement in the slip at the interface and Weissenberg number, the radial and azimuthal velocities enhances close to the disk, whereas they observe two diverse trends for the power-law index. Also, temperature escalates for radiation parameter, and this escalation is prominent for nonlinear radiation.

Here, we addressed comparative investigation of hybrid nanofluid (MWCNTs + Cu + Water), nanofluid (MWCNTs + Water) and base fluid (Water). Flow is due to curved sheet. Flow via slip boundary condition is examined. Heat transport analysis is carried out in the presence of viscous dissipation, mixed convection and convective boundary condition. Transformation procedure is adopted for converting PDEs (continuity eq., momentum eq., energy eq. and boundary conditions) into ODEs. These nonlinear coupled ODEs are solved via shooting method with RK-4 algorithms (bvp4c). Behaviors of involved parameters on flow, Nusselt number (heat transfer rate), temperature and skin friction coefficient are analyzed graphically. Velocity of fluid enhances with increment in nanoparticles volume fraction for multi-walled CNTs, nanoparticle volume fraction for Cu and mixed convection parameter, while it can be reduced via higher estimations of velocity slip parameter. Temperature of the fluid varies directly with an increase in nanoparticles volume fraction for multi-walled CNTs, nanoparticle volume fraction for Cu, Eckert number and thermal Biot number. Skin friction coefficient is reduced via higher mixed convection as well as velocity slip parameters. Nusselt number intensifies with increment in nanoparticles volume fraction for multi-walled CNTs, nanoparticle volume fraction for Cu, Eckert and thermal Biot numbers. During comparative study amongst hybrid nanomaterial, nanomaterial and base fluid, efficient behavior is noted for hybrid nanomaterial. [ABSTRACT FROM AUTHOR]

In this article, the steady two‐dimensional axisymmetric flow of an incompressible viscous fluid in a porous medium under the influence of a uniform transverse magnetic field with slip boundary condition is analyzed and solved by using a new iterative method (NIM). This NIM depends mainly on the coefficients of a power series resulting from integrating the nth order differential equation with nonlinear operator derivatives. NIM has been used to obtain the approximate analytical solution by varying the pertinent flow parameters. The influence of different parameters on the present solution of the flow is shown through graphs with a discussion. These graphs refer to fact that increasing numbers of Reynolds (Re) and Hartmann (Me) give the attribute of decreasing velocity, with the addition that with an increase in Hartmann number and decrease in Reynolds number the streamlines are stretched towards the x‐axis. Finally, the NIM is effective and highly able to obtain an excellent result. [ABSTRACT FROM AUTHOR]

In this paper, the nonconforming virtual element method is studied to solve a hemivariational inequality problem for the stationary Stokes equations with a nonlinear slip boundary condition. The nonconforming virtual elements enriched with polynomials on slip boundary are used to discretize the velocity, and discontinuous piecewise polynomials are used to approximate the pressure. The inf-sup condition is shown for the nonconforming virtual element method. An error estimate is derived under appropriate solution regularity assumptions, and the error bound is of optimal order when lowest-order virtual elements for the velocity and piecewise constants for the pressure are used. A numerical example is presented to illustrate the theoretically predicted convergence order. [ABSTRACT FROM AUTHOR]

Effective boundary conditions, correct to third order in a small parameter ϵ , are derived by homogenization theory for the motion of an incompressible fluid over a rough wall with periodic micro-indentations. The length scale of the indentations is l, and ϵ = l / L ≪ 1 , with L a characteristic length of the macroscopic problem. A multiple scale expansion of the variables allows to recover, at leading order, the usual Navier slip condition. At next order the slip velocity includes a term arising from the streamwise pressure gradient; furthermore, a transpiration velocity O (ϵ 2) appears at the fictitious wall where the effective boundary conditions are enforced. Additional terms appear at third order in both wall-tangent and wall-normal components of the velocity. The application of the effective conditions to a macroscopic problem is carried out for the Hiemenz stagnation point flow over a rough wall, highlighting the differences among the exact results and those obtained using conditions of different asymptotic orders. [ABSTRACT FROM AUTHOR]

Considering the slip boundary condition, the rotating electro-osmotic flow of a third grade fluid in a channel formed by two parallel plates is investigated in the present study. The charge distribution is treated with the Debye–Hückel approximation analytically. Based on the finite difference method, the velocity profile for rotating electro-osmotic flow of third grade fluid is obtained numerically. It is shown that the non-Newtonian parameter of third grade fluid and the velocity slip factor play the important roles for the rotating electro-osmotic flow. The increasing non-Newtonian parameter slows down the flow and decreases the velocity magnitude, and the increasing slip parameter β has the similar influence on the velocity profile. Furthermore, the effect of the inclusion of third grade on the velocity profile is more conspicuous in the area near the walls. [ABSTRACT FROM AUTHOR]