Order Slip Boundary Conditions Research Articles

Exact solution for heat transport of Newtonian fluid with quadratic order thermal slip in a porous medium

In this communication, an analytical solution for the thermal transfer of Newtonian fluid flow with quadratic order thermal and velocity slips is presented for the first time. The flow of a Newtonian fluid over a stretching sheet which is embedded in a porous medium is considered. Karniadakis and Beskok’s quadratic order slip boundary conditions are taking into account. A closed form of analytical solution of momentum equation is used to derive the analytical solution of heat transfer equation in terms of confluent hyper-geometric function with quadratic order thermal slip boundary condition. Accuracy of present results is assured with the numerical solution obtained by Iterative Power Series method with shooting technique. The impacts of porous medium parameter, tangential momentum accommodation coefficient, energy accommodation coefficient on velocity and temperature profiles, skin friction coefficient and reduced Nusselt number are discussed. The Nusselt number increases with the higher estimations of tangential momentum and energy accommodation coefficients.

Investigation of the effect of MHD slip flow on heat transfer between parallel plates with second order slip boundary conditions

Investigation of the effect of MHD slip flow on heat transfer between parallel plates with second order slip boundary conditions

Numerical and scale analysis of non-Newtonian fluid (Eyring-Powell) through pseudo-spectral collocation method (PSCM) towards a magnetized stretchable Riga surface

A numerical simulation of a two-dimensional Eyring-Powell nanofluid flow is carried out through a Riga plate with radiative walls. Simultaneous contributions of chemical reaction and activation energy have also been taken into account on the mixed convection flow. Lubrication effects are applied by considering second order slip boundary condition in order to reduce the role of skin friction. A set of highly nonlinear and coupled differential equations are tackled with the help of “Pseudo-spectral collocation method” by developing numerical code on MATLAB. A detailed parametric study is performed that reveals that more energy adds to the system in response to slip parameter and Brownian motion of the nano-particles. Numerical results have also been compiled and tabulated. This is worth noticing that skin friction coefficient reduces due to modified Hartmann factor while heat flux is an increasing function of Prandtl number. However, there is a prominent increase in mass flux against all parameter, except activation energy.

Dusty Nanofluid Past a Centrifugally Stretching Surface

This communication reports, the flow of Cu-water dusty nanofluid past a centrifugally stretching surface under the effect of second order slip and convective boundary conditions. The coupled nonlinear ordinary differential equations are get hold of from the partial differential equations which are derived from the conservation of momentum and energy of both nanofluid and dusty phases. Then, using apt resemblance transformation these ordinary differential equations were altered into a dimensionless form and then solved by bvp5c solver in Matlab software. The variation in velocity and temperature profiles of fluid and dusty phases for different parameters are thrash out in depth by figures and tables. The outcomes exhibit that the velocity profile of both fluid and dusty phases boot as the values of the dust particle volume fraction parameter is enlarged. Besides, the magnetic field parameter has similar effect on the velocity profile of both fluid and dusty phases. Also, the results illustrated that temperature profile of both Cu-water nanofluid and dusty particle phases are improved within an enhancement in the values of the temperature relaxation parameter, Cu-particle volume fraction, and Biot number. The results also confirm that for greater values of the magnetic field parameter the values of skin friction coefficient are enlarged for all values of the velocity ratio parameter.

Effect of heat transfer and geometry on micro-thruster performance

Coupled Navier-Stokes and Direct Simulation Monte Carlo (NS-DSMC) simulations of gas flow in micro-nozzles for various wall thermal conditions and geometrical aspects are presented. Micro-thrusters employed in miniature spacecraft and microsatellites experience substantial changes in wall thermal conditions. This can influence the internal boundary layer development and the exit plume structure of a micro-nozzle. These changes in flow physics differ with the nozzle divergence angle and the proximity of a similar nozzle in the cluster. Continuum solvers often fail to analyze the micro-nozzles operating in vacuum conditions as the flow in micro-nozzles experiences continuum, transitional, and rarefied regimes. Coupled NS-DSMC solver is an effective alternative that can simulate the non-equilibrium effects in a micro-nozzle flow field. A steady solution of the entire flow field has been obtained using a non-linear Harten-Lax-van Leer-Contact (HLLC) scheme based finite volume solver with a higher order slip boundary condition. Continuum breakdown regions are identified based on the gradient-length local (GLL) Knudsen number condition. This initial steady solution on the flow transition boundary is implemented in the DSMC solver as a Dirichlet boundary condition. The present computations are useful in calibrating micro-propulsion controllers to adapt to the substantial momentum changes associated with various nozzle wall thermal conditions and the proximity of similar nozzles in the cluster.

Numerical investigation of the effect of second order slip flow conditions on interfacial heat transfer in micro pipes

Heat transfer that occurs in micro scale devices has a very important place among the engineering applications that cooling or heating. This heat transfer mechanism in devices having dimensions at micron level is a completely different problem in the macro level analysis. Therefore, in the calculations made, the flow events and heat transfer in micron scale pipes are calculated by using more realistic expressions. For this reason, in this study heat transfer in a circular micro pipe with wall and fluid conjugation for laminar rarefied gas flow in transient regime is investigated under the second order slip boundary conditions at the interface. Patankar’s control volume method is used here to solve the problem numerically. This analysis includes of axial conduction, viscous dissipation and rarefaction effects which are indispensable in micro-flow structure. From the results, it is seen that the values that are indicating heat transfer are excessively affected by wall thickness, viscous heating and gas rarefaction especially in transient regime.

Effects of Second Order Slip Boundary Condition on Magnetohydrodynmaics Boundary Layer Flow and Heat Transfer of Nanofluid Over a Stretching Sheet

Effects of Second Order Slip Boundary Condition on Magnetohydrodynmaics Boundary Layer Flow and Heat Transfer of Nanofluid Over a Stretching Sheet

Heat transfer in rough microchannels under rarefied flow conditions

A hybrid solver dynamically coupling the kinetic solution computed in local rarefied areas and the Navier–Stokes solution in the rest of the flow is used for the investigation of heat transfer in microchannels with rough wall surfaces. Roughness is modeled as a series of triangular obstructions with a relative roughness height from 0 up to 5% of the channel height. A nearly incompressible gas flow (low Mach number) for a range of Knudsen numbers from 0.01 up to 0.1 is considered. The competition between roughness, rarefaction and heat transfer effects is discussed in terms of average Nusselt and Poiseuille numbers and mass flow rate. The discrepancy between the full Navier–Stokes and hybrid solutions is analyzed, assessing the range of applicability of the first order slip boundary condition for rough geometries in the presence of heat transfer.

Coal permeability: Gas slippage linked to permeability rebound

The main factors/mechanisms that influence coal permeability are effective stress, swelling, shrinkage, deformation and gas slippage. After an extended period of gas production, coal can have a rebound phenomenon where permeability increases with increasing effective stress. This rebound can have a significant impact on gas recovery during the late stages of a reservoir life cycle. This paper aims to characterise coal permeability by combining laboratory measurements with a simple gas slippage model that explains the rebound phenomenon. Gas and Klinkenberg corrected permeabilities of coal are measured at (1) constant confining pressure and (2) constant effective stress. We estimate the length scales relevant to gas flow using mercury intrusion, a permeability slip model, and the kinetic theory of gases, which allows us to estimate the Knudsen number for gas flow. Results show a linear relationship between slip length and the mean free path of gas for all of the tested mean pore pressures. This result suggests that a first order slip boundary condition is sufficient to explain the momentum exchange at the gas/solid boundary during flow under normal reservoir conditions. A correlation between Knudsen number and increased permeability is developed, which further demonstrates that slippage cannot be neglected in coals when Knudsen number is greater than 0.1. Overall, we present a simple model that explains permeability rebound in coal by considering only gas slippage. We do not discredit the mechanism of coal shrinkage, which could also influence coal permeability. We confirm that gas slippage should be considered in coal permeability models.

Numerical investigations on the effect of fin shape and surface roughness on hydrothermal characteristics of slip flows in microchannels with pin fins

Accurate modelling of convective heat transfer in micro/nano gas flows is critical for many applications such as temperature/mass flow micro-sensors and mixing/separation analysis of gases in micro-systems. This study numerically investigates the effect of pin fin shape and wall roughness on heat transfer and flow field of gas flows in rough microchannels in the presence of the second order slip boundary condition. The hydrothermal characteristics of flows were obtained for a smooth microchannel under slip boundary conditions for the constant wall temperature (330 K) and constant wall heat flux (10 kW/m2) boundary conditions. Afterwards, four types of pin fin shapes (rectangular, diamond, oblong, and elliptic) and two types of surface roughness (regular and random rough elements) were taken into account. According to the obtained results, although velocity slip raises Nusselt number, temperature jump tends to reduce it. It was shown that the generated recirculating flows between the roughness elements reduce Nusselt number and increase friction factor. Furthermore; it was found that the effect of pin fin shape diminishes with surface roughness.

Further Study on Second-Order Slip Flow Models in Channels of Various Cross Sections

ABSTRACTThe transition flow regime is the most complicated domain for modeling since the transition regime is a varying mixture of different transport mechanisms. Modeling the Navier-Stokes equations with the appropriate second order slip boundary condition at high Knudsen numbers is further explored in this paper. The general models proposed by one of the authors are validated against extensive numerical and experimental data in the literature. It is noteworthy that the geometry and the partial momentum accommodation are taken into account by the models. The present work provides a markedly simple way to compute quantities of engineering interest in microchannels and nanochannels of various cross sections over the full transition flow regime that is estimated to be accurate to 10%, with most data within 5%. As in the transition regime no data exist for most various geometries, this proposed engineering model fills this void. This paper further explores and validates working engineering macro models that can be easily implemented in applications.

Magnetohydrodynamics (MHD) flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition

This article presents the effect of thermal radiation on magnetohydrodynamic flow of tangent hyperbolic fluid with nanoparticle past an enlarging sheet with second order slip and convective boundary condition. Condition of zero normal flux of nanoparticles at the wall is used for the concentration boundary condition, which is the current topic that have yet to be studied extensively. The solution for the velocity, temperature and nanoparticle concentration is governed by parameters viz. power-law index (n), Weissenberg number We, Biot number Bi, Prandtl number Pr, velocity slip parameters δ and γ, Lewis number Le, Brownian motion parameter Nb and the thermophoresis parameter Nt. Similarity transformation is used to metamorphosed the governing non-linear boundary-value problem into coupled higher order non-linear ordinary differential equation. The succeeding equations were numerically solved using the function bvp4c from the matlab for different values of emerging parameters. Numerical results are deliberated through graphs and tables for velocity, temperature, concentration, the skin friction coefficient and local Nusselt number. The results designate that the skin friction coefficient Cf deplete as the values of Weissenberg number We, slip parameters γ and δ upturn and it rises as the values of power-law index n increase. The local Nusselt number -θ′(0) decreases as slip parameters γ and δ, radiation parameter Nr, Weissenberg number We, thermophoresis parameter Nt and power-law index n increase. However, the local Nusselt number increases as the Biot number Bi increase.

Diffusion of chemically reactive species of a Maxwell fluid due to an unsteady stretching sheet with slip effect

The influence of a first order slip boundary condition on the diffusion of chemically reactive species in an unsteady MHD boundary-layer flow of a non-Newtonian Maxwell fluid over a vertical permeable linearly stretching or shrinking sheet is considered. The whole analysis is performed taking the first order chemical reaction and linearly varying wall concentration. Choosing appropriate similarity variables, the governing equations of the problem are transformed into a set of non-linear coupled self-similar equations, which are then solved numerically by the shooting technique. A comparison with previously published results is performed for certain cases and the results are found to be in excellent agreement. The flow features and mass transfer characteristics for different values of the governing parameters are graphically presented and discussed in detail.

Compressibility Effects in 2d Wall Heating Microchannel flow

In this article we present a numerical solution of the Navier-Stokes equations and energy equation in parallel plate microchannels with the first order slip boundary conditions on the walls, adopting control volume scheme of CFD technique. Wall heating condition was considered on the walls. Noslip boundary conditions for compressible and incompressible flows were also solved to compare the effect of slip conditions. Compressibility effects were also investigated for compressible slip and compressible noslip flow conditions. A series of simulations were performed for different heights and lengths of channels and pressure ratios. Results are presented in graphs and tables and are compared with the available analytical and experimental results. It was found that the friction constants are the highest for noslip compressible flow and lowest for the slip flow against pressure ratio and mach numbers. Friction constant decreases continuously for compressible slip flow but it approaches to an asymptotic value of 96 for compressible noslip flow for the decrease of aspect ratio.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 57-71

Slip Velocity Distribution in a Parallel Plate Channel on Asymmetric Thermal Boundary Condition

Steady, laminar and fully developed flows in parallel plate microchannel with asymmetric thermal wall conditions are solved by control volume technique. In order to examine the influence of Reynolds number and Knudsen number on the velocity distributions, a series of simulations are performed for different Reynolds and Knudsen numbers. Nitrogen gas is used as working fluid and we assume the fluid as continuum but employ the slip boundary conditions on the walls. The Navier-Stokes and energy equations are solved simultaneously. The results are found in good agreement with those predicted by analytical solutions in 2D continuous flow model employing first order slip boundary conditions. It is concluded that the rarefaction flattens the velocity distribution. If the product of Reynolds numbers and Knudsen numbers is fixed, the cross sectional average velocity is fixed for incompressible flow.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 113-126

Rarefaction effects on the flow characteristics in microchannels on asymmetric wall thermal condition

Rarefaction effects on the flow characteristics in 2D microchannels on asymmetric wall thermal conditions are investigated by control volume technique. In order to examine the influence of Knudsen numbers on the flow characteristics, a series of simulations for both compressible and incompressible flow with different Reynolds and Knudsen numbers are performed. Nitrogen gas is used as working fluid and the slip boundary conditions are used on the walls. The Navier-Stokes and energy equations are solved simultaneously. The results are found in good agreement with those predicted by analytical solutions in 2D continuous flow model employing first order slip boundary conditions. It is shown here that the Knudsen number has effects on velocity and temperature distribution for both the compressible and incompressible flows. It causes the velocity slip on the wall and causes the temperature difference between the wall temperature and the gas temperature on the wall.

Modeling Cone Self-acting Gas Lubrication Bearing Dynamics

An analysis of static and clearance flow characteristics for the cone self-acting gas lubrication bearing of high precision gyro motor and deduction of first order slip boundary conditions were presented. The computation fluid dynamics based on boundary conditions of no slip and the first-order slip were used to make numerical simulation of clearance complex flow for the bearing. Static loads, stiffness, powers and offset angles with different eccentricity of the bearing, the results were compared with experiment results of the bearing, exterior load stiffness measurement system. The results show that: the experiment measurement results stiffness of the bearing in 1g were between the results of numerical simulation with no slip and first-order slip boundary conditions, experiment data less than about 10% with the calculation results from no slip boundary conditions, and more than 44% with the calculation results from first slip boundary conditions, which showed that slip phenomenon partly existed in clearance flow of the bearing. The offset angle orientation of numerical simulation were consistent with the experiment data, differences of 4° and 2.6° respectively.

Magnetic field effect on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect

Abstract The magnetic field effect on a steady two dimensional laminar radiative flow of an incompressible viscous water based nanofluid over a stretching/shrinking sheet with second order slip boundary condition is investigated both analytically and numerically. The governing partial differential equations are reduced to nonlinear ordinary differential equations by means of Lie symmetry group transformations. The dimensionless governing equations for this investigation are solved analytically using hyper-geometric function and numerically by the fourth order Runge–Kutta method with the shooting technique. A unique exact solution exists for momentum equation in stretching sheet case and dual solutions are obtained for shrinking sheet case which has upper and lower branches. It is found that the lower branch solution vanishes in the presence of higher magnetic field. The velocity and temperature profiles, the local skin friction coefficient and the reduced Nusselt number are examined and discussed for different spherical nanoparticles such as Au, Ag, Cu, Al, Al 2 O 3 and TiO2. A comparative study between the previously published results and the present analytical and numerical results for a special case is found to be in good agreement.

Pressure Distribution of the Gaseous Flow in Microchannel: A Lattice Boltzmann Study

AbstractIn this paper the pressure distribution of the gaseous flow in a microchannel is studied via a lattice Boltzmann equation (LBE) method. With effective relaxation times and a generalized second order slip boundary condition, the LBE can be used to simulate rarefied gas flows from slip to transition regimes. The Knudsen minimum phenomena of mass flow rate in the pressure driven flow is also investigated. The effects of Knudsen number (rarefaction effect), pressure ratio and aspect ratio (compression effect) on the pressure distribution are analyzed. It is found the rarefaction effect tends to the curvature of the nonlinear pressure distribution, while the compression effect tends to enhance its nonlinearity. The combined effects lead to a local minimum of the pressure deviation. Furthermore, it is also found that the relationship between the pressure deviation and the aspect ratio follows a pow-law.

Gaseous slip flow forced convection through ordered microcylinders

This is a theoretical study dealing with longitudinal gaseous slip flow forced convection between a periodic bunch of microcylinders arranged in regular array. The selected geometry has applications in microscale pin fin heat sinks used for cooling of microchips. The flow is considered to be hydrodynamically and thermally fully developed. The two axially constant heat flux boundary conditions of H1 and H2 are considered in the analysis. The velocity and temperature discontinuities at the boundary are incorporated into the solutions using the first order slip boundary conditions. The method considered is mainly analytical in which the governing equations and three of the boundary conditions are exactly satisfied. The remaining symmetry condition on the right-hand boundary of the typical element is applied to the solution through the point matching technique. The results show that both the Poiseuille number and the Nusselt number are decreasing functions of the degree of rarefaction characterized by the Knudsen number. While an increase in the blockage ratio leads to a higher Poiseuille number, the functionality of the Nusselt number on this parameter is not monotonic. At small and moderate values of the blockage ratio, the Nusselt number is higher for a higher blockage ratio, whereas the opposite may be right for higher values of this parameter. It is also observed that the angular variations of the parameters are reduced at smaller blockage ratios. Accordingly, the H1 and H2 Nusselt numbers are the same for small and moderate blockage ratios.