Forced convection of gaseous slip-flow in porous micro-channels under Local Thermal Non-Equilibrium conditions

Steady laminar forced convection gaseous slip-flow through parallel-plates micro-channel filled with porous medium under Local Thermal Non-Equilibrium (LTNE) condition is studied numerically. We consider incompressible Newtonian gas flow, which is hydrodynamically fully developed while thermally is developing. The Darcy–Brinkman–Forchheimer model embedded in the Navier–Stokes equations is used to model the flow within the porous domain. The present study reports the effect of several operating parameters on velocity slip and temperature jump at the wall. Mainly, the current study demonstrates the effects of: Knudsen number (Kn), Darcy number (Da), Forchheimer number (Γ), Peclet number (Pe), Biot number (Bi), and effective thermal conductivity ratio (K _R) on velocity slip and temperature jump at the wall. Results are given in terms of skin friction (C _f Re ^*) and Nusselt number (Nu). It is found that the skin friction: (1) increases as Darcy number increases; (2) decreases as Forchheimer number or Knudsen number increases. Heat transfer is found to (1) decreases as the Knudsen number, Forchheimer number, or K _R increases; (2) increases as the Peclet number, Darcy number, or Biot number increases.     

Variational iteration method for two-dimensional steady slip flow in micro-channels

In this paper, the two-dimensional steady slip flow in microchannels is investigated. Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). The Navier-Stokes equations for two-dimensional steady slip flow in microchannels are reduced to a nonlinear third-order differential equation by using similarity solution. The variational iteration method (VIM) is used to solve this nonlinear equation analytically. Comparison of the result obtained by the present method with numerical solution reveals that the accuracy and fast convergence of the new method.     

Forced convection flow over a microsphere

Forced convection heat transfer characteristics around a microsphere subjected to uniform heat flux boundary condition is numerically investigated in this study. Moderate to high values of Reynolds number and a wide range of Prandtl number are considered. The analysis assumes that the continuity assumption is valid and hence the Navier–Stokes equations are solved for the range of Knudsen number of 0.001 ≤ Kn ≤ 0.1. The appropriate boundary conditions at the surface of the microsphere; the velocity slip and temperature jump are applied. The effect of the flow parameters: Re, Pr and Kn on the velocity and temperature distribution is presented and hence a better control on the boundary layer thickness can be achieved in the microscale level. Furthermore, the effect of the controlling parameters on the delay of flow separation, reduced shear stress, drag coefficient and on the Nusselt number profiles is also presented in the results.     

Forced convection with laminar pulsating flow in a tube

The heat transfer model of laminar pulsating flow in a tube in rolling motion is established. The correlations of velocity, temperature and Nusselt number are obtained. The effects of several parameters on Nusselt number are investigated. The theoretical results are consistent with experimental data. Then the results are evaluated with Nield and Kuznetsov’s results. It is found that Nield and Kuznetsov’s results are not applicable for the laminar pulsating flow in nuclear power systems in ocean environments.     

Forced convection film condensation of flowing vapour on an inclined circular tube

In this work, an analytical study is made on the process of forced-convection-dominated laminar-film condensation on an inclined circular tube for a pure saturated vapour with vertical downflow. The resultant partial differential equation for the local film thickness has been solved analytically by applying the method of characteristics. Thus, an explicit analytical solution; in closed-form, has been obtained for calculating the local and average Nusselt numbers. The general solution yields, in special cases, the known analytical solutions of vertical and horizontal tubes. The results show that an optimum inclination of the tube, at which the rate of condensation on the whole surface is maximum, is a function of the ratio L/D. Received on 14 April 1999     

Forced convection of a power-law fluid in a porous channel – numerical solutions

A detailed numerical study of laminar forced convection in a porous channel which contains a fibrous medium saturated with a power-law fluid was performed. Hydrodynamic and heat transfer results are presented for a configuration that has uniform heat flux or uniform temperature heating at the walls. The flow in the porous medium was modeled using the modified Brinkman-Forchheimer-extended Darcy model for power law fluids in which the non-Darcy effects of inertia and boundary were considered. Parametric studies were conducted to examine the effects of Darcy number, power law index, inertia parameter and Prandtl number. The results indicate that when the power law index is decreased, the velocity gradient near the walls increases but these effects are reduced gradually as the Darcy number decreases until the Darcy regime (Da≤10⁻⁶) is reached in which case the effects of power law index become negligible. As the power law index is decreased, the thermal boundary layer thickness decreases significantly only in the non-Darcy regime. Consequently, as the power law index decreases, the fully developed Nusselt number increases considerably in the non-Darcy regime whereas in the Darcy regime the change in Nusselt number is very small. As the Prandtl number increases, the local Nusselt number increases and this effect is more significant for shear thinning fluids (n<1.0). Received on 2 March 1998     

Magnetogasdynamic flow past an infinite porous flat plate in slip flow regime

This paper presents an exact solution for the flow of a rarefied ionized gas over an infinite porous plate in the presence of a transverse magnetic field, by using the well known continuum approach. An attempt is made to bring out the salient features of the interaction between the applied magnetic field and the flow of a rarefied conducting gas. The analysis reveals that the skin friction, and the heat transfer into the plate are reduced due to gas rarefaction.     

Forced convection from an isolated heat source in a channel with porous medium

A numerical study is made of flow and heat transfer characteristics of forced convection in a channel that is partially filled with a porous medium. The flow geometry models convective cooling process in a printed circuit board system with a porous insert.The channel walls are assumed to be adiabatic. Comprehensive numerical solutions are acquired to the governing Navier-Stokes equations, using the Brinkman-Forchheimer-extended Darcy model for the regions of porous media. Details of flow and thermal fields are examined over ranges of the principal parameters; i.e., the Reynolds number Re, the Darcy number Da (≡K/H²), the thickness of the porous substrate S, and the ratio of thermal conductivities R_k (≡k_eff/k). Two types of the location of the porous block are considered. The maximum temperature at the heat source and the associated pressure drop are presented for varying Re, Da, S, and R_k. The results illustrate that as S increases or Da decreases, the fluid flow rate increases. Also, as R_k increases for fixed Da, heat transfer rates are augmented. Explicit influences of Re on the flow and heat transport characteristics are also scrutinized. Assessment is made of the utility of using a porous insert by cross comparing the gain in heat transport against the increase in pressure drop.     

On hydromagnetic fluctuating flow past an infinite porous plate in slip flow regime

The effect of a uniform external magnetic field on the laminar, incompressible rarefied gas flow along an infinite porous flat plate is studied under the following conditions: 1) there is uniform suction, 2) the external flow velocity varies periodically with time in magnitude but not in direction, 3) the magnetic Reynolds number is small and 4) the current occurs under slip flow boundary conditions. Expressions for the velocity and temperature fields in the boundary layer are obtained. The response of skin friction, and heat transfer to the fluctuating stream is studied for variations in the rarefaction parameter h ₁, the magnetic field parameter M, and the frequency of the fluctuating stream.Nomenclature c _p specific heat of the gas - f ₁ Maxwells reflection coefficient - f ₂ thermal accommodation coefficient - G as defined in (36) - h ₁ rarefaction parameter (L ₁ v ₀/) - h ₂ nondimensional temperature jump coefficient (L ₂ v ₀/) - H amplitude of the skin friction - k thermal conductivity - K _n Knudsen number - L mean free path - L ₁ (2–f ₁/f ₁) L - L ₂ - M magnetic field parameter ( ₀ B ₀ ² /v ₀ ² ) - m 1/2[1+(1+4M+4i)^1/2], m _r+im _i - n ₁ 1/2[1+(1+4M)^1/2] - q heat flux - R suction Reynolds number - T temperature - x, y coordinates along and perpendicular to the plates - u, v velocity components along x, y-directions - density - kinematic viscosity - ₀ electrical conductivity - Prandtl number - frequency of the fluctuating stream - nondimensional frequency parameter (/v ₀ ² ) - nondimensional distance from wall (v ₀ y/) - phase lead - U _{0 0} mean velocity in the boundary layer - U _{0 1}, U _{0 2} amplitude of the velocity fluctuation in the boundary layer - specific heat ratio     

Free convection above a horizontal circular disk in a saturated porous medium

The boundary-layer flow above a horizontal impermeable circular disk embedded in a saturated porous medium is considered in the cases both when the disk is held at a constant temperature above ambient and when heat is supplied to the convective fluid by the disk at a constant rate. Series solutions are obtained based on the flat plate solution, which holds at the edge of the disk, as the leading order terms. These series solutions can then be used to describe the flow nearly all the way across the disk. A simple approximate solution, based on an integrated form of the energy equation, is also obtained and is shown, for the constant wall temperature case, to be useful in indicating how the solution behaves near the centre of the disk. The solutions asr0, wherer measures distance from the centre is discussed in both cases, and it is shown that the solution develops a singularity with the boundary layer having a thickness of 0[(–logr)^1/2].

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Freie Konvektion über einer waagerechten Kreisscheibe in einem gesättigten porösen Medium

Zusammenfassung Es wird die Grenzschicht über einer waagerechten undurchlässigen Kreisscheibe, die in einem gesättigten porösen Medium eingebettet ist, untersucht. Zwei Möglichkeiten werden betrachtet: bei der einen ist die Oberflächentemperatur der Scheibe konstant, aber größer als die der Umgebung, bei der anderen ist die Scheibe gleichförmig beheizt. Es werden Lösungen in Form von Reihenentwicklungen erzielt, wobei die ersten Glieder den Lösungen einer ebenen Platte entsprechen, die an dem Rand der Scheibe gültig sind. Danach werden diese Lösungen zur Beschreibung der Bewegung der Flüssigkeit über einen großen Bereich der Scheibe benutzt. Eine einfache Lösung für den Fall der konstanten Wandtemperatur wird durch Integration der Energiegleichung erhalten und zur Beschreibung des Verhaltens der Lösung in der Nähe der Scheibenmitte verwendet. Man untersucht die Lösung fürr0 in beiden betrachteten Fällen, wobeir der Abstand von der Scheibenmitte ist. Bei einer Dicke von 0[(–logr)^1/2] der Grenzschicht weist die Lösung eine Singularität auf.

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Nomenclature a radius of disk - g acceleration due to gravity - K permeability of the porous medium - k thermal conductivity - q prescribed wall heat flux - Q (non-dimensional) heat transfer coefficient - r, coordinate measuring distance from the centre of the disk - R _a,R _a ^* Rayleigh number - T temperature of the convective fluid - T ₀ ambient temperature - T ₁ prescribed wall temperature - u Darcy's law velocity in ther-direction - V _w (non-dimensional) wall velocity - w Darcy's law velocity in thez-direction - x coordinate measuring distance from the edge of the disk - z, coordinate measuring distance normal to the disk - equivalent thermal diffusivity - coefficient of thermal expansion - non-dimensional temperature - _w (non-dimensional) wall temperature - viscosity of the convective fluid - stream-function - stream function at the edge of the boundary layer     

Transient free convection about a horizontal circular cylinder in a porous medium

The method of matched asymptotic expansions is employed for investigating the growth of the free convection boundary-layer on a horizontal circular cylinder which is embedded in a porous medium. It is assumed that the Rayleigh number is large, but finite, and the time of investigation is short. It is shown that the solution contains terms that are absent from the solution based on the boundary-layer approximation and that vortices form at both sides of the cylinder. The development of the plume region near the top of the cylinder, as well as the local and average Nusselt numbers, are evaluated and presented in graphical form.     

Mixed convection boundary layer flow near stagnation-point on vertical surface with slip

This paper considers the steady mixed convection boundary layer flow of a viscous and incompressible fluid near the stagnation-point on a vertical surface with the slip effect at the boundary. The temperature of the sheet and the velocity of the external flow are assumed to vary linearly with the distance from the stagnation-point. The governing partial differential equations are first transformed into a system of ordinary differential equations, which are then solved numerically by a shooting method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Both assisting and opposing flows are considered. The results indicate that for the opposing flow, the dual solutions exist in a certain range of the buoyancy parameter, while for the assisting flow, the solution is unique. In general, the velocity slip increases the heat transfer rate at the surface, while the thermal slip decreases it.     

The effect of slip distribution on flow past a circular cylinder

A slip boundary has been shown to have a significant impact on flow past bluff bodies. In this work and using a circular cylinder as a model system, the effects of various slip configurations on the passing flow are investigated. A theoretical analysis using matched-asymptotic expansion is first performed in the small-Reynolds number regime following Stokes and Oseen. A slip boundary condition is shown to lead to only higher-order effects (~1/ln(Re)) on the resulting drag coefficient. For higher Reynolds numbers (100–500), the effects of five types of symmetric slip boundary conditions, namely, no slip, fore-side slip, aft-side slip, flank slip, and all slip on the flow field and pertinent parameters are investigated with numerical simulations. Detailed results on the flow structure and force distribution are presented. Flank slip is found to have the best effect for drag reduction with comparable coverage of slip area. For asymmetric slip distributions, torque and lift are found to generally occur.     

Forced convection of turbulent flow in triangular ducts with different angles and surface roughnesses

The experimental investigations were consisting of two parts. The first part was carried out to study the effect of corner geometry on the steady-state forced convection inside horizontal isosceles triangular ducts with sharp corners. The electrically-heated triangular duct was used to simulate the triangular passage of a plate-fin compact heat exchanger. The isosceles triangular ducts were manufactured with duralumin, and fabricated with the same length of 2.4 m and hydraulic diameter of 0.44 m, but five different apex angles (i.e. θ _a =15^∘,30^∘, 40^∘,60^∘, and 90^∘) respectively. The investigation was performed under turbulent flow condition covering a wide range of Reynolds number (i.e. 7000≤Re _D ≤20000). It was found that the best thermal performance is achieved with the apex angle of 60^∘. The second part was performed to investigate the effect of surface roughness on the forced convection of the same system. Horizontal equilateral triangular ducts with an apex angle of 60^∘ were fabricated with the same length and hydraulic diameter, but different average surface roughnesses of 1.2 m,3.0 m and 11.5 m respectively. It was concluded that the duct with a higher surface roughness will have a better heat transfer performance. Non-dimensional expressions for the determination of the heat transfer coefficient of the triangular ducts with different apex angles and surface roughnesses were also developed. Received on 15 December 1997     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

Numerical simulation of laminar flow past a circular cylinder with slip conditions

The no‐slip condition is an assumption that cannot be derived from first principles and a growing number of literatures replace the no‐slip condition with partial‐slip condition, or Navier‐slip condition. In this study, the influence of partial‐slip boundary conditions on the laminar flow properties past a circular cylinder was examined. Shallow‐water equations are solved by using the finite element method accommodating SU/PG scheme. Four Reynolds numbers (20, 40, 80, and 100) and six slip lengths were considered in the numerical simulation to investigate the effects of slip length and Reynolds number on characteristic parameters such as wall vorticity, drag coefficient, separation angle, wake length, velocity distributions on and behind the cylinder, lift coefficient, and Strouhal number. The simulation results revealed that as the slip length increases, the drag coefficient decreases since the frictional component of drag is reduced, and the shear layer developed along the cylinder surface tends to push the separation point away toward the rear stagnation point so that it has larger separation angle than that of the no‐slip condition. The length of the wake bubble zone was shortened by the combined effects of the reduced wall vorticity and wall shear stress which caused a shift of the reattachment point closer to the cylinder. The frequency of the asymmetrical vortex formation with partial slip velocity was increased due to the intrinsic inertial effect of the Navier‐slip condition. Copyright © 2011 John Wiley & Sons, Ltd.     

Three dimensional unsteady convection and mass transfer flow through porous medium

The problem of three dimensional unsteady convection flow through a porous medium, with effect of mass transfer bounded by an infinite vertical porous plate is discussed, when the suction at the plate is transverse sinusoidal and the plate temperature oscillates in time about a constant mean. Assuming the free stream velocity to be uniform, approximate solutions are obtained for the flow field, the temperature field, the skin-friction and the rate of heat transfer. The dependence of solution on Pr (Prandtl number), Gr (Grashof number based on temperature), Gc (modified Grashof number based on concentration difference), Sc (Schimdt number), the frequency and the permeability parameter is also investigated.