Entropy generation in parallel plate microchannels

In this study the entropy generation in microchannels in microdevices induced by the transient laminar forced convection in the combined entrance region between two parallel plates has been investigated numerically. The study considers the microscales in the region of Kn < 0.001. The effects of aspect ratio, Reynolds number, Prandtl number, Brinkman number, and the motion of the lower plate on the entropy generation during the simultaneously developing flow in a parallel-plates channel are investigated. The obtained results addressing all cases are thoroughly in good agreement with the expectations that the entropy generation has its highest value at channel with the smallest aspect ratio at counter motion of the lower plate with the highest Re, Pr and Br/Ω values considered in the problem. An erratum to this article can be found at     

Starting flow in a rotating parallel plate channel

The starting flow due to a suddenly applied pressure gradient in a parallel plate channel which is rotating as a system is studied. Exact analytic series solutions to the unsteady Navier-Stokes equations are found by both the Laplace transform method and the separation of parameters method, the latter is shown to be superior. Rotation not only induces a secondary transverse flow but also alters the character of the transient flow rate and velocity profiles. Back flow and inertial oscillations occur, especially at higher rotation rates.     

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

Temperature profiles in a controlled-stress parallel plate rheometer

Actual melt temperatures are rarely measured directly when conducting rheological characterisations of materials in non-ambient conditions despite the potentially large influence temperature may have on the rheological data. For rheometers that use only a temperature-regulated lower plate, it is likely that the set point and true melt temperatures differ, an effect that becomes significant when characterising melts or suspensions close to phase-change events like crystallisation. This work investigates the magnitude of these effects for a controlled-stress rheometer featuring a temperature-controlled lower plate. The lower plate was fitted with a serrated cover disc that was found to exacerbate temperature deviations from the desired set point. Steady-state radial and vertical temperature profiles within the sample were measured and compared with the predictions of a finite element analysis model. The deviations between set point and measured temperatures were successfully predicted by the simulation for two typical gap heights for a thermoplastic, ceramic paste. The non-ideal heat transfer characteristics were also investigated numerically for a representative polymer system that demonstrated the increased deviations from ideal values for lower thermal conductivity materials.

Rotating electroosmotic flows in soft parallel plate microchannels

We present a theoretical investigation of rotating electroosmotic flows(EOFs) in soft parallel plate microchannels. The soft microchannel, also called as the polyelectrolyte-grafted microchannel, is denoted as a rigid microchannel coated with a polyelectrolyte layer(PEL) on its surface. We compare the velocity in a soft microchannel with that in a rigid one for different rotating frequencies and find that the PEL has a trend to lower the velocities in both directions for a larger equivalent electrical double layer(EDL) thickness λFCL(λFCL = 0.3) and a smaller rotating frequency ω(ω 5).However, for a larger rotating frequency ω(ω = 5), the main stream velocity u far away from the channel walls in a soft microchannel exceeds that in a rigid one. Inspired by the above results, we can control the EOF velocity in micro rotating systems by imparting PELs on the microchannel walls, which may be an interesting application in biomedical separation and chemical reaction.     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

The steady streaming generated by a vibrating plate parallel to a fixed plate

A viscous fluid is sandwiched between two parallel plates. The top plate performs both normal and lateral small translatory oscillations. The unsteady Navier-Stokes equations are solved using perturbations. The nonlinear Reynolds stress causes a secondary steady streaming. It is found that the normal oscillations induce a steady lift while the interaction between normal and lateral oscillations causes a directional net flux. The system thus operates as a valveless pump with rigid walls.     

Convective heat transfer in a parallel plate channel with porous lining

The paper proposes a theoretical model for the study of flow and heat transfer in a parallel plate channel, one of whose walls is lined with non-erodible porous material, both the walls being kept at constant temperatures. The analysis uses Brinkman model in the porous medium and employs the velocity and temperature slips at the interface (the so called nominal surface). The influence of the thickness as well as the permeability of the porous medium on the flow field and Nusselt numbers at the walls is investigated.

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Konvektive Wärmeübertragung in einem Parallelplattenkanal mit porösem Überzug

Zusammenfassung Die vorliegende Arbeit befaßt sich mit dem Vorschlag eines theoretischen Modells, um die Wärmeübertragung in einem Parallelplattenkanal mit unauswaschbarem porösem Überzug zu studieren. Die Strömung innerhalb des porösen Überzugs ist mit Hilfe der Brinkmannschen Gleichung analysiert. An der Grenze (der sogenannten Nominalfläche) zwischen dem Überzug und der freien Strömung sind die Geschwindigkeitsgleitung und die Temperaturgleitung benutzt. Die Beeinflussung des Geschwindigkeitsfelds und die Nusseltschen Zahlen an den Wänden in Abhängigkeit von der Dicke und der Durchlässigkeit des porösen Überzugs ist untersucht.

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Nomenclature u streamwise velocity in Zone 1 (Fig. 1) - û streamwise velocity in Zone 2 (Fig. 1) - p pressure - coefficient of viscosity of the fluid - k absolute permeability of the material used for lining - density of the fluid - R Reynolds number - the average velocity in Zone 1 (Fig. 1) - T temperature in Zone 1 (Fig. 1) - T temperature in Zone 2 (Fig. 1) - K thermal conductivity in Zones 1 and 2 (Fig. 1) - M ₁ non-dimensional mass flow rate in Zone 1 (Fig. 1) - M ₂ non-dimensional mass flow rate in Zone 2 (Fig. 1) - (Nu)_U Nusselt number at the upper plate (Fig. 1) - (Nu) _L Nusselt number at the lower plate (Fig. 1) - _E experimental value of the temperature in the channel (with porous lining) at a specified point - _E/* experimental value of the temperature in the channel (without porous lining) at a specified point     

The steady streaming generated by a vibrating plate parallel to a fixed plate in a dusty fluid

A viscous dusty fluid between two parallel plates, when the top plate is performing both normal and lateral small translatory oscillations, is considered. The unsteady Navier-Stokes equations are solved using the perturbation method. It is observed that nonlinear Reynolds stress introduces a secondary steady streaming. Normal oscillations induce a steady lift while the interaction between normal and lateral oscillations causes a directional net flux. The system thus is seen to be operating as a valveness pump with rigid walls. The presence of dust particles has accelerated the rate of increase of the lift forces. Net pumping is computed for both the fluid and the dust, and certain significant conclusions are drawn for the case Re = 1.     

Visco-elastic flow due to a plate which starts oscillating in the presence of a parallel stationary plate

In this paper, the flow of a visco-elastic liquid between two parallel plates has been studied when one plate is stationary and the other plate suddenly starts oscillating. Both finite Fourier sine transform and Laplace transform technique have been employed to solve the basic differential equations. The flow phenomenon has been characterized by the parameters, and and the effects of these on the flow characteristics have been studied through several graphs.Late professor of the department, who died in an accident on 7th July 1978.     

Electroviscous effect on electromagnetohydrodynamic flows of Maxwell fluids in parallel plate microchannels

Considering the influence of the streaming potential and electroviscous effects, the analytical solutions for electromagnetohydrodynamic(EMHD) flows in parallel plate microchannels are obtained. The electrolyte solutions in the microchannels are taken as generalized Maxwell fluids, and slip boundary conditions are adopted. To accurately analyze the EMHD flow characteristics, the variation trends of the electroviscous effects with the corresponding parameters must be understood. The results show that the electroviscous effects increase with the increase in the relaxation time De, the slip coefficientα, and the wall zeta potential ■ψ0. However, the increase in the inverse of the electrical double-layer(EDL) thickness K, the electrical oscillating Reynolds number Re, and the ionic P′eclet number P e can decrease the electroviscous effects. We also demonstrate that the electroviscous effect on the EMHD flows of generalized Maxwell fluids is larger than that of Newtonian fluids. This work will be useful in designing EMHD flows in parallel plate microchannels.     

Diffusioosmotic micropolar liquid flows in parallel plate microchannels subject to boundary slip

Meccanica - Analytical solutions to the microrotation, linear velocity, and volume flow rate are developed for electrokinetic diffusioosmotic flows of micropolar liquids in slit microchannel...     

Transient freezing of liquids in forced laminar flow inside a parallel plate channel

A simple analytical approximative solution was given for calculating the time dependent development of the ice-layers at the cooled walls inside a parallel plate channel. By ignoring the effect of acceleration, resulting from converging ice-layers in the axial direction, an analytical solution for the variation of the ice-layer thickness with time and axial position could be obtained. The approximative solution was checked by numerical calculations and good agreement was found.Es wurde ein analytisches Näherungsverfahren entwickelt, das es ermöglicht, die zeitliche Entwicklung der Erstarrungsfronten im gekühlten, ebenen Kanal zu bestimmen. Die Methode liefert unter Vernachlässigung der Beschleunigungsterme durch die konvergenten Eisschichten eine exakte Lösung der Phasengrenzbeziehung. Das Näherungsverfahren wurde mittels numerischer Berechnungen überprüft und stimmt bis zu Wandunterkühlungsverhältnissen vonB=10 sehr gut mit der numerischen Lösung überein.     

Entrance effects in the channels of the parallel plate stack in oscillatory flow conditions

This paper considers an entrance flow into the channels formed by a stack of parallel plates, placed in an acoustic resonator that provides oscillatory flow forcing. Interesting complex flow phenomena around the extremity of the stack are observed, essentially due to the introduction of cross-sectional discontinuities: vortex formation and shedding during the fluid ejection from the channels and development of an entrance flow during the suction phase, when the fluid enters the channels from outside. It is the latter that is of particular interest in this study. Particle image velocimetry (PIV) is used to investigate the flow structures in the “entrance region”. Velocity profiles are measured as a function of phase angle within an oscillation period and the distance from the stack end into the channel. Using the data obtained, an “entrance length” defined by analogy with existing fluid mechanical definitions, is estimated. The experiments are supplemented by CFD calculations to improve the understanding of such entrance flows.     

Thermophoretic augmentation of particle deposition in natural convection flow through a parallel plate channel

Present paper deals with temperature driven mass deposition rate of particles known as thermophoretic wall flux when a hot flue gas in natural convection flow through a cooled isothermal vertical parallel plate channel. Present study finds application in particle filters used to trap soot particles from post combustion gases issuing out of small furnaces with low technical implications. Governing equations are solved using finite difference marching technique with channel inlet values as initial values. Channel heights required to regain hydrostatic pressure at the exit are estimated for various entry velocities. Effect of temperature ratio between wall and gas on thermophoretic wall flux is analysed and wall flux found to increase with decrease in temperature ratio. Results are compared with published works wherever possible and can be used to predict particle deposition rate as well as the conditions favourable for maximum particle deposition rate.     

Development of instability in the wake of a plate positioned parallel to the flow

The data of systematic calculations of the development of instability in a wake under the influence of given external disturbances are presented. The evolution (linear and nonlinear) of a disturbance of given frequencyf of each mode and, moreover, the intermodal interaction are studied. Considerable attention is given to the investigation of the instability of the wake under the influence of a pair of disturbances: fundamental and subharmonic. In the calculations the amplitudes and phases of the disturbances were varied over a broad range.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 26–32, January–February, 1992.     

Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

Summary The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.Nomenclature A cross-sectional area of duct - c mean value of random molecular speed - d diameter of tube - f _p - f _t - h half height of parallel plate channel - Kn Knudsen number - L molecular mean free path - n directional normal of solid boundary - p pressure - p ₀ pressure at inlet - r radial co-ordinate - r _t radius of tube - R non-dimensional radial co-ordinate - Re _p 4hU/ - Re _t 2r _t U/ - s direction along solid boundary - T absolute temperature - u velocity in x direction - u* non-dimensional velocity - U bulk velocity = (1/A) _A u dA - v velocity in y direction - x axial co-ordinate - x* stretched axial co-ordinate - X non-dimensional axial co-ordinate - X* non-dimensional stretched axial co-ordinate - Y non-dimensional channel co-ordinate - eigenvalue in parallel plate channel - stretching factor - eigenvalue in tube - density - kinematic viscosity - i index - p parallel plate - t tube - v velocity vector - gradient operator - ² Laplacian operator