Parallel-plate Channel Research Articles

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan–Majda model and Taylor–Aris Dispersion

We study the long time behavior of an advection–diffusion equation with a random shear flow which depends on a stationary Ornstein–Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary N-point correlator using the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). In turn, appealing to the conclusion of the Hausdorff moment problem (Shohat and Tamarkin, 1943), we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability density function at long times. The strategy we presented is not only restricted to the parallel-plate channel domain. The same methods can derive effective equations for a straight channel with uniform arbitrary cross-section. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar’s long time limiting probability density function (PDF) for different types of initial conditions (e.g. deterministic and random).

Investigation of transition in a natural convection flow in an inclined parallel Plate Channel

In an inclined parallel plate channel, the influence of the angle of inclination on the transition of the free convection flow has been explored. Three dimensional large eddy simulations have been carried out employing the wall adaptive local eddy viscosity sub-grid scale model. The inclination angles of 15°, 30°, and 60° with respect to the horizontal are considered. The upper wall of the parallel plates is kept at constant heat flux, while the bottom one is adiabatic. The Grashof number, GrL considered in the present study is 8.22 × 1012. The variation of the transition zone of the flow with variation in inclination angle is shown using thermal boundary layer formation, variation of velocity fluctuations, Nusselt number variation, and Q criterion. Investigation of the thermal boundary layer shows the formation of flume structure at the early stage of the flow in the case of lower inclination of the channel. As the degree of inclination increases, the transition is delayed, and the transition region shifts downstream.

A fully analytical solution of convection in ferrofluids during Couette-Poiseuille flow subjected to an orthogonal magnetic field

We report the transport of thermal energy during a hydrodynamically as well as thermally developed Couette-Poiseuille flow of ferrofluid (FF) through parallel-plate channels subjected to a constant magnetic field, orthogonal to the channel axis. The fluid motion is imparted by a combination of tangential stress, attributable to the motion of one of the plates, and a constant pressure gradient present in the direction of the channel axis. Apart from imparting a body force and torque, the application of an external magnetic field increases both the dynamic viscosity and effective thermal conductivity of the FF. As a result, thermal energy transport is uniquely affected. The complex coupling of magnetics with hydrodynamics results in implicit governing equations, which cannot be solved analytically in their full form. On the basis of logical reasoning, the governing equations are simplified to generate closed-form solutions of temperature profiles and Nusselt numbers. For a comprehensive study, we consider three different sets of thermal boundary conditions amenable to analytical solutions. Till the saturation of magnetization, we observe an increase in the local temperature as the strength of the applied magnetic field increases. The effects of particle loading in FF, Brinkman number, and heat flux ratio on the temperature field are also discussed.

Flow characteristics in a parallel-plate porous channel under convective boundary conditions and triple diffusion for the non-Darcy porous matrix

This article inspects the effect of triple diffusion in a vertical conduit encapsulated with porous matrix and subjected to third kind boundary conditions. Third kind boundary condition is a combination of Dirichlet and Neumann boundary conditions which specifies a linear combination of function and its derivative values on the boundary. Homogeneous chemical reaction along with viscous and Darcy dissipation effects are included. Adapting the Boussinesq approximation, the soultal buoyancy effects due to concentration gradients of the dispersed components are taken into account. Applying suitable transformations, the conservation equations are reduced into dimensionless form and the dimensionless parameters evolved are thermal Grashof number (0≤Λ1≤20), solutal Grashof number (for species 1 and 2, 0≤Λ2,Λ3≤20), porous (2≤σ≤8) and inertial parameters (0≤I≤6), Biot numbers (at the left and right walls, 1≤Bi1,Bi2≤10), Brinkman number (0≤Br≤1), Schmidt numbers (0≤Sc1,Sc2≤6), Soret numbers (Sr1=Sr2=1) and temperature difference ratio (RT=1). Adopting perturbation technique, the analytical solutions which are applicable only when the Brinkman number is less than one is appraised. However for any values of the Brinkman number, Runge-Kutta shooting method is operated. The impact of selected parameters on the momentum, heat and dual species concentration fields are presented in the form of pictures. The solutions computed by numerical method are justified by comparing with the analytical method. The numerical and analytical solutions are equal in the absence of Darcy and viscous dissipations and the discrepancy advances as the Brinkman number expands. Further the solutions obtained are also justified by comparing the results with Zanchini [1] in the absence of chemical reaction for clear fluid. The thermal field is augmented with the Brinkman number for symmetric and asymmetric Biot numbers. However the profiles are highly distinct at the cold plate for unequal Biot numbers in comparison with equal Biot numbers. The conclusions are admissible to materials processing and chemical transport phenomena.

Unsteady MHD Plane Couette-Poiseuille Flow of Fourth-Grade Fluid with Thermal Radiation, Chemical Reaction and Suction Effects

This study investigates the unsteady MHD flow of a fourth-grade fluid in a horizontal parallel plates channel. The upper plate is oscillating and moving while the bottom plate is stationary. Solutions for momentum, energy and concentration equations are obtained by the He-Laplace scheme. This method was also used by Idowu and Sani [12] and there is agreement with our results. The effect of various flow parameters controlling the physical situation is discussed with the aid of graphs. Significant results from this study show that velocity and temperature fields increase with the increase in the thermal radiation parameter, while velocity and concentric fields decrease with an increase in the chemical reaction parameter. Furthermore, velocity, temperature and concentric fields decrease with an increase in the suction parameter. It is also interesting to note that when <i>S4</i> = <i>0</i>, our results will be in complete agreement with Idowu and Sani [12] results. The results of this work are applicable to industrial processes such as polymer extrusion of dye, draining of plastic films etc.

Enhanced transport of ions by tuning surface properties of the nanochannel.

Motivated by recent observations of anomalously large deviations of the conductivity currents in confined systems from the bulk behavior, we revisit the theory of ion transport in parallel-plate channels and also discuss how the wettability of a solid and the mobility of adsorbed surface charges impact the transport of ions. It is shown that depending on the ratio of the electrostatic disjoining pressure to the excess osmotic pressure at the walls two different regimes occur. In the thick channel regime this ratio is small and the channel effectively behaves as thick, even when the diffuse layers strongly overlap. The latter is possible for highly charged channels only. In the thin channel regime the disjoining pressure is comparable to the excess osmotic pressure at the wall, which implies relatively weakly charged walls. We derive simple expressions for the mean conductivity of the channel in these two regimes, highlighting the role of electrostatic and electrohydrodynamic boundary conditions. Our theory provides a simple explanation of the high conductivity observed experimentally in hydrophilic channels, and allows one to obtain rigorous bounds on its attainable value and scaling with salt concentration. Our results also show that further dramatic amplification of conductivity is possible if hydrophobic slip is involved, but only in the thick channel regime provided the walls are sufficiently highly charged and most of the adsorbed charges are immobile. However, for weakly charged surfaces the massive conductivity amplification due to hydrodynamic slip is impossible in both regimes. Interestingly, in this case the moderate slip-driven contribution to conductivity can monotonously decrease with the fraction of immobile adsorbed charges. These results provide a framework for tuning the conductivity of nanochannels by adjusting their surface properties and bulk electrolyte concentrations.

Leveque-Type Similarity Transformation for a Thermally Developing Viscous Dissipative Flow in a Parallel Plate Channel

Compared to the existing more elaborate eigenvalues-eigenfunction analytical solution where the solution of a thermally developing forced convection problem converges very slowly at the beginning of thermal entrant region, Leveque-type similarity transformation method provides a more convenient way to look into the insights of the problem. Assuming that the wall heat flux and viscous dissipation only has an effect within the thin thermal boundary layer at the beginning of the thermal entrance region, this study intends to solve the governing thermal energy equation for a thermally developing flow in a parallel plate channel, subjected to uniform heat flux, by means of Leveque-type similarity transformation. The resulting ordinary differential equation, is subsequently solved by a fourth order Runge Kutta method. A comparison of the Nusselt number along the axial direction, at the beginning of the thermally developing region with the literature, reveals less than 10% discrepancy for Brinkman number less than one, which is a commonly acceptable range for practical applications. Although its accuracy depletes downstream the channel, similarity transformation provides sufficiently accurate temperature distribution, and captures the heat transfer insights for a thermally developing viscous dissipative forced convection.

Density Modelling in Mixed Convection Flow in a Vertical Parallel Plate Channel

In this paper, a novel way of modelling the density in buoyancy term of mixed convection flow problem is presented using equation of state and Boussinesq approximation without first-order approximation of density with respect to temperature. The presented density model is used to investigate the laminar mixed convection flow in a vertical parallel plate channel under symmetric constant wall heat flux. The results obtained are compared with the results obtained using first-order approximation of density with Boussinesq approximation, and also compared with the results obtained using variable thermophysical properties with negligible viscous dissipation. Investigation is performed on the basis of flow and thermal fields for Re=150 and 300, Ri=0.1 to 25. It is found that the presented density model produces relatively better results, which is able to describe the case of developing flow under constant heat flux condition that is not evident if Boussinesq approximation with first-order approximation of density is used. An appearance of recirculatory cells when reverse flow takes place is also witnessed in vertical channel flow with constant heat flux boundary condition which was not reported earlier.

Analysis of thermo-hydraulic and entropy generation characteristics for flow through ribbed-wavy channel

PurposeThe purpose of this study is to analyze the thermal, hydraulic and entropy generation characteristics for laminar flow of water through a ribbed-wavy channel with the top wall as wavy and bottom wall as flat with ribs of three different geometries, namely, triangular, rectangular and semi-circular.Design/methodology/approachThe finite element method-based numerical solver has been adopted to solve the governing transport equations.FindingsA critical value of Reynolds number (Recri) is found beyond which, the average Nusselt number for the wavy or ribbed-wavy channel is more than that for a parallel plate channel and the value of Recri decreases with the increase in a number of ribs and for any given number of ribs, it is minimum for rectangular ribs. The performance factor (PF) sharply decreases with Reynolds number (Re) up to Re = 50 for all types of ribbed-wavy channels. For Re > 50, the change in PF with Re is gradual and decreases for all the ribbed cases and for the sinusoidal channel, it increases beyond Re = 100. The magnitude of PF strongly depends on the shape and number of ribs and Re. The relative magnitude of total entropy generation for different ribbed channels varies with Re and the number of ribs.Practical implicationsThe findings of the present study are useful to design the economic heat exchanging devices.Originality/valueThe effects of shape and the number of ribs on the heat transfer performance and entropy generation have been investigated for the first time for the laminar flow regime. Also, the effects of shape and number of ribs on the flow and temperature fields and entropy generation have been investigated in detail.

Viscous dissipation effect on CuO-Water nanofluid-cooled microchannel heat sinks

While a microchannel heat sink is known to regulate the temperatures of microelectronics devices, viscous dissipation that originates from work done by viscous forces, might have an impact on the heat convection coefficient from a heated boundary. In this study, the exact solution of the temperature field is established for a fully developed forced convection in a parallel plate channel saturated with a porous medium, subjected to uniform heating at one boundary. Applying the new temperature field in a microchannel heat sink, modelled as a porous medium, this study gauges the effect of viscous dissipation on the electronics cooling device; utilizing a 4% volume fraction CuO-Water nanofluid as a coolant. Assuming a homogeneous model whereby the concentration of the nanofluid is taken as uniform spatially, the two-dimensional temperature field, computed based on the effective thermophysical properties of the nanofluid, reveals that viscous dissipation affects the heat convection at the heated boundary adversely, lowering the heat transfer coefficient by 15.3%, for Reynolds number, Re=1500 and elevates the temperature gradient in the vicinity of the insulated boundary.

Numerical investigation of combined heat transfer (mixed convection-radiation) in 2D channel using the LBM

In this paper, the problem of combined convection (forced and natural) and radiation heat transfer between two parallel plates (channel) with isothermal walls has been numerically studied. For this purpose, the equations of momentum, energy and radiation transfer must be solved. In this study, lattice Boltzmann method is used to solve all of the equations simultaneously, that, to increase the stability for solving the energy equation by lattice Boltzmann method, a modern approach called the two relaxation time model was used. The medium inside the channel was considered as a participating medium to solve the radiation transfer equation. Finally, after the formulation of governing equations and boundary conditions for the present problem in the lattice Boltzmann method, validating the results and simultaneously solving the equations, the effects of some parameters such as optical thickness, Planck number and Richardson number with changing the angle of the channel are studied on heat transfer, flow and temperature distribution.The results showed that the radiation parameters have a significant effect on the problem and can change the velocity and temperature distribution of the channel. In between these parameters, the Planck number has the most effect on the temperature and velocity of the fluid flow. Increasing the angle of the channel relative to the horizon increases the effect of buoyancy force, fluid velocity variations and fluid temperature values.

Influence of conjugate heat transfer on the minimization of entropy generation for forced convective flow through parallel plate channel filled with porous material

AbstractA fluid–solid conjugate heat transfer model is developed to analyze the characteristics of entropy generation for forced convective steady hydrodynamically fully developed laminar flow of a Newtonian fluid through a parallel plate channel filled with porous material by modulating the following parameters: substrate thickness, the ratio of thermal conductivity of wall to fluid, Biot number, the axial temperature gradient in the fluid, and Peclet number. The exteriors of both the walls are subjected to the thermal boundary conditions of the third kind. The mass and Brinkman momentum conservation equations in the fluidic domain and the coupled energy conservation in both the solid and fluidic domain are solved analytically using the local thermodynamic equilibrium model, so as to derive closed‐form expressions for the velocity in the fluid and the temperature both in the fluid and solid walls in terms of relevant parameters. Suitable combinations of influencing factors, namely the geometric parameters of the system, fluid, flow, and substrate properties are identified for which global entropy generation rate is minimized. The findings may be helpful in the design of thermal systems frequently used in diverse engineering applications having heat transfer in the solid wall being a crucial parameter.

Mesoscopic model framework for liquid slip in a confined parallel-plate flow channel

We investigated the liquid slip in a planar confined flow channel by proposing an exponentially decaying interaction force between fluid particles and two flat walls in the mesoscopic lattice Boltzmann model framework. In this way, we can explicitly link density profile, velocity profile, and apparent slip length with the mesoscale interaction parameters (force strength and decay length), and formulate the permeability-enhancement ratio as a function of two dimensionless numbers that indicate the role of interaction strength and interaction distance (relative to gap size of flow channel) in the slip-flow system.

Heat transfer from a ferrofluid during generalized Couette flow through parallel plates in the presence of an orthogonal magnetic field

In the present work, we report heat transfer during the unidirectional laminar flow of ferrofluids in a parallel plate channel in the presence of a homogeneously applied magnetic field perpendicular to the flow direction. We have considered generalized Couette flow which results from the combination of shear stress, due to the relative motion between the parallel plates, and applied pressure gradient. For a comprehensive investigation, four different thermal boundary conditions have been considered. The application of magnetic field modifies the viscosity of the colloidal ferrofluid and thereby causes variations in the local flow field. Further, it also modifies the thermal conductivity of the fluid by changing particle orientation. As a consequence, the temperature distribution between the parallel plate channel is uniquely modified. The complex interplay between the hydrodynamics and the magnetics results in implicitly coupled governing differential equations, which have been solved numerically. We report that the local temperature in the domain could be reduced by increasing the applied field strength irrespective of the imposed thermal boundary conditions. However, the influence of the applied field finally gets saturated. We also discuss the effect of Brinkman number, loading of magnetic particles and pressure gradient on temperature distribution. Finally, analytical solutions, which presents closed-form expressions of the temperature profiles for four different types of boundary conditions, have been proposed.

Electrokinetics of polymeric fluids in narrow rectangular confinements.

The flow of polymeric liquids in narrow confinements with a rectangular cross section, in the presence of electrical double layers is analyzed here. Our analysis is motivated by the fact that many of the previous studies on the flow of complex fluids tend to focus on highly idealized parallel plate channels, which are markedly different from the rectangular ducts, used in many experiments and devices. We consider the combined electroosmotic and pressure driven flows as well as the streaming potential resulting from a mechanically driven flow. We use two distinct constitutive relations to model the polymeric liquids, namely the simplified exponential Phan-Thien-Tanner (sePTT) model and the Giesekus model, both of which are non-linear viscoelastic models, capable of capturing the shear thinning behavior. We establish that the applied electric field may have a strong influence on the overall flow rate, which rapidly increases with the field strength as well as the extent of viscoelasticity of the fluid. Viscoelasticity and shear thinning behavior also enhance the streaming potential by several fold as compared to a Newtonian medium. We demonstrate that the aspect ratio of a channel has a bigger influence on the net throughput and the streaming potential, when the extent of viscoelasticity is relatively large. We illustrate that for sePTT fluids, the flow is strictly unidirectional, while for Giesekus fluids, secondary flows are inevitably present on account of their non-zero second normal stress coefficient. Although the electric field does not change the overall patterns of these secondary flows, their magnitude does depend on the imposed field strength for combined flows.

Effects of Second-Order Velocity Slip and the Different Spherical Nanoparticles on Nanofluid Flow

The paper theoretically investigates the heat transfer of nanofluids with different nanoparticles inside a parallel-plate channel. Second-order slip condition is adopted due to the microscopic roughness in the microchannels. After proper transformation, nonlinear partial differential systems are converted to ordinary differential equations with unknown constants, and then solved by homotopy analysis method. The residual plot is drawn to verify the convergence of the solution. The semi-analytical expressions between NuB and NBT are acquired. The results show that both first-order slip parameter and second-order slip parameter have positive effects on NuB of the MHD flow. The effect of second-order velocity slip on NuB is obvious, and NuB in the alumina–water nanofluid is higher than that in the titania–water nanofluid. The positive correlation between slip parameters and Ndp is significant for the titania–water nanofluid.

Electroosmotic flow in small-scale channels induced by surface-acoustic waves

Numerical simulations of the Navier-Stokes, Nernst-Planck, and the Poisson equations are employed to describe the transport processes in an aqueous electrolyte in a parallel-plate nanochannel, where surface-acoustic waves (SAWs) are standing or traveling along (piezo-active) channel walls. It is found that -- in addition to the conventional acoustic streaming flow -- a time-averaged electroosmotic flow is induced. Employing the stream function-vorticity formulation, it is shown that the Maxwell stress term causes an electroosmotic propulsion that is qualitatively identical to the one discussed in the context of alternating current (AC) electroosmosis (EOF). Differences arise mainly due to the high actuation frequencies of SAWs, which are in the MHz range rather than in the kHz regime typical for ACEOF. Moreover, the instantaneous spatial periodicity of the EOF in the travel direction of the SAW is intrinsically linked to the dispersion relation of the latter rather than a free geometric parameter. This leads to a specific frequency band where an EOF of sizable magnitude can be found. On the low frequency end, the ratio between the electric double layer (EDL) thickness and the SAW wavelength becomes extremely small so that the net force leading to a non-vanishing time-averaged flow becomes equally small. On the high frequency end, the RC time of the EDL is much larger than the inverse of the SAW frequency leading to a vanishing effective charge density of the EDL. For a parallel-plate channel, the EOF can be maximized by using two SAWs on both channel walls that have the same frequency but are phase-shifted by $180^\circ$. It appears that the SAW-EOF is the dominant pumping mechanism for such a scenario. The proposed actuation might be a viable alternative for driving liquid electrolytes through narrow ducts and channels, without the need for electric interconnects and electrodes.

Two-phase MHD Flow through Porous Medium with Heat Transfer in a Horizontal Channel

The present study deals with two layered MHD immiscible fluid flow through porous medium in presence of heat transfer through parallel plate channel. The fluids are incompressible, and flow is fully developed. The fluids are of different viscosities and thermal conductivities so flowing without mixing each other. Two different phases are accounted for study and are electrically conducting. Temperature of the walls of parallel plate channel is constant. Rheological properties of the immiscible fluids are constant in nature. The flow is governed by coupled partial differential equations which are converted to ordinary differential equations and exact solutions are obtained. The velocity profile and temperature distribution are evaluated and solved numerically for different heights and viscosity ratios for the two immiscible fluids. The effect of magnetic field parameter M and porosity parameter K is discussed for velocity profile and temperature distribution. Combined effects of porous medium and magnetic fields are accelerating the flow which, can be helpful in draining oil from oil wells.

The Effect of MHD and Brinkman Number on Laminar Mixed Convection of Newtonian Fluid Between Vertical Parallel Plates Channel

This study investigates MHD and Brinkman number on mixed convection flow in a two parallel-plates vertical channel with reference to laminar, thermal and hydrodynamical developing flow of Newtonian fluid. The boundaries are considered to be isothermal with equal temperatures. The governing equations are solved numerically. Also, their dependence upon certain material parameters have been studied. Velocity, temperature, pressure gradient and Nusselt number profiles have also been presented.

Hybrid solutions for thermally developing flows in channels partially filled with porous media

A new approach based on the Generalized Integral Transform Technique is advanced to deal with convective heat transfer in partially porous channels under local thermal nonequilibrium (LTNE) formulation. A semi-infinite partially porous parallel plates channel configuration is used to illustrate the semi-analytical solution methodology. The proposed eigenfunction expansion is based on a novel coupled multilayer eigenvalue problem which results in similar accuracy and computational performance in comparison to the expansions employed in solving the formulation under the local thermal equilibrium (LTE) hypothesis. Results from the LTE and LTNE formulations are then critically compared to explore the validity of the former in the present configuration. It is observed that the LTE formulation results may significantly overestimate the Nusselt number, leading to overly optimistic heat transfer predictions. Finally, an analysis of the effect of each parameter in the difference between the temperatures in the solid and fluid phases of the porous medium and in the Nusselt number is carried out, offering guidance for the design of heat transfer devices.