In this work we address the quantification of Brinkman’s effective viscosity, which arises due to the presence of
the porous medium and its effect on increasing or decreasing the fluid viscosity in relation to the base fluid
viscosity. This work is motivated in the main part by the lack of consensus in the available literature on the
ranges of effective viscosity. To this end, this work provides a model of quantifying the effective viscosity by
incorporating the porous microstructure in the volume-averaged Navier-Stokes equations. Extensive analysis and
testing are provided in the current work which considers five different porous microstructures, and the effective
viscosity is quantified in each case under Poiseuille flow
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
MHD Mixed Convection Flow of casson Nanofluid over a NonLinear Permeable Stre...inventionjournals
In this paper, numerical analysis has been carried out on the problem of magnetic hydro dynamic mixed convection flow of a Casson nanofluid past a nonlinear permeable stretching sheet with viscous dissipation in the presence of double stratification and heat generation or absorption. The governing partial differential equations were transformed into a system of ordinary differential equations using suitable similarity transformations. The resultant ordinary differential equations were then solved using Bvp4c mat lab solver. Effects of the physical parameters on the velocity, temperature and concentration profiles as well as the local skin friction coefficient, the heat and mass transfer rates are depicted in tabular form and discussed. The results indicate that the local Nusselt number decreases with an increase in both Brownian motion parameter (Nb) and the thermophoresis parameter (Nt). However the local Sherwood number (??) increases with an increase in the parameter Nb. But it decreases as the values of Nt increases. Besides it was found that the surface temperature of a sheet increases with an increase in the heat generation (Q>0) or absorption parameter (Q<0). Comparison of present results with previous reported results has been found in excellent agreement.
A Note on the Beavers and Joseph Condition for Flow over a Forchheimer Porous...IJRESJOURNAL
ABSTRACT : In thiswork, an attemptis made to relate the slip parameter in the Beavers and Joseph condtionat the interface between a Darcy layer and a Navier-Stokes channel, to the slip parameter to beusedwhen the porous layer is a Forchheimer layer.
This is related to properties of fluids in Fluid mechanics basically helpful for the Mechanical Engineering students.Most of the part is covered in this regarding the basic properties of fluids and about the meaning of fluid.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
MHD Mixed Convection Flow of casson Nanofluid over a NonLinear Permeable Stre...inventionjournals
In this paper, numerical analysis has been carried out on the problem of magnetic hydro dynamic mixed convection flow of a Casson nanofluid past a nonlinear permeable stretching sheet with viscous dissipation in the presence of double stratification and heat generation or absorption. The governing partial differential equations were transformed into a system of ordinary differential equations using suitable similarity transformations. The resultant ordinary differential equations were then solved using Bvp4c mat lab solver. Effects of the physical parameters on the velocity, temperature and concentration profiles as well as the local skin friction coefficient, the heat and mass transfer rates are depicted in tabular form and discussed. The results indicate that the local Nusselt number decreases with an increase in both Brownian motion parameter (Nb) and the thermophoresis parameter (Nt). However the local Sherwood number (??) increases with an increase in the parameter Nb. But it decreases as the values of Nt increases. Besides it was found that the surface temperature of a sheet increases with an increase in the heat generation (Q>0) or absorption parameter (Q<0). Comparison of present results with previous reported results has been found in excellent agreement.
A Note on the Beavers and Joseph Condition for Flow over a Forchheimer Porous...IJRESJOURNAL
ABSTRACT : In thiswork, an attemptis made to relate the slip parameter in the Beavers and Joseph condtionat the interface between a Darcy layer and a Navier-Stokes channel, to the slip parameter to beusedwhen the porous layer is a Forchheimer layer.
This is related to properties of fluids in Fluid mechanics basically helpful for the Mechanical Engineering students.Most of the part is covered in this regarding the basic properties of fluids and about the meaning of fluid.
Project for engineering and school students.
In this project, you can find everything related to the concept of viscosity :- Definition, Derivation, Units, Kinematic viscosity. Newton's law of viscosity, variation of viscosity with temperature, types of fluids are also included. If you find it helpful to you, give some feedback.
Thank You very much :)
The New Capillary Number Parameterization for Simulation in Surfactant FloodingPremier Publishers
The Capillary number hypothesis is very empirical in Surfactant flooding Enhanced Oil Recovery (EOR) method. which is of modest experience on the North Sea and many other offshore platforms. The capillary number drives the force wetting processes, which is controlled by the balance between capillary and viscous forces. The mobilization of oil trapped in pores of water-wet rock is steered by capillary number that is typically within specific ranges (〖10〗^(-5) to〖 10〗^(-4)). There is high uncertainty and confusion in the parameterization of capillary number formula, as every quantity is given on a macroscale level. As demonstrated herein, a new microscopic capillary number parameterization was proposed. This paper is written to improve the numerical formulation of capillary number in surfactant flooding model. The new formula for capillary number was derived based on existing equations as a function of residual oil saturation and tested. Thus, the proposed mobility mechanism easily accounts for a broader critical range of capillary number (〖10〗^(-6) to〖 10〗^(-4)) in comparison with available models with a critical capillary number (〖10〗^(-5) to〖 10〗^(-4)). We used an existing model to quantify the effect of capillary number on a miscible and immiscible relative permeability curves by computing the interpolation parameter F_kr as a tabulated function of the Logarithm (base 10) of the capillary number using a new capillary number formulation.
Surface Tension is defined as the tension of the surface film of a liquid caused by the attraction of the particles in the surface layer by the bulk of the liquid, which tends to minimize surface area.
It is due to the phenomena of surface tension that the drops of water tend to assume a spherical shape to attain minimum surface area. the presentation gives a brief description of the methods to measue this important property of the interface of two fluid.
Fluidized Bed Reactor - Basic Mechanism, Mass Transfer in Fluidized Beds, Reaction Behaviour in a Fluidized Bed, Mole Balance on the Bubble, the Cloud, and the Emulsion, Advantages & Disadvantages, Current Applications of FBR.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Energy Saving and Water Cooling Facility in an Industry Using Prgrammable Log...IJERA Editor
This project gives the idea about how to reduce the power loss in an industry, which are being wasted by the
fans and pumps in operating condition as well as to get water as per the necessity by an industry with proper
quality. This can be done by proper switching and speed control of motors used in fans and pumps. As the
power consumed is directly proportional to the cube of speed so by reducing the speed energy can be saved. An
attempt is also made to connect the power factor improving capacitor for energy saving. By using this concept
the industry can save huge amount of energy and can be able to reduce the electricity bill.
Room Transfer Function Estimation and Room Equalization in Noise EnvironmentsIJERA Editor
Audio quality in listening situation is degraded by indoor room reverberation. Room equalization can be used to
increase the audio quality by applying the inverse transfer function to the input audio signals. In noise
environments, however, it is hard to exactly measure the room transfer function. In this work, we developed the
techniques to measure the room transfer function in indoor noise environments and to enhance the audio quality
by room equalization. From the experimental results, we showed that the proposed techniques can be
successfully used in indoor noise environments
Project for engineering and school students.
In this project, you can find everything related to the concept of viscosity :- Definition, Derivation, Units, Kinematic viscosity. Newton's law of viscosity, variation of viscosity with temperature, types of fluids are also included. If you find it helpful to you, give some feedback.
Thank You very much :)
The New Capillary Number Parameterization for Simulation in Surfactant FloodingPremier Publishers
The Capillary number hypothesis is very empirical in Surfactant flooding Enhanced Oil Recovery (EOR) method. which is of modest experience on the North Sea and many other offshore platforms. The capillary number drives the force wetting processes, which is controlled by the balance between capillary and viscous forces. The mobilization of oil trapped in pores of water-wet rock is steered by capillary number that is typically within specific ranges (〖10〗^(-5) to〖 10〗^(-4)). There is high uncertainty and confusion in the parameterization of capillary number formula, as every quantity is given on a macroscale level. As demonstrated herein, a new microscopic capillary number parameterization was proposed. This paper is written to improve the numerical formulation of capillary number in surfactant flooding model. The new formula for capillary number was derived based on existing equations as a function of residual oil saturation and tested. Thus, the proposed mobility mechanism easily accounts for a broader critical range of capillary number (〖10〗^(-6) to〖 10〗^(-4)) in comparison with available models with a critical capillary number (〖10〗^(-5) to〖 10〗^(-4)). We used an existing model to quantify the effect of capillary number on a miscible and immiscible relative permeability curves by computing the interpolation parameter F_kr as a tabulated function of the Logarithm (base 10) of the capillary number using a new capillary number formulation.
Surface Tension is defined as the tension of the surface film of a liquid caused by the attraction of the particles in the surface layer by the bulk of the liquid, which tends to minimize surface area.
It is due to the phenomena of surface tension that the drops of water tend to assume a spherical shape to attain minimum surface area. the presentation gives a brief description of the methods to measue this important property of the interface of two fluid.
Fluidized Bed Reactor - Basic Mechanism, Mass Transfer in Fluidized Beds, Reaction Behaviour in a Fluidized Bed, Mole Balance on the Bubble, the Cloud, and the Emulsion, Advantages & Disadvantages, Current Applications of FBR.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Energy Saving and Water Cooling Facility in an Industry Using Prgrammable Log...IJERA Editor
This project gives the idea about how to reduce the power loss in an industry, which are being wasted by the
fans and pumps in operating condition as well as to get water as per the necessity by an industry with proper
quality. This can be done by proper switching and speed control of motors used in fans and pumps. As the
power consumed is directly proportional to the cube of speed so by reducing the speed energy can be saved. An
attempt is also made to connect the power factor improving capacitor for energy saving. By using this concept
the industry can save huge amount of energy and can be able to reduce the electricity bill.
Room Transfer Function Estimation and Room Equalization in Noise EnvironmentsIJERA Editor
Audio quality in listening situation is degraded by indoor room reverberation. Room equalization can be used to
increase the audio quality by applying the inverse transfer function to the input audio signals. In noise
environments, however, it is hard to exactly measure the room transfer function. In this work, we developed the
techniques to measure the room transfer function in indoor noise environments and to enhance the audio quality
by room equalization. From the experimental results, we showed that the proposed techniques can be
successfully used in indoor noise environments
Protons Relaxation and Temperature Dependence Due To Tunneling Methyl GroupIJERA Editor
Tunneling frequency and temperature dependence of proton spin lattice relaxation time T1, are depend upon the
height and the shape of the hindering barrier of methyl rotation and carry information on the group is molecular
environment are reported for some samples containing tertiary-butyl group.The temperature rang was 4-
300k.Data has been analyzed to provide estimates for the magnitude of the three fold potential barrier to
reorientation of all methyl groups in these materials. At low temperature the motion of the tertiary-butyl protons
can usually be neglected. All protons of the samples relax as a single system.In one or two cases tunneling is
observed for the first time in Tert-butyl. The T1 results are used to evaluate tunnel frequency in other cases. The
result suggest the importance of collective motion of methyl group in tert-butyl
Influence of variable water-soluble soy extract and inulin contents on the rh...IJERA Editor
The present study aimed to evaluate the influences of the partial substitution of caprine milk for water-soluble
soy extract (WSSE) and the addition of inulin on the rheological, technological and sensory properties of grapeflavored
yogurt-like beverages. For this purpose, a Central Composite Design (CCD) in conjunction with
Response Surface Methodology (RSM) was employed. WSSE and inulin influenced the overall acceptability of
the product, whereas syneresis, water holding capacity and rheological properties (the consistency index and the
flow behavior index) were influenced only by the WSSE content. RSM was shown to be an adequate statistical
tool that can be used for the development of formulations with specified properties in the range of the ingredient
concentrations studied.
Improving the Reliability of Synthetic S-Wave Extraction Using Biot-Gassman F...IJERA Editor
Usage of Castagna’s relation by mean the P-wave log directly in estimation of the S-wave log gives a large error
to the original S-wave log so that the result is not reliable for further analysis. This paper offers new method for
S-wave log estimation based Castagna’s relation which is combined with fluid replacement modeling method
based on Biot-Gassman substitution and the use of petro physical data as input. The S-wave log result of the
offered estimation method has a small error to the original S-wave log so that more reliable and accurate for
further analysis. The offered method could be used for S-wave log estimation in various litology such as sand,
limestone, dolomite and shale. The S-wave log result of the offered estimation method has successfully used for
cross plot analysis of Vp/Vs ratio as function of acoustic impedance and Gamma Ray for delineation of
sandstone bearing hydrocarbon from two different field.
A Study on Effective Use of Rice Husk Ashes in Geotechnical ApplicationsIJERA Editor
RHA is an agricultural industrial waste which is nearly 100 million tons, requires huge quantities of land for its
disposal. To utilize these huge quantities in several civil engineering applications, physical and engineering
properties has been identified to study the quality of RHA. In this connection Gradation, Seepage, compaction,
strength and chemical composition tests were conducted. Test results shows there are non-plastic, porous, elongated
light weight materials and dominated by siliceous materials. These have low dry densities at high moisture contents
achieved high values of shear strength (φ> 36o
) and CBR (>7) can be use as Embankment, structural fill and sub
grade material. Pozzolanic nature of Rice Husk Ashes helps to achieve high strengths with chemical addictives
exposed to curing
Amplifying Your Voice - Social Media 201Mark Anderson
These are the slides created by Monica Burns (@ClassTechTips), Kelly Grogan (@KellyEd121), Kristin Ziemke (@KristinZiemke) and me, Mark Anderson @ICTEvangelist. They come from our recent session together on 'Amplifying your voice' aka Social Media 201 for Educators.
Sediment Yield Studies For Selected Catchments in Visakhapatnam District, And...IJERA Editor
Sediment yield studies determine the amount of sediment that leaves a basin for an event or over a period of
time. The sediment yield provides the necessary input to determine sedimentation impacts on a reservoir. Each
reservoir project requires a sediment yield analysis to determine the storage depletion resulting from the
deposition of sediment during the life of the project. The main object of the present study is to identify the
particular suitable theoretical method to predict the sediment rate for selected catchments by comparing with the
hydrographic survey results.Literature review has enabled the identification of three methods for determining
the rate of sediment yield. These methods are (a) Grade’s method (b) Khosla’s method (c)
Joglekar’smethod.Rate of sediment yield obtained from Khosla’s method is close to the rate computed form the
hydrographic survey data for Thandava and KonamReservoirs. For the reservoir on Raiwada Reservoir,
Joglekar’s method is found to give sediment rate closer to the value arrived from hydrographic survey data
Broadband Slotted Rectangular Shaped Microstrip Antenna For WI-Max ApplicationsIJERA Editor
Many applications require very broadband antenna, but the narrow bandwidth of a microstrip antenna restricts
its wide usage. The aim of this paper is to enhance the bandwidth of rectangular microstrip patch antenna. For
this purpose, we cut four slots in the proposed antenna. The dielectric substrate material of the antenna is glass
epoxy FR4 having εr=4. 4 and loss tangent 0.025. The performance of the final modified antenna is compared
with that of a conventional rectangular microstrip antenna. The designed antenna has two resonant frequencies
5.42 GHz and 5.70 GHz. So this antenna is best suitable for the Wi-Max applications. The designed antenna
offers much improved impedance bandwidth 10.45 %. This is approximately two times higher than that in a
conventional rectangular patch antenna (Bandwidth= 5.34%) having the same dimensions.
Preparation Effect of Mould Systems on Microstructure and Mechanical Properti...IJERA Editor
This study is based on evaluation of microstructure and mechanical properties such as tensile strength, Brinell
hardness and Charpy impact test of as-cast spheroidal graphite iron using sandwich techniques in different mould
systems viz. green sand mould, dry sand mould and CO2 sand mould under varying cooling rates
Dynamic Response of Rcc and Composite Structure with Brb Frame Subjected To S...IJERA Editor
Concrete structures impart more seismic weight and less deflection whereas Steel structures instruct more
deflections and ductility to the structure, which is beneficial in resisting earthquake forces. Composite
Construction combines the better properties of both steel and concrete. Buckling restrained braced frames
(BRBFs) are primarily used as seismic-force resisting systems for buildings in seismically-active regions.The
objective of the present work is to compare a twenty storied RCC and composite framed structure with BRB
frame subjected to Seismic and different Temperatureloadings using Non-Linear Time History Analysis. Three
dimensional modeling and analysis of the structure is carried out with the help of SAP-2000 v16 software. It is
observed that the storey displacements were decreased by 36% for twenty storey RCC building and for
composite buildings it was decreased by 45% for twenty storeys suggesting the effectiveness of Buckling
restrained brace frame. The overall results suggested that BRB were excellent seismic control device for
composite building as the roof displacement is reduced by 40% but whereas for RCC it is reduced only by 25%.
For Seismic prone areas composite building with BRB frame is more effective. Under Temperature loading
RCC building is more effective than composite structure.
Antibacterial Resistance in the Muscles of Chicken, Pig and Beef IJERA Editor
Though antibiotic drugs are known to improve the health and welfare of food animals , there is parallel risk due
to the development of resistant microorganisms in the body of target animals. Seven meat samples were
procured from wet market in Old Town,Petaling Jaya, Malaysia and assessed for the presence of antibiotic
residues. The samples chosen were chicken parts (skin, muscle and liver) , pig parts (liver, muscle and
intestine) and beef muscle. The results indicated that chicken skin had high level of antibioticresidues which
positively resisted the presence of gram positive, Staphylococcus aureus, S. epidermidisand B. cereus as known
by the zone of inhibition.The beef muscle also held residue which resisted S. aureusChosenbacteriaalong with
the extracts of chicken skin, pig intestine and beef muscle were observed to be resistant totetracycline
hydrochloride, ciprofloxacin hydrochloride monohydrate and their combinations when tested at a concentration
of 1 percent
Emulation OF 3gpp Scme CHANNEL MODELS USING A Reverberation Chamber MEASUREME...IJERA Editor
The MIMO OTA way forward was approved at the 3GPP RAN4 #62bis meeting in Jeju (Korea) [1], and among
the remaining works to be accomplished it was highlighted that the channel model had to be validated across
methods in order to ensure that a minimal number of artifacts are inserted into the channel model by any given
methodology and that the different methods reproduce, and the DUT experiences, the same radio conditions
regardless of the methodology.The objective of this contribution is to validate the power delay profile (PDP) for
the 3GPP SCME Urban macro (UMA) and Urban micro (UMI) channel models emulated in a mode-stirred
reverberation chamber for the reverberation chamber
Development of Aluminum in ArchitectureIJERA Editor
Generally speaking, the task of the architecture is to create suitable atmosphere for performing a certain function
of the human being. Though today, the interior and exterior space are shaped with equal responsibility and
seriousness, the interior ambient is the one in which man spends most of his time. A building as a closed space
provides protection from outside climate and weather changes and provides necessary conditions for work and
living no matter the outside weather and climate conditions, sounds and noises. The outside walls with the
windows are the vital shell under whose protection many useful human activities take place. The evolution of
the window is as old as the architecture itself. Since the beginnings of architecture, man has been working on
developing the window. One of the features of that evolution of the window is a relation of the surface of the
hole of the window towards the outside walls. The advancing of the technology had its influence over the range
of capacity of the construction which on the other hand had its influence over the enlarging the window's
dimensions. From this we conclude that windows and window openings were and still are important instrument
of expressing in the architecture. Besides providing daylight and air in the rooms, the windows also are a visual
contact with the nature and underline the role of the building as a human being habitat like no other element of
the building
Implementing an Android Tool for Visually Impaired Students of E-LearningIJERA Editor
This article aims to describe the process of learning and development of an educational tool designed for mobile
devices (smartphones) with Android technology. In summary, the application was developed based on the virtual
learning environment Moodle and aims to develop a learning environment that supports the visually impaired
students of e-learning, allowing them to ask questions, discuss and share ideas through forums and use chat
rooms in real time. The fundamental purpose of this application is to cooperate with scientific knowledge in the
sense that this is a representation of technological advance on the accessibility tools in distance education mode
and provide comfort, flexibility and accessibility for the visually impaired students, realizing that education
should always be inclusive.
Database Security Two Way Authentication Using Graphical PasswordIJERA Editor
As data represent a key asset for today's organizations. The problem is that how to protect this data from
attackers, theft and misuse is at the forefront of any organization’s mind. Even though today several data
security techniques are available to protect database and computing infrastructure, many such as network
security and firewalls tools are unable to prevent attacks from insider. Insider is a person working in
organization who can try to access the sensitive data. This paper proposes a two-way authentication method
which fuses knowledge-based secret and personal trait information.
SVM Based Identification of Psychological Personality Using Handwritten Text IJERA Editor
Identification of Personality is a complex process. To ease this process, a model is developed using cursive
handwriting. Area based, width based and height based thresholds are set for only character selection, word
selection and line selection. The rest is considered as noise. Followed by feature vector construction. Slope
feature using slope calculation, shape features and edge detection done using Sobel filter and direction
histogram is considered. Based on the direction of handwriting the analysis was done. Writing which rises to
the right shows optimism and cheerfulness. Sagging to the right shows physical or mental weariness. The lines
which are straight, reveals over-control to compensate for an inner fear of loss of control.The analysis was done
using single line and multiple lines. Simple techniques have provided good results. The results using single line
were 95% and multiple lines were 91%.The classification is done using SVM classifier.
Effects of Several Purple Potato Additions on Bread QualityIJERA Editor
Potato cultivars with purple flesh represent an efficient and natural source of antioxidants, this vegetable having
high content in polyphenols (especially anthocyanin pigments). The research goal of this work was to evaluate
the anthocyanin and polyphenols content of several Romanian potato varieties (Albastru-Violet de Gălănești and
Christian) and the effects of these potatoes (add to dough in different proportions) on several bread quality
indicators. The bread quality depends on physical and chemical properties and on several signs like: flavor and
taste, external appearance, crumb porosity and texture, bread’s volume. In this research experiment, beside the
total polyphenols and anthocyanin content, the analysis performed on bread (prepared using different potatoes
addition 5%, 15% and 30%) were sensorial and physic chemical analysis (product volume, crumb porosity,
height/diameter ratio, moist and acidity). Experimental results indicated that 15% purple potato cultivar added on
the dough was the most indicate proportion to be used in bread processing.
Carbon Nanotubes On Unsteady MHD Non-Darcy Flow Over A Porous Wedge In The Pr...IJERA Editor
Thermal radiation energy technologies are clean sources of energy that have a much lower environmental
impact than conventional energy technologies. The objective of the present work is to investigate theoretically
the effect of copper nanoparticles and carbon nanotubes in the presence of base fluid (water) with variable
stream condition due to thermal radiation energy. Single wall carbon nanotubes (SWCNTs) in the presence of
base fluid flow over a porous wedge plays a significant role compared to that of copper nanoparticles on
absorbs the incident solar radiation and transmits it to the working fluid by convection.
The deposition of the flow of suspended particles has always been a problematic case in the process of flow transmission through sewers. Deposition of suspended materials decreases transmitting capacity. Therefore, it is necessary to have a method capable of precisely evaluating the flow velocity in order to prevent deposition. In this paper, using Gene-Expression Programming, a model is presented which properly predicts sediment transport in the sewer. In order to present Gene-Expression Programming model, firstly parameters which are effective on velocity are surveyed and considering every of them, six different models are presented. Among the presented models the best is being selected. The results show that using verification criteria, the presented model presents the results as Root Mean Squared Error, RMSE=0.12 and Mean Average Percentage Error, MAPE=2.56 for train and RMSE=0.14 and MAPE=2.82 for verification. Also, the model presented in this study was compared with the other existing sediment transport equations which were obtained using nonlinear regression analysis.
A Revisit To Forchheimer Equation Applied In Porous Media FlowIJRES Journal
A brief reference to various non-linear forms of relation between hydraulic gradient and velocity of
flow through porous media is presented, followed by the justification of the use of Forchheimer equation. In
order to study the nature of coefficients of this equation, an experimental programme was carried out under
steady state conditions, using a specially designed permeameter. Eight sizes of coarse material and three sizes
of glass spheres are used as media with water as the fluid medium. Equations for linear and non-linear
parameters of Forchheimer equation are proposed in terms of easily measurable media properties. These
equations are presented in the form of graphs as quick reckoners.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Mesoscopic simulation of incompressible fluid flow in porous mediaeSAT Journals
Abstract
Lattice Boltzmann method is used to simulate cavity driven fluid flow in porous media. A square cavity is considered with the top
lid moving with uniform velocity and other sides kept stationary. Simulation is carried out for values of Darcy number ranging
from 10-6 to10-2 at Reynolds number 10 and 100. Influence of Darcy number and Reynolds number is investigated on velocity
profiles and the streamline plots. Half-way bounce back boundary conditions are employed in the numerical simulation. The
numerical code is first verified with the results available in the literature and then used to simulate the Newtonian fluid flow in
porous media. The Darcy number and the Reynolds number were observed to have great influence on the flow properties and the
location of the primary vortex. Simulation was carried out for a 100100 mesh grid and a fine agreement is established theories
in incompressible fluid flow.
Keywords: Lattice Boltzmann method, incompressible flow, porous media
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Fluid structure and boundary slippage in nanoscale liquid filmsNikolai Priezjev
During the last ten years, there has been enormous interest in understanding transport phenomena in micro and nanofluidic systems and, in particular, in accurate prediction of fluid flows with slip boundary conditions at liquid-solid interfaces. In this chapter, we discuss recent results obtained from molecular dynamics simulations of fluids that consist of monomers or linear polymer chains confined by flat crystalline surfaces. The effects of shear rate and wall lattice orientation on the slip behaviour are studied for a number of material parameters of the interface, such as fluid and wall densities, wall-fluid interaction energy, polymer chain length, and wall lattice type. A detailed analysis of the substrate-induced fluid structure and interfacial diffusion of fluid molecules is performed to identify slip flow regimes at low and high shear rates.
The sun is the source of solar energy and the solar energy is pollution free and available in ample amount which can be used for heating, electricity generation and coking purpose.The position of sun constantly varies so its intensity also varies which affect the heat quantity incident upon the system. The aim of present work is overcome this difficultly using wax type phase change material which behaves like heat storage medium as position of sun changes. This phase change material based solar water heater is fabricated and thermal performance evaluation can be carried out using K type thermocouple.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
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Investigations in Effective Viscosity of Fluid in a Porous Medium
1. W.S. Almalki. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 4) April 2016, pp.41-51
www.ijera.com 41|P a g e
Investigations in Effective Viscosity of Fluid in a Porous Medium
W.S. Almalki*, M.H. Hamdan**
*(Department Of Mathematics, Umm Al-Qura University, Saudi Arabia,)
** (Department Of Mathematical Sciences, University Of New Brunswick, Saint John, Canada E2L 4L5)
ABSTRACT
In this work we address the quantification of Brinkman’s effective viscosity, which arises due to the presence of
the porous medium and its effect on increasing or decreasing the fluid viscosity in relation to the base fluid
viscosity. This work is motivated in the main part by the lack of consensus in the available literature on the
ranges of effective viscosity. To this end, this work provides a model of quantifying the effective viscosity by
incorporating the porous microstructure in the volume-averaged Navier-Stokes equations. Extensive analysis and
testing are provided in the current work which considers five different porous microstructures, and the effective
viscosity is quantified in each case under Poiseuille flow.
Keywords - Effective Viscosity, Brinkman Equation
I. INTRODUCTION
In spite of the popularity of Darcy’s law in
the study of groundwater flow, it suffers some
limitations that include not accounting for
microscopic inertia that arises due to tortuosity of
the flow path, [1], [2], [3]. Furthermore, the absence
of viscous shear term in Darcy’s law limits its ability
to account for viscous shear effects that are
important when a macroscopic, solid boundary is
encountered and on which the no-slip condition is
imposed. Darcy’s law has been argued to be valid in
low permeability, low porosity media where
variations at the microscopic, pore-level length scale
are negligible at the macroscopic (say, thickness of a
porous layer) length scale.
In 1947, Brinkman [4] introduced an
extension to Darcy’s law by including a viscous term
to account for the viscous shear effects that are
important in the thin boundary layer near a
macroscopic boundary. Accordingly, equations
governing the steady flow of a Newtonian fluid
through a porous medium composed of a swarm of
particles fixed in space are given by the following
equation of continuity, and linear momentum
equation that is due to Brinkman [4]:
Conservation of Mass:
.0 u
…(1)
Conservation of Momentum (Brinkman’s Equation):
0
2*
u
k
up
…(2)
whereu
is the ensemble-averaged velocity, is the
base fluid viscosity,
*
is the viscosity of the fluid
saturating the porous medium (i.e. effective
viscosity), k is the permeability and p is the
pressure. The term u
k
is the Darcy resistance, and
the term u
2*
is the viscous shear term that
facilitates imposition of a no-slip velocity condition
on solid boundaries. It is clear that when 0
*
,
Brinkman’s equation (2) reduces to Darcy’s law.
It has been long realized and documented
that there is an increased effective shear viscosity in
constricted geometries (such as when a dilute system
is present, or as in the pore structure of a porous
medium), and close to solid surfaces. Experimental
evidence has been summarized and reported in the
work of Henniker, (1949), (cf. [5] for reference). The
literature also reports on experimental and
theoretical investigations that provide estimates for
*
from less than unity to as high as 10, and
employ a number of configurations that include flow
over spheres, flow in the presence of bounding
walls, flow over arrays of circular or elliptic
cylinders, and flow through channel-like porous
media. Other authors believe that the effective
dynamic viscosity arises due to volume averaging
and provided detailed theoretical analysis of its
quantification. Alas; the reported studies provide no
definite answer as to what the value of should
be. However, many agree that the presence of solid
walls bounding the porous flow domain, and the
geometric configurations of the porous material play
an important part in the value of , (cf. [6], [7], [8],
[9], [10], [11], and the references therein).
A number of studies have investigated the validity of
Brinkman’s equation and a number of excellent and
sophisticated analyses have been carried out to
quantify the effective viscosity,
*
. Some argue that
the viscosity factor, or the ratio, , could be greater
than unity or less than unity, (cf. [1], [12] and the
references therein), or could be taken as unity (as
RESEARCH ARTICLE OPEN ACCESS
2. W.S. Almalki. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 4) April 2016, pp.41-51
www.ijera.com 42|P a g e
favoured by Brinkman [4]). Brinkman initially used
Einstein’s formula for the viscosity of suspension,
[13], [14], to relate fluid viscosity and the effective
viscosity, namely
)]1(
2
5
1[
*
…(3)
where is the effective porosity of the medium
(i.e. ratio of volume of interconnected pores to the
bulk volume of the medium). Table 1, below, is
produced using equation (3).
*
= )1(
2
5
1
0 5.3
0.01 3.475
0.5 2.25
0.9 1.25
0.99 1.025
1 1
Table 1. Viscosity Factor Based on Einstein’s
Law for Viscosity of a Suspension
Table 1 illustrates that when 1 , then 1 , or
*
. This is the Navier-Stokes flow limit, as this
case corresponds to the flow of a Newtonian fluid in
free-space, that is, in the absence of the porous
matrix. When 0 ,
2
7*
. Clearly, this
should be the Darcy flow limit. However,
Brinkman’s equation is supposed to reduce to
Darcy’s law when porosity is small and 0
*
.
This could be an indication that equation (3) does
not provide a good measure of the effective viscosity
in porous media.
Many investigators have attempted to
resolve the question of effective viscosity. Some
argue that equation (3) is valid for high porosity
media (that is, porosity being close to unity). As
porosity of the medium decreases, viscosity of the
saturating fluid increases. Indeed, the literature
includes studies and experiments that report
viscosity factors ranging from less than unity to
approximately 10, (cf. [7], [10], [11] and the
references therein). In a recent work, Breugem [12]
provided elegant and thorough analysis of the
effective viscosity of fluid in a channel-type porous
medium and provided the following effective
viscosity estimate, referred to here as Breugem’s
Estimate:
7
3
;0
7
3
;)
7
3
(
2
1
*
…(4)
hence, the viscosity factor takes the form:
).
7
3
(
2
1
*
…(5)
We produce Table 2 of viscosity factor, below,
using equation (5):
*
= )
7
3
(
2
1
0.01 0
0.4 0
0.429 0.0002142857
0.5 0.0357142857
0.95 0.2607142857
0.99 0.2807142857
1 0.2857142857
Table 2. Viscosity Factor Based on Breugem’s
Estimate
Table 2 shows that the effective viscosity is
always less than the base fluid viscosity. When
1 , Breugem’s formula yields an effective
viscosity that is less than 30% of the base fluid
viscosity. However, one expects that when 1 the
Navier-Stokes flow limit should be reached and
*
.
Liu et al. [10] studied theoretically and
numerically the effects of bounding solid walls on
slow flow over regular, square arrays of circular
cylinders between two parallel plates. Their results
indicate that, between the two limits of the Darcian
porous medium and the viscous flow, the magnitude
of the viscosity factor needs to be close to unity
in order to satisfy the non-slip boundary conditions
at the bounding walls.
In summary, there seems to be no
consensus as to what the viscosity factor should be,
nor as to how the effective viscosity could be
computed, and how it depends on the porosity of the
medium. However, it is agreed upon that the
effective viscosity depends on factors that include
the type of fluid, the speed of the flow, porosity and
permeability of the medium, and the presence of
bounding solid walls to the medium.
The above lack of consensus motivates the
current work in which we provide analysis of the
effective viscosity in flow through porous media.
This will be accomplished by analyzing a general
model of flow through isotropic porous media that
was developed by Du Plessis and coworkers, [1],
[2], [15], [16], based on intrinsic volume averaging.
We devise a method for evaluating the effective
viscosity under Poiseuille flow based on analyzing
the geometric factors (porosity functions) of five
different types of porous media. Expressions for the
effective viscosity, mean velocity, and maximum
velocity will be obtained. We also evaluate the
viscous flow limit and the Darcy limit, and derive a
3. W.S. Almalki. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 4) April 2016, pp.41-51
www.ijera.com 43|P a g e
threshold for the transition from Darcy’s to
Brinkman’s flow regimes. Our goal is to analyze the
effect of porosity function on the effective viscosity,
with the following objectives:
1) Study the viscosity factor under Poiseuille flow.
2) Analyze the effect of the porous microstructure
on the viscosity ratio by considering different
types of porous media.
3) Quantify the effects of porosity function on the
viscosity ratio.
4) Derive expressions for the velocity
distributions, the mean and maximum velocities
attained, flow rates, and the necessary driving
pressure gradients for the cases of flow through
porous media and through free-space.
5) Derive expressions for the viscous flow limit
and the Darcy flow limit.
II. MODEL EQUATIONS
A number of authors have derived
equations of flow through porous media by
averaging the Navier-Stokes equations over a
representative elementary volume, REV, and
quantifying the frictional forces exerted by the
porous medium on the traversing fluid through an
idealized description of the porous microstructure,
[1], [2], [15], [16]. Letting V be the bulk volume of
an REV (that is, the combined volume of the solid
and pore space), and V the volume of pore space,
then porosity of the medium is defined by
V
V
. …(6)
Now, for a fluid quantity of interest, G, the
volume average of G, that is the average of G over
the bulk volume, V, is defined by:
V
GdV
V
G
1
…(7)
and the intrinsic volume average of G, that is the
average of G over the pore space, V , is defined by
V
GdV
V
GG
1
. …(8)
Deviation of the average of G from the true quantity
G is defined by:
GGG
…(9)
and the relationship between the volume average and
the intrinsic volume average of G is given by:
GGG … (10)
The Darcy resistance term can be argued to
be dependent on porosity function in order to
account for different types of porous structures.
What gives rise to the Darcy resistance term in the
process of intrinsic volume averaging is a surface
integral of the form, [1], [2]:
S
dSnunp
V
][
1
…(11)
where
p is the pressure deviation, n
is a unit normal
pointing into the solid, and S is the surface of the
solid within the REV that is in contact with the fluid.
Through their concept of representative unit
cell, RUC, Du Plessis and Masliyah, [1], [2],
quantified the integral in (11) with the help of a
geometric porosity function. In the absence of
inertial and gravitational effects, Du Plessis and
Masliyah’s averaged equations take the following
form for constant porosity media, written in terms of
the specific discharge, q
:
0
2
qFqp
…(12)
where uq
and u
is the intrinsic averaged
velocity, p is the intrinsic averaged pressure, F
is a porosity function that depends on the porous
microstructure and tortuosity of the medium. Du
Plessis and Masliyah, [1], [2], argue that
hydrodynamic permeability inclusive of nonlinear
microscopic inertial effects is given by
F
k
.
Hydrodynamic permeability is the same as the
velocity-independent Darcy permeability for low
Reynolds number. Upon using
F
k
in (12), we
obtain
0
2
q
k
qp
. …(13)
Now, comparing (13) with (2), we see that if we can
identify the Brinkman ensemble averaged velocity,
u
, with the specific discharge, q
and the Brinkman
pressure, p , with the intrinsic averaged pressure,
p , then
*
. However, this may not be the
case since Brinkman’s velocity is an intrinsic
velocity. We therefore write (13) in terms of the
intrinsic averaged velocity,
u
, as:
0
2
2
u
k
up …(14)
Removing the subscript , and dividing equation
(14) by the constant porosity , we obtain:
0
2
u
k
up
…(15)
or in the form
0
2
uFup …(16)
4. W.S. Almalki. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 4) April 2016, pp.41-51
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Brinkman’s intrinsic velocity and pressure
in equation (2) can thus be identified with the
intrinsic velocity and pressure of equation (16). We
point out that while equation (2) involves the
effective viscosity,
*
, equation (16) does not
include
*
explicitly. However, equation (16)
includes a porosity function, F. Porosity function (or
geometric factor) F is dependent on factors such as
the porous microstructure, the constituents of the
porous matrix (diameter of solid particles), the pore
diameter, the porosity of the medium (defined in
equation (6)), and tortuosity, T, of the medium
(defined as the ratio of the pore volume to the pore
area). A number of expressions for F are available in
the literature. Some of these porosity functions are
listed in Table 3, below, which also gives the
corresponding expressions for the hydrodynamic
permeability, [15], [16].
Table 3. Geometric Factors and Hydrodynamic
Permeability.
p
d is the average pore diameter in a channel-like
porous material.
d is a microscopic characteristic length.
md is a median diameter of spherical particles.
In the analysis to follow, we will study the
viscosity factor under Poiseuille flow then recover
the effective viscosity and view it through analysis
of different porosity functions.
III. PLANE POISEUILLE FLOW
Consider the flow driven by a constant
pressure gradient, between parallel plates located at
y = -h and y = h, as shown in Figure 1.
Figure 1. Representative Sketch
When the channel configuration is filled
with a fluid-saturated porous material and viscous
shear effects are taken into account, the flow is
governed by equation (16), which reduces to the
following form in which we use
dx
dp
p x :
u xpFu
1
. …(17)
Boundary conditions are no-slip on solid walls,
namely
0)()( huhu …(18)
We non-dimensionalize equations (17) and (18) with
respect to h, and with respect to a characteristic
velocity (the mean velocity of the flow, m
u ) by
defining:
K
f
h
f
Khk
FKhkhYyUuu m
;;;;
22
2
…(19)
where k is the permeability and
2
h
k
K is the
Darcy number, Da, (or dimensionless permeability).
In addition, we define:
.
2
m
x
u
ph
P …(20)
Equations (17) and (18) are thus expressed in the
following dimensionless form, respectively:
U PfU …(21)
U(-1)=U(1)=0. …(22)
Factor Fhf
2
that appears in (19) and
(21) is critical in the analysis to follow. At the
outset, we provide Tables 4(a)-4(e) of its values, for
a range of porosities, using the expressions for the
geometric factor F of Table 3. These tables provide
values for f as a function of h/d, then specific values
are provided when h/d takes on the values 10 and 20,
for the sake of illustration.
It is clear from Tables 4(a)-(e) that f
increases with increasing h/d, and decreases with
increasing porosity, until it reaches a value of zero
when 1 (a case that corresponds to flow in free-
space, that is in the absence of a porous matrix). In
the analysis to follow, the value f = 0 corresponds to
the Navier-Stokes (viscous flow) limit.
5. W.S. Almalki. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 4) April 2016, pp.41-51
www.ijera.com 45|P a g e
Table 4(a). Values of f for a range of porosities
for granular media.
Table 4(b). Values of f for a range of porosities
for consolidated media.
*Approximated using 3T = 1+2
Table 4(c). Values of f for a range of porosities
for unidirectional fibre bed.
Table 4(d). Values of f for a range of porosities
using Ergun’s equation.
Table 4(e). Values of f for a range of porosities
using Kozeny-Carman relation.
Brinkman Velocity Profile
General solution to (21) is given by:
f
P
YfcYfcU sinhcosh 21 . …(23)
Using (22) in (23) we obtain
0;
cosh
21 c
ff
P
c and (23) becomes:
1
cosh
cosh
f
Yf
f
P
U . …(24)
In terms of the dimensional variables, (24) takes the
following form in which we leave f dimensionless
and denote the Brinkman velocity through the
porous medium by, Bu :
f
hyf
f
ph
u
x
B
cosh
/cosh
1
2
. …(25)
The maximum velocity, maxu , occurs at y = 0 and is
given by:
ff
ph
u
x
B
cosh
1
1)(
2
max
. …(26)
If the dimensional form of f is used, that is
k
h
f
2
, equation (25) takes the form:
h
k
y
kkp
u
x
B
cosh
cosh
1 . …(27)
Darcy Velocity
When the flow is of the seepage type, then it is
governed by Darcy’s law. In this case, viscous shear
effects are ignored and the no-slip on the solid plates
is no longer valid. We can obtain the dimensionless
Darcy velocity from (21) by setting to zero the
viscous shear term, as:
f
P
U . …(28)
Now, from Uuu mD , where Du is the
dimensional Darcy velocity, and equation (28), we
get:
f
ph
f
P
uUuu
x
mmD
2
. …(29)
Using khf /
2
in (29), we obtain:
x
D
kp
u …(30)
6. W.S. Almalki. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 4) April 2016, pp.41-51
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and upon dividing (30) by m
u , we obtain:
1/
.
/
2
ff
uf
ph
uu
m
x
mD
. …(31)
Equation (31) provides us with the following
observation.
Observation 1: Darcy velocity Du is the same as
the mean velocity, m
u . This is a constant, uniform
velocity across the channel. Furthermore, the
maximum Darcy velocity max)( Du is also the mean
velocity.
Navier-Stokes Velocity
If 0f , that is if k , equation (21)
reduces to the Navier-Stokes flow under Poiseuille
conditions. Now, taking 0f in (21), we obtain
the following Navier-Stokes solution satisfying U(-
1)=U(1)=0, written in terms of the dimensional
variables, wherein we denote the Navier-Stokes
velocity by N
u :
22
2
yh
p
u
x
N
…(32)
The maximum velocity, max)( Nu , occurs at y = 0
and is given by:
.
2
)(
2
max
x
N
ph
u …(33)
IV. VOLUMETRIC FLOW RATES AND
EFFECTIVE VISCOSITY
The Brinkman volumetric flow rate through
the porous medium is denoted here by BQ , and
defined by:
f
f
f
ph
dy
f
hyf
f
ph
dyuQ
x
h
h
x
h
h
BB
tanh
1
2
cosh
/cosh
1
32
. …(34)
The Darcian volumetric flow rate through
the porous medium is denoted here by DQ , and
defined by:
f
ph
dy
f
ph
dyuQ
x
h
h
x
h
h
DD
32
2.
. …(35)
Upon using (31) in (35), we obtain
hu
hkp
Q D
x
D 2
2
. …(36)
The Navier-Stokes volumetric flow rate is denoted
by NQ , and given by:
3
2
2
3
22 x
h
h
x
h
h
NN
ph
dyyh
p
dyuQ
. …(37)
Brinkman’s Effective Viscosity in Relation to
Base Fluid Viscosity:
In order to find the effective viscosity,
*
B , we replace by
*
B and replace NQ in (37)
by BQ of (34). This is equivalent to saying: If the
Navier-Stokes volumetric flow rate NQ across the
channel is replaced by the Brinkman volumetric
flow rate under the same driving pressure gradient
and channel depth, what is the corresponding fluid
viscosity,
*
B
?
We thus have:
.
tanh33tanh
33
3
2
2/33
*
ff
f
f
f
f
Q
ph
B
x
B
…(38)
Brinkman’s viscosity factor, BN , with
respect to Navier-Stokes base fluid is defined as:
.
tanh33tanh
33
2/3*
ff
f
f
f
fB
BN
…(39)
Darcy’s Effective Viscosity in Relation to Base
Fluid Viscosity:
In order to find Darcy’s effective viscosity,
*
D , we replace by
*
D and replace NQ in (40)
by DQ of (37).
33
2
3
*
f
Q
ph
D
x
D . …(40)
Darcy’s viscosity factor, DN , with respect to
Navier-Stokes base fluid, is thus obtained by
dividing (40) by , namely:
.
33
2
*3
f
Q
ph D
D
x
DN
…(41)
In terms of Du , *
D takes the form, obtained by
substituting (36) in (40):
D
x
D
u
ph
2
*
.
An expression for the Darcy pressure gradient is
obtained from (40) or (41) and is of the form:
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.
3
22
*
h
uf
h
u
p
mDD
x
…(42)
V. FLOW LIMITS AND MEAN
VELOCITIES
Viscous Flow Limit
Viscous flow limit corresponds to 1BN ,
or
*
B
. It is reached when 0f , or when the
permeability approaches infinity, k . When
0f , the flow approaches the Navier-Stokes flow
and the viscous flow limit is reached. In this case, as
,0f
3
tanh
ff
ff and
1
3
33
tanh33
2/32/3
ff
ff
f
ff
f
.
…(43)
Thus, .1
*
BN
Darcian Flow Limit
The Darcian flow limit corresponds to
0BN , or 0
*
. It is reached when the
permeability is small, or .0k This corresponds to
large values of f. The pressure gradient term can be
written as:
f
f
f
ff
ff
u
ph
m
x
tanh
1
tanh
2
…(44)
Now, for large f , 0
tanh
f
f
and the
pressure gradient takes the form
.
2
f
u
ph
m
x
…(45)
This Darcian flow limit depends on Darcy number
(Da) and the porosity, , since
./
2
KDa
khf
Equation (45) can thus be
written in the form
k
u
p
m
x
/
…(46)
and the Darcy permeability may be expressed in the
following form:
x
m
p
u
k
…(47)
while the Darcy number can be expressed as:
x
m
ph
u
DaK
2
…(48)
It is clear from (47) and (48) that the
permeability and Darcy number, respectively, are
tied to the medium properties of porosity and
channel depth, the driving pressure gradient, and the
fluid viscosity. However, the question of how high
or low the Darcy number can be chosen in order to
make Darcy’s law valid, remains to be answered. In
what follows we attempt to provide a partial answer
in terms of the factor f.
Following Bear [4], we express the permeability in
the form
3
2
h
k
…(49)
From (47) and (49) we have:
x
m
p
uhk
3
2
…(50)
hence we obtain the following expression for the
fluid viscosity:
.
3
2
m
x
u
ph
…(51)
Now, from (50) and the knowledge that khf /
2
,
we obtain
3
3
/
2
2
2
h
h
khf . …(52)
Upon using f = 3 in equation (40), namely,
33
2
3
*
f
Q
ph
D
x
D , we obtain .
*
D
The above analysis furnishes the following
observation.
Observation 2: In Darcy’s flow, the concept of
effective viscosity is irrelevant, or the “effective
viscosity” is the same as the base fluid viscosity.
Furthermore, the value of f = 3 represents the
threshold for validity of Darcy’s law. With this
knowledge, we can determine the value of porosity
below which we can assume Darcy’s law to be valid.
Mean Velocity
The mean velocity in the channel is the
volumetric flow rate per unit depth of the channel.
That is,
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h
Q
u m
2
. …(53)
For Navier-Stokes flow, we have
32
3
2
2
2
3
x
x
N
m
ph
h
ph
h
Q
u
…(54)
and the Navier-Stokes pressure gradient is expressed
as:
3
2
m
x
u
ph
. …(55)
For porous medium and the Brinkman flow, we
have:
.
tanh
33
32
2
f
f
f
ph
h
Q
u
xB
m
…(56)
The pressure gradient is expressed as:
f
f
f
ff
ff
u
ph
m
x
tanh
1
3
tanh
3
2
. …(57)
Now, using equation (43), we see that as
1
tanh
2/3
ff
f
, 3
tanh
3
2/3
ff
f
.
Hence, 3
2
m
x
u
ph
when 0f ; and
3
2
x
m
p
u
h
. …(58)
This emphasizes the fact that equation (57) reduces
to (55) as 0f , and the Brinkman flow
approaches the Navier-Stokes flow.
The Darcian mean velocity has been obtained from
equation (29), and is equal to the Darcy velocity, D
u
namely
D
x
m u
f
ph
u
2
…(59)
and the Darcy pressure gradient is thus expressed as:
f
u
ph
m
x
2
. …(60)
Upon using the threshold of validity of
Darcy’s law, that is f = 3, we obtain transition to
Brinkman’s flow, and equation (59) yields
3
2
m
x
u
ph
. … (61)
The above analysis furnishes the following
observation.
Observation 3: Equations (55), (58) and (61)
suggest that the Brinkman pressure gradient (with
appropriate scaling with respect to the relevant mean
velocity and channel depth) approaches the Navier-
Stokes pressure gradient when f = 0, while the Darcy
pressure gradient approaches the Brinkman pressure
gradient when f = 3.
VI. RESULTS AND DISCUSSION
Tables 6(a) and 6(b) provide a listing of the
values of max2
)(
)(
D
x
u
ph
for h/d=10 and h/d=20,
respectively. The Tables demonstrate the expected
increase inthis quantity with increasing porosity,
thus indicating an increase in max
)( D
u with
increasing porosity for a given combination of
)(
2
xph
. The maximum Darcy velocity values
consistently decrease with increasing h/d. These
trends persist for all geometric factors considered.
Tables 7(a) and 7(b) provide a listing of the values
of max2
)(
)(
B
x
u
ph
for h/d=10 and h/d=20,
respectively. These Tables also demonstrate the
expected increase in this quantity with increasing
porosity, thus indicating an increase in max)( Bu with
increasing porosity for a given combination of
)(
2
xph
. The maximum Brinkman velocity values
consistently decrease with increasing h/d. These
trends persist for all geometric factors considered.
Results obtained when using Ergun’s equation are
close in values to those obtained when using the
Kozeny-Carmen relation.
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Table 6(a) Values of 1/f or max2
)(
)(
D
x
u
ph
when h/d = 10
Table 6(b) Values of 1/f or max2
)(
)(
D
x
u
ph
when h/d = 20
Table 7(a) Values of
ff cosh
1
1
1
or
max2
)(
)(
B
x
u
ph
when h/d = 10
Table 7(b) Values of
ff cosh
1
1
1
or
max2
)(
)(
B
x
u
ph
when h/d = 20
Tables 8(a) and 8(b) provide a listing of the values
of B
x
Q
ph )2(
3
for h/d=10 and h/d=20,
respectively. These Tables demonstrate an increase
in this quantity with increasing porosity, thus
indicating an increase in the volumetric flow rate
BQ with increasing porosity for a given combination
of
)2(
3
xph
. The maximum Brinkman volumetric
flow rate values consistently decrease with
increasing h/d. These trends persist for all geometric
factors considered. Results obtained when using
Ergun’s equation are close in values to those
obtained when using the Kozeny-Carmen relation.
The viscosity ratio
f
f
fB
BN
tanh
33
*
is
illustrated in Tables 9(a) and 9(b) for h/d=10 and
h/d=20, respectively. Qualitative behavior of the
viscosity ratios is illustrated in Figures 2 and 3.
Tables 9(a) and 9(b) demonstrate an increase in the
viscosity ratio with increasing h/d, and a decrease
with increasing porosity, for all geometric factors
considered. When using Ergun’s equation and the
Kozeny-Carmen relation, the viscosity ratio gets
closer and closer to unity as porosity gets closer and
closer to unity (the viscous flow limit). When using
other geometric factors, the viscosity ratio is still far
from unity for the values of h/d tested. This points
out the need to further decrease the value of h/d
when these porosity functions are employed. While
the case of consolidated media renders moderate
results for the viscosity ratio, the value remains close
to 2 even when the porosity is close to unity.
In terms of the transition from a Darcy regime to a
Brinkman regime, the value of f = 3 is reached when
the porosity is between 0.98 and 0.985 for the
Ergun’s equation and the Kozeny-Carmen relation,
when h/d = 10, and a porosity between 0.991 and
0.992 for h/d = 20. For consolidated media, f = 3 is
reached between the porosity values of 0.998 and
0.999, for h/d = 10. It is not reached for h/d = 20.
For granular media and unidirectional fibres, f = 3
cannot be reached for the values h/d = 10 or 20. This
may be an indication that for these porous structures,
transition from Darcy to Brinkman regime occurs at
values higher than f = 3. Furthermore, these results
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emphasize the importance of the geometric factors in
determining the viscosity ratio and describing the
Brinkman flow.
Table 8(a) Values of
f
f
f
tanh
1
1
or
BQ
dx
dp
h )2(
3
when h/d = 10
Table 8(b) Values of
f
f
f
tanh
1
1
when h/d =
20
Table 9(a) Values of
ff
ffB
BN
tanh33
*
when h/d = 10
Table 9(b) Values of
ff
ffB
BN
tanh33
*
when h/d = 20
Figure 2 Brinkman Viscosity Ratio, h/d=10
Figure 3 Brinkman Viscosity Ratio, h/d=20
VII. CONCLUSION
In conclusion, we have provided in this
work a method of estimating the viscosity factor
(hence Brinkman’s effective viscosity) based on
using different geometric factors (porosity functions)
under Poiseuille flow. The most informative results
are obtained when using Ergun’s equation and the
Kozeny-Carmen relation, where results indicate that
the viscosity factor approaches unity (or the
effective viscosity approaches the base viscosity) as
porosity approaches unity (the viscous flow limit). In
the analysis we have also determined a threshold
value of f = 3 to serve as the transition from Darcy
to Brinkman flow regime. This value corresponds to
value of porosity higher than 98%, thus emphasizing
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that Brinkman’s equation is possibly valid for high
values of porosity.
REFERENCES
[1]. J.P.Du Plessis and J.H.Masliyah,
Mathematical modeling of flow through
consolidated isotropic porous media,
Transport in Porous Media, 3 (1988), 145-
161.
[2]. J.P.Du Plessis and J.H.Masliyah, Flow
through isotropic granular porous media,
Transport in Porous Media, 6 (1991), 207-
221.
[3]. J.L. Auriault, On the domain of validity of
Brinkman’s equation, Transport in Porous
Media, 79 (2009), 215–223.
[4]. H.C.Brinkman, A Calculation of the
viscous force exerted by a flowing fluid on
a dense swarm of particles, Applied
Scientific Research, A1 (1947), 27-34.
[5]. S.C. Cowin, The theory of polar fluids,
Advances in Applied Mechanics, 14
(1974), 279–347.
[6]. J. Francisco, J. Valdes-Parada, Alberto
Ochoa-Tapia and Jose Alvarez-Ramirez,
On the effective viscosity for the Darcy–
Brinkman equation, Physica A, 385 (2007),
69–79.
[7]. R.C. Givler and S.A. Altobelli, A
determination of the effective viscosity for
the Brinkman-Forchheimer flow model,
Journal of Fluid Mechanics, 258 (1994),
355-370.
[8]. J.A.Kolodziej, Influence of the porosity of
a porous medium on the effective viscosity
in Brinkman’s filtration equation, Acta
Mechanica, 75 (1988), 241-254.
[9]. J. Koplik, H. Levine and A. Zee, Viscosity
renormalization in the Brinkman equation,
Physics of Fluids, 26 (1983), 2864-2870.
[10]. H.Liu, P.R.Patil and U. Narusawa, On
Darcy-Brinkman equation: viscous flow
between two parallel plates packed with
regular square arrays of cylinders, Entropy,
9 (2007), 118-131.
[11]. N. Martys, D.P. Bentz and E.J. Garboczi,
1994 Computer simulation study of the
effective viscosity in Brinkman’s equation,
Physics of Fluids, 6(4) (1994), 1434-1439.
[12]. W.-P. Breugem, The effective viscosity of a
channel-type porous medium, Physics of
Fluids, 19(10) (2007), 103103-1- 103104-8.
[13]. A.Einstein, EineneueBestimmung der
Molek¨uldimensionen, Annalen der Physik,
19 (1906), 289-306.
[14]. A.Einstein, 1911 Berichtigungzumeiner
Arbeit: “Eineneue Bestimmungder
Molek¨uldimensionen.”, Annalender
Physik, 34 (1911), 591-592.
[15]. J.P.Du Plessis, Analytical quantification of
coefficients in the Ergun equation for fluid
friction in a packed bed, Transport in
Porous Media, 16 (1994), 189-207.
[16]. J.P. Du Plessis and G.P.J.Diedericks, Pore-
scale modeling of interstitial phenomena. In
“Fluid Transport in Porous Media”, J.P.Du
Plessis, ed., Computational Mechanics
Publications, 1997, 61-104.