The document summarizes research on modeling oscillatory flow and particle suspension in a fluid through an elastic tube. It begins by reviewing previous work in this area by Womersley. The authors then present a new three-phase model that accounts for the fluid, particulate, and elastic tube phases. Governing equations are derived for each phase using assumptions of incompressibility, negligible inertia terms, and harmonic pressure waves. Solutions are obtained using Bessel functions, with the frequency equation having a different structure than Womersley's solution to account for the particulate phase. Removing the particle density term causes the new solution to collapse back to Womersley's original equation.
Effect Of Elasticity On Herschel - Bulkley Fluid Flow In A TubeIJARIDEA Journal
Abstract— The impact of flexibility on Herschel-Bulkley liquid in a tube is researched. The issue is unraveled scientifically
for two unique sorts: one flux is computed taking the worry of the flexible tube into thought and the other flux is acquired by
considering the weight range relationship. Speed of the inelastic tube is additionally considered. The impact of various
parameters on flux and speed are talked about through charts. The outcomes acquired for the stream attributes uncover
many intriguing practices that warrant additionally contemplate on the non-Newtonian liquid stream wonders, particularly
the shear-diminishing marvels. Shear diminishing decreases the divider shear stretch.
Keywords— Elastic tube, Herschel-Bulkley Fluid, Inlet pressure, Outlet Pressure, Yield Stress.
Effect of an Inclined Magnetic Field on Peristaltic Flow of Williamson Fluid ...QUESTJOURNAL
ABSTRACT: This paper deals with the influence ofinclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
Casson flow of blood through an arterial tube with overlapping stenosisiosrjce
The objective of the present analysis is to study the effect of overlapping stenosis on blood flow
through an artery by taking the blood as Casson type non-Newtonian fluid. The expressions for flux and
resistance to flow have been studied here by assuming the stenosis is to be mild. The results are shown
graphically for different values of yield stress, stenosis length, stenosis height and discussed.
MHD convection flow of viscous incompressible fluid over a stretched vertical...IJERA Editor
The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
Influence of MHD on Unsteady Helical Flows of Generalized Oldoyd-B Fluid betw...IJERA Editor
Considering a fractional derivative model for unsteady magetohydrodynamic (MHD)helical flows of an
Oldoyd-B fluid in concentric cylinders and circular cylinder are studied by using finite Hankel and Laplace
transforms .The solution of velocity fields and the shear stresses of unsteady magetohydrodynamic
(MHD)helical flows of an Oldoyd-B fluid in an annular pipe are obtained under series form in terms of
Mittag –leffler function,satisfy all imposed initial and boundary condition , Finally the influence of model
parameters on the velocity and shear stress are analyzed by graphical illustrations.
Determination of the Probability Size Distribution
of Solid Particles in a Technical Water by Tomantschger Kurt W*, Petrović Dragan V and Radojević Rade L in Evolutions in Mechanical Engineering
Effect Of Elasticity On Herschel - Bulkley Fluid Flow In A TubeIJARIDEA Journal
Abstract— The impact of flexibility on Herschel-Bulkley liquid in a tube is researched. The issue is unraveled scientifically
for two unique sorts: one flux is computed taking the worry of the flexible tube into thought and the other flux is acquired by
considering the weight range relationship. Speed of the inelastic tube is additionally considered. The impact of various
parameters on flux and speed are talked about through charts. The outcomes acquired for the stream attributes uncover
many intriguing practices that warrant additionally contemplate on the non-Newtonian liquid stream wonders, particularly
the shear-diminishing marvels. Shear diminishing decreases the divider shear stretch.
Keywords— Elastic tube, Herschel-Bulkley Fluid, Inlet pressure, Outlet Pressure, Yield Stress.
Effect of an Inclined Magnetic Field on Peristaltic Flow of Williamson Fluid ...QUESTJOURNAL
ABSTRACT: This paper deals with the influence ofinclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
Casson flow of blood through an arterial tube with overlapping stenosisiosrjce
The objective of the present analysis is to study the effect of overlapping stenosis on blood flow
through an artery by taking the blood as Casson type non-Newtonian fluid. The expressions for flux and
resistance to flow have been studied here by assuming the stenosis is to be mild. The results are shown
graphically for different values of yield stress, stenosis length, stenosis height and discussed.
MHD convection flow of viscous incompressible fluid over a stretched vertical...IJERA Editor
The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
Influence of MHD on Unsteady Helical Flows of Generalized Oldoyd-B Fluid betw...IJERA Editor
Considering a fractional derivative model for unsteady magetohydrodynamic (MHD)helical flows of an
Oldoyd-B fluid in concentric cylinders and circular cylinder are studied by using finite Hankel and Laplace
transforms .The solution of velocity fields and the shear stresses of unsteady magetohydrodynamic
(MHD)helical flows of an Oldoyd-B fluid in an annular pipe are obtained under series form in terms of
Mittag –leffler function,satisfy all imposed initial and boundary condition , Finally the influence of model
parameters on the velocity and shear stress are analyzed by graphical illustrations.
Determination of the Probability Size Distribution
of Solid Particles in a Technical Water by Tomantschger Kurt W*, Petrović Dragan V and Radojević Rade L in Evolutions in Mechanical Engineering
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...IOSR Journals
The goal of the paper is to examine the concentration of the longitudinal dispersion phenomenon arising in fluid flow through porous media. The solution of the Burger’s equation for the dispersion problem is
presented by approach to Sumudu transformation. The solution is obtained by using suitable conditions and is
more simplified under the standard assumptions.
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...IJERA Editor
This paper presents a research for magnetohydrodynamic (MHD) flow of an incompressible generalized
Burgers’ fluid including by an accelerating plate and flowing under the action of pressure gradient. Where the
no – slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is
introduced to establish the constitutive relationship of the generalized Burgers’ fluid. By using the discrete
Laplace transform of the sequential fractional derivatives, a closed form solutions for the velocity and shear
stress are obtained in terms of Fox H- function for the following two problems: (i) flow due to a constant
pressure gradient, and (ii) flow due to due to a sinusoidal pressure gradient. The solutions for no – slip condition
and no magnetic field, can be derived as special cases of our solutions. Furthermore, the effects of various
parameters on the velocity distribution characteristics are analyzed and discussed in detail. Comparison between
the two cases is also made.
In this paper, the unsteady motion of a spherical particle rolling down an inclined tube in a
Newtonian fluid for a range of Reynolds numbers was solved using a simulation method called
the Differential Transformation Method (DTM). The concept of differential transformation is
briefly introduced, and then we employed it to derive solution of nonlinear equation. The
obtained results for displacement, velocity and acceleration of the motion from DTM are
compared with those from numerical solution to verify the accuracy of the proposed method.
The effects of particle diameter (size), continues phase viscosity and inclination angles was
studied. As an important result it was found that the inclination angle does not affect the
acceleration duration. The results reveal that the Differential Transformation Method can achieve suitable results in predicting the solution of such problems.
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...eSAT Journals
Abstract This paper describes an analytical approach to describe the areas where Pipes (used for flow of fluids) are mostly susceptible to damage and tries to visualize the flow behaviour in various geometric conditions of a pipe. Fluent software was used to plot the characteristics of the flow and gambit software was used to design the 2D model. Two phase Computational fluid dynamics calculations, using K-epsilon model were employed. This simulation gives the values of pressure and velocity contours at various sections of the pipe in which water as a media. A comparison was made with the sudden and gradual change of pipe diameter (i.e., expansion and contraction of the pipe). The numerical results were validated against experimental data from the literature and were found to be in good agreement. Index Terms: gambit, fluent software.
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 TechnologyIJRET : 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
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...IJMER
This paper presents an analytical study for the magnetohydrodynamic (MHD) flow of a
generalized Burgers’ fluid in an annular pipe. Closed from solutions for velocity is obtained by using finite
Hankel transform and discrete Laplace transform of the sequential fractional derivatives. Finally, the
figures are plotted to show the effects of different parameters on the velocity profile.
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...IJERA Editor
Three exact solutions to the Darcy-Lapwood-Brinkman equation with variable permeability are obtained in this
work. Solutions are obtained for a given vorticity distribution, taken as a function of the streamfunction.
Classification of the flow field is provided and comparison is made with the solutions obtained when
permeability is constant. Interdependence of Reynolds number and variable permeability is emphasized.
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
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability
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.
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...IOSR Journals
The goal of the paper is to examine the concentration of the longitudinal dispersion phenomenon arising in fluid flow through porous media. The solution of the Burger’s equation for the dispersion problem is
presented by approach to Sumudu transformation. The solution is obtained by using suitable conditions and is
more simplified under the standard assumptions.
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...IJERA Editor
This paper presents a research for magnetohydrodynamic (MHD) flow of an incompressible generalized
Burgers’ fluid including by an accelerating plate and flowing under the action of pressure gradient. Where the
no – slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is
introduced to establish the constitutive relationship of the generalized Burgers’ fluid. By using the discrete
Laplace transform of the sequential fractional derivatives, a closed form solutions for the velocity and shear
stress are obtained in terms of Fox H- function for the following two problems: (i) flow due to a constant
pressure gradient, and (ii) flow due to due to a sinusoidal pressure gradient. The solutions for no – slip condition
and no magnetic field, can be derived as special cases of our solutions. Furthermore, the effects of various
parameters on the velocity distribution characteristics are analyzed and discussed in detail. Comparison between
the two cases is also made.
In this paper, the unsteady motion of a spherical particle rolling down an inclined tube in a
Newtonian fluid for a range of Reynolds numbers was solved using a simulation method called
the Differential Transformation Method (DTM). The concept of differential transformation is
briefly introduced, and then we employed it to derive solution of nonlinear equation. The
obtained results for displacement, velocity and acceleration of the motion from DTM are
compared with those from numerical solution to verify the accuracy of the proposed method.
The effects of particle diameter (size), continues phase viscosity and inclination angles was
studied. As an important result it was found that the inclination angle does not affect the
acceleration duration. The results reveal that the Differential Transformation Method can achieve suitable results in predicting the solution of such problems.
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...eSAT Journals
Abstract This paper describes an analytical approach to describe the areas where Pipes (used for flow of fluids) are mostly susceptible to damage and tries to visualize the flow behaviour in various geometric conditions of a pipe. Fluent software was used to plot the characteristics of the flow and gambit software was used to design the 2D model. Two phase Computational fluid dynamics calculations, using K-epsilon model were employed. This simulation gives the values of pressure and velocity contours at various sections of the pipe in which water as a media. A comparison was made with the sudden and gradual change of pipe diameter (i.e., expansion and contraction of the pipe). The numerical results were validated against experimental data from the literature and were found to be in good agreement. Index Terms: gambit, fluent software.
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 TechnologyIJRET : 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
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...IJMER
This paper presents an analytical study for the magnetohydrodynamic (MHD) flow of a
generalized Burgers’ fluid in an annular pipe. Closed from solutions for velocity is obtained by using finite
Hankel transform and discrete Laplace transform of the sequential fractional derivatives. Finally, the
figures are plotted to show the effects of different parameters on the velocity profile.
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...IJERA Editor
Three exact solutions to the Darcy-Lapwood-Brinkman equation with variable permeability are obtained in this
work. Solutions are obtained for a given vorticity distribution, taken as a function of the streamfunction.
Classification of the flow field is provided and comparison is made with the solutions obtained when
permeability is constant. Interdependence of Reynolds number and variable permeability is emphasized.
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
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability
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.
C.E. Ministerio Un Toque de Dios is reaching the community thru Operation Rescue Mission. We are please to show this short slide show and realize that there families in need. Help us make a difference. www.ToquedeDios.com
Evolución político-territorial de la República MexicanaChristianovl
Al correr la historia de nuestro país, han sucedido diversos acontecimientos históricos, los cuales fueron presentando importantes modificaciones en la división político-administrativa.
En la Constitución de 1824, la división territorial de México estaba constituida por diecisiete estados y tres territorios, destacando por la extensión que ocupaba el estado Interno del Norte, el de Occidente Y el de Oriente Y por estar divididos el estado de Veracruz y el de Zacateca.
Peristaltic flow of blood through coaxial vertical channel with effect of mag...ijmech
The present paper investigates the effects of peristaltic flow of blood through coaxial vertical channel with
effect of magnetic field: blood flow study.The effects of various physical parameters on axial velocity and
pressure gradient have been computed numerically. It is observed that the maximum velocity increases
with increase in Magnetic field (M) even though for phase shiptııııı/ 4 for all the two cases
= - 0.5,
= -1. However, opposite effects are noticed for
= 0.5,
= 1.
Non- Newtonian behavior of blood in very narrow vesselsIOSR Journals
The purpose of the study is to get some qualitative and quantitative insight into the problem of flow in vessels under consideration where the concentration of lubrication film of plasma is present between each red cells and tube wall. This film is potentially important in region to mass transfer and to hydraulic resistance, as well as to the relative resistance times of red cells and plasma in the vessels network.
An exact solution of einstein equations for interior field of an anisotropic ...eSAT Journals
Abstract
In this paper, an anisotropic relativistic fluid sphere with variable density, which decreases along the radius and is maximum at
the centre, is discussed. Spherically symmetric static space-time with spheroidal physical 3-space is considered. The source is an
anisotropic fluid.
The solution is an anisotropic generalization of the solution discussed by Vaidya and Tikekar [1]. The physical three space
constant time has spheroidal solution. The line element of the solution can be expressed in the form Patel and Desai [2]. The
material density is always positive. The solution efficiently matches with Schwarzschild exterior solution across the boundary. It is
shown that the model is physically reasonable by studying the numerical estimates of various parameters. The density vs radial
pressure relation in the interior is discussed numerically. An anisotropy effect on the redshift is also studied numerically.
Key Words: Cosmology, Anisotropic fluid sphere, Radial pressure, Radial density, Relativistic model.
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014Cristina Staicu
Cristobal Bertoglio, David Barber, Nicholas Gaddum, Israel Valverde, Marcel Rutten, et al.. Identification of artery wall stiffness: in vitro validation and in vivo results of a data assimilation procedure applied to a 3D fluid-structure interaction model. Journal of Biomechanics, Elsevier, 2014, 47 (5),
pp.1027-1034. 10.1016/j.jbiomech.2013.12.029 . hal-00925902v2
I received explicit thank you from the INRIA team for my support in the Sheffield team.
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...IOSRJM
This paper deals with the influence of magnetic field on peristaltic flow of an incompressible Williamson fluid in a symmetric channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
Fluctuating Flow of Vescoelastic Fluids between Two Coaxial Cylindersinventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...IOSR Journals
Aim of this study is to investigate the effect of magnetic field on blood flow in cylindrical artery through porous medium. In this paper blood is considered elastico viscous, Non Newtonian fluid and flow is assumed as fully developed and laminar. Laplace transforms and Finite Hankel Transforms are used to obtain the analytical expression for velocity profile, flow rate and fluid acceleration. The effect of magnetic field on velocity and fluid acceleration has been discussed with the help of graphs. It is found that velocity distribution, flow rate and fluid acceleration of blood in cylindrical artery decrease as magnetic field increases.
The current study examines the generation and propagation of a Third order solitary water wave along
the channel. Surface displacement and wave profi le prediction challenges are interesting subjects in the
fi eld of marine engineering and many researchers have tried to investigate these parameters. To study the
wave propagation problem, here, fi rstly the meshless Incompressible Smoothed Particle Hydrodynamics
(ISPH) numerical method is described. Secondly,
Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers IJAEMSJORNAL
This flow in rivers is concerned with unsteady flow in open channel and it mathematically governed by the Saint Venant equation, using a four-point implicit finite difference scheme. For a one-dimensional applications, the relevant flow parameters are functions of time, and longitudinal positions. Considering the equations for the conservation of mass (continuity) and conservation of momentum. The mathematical method is empirical with the computer revolution, numerical methods are now effective to develop the hydraulic model. Conventional methods used to compute discharge, such as stage-discharge and stage-fall discharge relationships has been inadequate. Further research is recommended to include tributaries in rivers.
Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...IJERA Editor
The aim of the present paper is to study the axially symmetric, laminar, steady, one-dimensional flow of blood through narrow stenotic vessel. Blood is considered as Bingham plastic fluid. The analytical results such as pressure drop, resistance to flow and wall shear stress have been obtained. Effect of yield stress and shape of stenosis on resistance to flow and wall shear stress have been discussed through tables and graphically. It has been shown that resistance to flow and the wall shear stress increase with the size of stenosis but these increase are, however, smaller due to non-Newtonian behaviour of the blood.
This project aims at simulating lid driven cavity flow problem using package MATLAB. Steady Incompressible Navier-Stokes equation with continuity equation will be studied at various Reynolds number. The main aim is to obtain the velocity field in steady state using the finite difference formulation on momentum equations and continuity equation. Reynold number is the pertinent parameter of the present study. Taylor’s series expansion has been used to convert the governing equations in the algebraic form using finite difference schemes.
Possible methods of providing further (and perhaps better) alternative solutions for the exponential
integral of aquifer parameter evaluation are investigated. Three known mathematical methods of approach
(comprising self-similar, separable variable and travelling wave) are applied, providing three relevant solutions.
Further analysis of the self-similar solution reveals that this provides an alternative solution involving normal
graph of drawdown versus the measurement intervals. The geomathematical relevance of this method is assessed
using data from aquifers from two chronologically different hydrogeological units – the Ajalli Sandstone and
Ogwashi-Asaba Formation. The results indicate good functional relationship with satisfactory transmissivity
values
Possible methods of providing further (and perhaps better) alternative solutions for the exponential
integral of aquifer parameter evaluation are investigated. Three known mathematical methods of approach
(comprising self-similar, separable variable and travelling wave) are applied, providing three relevant solutions.
Further analysis of the self-similar solution reveals that this provides an alternative solution involving normal
graph of drawdown versus the measurement intervals. The geomathematical relevance of this method is assessed
using data from aquifers from two chronologically different hydrogeological units – the Ajalli Sandstone and
Ogwashi-Asaba Formation. The results indicate good functional relationship with satisfactory transmissivity
values
Application of DRP scheme solving for rotating disk-driven cavityijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
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Oscillatory flow and particle suspension in a fluid through an elastic tube
1. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
www.iiste.org
Oscillatory Flow and Particle Suspension in a Fluid Through an
Elastic Tube
P. N. Habu* M. A. Mbah
Department of Mathematics, Federal University Lafia, P. M. B 146, Lafia, Nigeria.
* E-mail: peterhabu68@gmail.com
Abstract
Womersley gave a solution for the case of a thin-walled elastic tube, it being assumed that the effect of the
inertia term in the equations of viscous fluid motion can be neglected. He did not consider the presence of
particles, to account for the blood cells in the blood, within the viscous flow through the tube (artery).
In this paper, the corresponding solution for an oscillatory flow and particle suspension in a fluid (blood), to
account for blood cells, through an elastic tube is obtained. This solution is the frequency equation as it was
obtained by Womersley but it has a different structure. If the volume fraction particle density , is removed from
this solution it collapses to give the same equation as Womersley’s case, without particles.
Keywords: Oscillatory flow, Particle suspension, Elastic tube, Periodic function
1. Introduction:
The problem of blood flow and wave propagation in the arterial system has stimulated the interest of
physiologists and mathematicians for years. Its fundamental importance for present day research can be traced to
Witzig and later to the works of Womersley [10]. The contributions of Womersley [10] represent the best
attempt to date in developing a complete, practical, unified theory of arterial blood flow and pressure
propagation [6].
A survey of the published literature on the propagation of waves in the arterial system will show a large variation
of assumptions on the nature of flow conditions, fluid properties, and types of vessel walls. The analytical
solutions for pulsatile flow in rigid tubes are therefore based on a combination of different assumptions, just the
same as those of oscillatory flow through elastic tube.
The model used in this paper consists of three phases as against two phases by Womersley. The three phases are
the fluid phase, particulate phase and the equations of motion of the tube.
According to Womersley, the simple solution for the oscillatory motion of viscous liquid in a rigid tube, under a
simple-harmonic pressure gradient, was given by Lambossay who gave the formula for velocity and viscous drag
solely concerned with the effect of the viscous drag on the frequency response of pressure recording instruments.
Womersley obtained the same result independently, in a different form and derived the expression for the rate of
flow.
For the equations of the elastic tube, we adopt Oslen J.W et al [7] and Womersley [10]. All the assumptions by
Womersley are adopted and applied in this paper alongside his method of solution. The only variation between
Womersley and this approach is the introduction of the particulate phase, with , as the volume fraction particle
density, so that we now simulate blood properly, with proper consideration to the existence of blood cells, as the
particles.
2. Methods
2.1 Assumptions and Nomenclature
( , ) denote fluid phase velocities, ( , ) denote particulate phase velocities,
and,
are the actual
is the fluid phase
densities of the materials constituting fluid and particulate phase respectively, (1 − )
density,
the particulate phase density, P denotes the pressure, denotes the volume fraction density of the
particles,
is the particle fluid mixture viscosity and S is the drag coefficient of interaction for the force
exerted by one phase on the other. Inertia terms of the equations of motion are neglected, diffusivity terms are
also neglected. is chosen as constant, and we assume that the pressure wave is harmonic in time having
frequency n and wave velocity c.
2.2 Formulation of the Model
A porous tapered elastic tube filled with a viscous fluid is considered as a model of a vascular bed in which
successive branching of the blood cells leads to a rapid decrease in diameter with distance. The porosity of the
tapered tube is adjusted to simulate the effect of branching
The motion of the fluid is assumed to be cycisymetric and governed by the Navier-Stokes equations for an
incompressible fluid, now modified to simulate the effect of particle suspension as follows:
(1− )[
+
+
] = −(1− )
+(1− ) [
109
+
+
]+
(
−
)
(1)
2. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
(1− )[
+
] = −(1− )
With equation of continuity as,
www.iiste.org
+(1− ) [
+
−
+
]+
(
−
)
(2)
[(1− ) ]+ [(1− ) ]+
= 0,
(3)
where
!
are the axial and radial components of velocity, P is the pressure and and are the density
and viscosity of the fluid respectively.
The problem of determining the motion of a liquid in an elastic tube when subjected to a pressure – gradient
which is a periodic function of the time arises in connection with the flow of blood in the larger arteries. The
equations of motion of the tube from [6] are:
(
"
+
=
=
#$ "
#
%&
−
,
%#
[
'
.
-
+
* "
#
&
& /
+
+
&
]'( +
0
)
)
#
[
"
+
* +
&
]
(4)
(5)
Together with the boundary conditions for the motion of the fluid,
+
=
12=1
=
12=1
The particulate phase: The equations of motion of the particles are:
"
[
+
3
3
+
3
]=−
[
+
+
]=−
and their equation of continuity is:
[
]+ [
]+ 3 = 0
With boundary conditions
+
=
12=1
3
3
+
(
+
(
−
−
)
(7)
)
(8)
(9)
12=1
"
=
3
(6)
(10)
A
3. Solution: The expression for the drag coefficient for the present study is selected as,
=
4 6$ 9
8 (:)
57
, where 89 (:) =
;<=>?; =
[5 = ]
@
+3
Where C is the fluid viscosity, and ‘a’ is the radius of the particle. Relation where 89 (:) represents the classical
Stokes’ drag for small particle Reynolds number, modified to account for the finite particulate fractional volume
through the function 89 (:), obtained by [9].
In order to investigate the Pulsatile flow along the axis of the tube, we assume that the pressure wave is harmonic
in time having frequency n and wave velocity c. We therefore assume that,
(F), 5 (F), 5 (F)] exp [in (t− )]
(D, , , , , )= [E (F), (F),
(11)
G
We shall be concerned with the motion in which
H&
,
, 3 , 3 , are small, i.e when the wave velocity, c is very large, as compared to , , ,
, and nR.
G
G
G
G
G
From (11) we can write
P = E (r)exp[in(t− )]
I =
J=
I =
J =
=
G
G
(r)exp[in(t− )]
=
(12)
5 (r)exp[in(t− G )]
5 (r)exp[in(t− G )]
K
=0
KL
= −
=
G
(r)exp[in(t− )]
A
G
in exp[in(t− )]
A
G
G
exp[in(t− )]
A
G
exp[in(t− )]
(13)
G
exp[in(t− )]
110
3. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
=
A
=
A NH
G
=
=
G
exp[in(t− )]
M exp[in(t− )]
=−
=
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A
G
A
G
G
exp[in(t− )]
G
(13 cont.)
exp[in(t− )]
A
G
exp[in(t− )]
G
exp[in(t− )]
=−
in exp[in(t− )]
G
G
Neglecting the inertia terms in (1), we can write equation (1) as
= −(1− )
(1− )
+(1− ) [
+
+
]+
(
−
)
(14)
Now substituting equations (12) and (13) into equation (14) and neglecting the diffusivity terms
Equation (14) simplifies to
'
A
+
'
'
+
'
'
A
−
=−
NH&
Let P = Q( ) R5 ⇒ P 5 =
"
"
Substituting (16) in (15), we get
'
A
H
A
− MP 5
=−
−
& ,A NH
6G
& H
"
& ,A NH
6G
−
&
O
6(
&
6(
O
)
)
[
[
−
]
−
]
5
5
(15)
(16)
(17)
(1− )
=−(1− ) +(1− ) [
+
− +
Now substituting equations (12) and (13) into (18), we get
]+
Similarly in equation (2), neglecting inertia terms, can be written as
M =−
Let 2 =
⇒
A
'
&
+
,A
#
+
"
A
+
"
A
−P
+
# (
O
)
[
5
−
]
T
−
U
⇒ F = 2Q, therefore equation (18) can be written as
A
'
− MP 5
'
& H
−
'
A
=
& ,A
6
'
−
&
6(
V
)
[
5
we have,
−
]
(18)
(19)
(20)
Where P 5 =
"
For equations (19) and (20), the conservation of mass equation can be written using (12) and (13) and
simplifying, we get
( A ')
NH& A
=
(21)
'
G
'
Equations (19) and (20) are now the equations of motion of the fluid and equation (21) is the conservation of
mass equation.
3.1 Evaluating the particulate phase
Now neglecting the inertia terms in equation (7), we can write equation (7) as
[ 3] = −
+ + ( − )
Now substituting (12) and (13) into (22), we have
V A
A
=
−
&[#3 NH<V]
#3 NH<V
(22)
'
(23)
=−
+ ( − )
Substituting equations (12) and (13) into (24), we get
V A
A NH
=
+
Also equation (8) can similarly be written as
3
G[#3 NH<V]
(24)
#3 NH<V
(25)
The equation of conservation of mass of the particles can be written using (12) and (13) as
( ')
NH&
=
'
'
G
(26)
We can now re-arrange the equations of motion of the particles to be written from (23), (25) and (26) to have:
V A
A
=
−
#3 NH<V
V A
=
&>#3 NH<V@
A NH
G[#3 NH<V]
+
#3 NH<V
'
(27)
with the equation of continuity as
111
4. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
(
'
')
'
=
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NH&
G
Now to solve equations (19) and (20) [i.e. equations of motion of the fluid]
From equation (23), equation (19) can be written as
+
A
'
+
A
'
'
'
'
−
'
−
+ WM = P 5 +
A
+ M =P 5Y5
6(
&
=
Equation (28), can be written as
A
'
A
'
A
where Y 5 = 1 +
+
−
'
−
'
&
N Z "A (
O
&
6&
)T#3 NH<VU
+ M =[5
=
Equation (29) can be written as
A
'
A
'
'
A
Where [ = P Y
O
)T#3 NH<VU
&
6&
!] =1+(
[1 +
−
−
6(
& V
N Z "A (
A
'
& V
6(
)
V
X
)
]
)(#3 NH<V)
=
&
6&
A
[1 +
6(
V
)
−
#3 NH<V
]
A
'
(28)
'
(29)
(30)
V
)(#3 NH<V)
Using Bessel function, the solution of equation (30), expressing E = ^ _C (`2), where k is to be determined.
Then A = ^ a_ (a2), therefore equation (29) can be written as
+
A
'
=
'
A
'
'
+ M =[5
A
−
bA c$ (N Z d ')
=
&
6&
]^ a_ (a2)
With the solution
complementary and particular solutions respectively, giving,
=
c$ (N Z d )
b c$ (N Z d ')
−
& efA cA (e')
& 6 NZd
=
G
+
, where
G
!
e
are the
(31)
Similarly equation (20) has the solution,
c$ (N Z d )
NH& fA c$ (e')
NZd
e
(32)
Now using the approximation, Womersley [10], to equations (31) and (32), where from the equation of
continuity of the fluid we get the identities,
GA
NH&
=
G
d N G
N H & Z gfA
=
G 6
NH&gA⁄
and
&e fA
6
⇒a=
, so that _C (a2) becomes _C i
G
Also, we use the approximation
= _C i
H&gA⁄
G
H&g A⁄
G
NH&gA⁄
j=1
G
2j = 1 and _ (a2) becomes M_ i
H&gA⁄
G
2j =
NH&'gA⁄
5G
And
=_ i
j.
5G
G
Inserting these approximations in equations (31) and (32) we get
H&'g A⁄
=
=
−
bA cA (N Z d ')
c$ (N Z d )
b c$ (N Z d ')
c$ (N Z d )
−
H&g A⁄
& efA cA (e')
& 6 NZd
N& HgfA
6GN Z d
H&'g A⁄
Since _C (a2) = _C i
=
b c$ (N Z d ')
c$ (N Z d )
+
G
gfA
# G-
e
. On further simplification we get
j = 1 and M = [ 5 − a 5 = M = [ 5 similarly,
, after simplification
At the inner surface of the tube, i.e. when y=1, equation (33) and (34) become respectively,
NH&
NH& gfA
=
k5 l C ([) +
5 G
5 G # G-
Where, l C ([) =
And
=
5cA (N Z d )
d N Z c$ (N Z d )
gfA
= k5 +
[
# G-
+
=
#$ "
#
%#
%&
−
)
#
[
'
+
&
]'( +
* "
&
+
+
&
]
(34)
(35)
(36)
[
+
The equations of motion of the tube are: "
(33)
)
#
"
* +
&
]
(37)
(38)
Together with the boundary conditions for the motion of the liquid,
+
=
12=1
112
5. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
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=
12=1
If it is now assumed that from (12) and (13)
8 = m exp [M (1 − q⁄k )] and P = r exp [M (1 − q⁄k )], where m and r are arbitrary constants, the boundary
conditions for
and , using the boundary conditions for the motion of the liquid we have:
+
= m inexp[M (1 − q⁄k )] =
(39)
But
= m exp [M (1 − q⁄k )] from (39)
Therefore = m M
(40)
From (36), we can write (37) as
NH&
gfA
M m =
[k5 l C ([) +
]
(41)
"
5 G
# G-
Also = r inexp[M (1 − q⁄k )] =
But from (12),
= exp [M (1 − q⁄k )], this implies
"
M r =
= k5 +
gfA
# G-
Therefore M r = k5 +
]
gfA
# G-
]
(43)
From (37) and (40), the equations of the tube become
f
) c
NHu
v
m 5 = A − [ i− Aj + A
−r
(42)
%#
# &
G
&
# "
5
= 5
=
W− M [ k5 l C ([)
%#&
5
+
H & g
G # -
X + .−
)
#
H uA
G
+ i−
c
&
NHvA
G
(44)
j0,
(45)
Equations (41), (43), (44) and (45) are four homogeneous equations in the arbitrary constants w, ^ ,k5 , m and r .
Eliminating them will give a frequency equation, which will determine the wave velocity c, in terms of the
elastic properties of the tube and the non-dimensional parameter [. The result of the elimination is:
x
kY 5
1
M Ql C
2k
M Qx
2k 5 Y 5
−[
P
5
1
ℎ
−M
5
0
Q5 x + 5 Px 5 Q
]
2k = Y 5 ℎQ
−M
0
N# &H-
−
A$
5%#
−
0
)
#&
NHc{
#&G
NHc{
5
(1 −
#&G
)
#G
= 0 (46)
)
H &
Operating on the rows and columns of equation (46) and neglecting
and continuing operating on rows and
G d
columns until we get
x
1
0
1
g
A$
−1
0 |
5
5
|
}
*}
=0
(47)
0
|e
|
e
e
*}
}
1−
0
− l CY5
where a =
%#
# &
5
,
!L=
e)
# G
Equation (47) can be written as
x
1
x
lC
| 5
0
| Y
e
0
−2
−L
−~L
e
1
0
|
−~L | = 0
(a − L)
0
5
Whose solution is
g
[x(1 − ~ 5 )(1 − l C )]L 5 − .2Y 5 + ax(1 − l C ) + l C Y 5 i − ~ − ~xj0 L + 2aY 5 + l C Y ; = 0
5
This reduces to
A$ - e
113
(48)
(49)
6. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
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(1 − ~ 5 )(1 − l C )L 5 − .2 + a(1 − l C ) + l C i − 2~j0 L + 2a + l C = 0
5
I.e. when x = 1 and Y 5 = 1 in (48), i.e., when = 0, since
x=(
< V
)(#3 NH<V)
and Y 5 =
G Z "A (
&
V
)(#3 NH<V)
−
(
(50)
& V
)N Z "A
Using Womersley [10], the roots of equation (49) are given by
(1 − ~ 5 )L = • ± [• 5 − (1 − ~ 5 )•] R5
where • =
‚ A
W ƒ <„ *X
(
A$ )
+ +
e
5
*W
When Y 5 = 1 and x = 1, i.e.
A$ ‚ i < A jX AT
ƒ
„
(
A$ )
(51)
A$ - U
= 0 equation (51) reduces to • =
[10] solution for the case without suspension and the same as [6].
4. Conclusion
In terms of the notation used in the case of the solid case,
…A$
A
. < *0
„
(
A$ )
(52)
+ + ~ − , which is Womersley’s
e
5
;
= exp(−M†) /]9 C so that the quantities required
to compute the roots of equation (49) are already available. When ˆ C ([) is complex, L is always complex and
the motion is either damped or unstable.
If we write ((1 − ~ 5 ) LR2) R5 = ‰ − MŠ and denote by wC the velocity for the perfect fluid, then if w is the
w
wave-velocity, CRw = ‰ and over a distance of one wavelength the amplitude will be reduced to the ratio
exp (−2‹ ŠR‰ ) [6] [10]. Since Womersley’s case without suspension is considered as the first best description
of blood flow in arteries, we have now further proved that our model is in line with this best description of blood
flow and the propagation of waves in the arterial system.
Acknowledgement
We wish to acknowledge the contribution of Dr. L. M. Srivastava towards his guide in the M.Sc thesis (1986),
“Flow of Particle-Fluid Suspension in Elastic Tube”, at the Department of Mathematics, A.B.U, Zaria, Nigeria.
This contribution is one of the two major sources, which made this publication possible.
Also, to Associate Professor M. E. Nja of the Mathematics Department, Federal University Lafia,
Nigeria. For his great motivation and encouragement for this research. We thank you all.
References
1. Atabek, H.B. (1996). Wave propagation through a viscous incompressible fluid contained in an initially
stressed elastic tube. Biophysical Journal volume 6, pp 481-501
2. Cox, R.H. (1968) “Wave propagation through a Newtonian fluid contained within a thick-walled,
viscoelastic tube”, Biophysical Journal, volume 8, pp 691-708.
3. Evans, R.L. “Pulsatile flow in vessels whose distinsibility and size vary with site”. Physics in Medicine
and Biology, volume 7, pp 105-115.
4. Iberall, S.A. (1964) “study of the General Dynamics of the physical-chemical systems in Mammals”,
NASA CR-120
5. Lambert, J. W. (1958). “On the non-linearities of fluid flow in nonrigid tubes”, J. Franklin inst. 266, 83.
6. Ngbo, P.H, (1986), “Flow of Particle-Fluid Suspension in Elastic Tube”, M.Sc Thesis (unpublished),
Department of Mathematics, A.B.U, Zaria, Nigeria.
7. Oslen, J.H. et al (1967) “Large- amplitude unsteady flow in liquid filled elastic tubes”, J. Fluid mech,
vol 33, pp 513.
8. Pennisi, L.L. (1976) “Elements of complex variables”, Holt, Rinehart and Winston, pp 17-23.
9. Tam, C.K.W. (1969) “The drag on a cloud of spherical particles in low Reynolds number flow”, J. Fluid
Mech, volume 38, pp 537-546.
10. Womersley, J.R. (1955) Oscillatory motion of a viscous liquid in a thin-walled elastic tube 1. The linear
approximation for long waves, philosophical mag, volume 46, pp 199-215.
114
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