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.
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.
The Interaction Forces between Two Identical Cylinders Spinning around their ...iosrjce
This paper derives the equations that describe the interaction forcesbetween two identical cylinders
spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, in-vortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing this
problem,and the system boundary conditions. It then determines the pressure field from the velocity field using
Bernoulli equation. Finally, the paper integrates the pressure around either cylinder-surface to find the force
acting on its axis.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such identical cylinders repel or attract each other in
inverse relation with separation between their axes, according to similar or opposite direction of rotation,
respectively.
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
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.
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.
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.
The Interaction Forces between Two Identical Cylinders Spinning around their ...iosrjce
This paper derives the equations that describe the interaction forcesbetween two identical cylinders
spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, in-vortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing this
problem,and the system boundary conditions. It then determines the pressure field from the velocity field using
Bernoulli equation. Finally, the paper integrates the pressure around either cylinder-surface to find the force
acting on its axis.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such identical cylinders repel or attract each other in
inverse relation with separation between their axes, according to similar or opposite direction of rotation,
respectively.
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
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.
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.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
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.
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
Abstract— A scientific investigation is created to examine the nonlinear unfaltering blended convection limit
layer stream and warmth exchange of an incompressible digression hyperbolicnon-Newtonian liquid from a
non-isothermal wedge in the nearness of attractive field. The changed preservation conditions are understood
numerically subject to physically fitting limit conditions utilizing a second-arrange precise verifiable limited
distinction Keller Box method. The numerical code is accepted with past studies. The impact of various rising
non-dimensional parameters, to be specific Weissenberg number (We), power law record (n), blended
convection parameter, weight angle parameter (m), Prandtl number (Pr), Biot number, attractive parameter
(M)and dimensionless extraneous direction on speed and temperature development in the limit layer
administration are inspected in subtle element. Moreover, the impacts of these parameters on surface warmth
exchange rate and nearby skin erosion are additionally examined. Approval with prior Newtonian studies is
introduced and amazing relationship accomplished. It is found that speed is lessened with expanding We,
while, temperature is increased. Expanding n improves speed yet diminishes temperature, a comparable
pattern was seen. An expanding M is found to decline speed however temperature increments.
Keywords— Magnetic parameter, Mixed Convection parameter, Non-Newtonian digression hyperbolic liquid,
power law index, Weissenberg number, Weight inclination parameter.
Boundry Layer Flow and Heat Transfer along an Infinite Porous Hot Horizontal ...IJERA Editor
An analysis is made for the two-dimensional laminar boundary layer flow of a viscous, incompressible
fluid,along an infinite porous hot horizontal continuous moving plate. The governing system of partial
differential equations was transformed into ordinary differential equations before being solve by Natural
transformation. The effects of various physical parameters, such as Eckert number Ec , Prandtlnumber Pr ,
plate velocity α and heat source/sink parameter Sare presented and discussed. It is found that the plate velocity
significantly effects the flow and heat transfer.
The Interaction Forces between Two non-Identical Cylinders Spinning around th...iosrjce
This paper derives the equations that describe the interaction forcesbetween two non-identical
cylinders spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, invortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing
this problem,and the system boundary conditions. It then determines the pressure field from the velocity field
using Bernoulli equation. Finally, the paper integrates the pressure around both cylinder-surface to find the
force acting on their axes.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such cylinders repel or attract each other in inverse
relation with separation between their axes, according to similar or opposite direction of rotation, respectively.
Methods to determine pressure drop in an evaporator or a condenserTony Yen
This articles aims to explain how one can relatively easily calculate the pressure drop within a condenser or an evaporator, where two-phase flow occurs and the Navier-Stokes equation becomes very tedious.
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.
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.
Heat Transfer in the flow of a Non-Newtonian second-order fluid over an enclo...IJMERJOURNAL
ABSTRACT : The problem of the heat transfer in the flow of an incompressible non-Newtonian second-order fluid over an enclosed torsionally oscillating discs in the presence of the magnetic field has been discussed. The obtained differential equations are highly non-linear and contain upto fifth order derivatives of the flow and energy functions. Hence exact or numerical solutions of the differential equations are not possible subject to the given natural boundary conditions; therefore the regular perturbation technique is applied. The flow functions 퐻, 퐺, 퐿 and 푀 are expanded in the powers of the amplitude (taken small) of the oscillations. The behaviour of the temperature distribution at different values of Reynolds number, phase difference, magnetic field and second-order parameters has been studied and shown graphically. The results obtained are compared with those for the infinite torsionally oscillating discs by taking the Reynolds number of out-flow 푅푚 and circulatory flow 푅퐿 equals to zero. Nusselt number at oscillating and stator disc has also been calculated and its behaviour is represented graphically.
Couette type mhd flow with suction and injection under constant pressure grad...eSAT Journals
Abstract The unsteady magnetohydrodynamic Couette-type flow of an electrically conducting, viscous and incompressible fluid bounded by two parallel non- conducting porous plates under the influence of a constant pressure gradient and a transversely applied uniform magnetic field is studied with heat transfer. A uniform suction on the upper plate and an injection on the lower plate are applied perpendicularly to the plates keeping the rates of suction and injection the same. The two plates are maintained at different but constant temperatures. The governing nonlinear partial differential equations are solved by both analytical as well as numerical methods. An exact solution for the velocity of the fluid has been obtained by Laplace transform method. The Crank-Nicholson implicit method is used to obtain the unsteady fluid velocity profile. The transient part of the fluid velocity tends to zero as the time t tends to infinity. The energy equation is solved by the finite difference method .The effect of the magnetic field coupled with suction and injection on the velocity and temperature distributions is examined graphically and discussed in the present work. Keywords: Magnetohydrodynamics, Transverse magnetic field, Suction and injection, Constant pressure gradient, Transient state flow, Crank-Nicholson implicit method, Finite difference method.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
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.
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
Abstract— A scientific investigation is created to examine the nonlinear unfaltering blended convection limit
layer stream and warmth exchange of an incompressible digression hyperbolicnon-Newtonian liquid from a
non-isothermal wedge in the nearness of attractive field. The changed preservation conditions are understood
numerically subject to physically fitting limit conditions utilizing a second-arrange precise verifiable limited
distinction Keller Box method. The numerical code is accepted with past studies. The impact of various rising
non-dimensional parameters, to be specific Weissenberg number (We), power law record (n), blended
convection parameter, weight angle parameter (m), Prandtl number (Pr), Biot number, attractive parameter
(M)and dimensionless extraneous direction on speed and temperature development in the limit layer
administration are inspected in subtle element. Moreover, the impacts of these parameters on surface warmth
exchange rate and nearby skin erosion are additionally examined. Approval with prior Newtonian studies is
introduced and amazing relationship accomplished. It is found that speed is lessened with expanding We,
while, temperature is increased. Expanding n improves speed yet diminishes temperature, a comparable
pattern was seen. An expanding M is found to decline speed however temperature increments.
Keywords— Magnetic parameter, Mixed Convection parameter, Non-Newtonian digression hyperbolic liquid,
power law index, Weissenberg number, Weight inclination parameter.
Boundry Layer Flow and Heat Transfer along an Infinite Porous Hot Horizontal ...IJERA Editor
An analysis is made for the two-dimensional laminar boundary layer flow of a viscous, incompressible
fluid,along an infinite porous hot horizontal continuous moving plate. The governing system of partial
differential equations was transformed into ordinary differential equations before being solve by Natural
transformation. The effects of various physical parameters, such as Eckert number Ec , Prandtlnumber Pr ,
plate velocity α and heat source/sink parameter Sare presented and discussed. It is found that the plate velocity
significantly effects the flow and heat transfer.
The Interaction Forces between Two non-Identical Cylinders Spinning around th...iosrjce
This paper derives the equations that describe the interaction forcesbetween two non-identical
cylinders spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, invortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing
this problem,and the system boundary conditions. It then determines the pressure field from the velocity field
using Bernoulli equation. Finally, the paper integrates the pressure around both cylinder-surface to find the
force acting on their axes.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such cylinders repel or attract each other in inverse
relation with separation between their axes, according to similar or opposite direction of rotation, respectively.
Methods to determine pressure drop in an evaporator or a condenserTony Yen
This articles aims to explain how one can relatively easily calculate the pressure drop within a condenser or an evaporator, where two-phase flow occurs and the Navier-Stokes equation becomes very tedious.
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.
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.
Heat Transfer in the flow of a Non-Newtonian second-order fluid over an enclo...IJMERJOURNAL
ABSTRACT : The problem of the heat transfer in the flow of an incompressible non-Newtonian second-order fluid over an enclosed torsionally oscillating discs in the presence of the magnetic field has been discussed. The obtained differential equations are highly non-linear and contain upto fifth order derivatives of the flow and energy functions. Hence exact or numerical solutions of the differential equations are not possible subject to the given natural boundary conditions; therefore the regular perturbation technique is applied. The flow functions 퐻, 퐺, 퐿 and 푀 are expanded in the powers of the amplitude (taken small) of the oscillations. The behaviour of the temperature distribution at different values of Reynolds number, phase difference, magnetic field and second-order parameters has been studied and shown graphically. The results obtained are compared with those for the infinite torsionally oscillating discs by taking the Reynolds number of out-flow 푅푚 and circulatory flow 푅퐿 equals to zero. Nusselt number at oscillating and stator disc has also been calculated and its behaviour is represented graphically.
Couette type mhd flow with suction and injection under constant pressure grad...eSAT Journals
Abstract The unsteady magnetohydrodynamic Couette-type flow of an electrically conducting, viscous and incompressible fluid bounded by two parallel non- conducting porous plates under the influence of a constant pressure gradient and a transversely applied uniform magnetic field is studied with heat transfer. A uniform suction on the upper plate and an injection on the lower plate are applied perpendicularly to the plates keeping the rates of suction and injection the same. The two plates are maintained at different but constant temperatures. The governing nonlinear partial differential equations are solved by both analytical as well as numerical methods. An exact solution for the velocity of the fluid has been obtained by Laplace transform method. The Crank-Nicholson implicit method is used to obtain the unsteady fluid velocity profile. The transient part of the fluid velocity tends to zero as the time t tends to infinity. The energy equation is solved by the finite difference method .The effect of the magnetic field coupled with suction and injection on the velocity and temperature distributions is examined graphically and discussed in the present work. Keywords: Magnetohydrodynamics, Transverse magnetic field, Suction and injection, Constant pressure gradient, Transient state flow, Crank-Nicholson implicit method, Finite difference method.
Effects on Study MHD Free Convection Flow Past a Vertical Porous Plate with H...IJMTST Journal
This paper deals with the combined soret effect of thermal radiation and heat generation on the MHD free
convection heat and mass transfer flow of a viscous incompressible fluid past a continuously moving infinite
plate. Closed form of solution for the velocity, temperature and concentration field are obtained and
discussed graphically for various values of the physical parameters present. In addition, expressions for the
skin friction and Sherwood number is also derived and finally discussed with the graphs.
The approximate bound state of the nonrelativistic Schrӧdinger equation was
obtained with the modified trigonometric scarf type potential in the framework of
asymptotic iteration method for any arbitrary angular momentum quantum number l
using a suitable approximate scheme to the centrifugal term. The effect of the screening
parameter and potential depth on the eigenvalue was studied numerically. Finally, the
scattering phase shift of the nonrelativistic Schrӧdinger equation with the potential
under consideration was calculated.
نموذج الدراسة :
Effect of convective conditions in a radiative peristaltic flow of pseudoplastic nanofluid through a porous medium in a tapered an inclined asymmetric channel
تأثير الشروط الحدودية على الجريان الشعاعي للمائع النانوي الكاذب خلال وسط مسامي في قناة مستدقة مائلة غير متماثلة
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.
GDQ SIMULATION FOR FLOW AND HEAT TRANSFER OF A NANOFLUID OVER A NONLINEARLY S...AEIJjournal2
This paper presents the generalized differential quadrature (GDQ) simulation for analysis of a nanofluid
over a nonlinearly stretching sheet. The obtained governing equations of flow and heat transfer are
discretized by GDQ method and then are solved by Newton-Raphson method. The effects of stretching
parameter, Brownian motion number (Nb), Thermophoresis number (Nt) and Lewis number (Le), on the
concentration distribution and temperature distribution are evaluated. The obtained results exhibit that
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Cosmetic shop management system project report.pdf
Influence of MHD on Unsteady Helical Flows of Generalized Oldoyd-B Fluid between Two Infinite Coaxial Circular Cylinders
1. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 58 | P a g e
Influence of MHD on Unsteady Helical Flows of Generalized
Oldoyd-B Fluid between Two Infinite Coaxial Circular
Cylinders
Sundos Bader ,Ahmed M, Abdulhadi
Department of Mathematic, College of Science ,University of Baghdad ,,Baghdad,, Iraq
Email:badersundos@gmail.com,ahm6161@yahoo.com
ABSTRACT
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.
I. Introduction
Recently , a considerable attention has been
devoted to the problem of how to predict the
behavior of non-Newtonian fluid .The main reason
for this is probably that fluids(such a molten
plastics ,pulps ,slurries ,elmulsions, ect..) which do
not obey the assumption of Newtonian fluid that
the stress tensor is directing proportional to the
deformation tensor ,are found in various
engineering applications .An important class of
non-Newtonian fluids is in the viscoelastic
Oldroyd-B fluid has been intensively studied , and
it has been widely applied to flow problems of
small relaxation and retardation times.
Rheological constitutive equations with fractional
derivatives have been proved to be a valuable tool
to handle viscoelastic properties . The fractional
derivative models of the viscoelastic fluids are
derived from the classical equations which are
modified by replacing the time derivative of an
integer order by precise non-integer order integrals
or derivatives.
In recent years , the Oldroyd –B has been acquired
a special status amongst the many fluids
of the rate type ,as it includes as special case the
classical Newtonian fluid and Maxwell fluid.As a
result of their wide implications ,a lot of papers
regarding
these fluids have been published in the last time .
The Oldroyd –B fluid model [11], which takes
into account elastic and memory effects exhibited
by most polymeric and biological liquids, has been
used quite widely [4] .Existence ,uniqueness and
stability results for some shearing motions of such
a fluid have been obtained in[13] . the exact
solution for the flow of an Oldroyd –B fluid was
established by Waters and Kings [14], Rajagopal
and Bhatnager [12],Fetecau [3], and Fetecau [2],
other analytical results were given by Georgious
[8] for small one- dimensional perturbations and
for the limiting case of zero Reynold number
unsteady( unidirectional and rotating )transient
flows of an Oldroyd –B fluid in an annular
areobtained by Tong[7] .thegeneral case of helical
flow of Oldroyd –B fluid due to combine action of
rotating cylinders(with constant angular velocities )
and a constant axial pressure gradient has consider
byWood [16].The velocity fields and the associated
tangential stresses corresponding to helical flows of
Oldroyd –B fluids using forms of series in term of
Bessel functions are given by Fetecau et al [1].
Recently , the velocity field ,shear stress and vortex
sheet of a generalized second –order fluid with
fractional , fractional derivative using to the
constitutive relationship models of Maxwell
viscoelastic fluid and second order ,and some
unsteady flows of a viscoelastic fluid and of second
order fluids between two parallel platesare
examined by Mingyo and wenchang
[17].Unidirectionalflows of a viscoelastics fluid
with the fractional Maxwell model helical flows of
a generalized Oldroyd-B fluid with fractional
calculus between two infinite coaxial circular
cylinders are investigated by Dengke [6].
In this paper ,we study the effect of MHD on the
helical flows of a generalized Oldroyd-B fluid with
fractional calculus between two infinite coaxial
circular cylinders. The velocity fields and the
resulting shear stresses are determined by means of
Laplace and finite Hankel transform and are
presented under integral and series forms in the
Mittag –leffler function.
RESEARCH ARTICLE OPEN ACCESS
E
2. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 59 | P a g e
II. Basic governing equations of Helcial flow
between concentric cylinders1
We consider here an unsteady helical flow
between two infinite coaxial cylinders located
at 𝑟 = 𝑅1 and 𝑟 = 𝑅2 (𝑅1 < 𝑅2) in the cylindrical
coordinates (𝑟 , 𝜃, 𝑧) , the helcial velocity is given
by
𝑽 = 𝑟 𝑣 𝑟, 𝑡 𝑒 𝜃 + 𝑤 𝑟, 𝑡 𝑒𝑧 (1)
is called helical , because its streamlines are helical
and𝑒 𝜃 and 𝑒𝑧 are the unit vectors in the 𝜃 and
𝑧 − directions , respectively . Since the velocity
field is independent of 𝜃 and 𝑧 and the constraint
of incompressibility is automatically satisfied .
The constitutive equation of generalized Oldroyd-
B (G Oldroyd-B) fluid has the form [1]
𝑺 + 𝜆1
𝛼
𝛿 𝛼
𝑺
𝛿𝑡 𝛼
= 𝜇 1 + 𝜆2
𝛽 𝛿 𝛽
𝛿𝑡 𝛽
𝑨1 (2)
Where 𝑺 is the extra stress tensor , 𝜇 is the
dynamic viscosity, 𝜆1and 𝜆2 are material time
constants referred to, the characteristic relaxation
and characteristic retardation times , respectively .
it is assumed that 𝜆1 ≥ 𝜆2 ≥ 0 .𝑨1 = 𝐿 + 𝐿𝑇
is the
first Rivlin –Erickscn tensor with 𝐿 the velocity
gradient , 𝛼 and 𝛽are fractional calculus parameters
such that 0 ≤ 𝛼 ≤ 𝛽 ≤ 1 and the fractional
operator
𝛿 𝛼 𝑺
𝛿 𝑡 𝛼 on any tensor 𝑺 is defined by
𝛿 𝛼
𝑺
𝛿𝑡 𝛼
= 𝐷𝑡
𝛼
𝑺 + 𝑽. ∇𝑺 − 𝐿𝑺 − 𝑺 𝐿𝑇
(3)
The operator 𝐷𝑡
𝛼
based on Caputo's fractional
differential of order 𝛼 is defined as
𝐷𝑡
𝛼
𝑦 𝑡 =
1
Γ 𝑛 − 𝑎
𝑓 𝑛
𝜏
𝑡 − 𝜏 𝛼−𝑛+1
𝑑𝜏 ,
𝑡
0
𝑛 − 1 < 𝛼 ≤ 𝑛 (4)
where Γ . denotes the Gamma function. This
model can be reduced to the ordinary Maxwell
fluid model when α = 1 ,to a generalized
theMaxwell fluid model when β = 1 and to an
ordinary Oldroyd-B model 𝛽 = α = 1.
𝑣 𝑟, 0 = 𝑤 𝑟, 0 = 0 and 𝑺 𝑟, 0 = 0 (5)
Here we assume that the generalized Oldroyd-B
fluid is incompressible then
∇. 𝑽 = 0 (6)
Substituting Eq.(1) into Eq.(2) and Eq.(3) and
taking into account (5), we find that 𝑺 𝑟𝑟 = 0 ,
𝝉 𝑟, 𝑡 = 𝑺 𝑟𝜃 𝑟, 𝑡 is the shear stress and
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝑺𝑟𝜃 = 𝜇 1 + 𝜆2
𝛽
𝐷𝑡
𝛽
𝑟
𝜕v
𝜕𝑟
(7)
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝑺 𝑟𝑧 = 𝜇 1 + 𝜆2
𝛽
𝐷𝑡
𝛽 𝜕𝑤
𝜕𝑟
(8)
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝑺 𝜃𝑧 = 𝜆1
𝛼
𝑟
𝜕v
𝜕𝑟
𝑺 𝑟𝑧 +
𝜕𝑤
𝜕𝑟
𝑺 𝑟𝜃
− 2𝜇 𝜆2
𝛽 𝜕𝑤
𝜕𝑟
𝑟
𝜕v
𝜕𝑟
(9)
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝑺 𝜃𝜃 = 2𝜆1
𝛼
𝑟
𝜕v
𝜕𝑟
𝑺 𝑟𝜃
− 2𝜇 𝜆2
𝛽
𝑟
𝜕v
𝜕𝑟
2
(10)
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝑺 𝑧𝑧 = 2𝜆1
𝛼
𝜕𝑤
𝜕𝑟
𝑺 𝑟𝑧 − 2𝜇 𝜆2
𝛽 𝜕𝑤
𝜕𝑟
2
(11)
III. Momentum and continuity equation
We will write the formula of the momentum
equation which governing the
magnetohydrodynamic as fallows :
𝜌
𝑑𝑽
𝑑𝑡
= −∇𝑝 + ∇. 𝑺 + 𝑱 × 𝑩 (12)
Where𝜌 is the density of the fluid , 𝑱 is the current
density and 𝑩 = 0, 𝛽0, 0 is the total magnetic field
.
In the absence of body forces and a pressure
gradient, the equation of motion reduce to the
relevant equations
𝜌
𝜕𝑤
𝜕𝑡
=
𝜕
𝜕𝑟
+
1
𝑟
𝑺 𝑟𝑧 − 𝜍𝛽0
2
𝑤 (13)
𝜌𝑟
𝜕v
𝜕𝑡
=
𝜕
𝜕𝑟
+
1
𝑟
𝑺 𝑟𝜃 − 𝜍𝛽0
2
𝑟v (14)
𝜕𝑝
𝜕𝑟
+
𝑺 𝜃𝜃
𝑟
= 𝜌𝑟v2
(15)
Where 𝜍 is the electric conductivity .
Eliminating 𝑺 𝑟𝑧 and 𝑺 𝑟𝜃 among Eqs(7),(8),(13)
and(14) ,we attain to the governing equations are
𝜌
𝜕𝑤
𝜕𝑡
= 𝜇
1 + 𝜆2
𝛽
𝐷𝑡
𝛽
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝜕2
𝑤
𝜕𝑟2
+
1
𝑟
𝜕𝑤
𝜕𝑟
− 𝜍𝛽0
2
𝑤 (16)
𝜌 𝑟
𝜕v
𝜕𝑡
= 𝜇
1 + 𝜆2
𝛽
𝐷𝑡
𝛽
1 + 𝜆1
𝛼
𝐷𝑡
𝛼 𝑟
𝜕2
v
𝜕𝑟2
+
1
𝑟
𝑟
𝜕v
𝜕𝑡
− 𝜍𝛽0
2
𝑟v (17)
Multiply above two equations by 1 + 𝜆1
𝛼
𝐷𝑡
𝛼
,we
get
𝜌 1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝜕𝑤
𝜕𝑡
= 𝜇 1 + 𝜆2
𝛽
𝐷𝑡
𝛽 𝜕2
𝑤
𝜕𝑟2
+
1
𝑟
𝜕𝑤
𝜕𝑟
− 𝜍𝛽0
2
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝑤 (18)
𝜌 𝑟 1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝜕v
𝜕𝑡
= 𝜇 1 + 𝜆2
𝛽
𝐷𝑡
𝛽
𝑟
𝜕2
v
𝜕𝑟2
+
1
𝑟
𝑟
𝜕v
𝜕𝑡
− 𝜍𝛽0
2
𝑟 1 + 𝜆1
𝛼
𝐷𝑡
𝛼
(19)
Divide Eq(18) by 𝜌 and divide Eq(19) by 𝜌 𝑟 ,we
get
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝜕𝑤
𝜕𝑡
= 𝜐 1 + 𝜆2
𝛽
𝐷𝑡
𝛽 𝜕2
𝑤
𝜕𝑟2
+
1
𝑟
𝜕𝑤
𝜕𝑟
−
𝜍𝛽0
2
𝜌
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝑤 (20)
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
𝜕v
𝜕𝑡
= 𝜐 1 + 𝜆2
𝛽
𝐷𝑡
𝛽 𝜕2
v
𝜕𝑟2
+
1
𝑟
𝜕v
𝜕𝑡
−
𝜍𝛽0
2
𝜌
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
v (21)
Where 𝜐 =
𝜇
𝜌
is the kinematic viscosity of the fluid.
The boundary conditions are expressed by
𝑤 𝑅1, 𝑡 = 𝑈1 , 𝑤 𝑅2, 𝑡 = 𝑈2 , 𝑡 > 0 (22)
And
v 𝑅1, 𝑡 = 𝛺1 , v 𝑅2, 𝑡 = 𝛺2 , 𝑡 > 0 (23)
The initial conditions are expressed by
𝑤 𝑟, 0 = v 𝑟, 0 = 0 (24)
𝜕𝑡 𝑤 𝑟, 0 = 𝜕𝑡v 𝑟, 0 = 0 (25)
3.1.Calculation of the velocity field
Making the change of unknown function
v 𝑟, 𝑡 =
𝑢(𝑟, 𝑡)
𝑟
(26)
substitute the value of velocity v 𝑟, 𝑡 in Eq. (21)
with initial and boundary conditions ,we get
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
1
𝑟
𝜕u
𝜕𝑡
=
𝜐
𝑟
1 + 𝜆2
𝛽
𝐷𝑡
𝛽 𝜕2
u
𝜕𝑟2
+
1
𝑟
𝜕u
𝜕𝑡
−
𝑢
𝑟2
−
𝜍𝛽0
2
𝜌𝑟
1 + 𝜆1
𝛼
𝐷𝑡
𝛼
u (27)
10. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 67 | P a g e
𝑤 𝑟, 𝑡
= 𝑊 𝑟
− 𝜋
𝐽0 𝑠1𝑛 𝑅1 𝜓1 𝑠1𝑛 𝑟 𝑈2 𝐽0 𝑠1𝑛 𝑅1 − 𝑈1 𝐽0 𝑠1𝑛 𝑅2
𝐽0
2
𝑠1𝑛 𝑅1 − 𝐽0
2
(𝑠1𝑛 𝑅2)
∞
𝑛=1
× 𝐺1 𝑠1𝑛 , 𝑡 (125)
where
𝐺1 𝑠1𝑛 , 𝑡 = (−1) 𝑚
∞
𝑚=0
𝑠1𝑛
2
𝜐 𝑐+𝑑
𝑐! 𝑑!
𝑐,𝑑≥0
𝑐+𝑑=𝑚
𝜆2
𝛽 𝑑
𝜆1
𝛼 𝑚+1
∗ 𝑡 𝛼𝑚+(𝛼−𝛿)−1
𝑡𝐸 𝛼,(𝛼−𝛿)
(𝑚)
−𝜆1
−𝛼
𝑡 𝛼
+ 𝜆1
𝛼
𝑡 𝛼+1
𝐸𝛼,(𝛼−𝛿)
(𝑚)
−𝜆1
−𝛼
𝑡 𝛼
Or
𝐺1 𝑠1𝑛, 𝑡 =
−1 𝑚
𝑚 − 1 !
𝑠1𝑛
2
𝜐
𝜆1
𝛼
𝑚∞
𝑚=0
×
𝑚
𝑘
𝜆2
𝛽 𝑘
𝑚
𝑘=0
𝑡(1+𝛼)𝑚−𝛽𝑘
𝐸𝛼,(𝛼+𝑚+1−𝛽𝑘)
(𝑚−1)
−𝜆1
−𝛼
𝑡 𝛼
And
𝑢 𝑟, 𝑡
= 𝑈 𝑟
− 𝜋
𝐽1 𝑠2𝑛 𝑅1 𝜓2 𝑠2𝑛 𝑟 𝑅2 𝛺2 𝐽1(𝑠2𝑛 𝑅1) − 𝑅1 𝛺1 𝐽1(𝑠2𝑛 𝑅2)
𝐽1
2
𝑠2𝑛 𝑅1 − 𝐽1
2
(𝑠2𝑛 𝑅2)
∞
𝑛=1
× 𝐺1 𝑠2𝑛, 𝑡 (126)
Where
𝐺1 𝑠2𝑛, 𝑡 = (−1) 𝑚
∞
𝑚=0
𝑠2𝑛
2
𝜐 𝑐+𝑑
𝑐! 𝑑!
𝑐,𝑑≥0
𝑐+𝑑=𝑚
𝜆2
𝛽
𝑑
𝜆1
𝛼 𝑚+1
∗ 𝑡 𝛼𝑚+(𝛼−𝛿)−1
𝑡𝐸 𝛼,(𝛼−𝛿)
(𝑚)
−𝜆1
−𝛼
𝑡 𝛼
+ 𝜆1
𝛼
𝑡 𝛼+1
𝐸 𝛼,(𝛼−𝛿)
(𝑚)
−𝜆1
−𝛼
𝑡 𝛼
Or
𝐺1 𝑠1𝑛, 𝑡 =
−1 𝑚
𝑚 − 1 !
𝑠2𝑛
2
𝜐
𝜆1
𝛼
𝑚∞
𝑚=0
𝑚
𝑘
𝜆2
𝛽 𝑘
𝑚
𝑘=0
𝑡 1+𝛼 𝑚−𝛽𝑘
𝐸𝛼,(𝛼+𝑚+1−𝛽𝑘)
(𝑚−1)
−𝜆1
−𝛼
𝑡 𝛼
2- Making the limit of Eqs(77) and (78) when
𝛼 ≠ 0 , 𝛽 ≠ 0 and
𝜍𝛽0
2
𝜌
= 𝑀 = 0 ,we can attain
the similar solution shear stress for unsteady
helical flows of a generalized Oldroyd –B fluid , as
obtained in [6] , thus shear stress reduces to
𝑺 𝑟𝑧 = 𝜌𝜋
𝐽0 𝑠1𝑛 𝑅1 𝜓3 𝑠1𝑛 𝑟
𝑠1𝑛 𝐽0
2
𝑠1𝑛 𝑅1 − 𝐽0
2
(𝑠1𝑛 𝑅2)
∞
𝑛=1
𝑈2 𝐽0 𝑠1𝑛 𝑅1
− 𝑈1 𝐽0 𝑠1𝑛 𝑅2
× 𝐺2 𝑠1𝑛, 𝑡 (127)
where
𝐺2 𝑠1𝑛 , 𝑡 = (−1) 𝑚
∞
𝑚=0
𝑠1𝑛
2
𝜐 2+𝑐+𝑑
𝑐! 𝑑!
𝑐,𝑑≥0
𝑐+𝑑=𝑚
𝜆2
𝛽 𝑑
𝜆1
𝛼 𝑚+1
∗ 𝑡 𝛼𝑚+(2𝛼−𝛿)−2
∗ 𝐸𝛼,(𝛼−𝛿)
(𝑚)
−𝜆1
−𝛼
𝑡 𝛼
+ 2𝜆2
𝛽
𝑡 𝛽
𝐸 𝛼,(𝛼−𝛿)
(𝑚)
−𝜆1
−𝛼
𝑡 𝛼
+ 𝜆2
2𝛽
𝑡2𝛽
𝐸 𝛼,(𝛼−𝛿)
(𝑚)
−𝜆1
−𝛼
𝑡 𝛼
𝐸𝛼,𝛼 −𝜆1
𝛼
𝑡 𝛼
Or
𝐺2 𝑠1𝑛 , 𝑡 =
−1 𝑚
𝑚 + 1 !
𝑠1𝑛
2
𝜐
𝜆1
𝛼
𝑚+2∞
𝑚=0
𝑚 + 2
𝑘
𝜆2
𝛽 𝑘
𝑚
𝑘=0
× 𝑡 1+𝛼 𝑚+2 −𝛽𝑘−1
𝐸𝛼,(𝛼+𝑚+2−𝛽𝑘)
(𝑚+1)
−𝜆1
−𝛼
𝑡 𝛼
𝑺 𝑟𝜃 = 𝜌𝜋
𝐽1 𝑠2𝑛 𝑅1 𝑟𝑠2𝑛 𝜓3 𝑠2𝑛 𝑟 − 𝜓2 𝑠2𝑛 𝑟
𝑟𝑠2𝑛
2
𝐽1
2
𝑠2𝑛 𝑅1 − 𝐽1
2
(𝑠2𝑛 𝑅2)
∞
𝑛=1
× 𝑅2 𝛺2 𝐽1(𝑠2𝑛 𝑅1)
− 𝑅1 𝛺1 𝐽1(𝑠2𝑛 𝑅2) 𝐺2 𝑠2𝑛, 𝑡 (128)
VI.Numerical result and conclusion
In this paper we established the effect of MHD
flow of the unsteadyhelical flows of an Oldroyd-B
fluid in concentric cylinders and circular cylinder.
The exact solutionof the unsteadyhelical flows of
an Oldroyd-B fluid in an annular for the velocity
field u , w and the associated shear stresses 𝜏1 , 𝜏2
are obtained by using Hankel transform and
Laplace transform for fractional calculus .
Moreover , some figures are plotted to show the
behavior of various parameters involved in the
expression of velocities w ,u ( Eqs(61 and 62)) ,
shear stresses 𝜏1 , 𝜏2(Eqs(85 and 86)) ,respectively
.
All theresult in this section are plotting graph by
using MATHEMATICA package .
Fig(1) is depicted to show the change of the
velocity w(r,t) with the non-integer fractional
parameter 𝛼 . the velocity is decreasing with
increase of 𝛼. Fig(2) is prepared to show the effect
of the variations of fractional parameter 𝛽 .the
velocity field w is increasing with
increasefractional parameter 𝛽 .Fig(3) represents
the variation of velocity w for different values of
relaxation 𝜆1 . The velocity is decreasing with
increase 𝜆1.Fig(4 and 5) are prepared to show the
effect of the variation of retardation 𝜆2 and
kinematic viscosity v on the velocity field .The
velocity field w is increasing with increase the
parameters 𝜆2 and v .Fig (6) is established value of
magnetic parameter M .the velocity field is
decreases with increasing of M .Fig(7) illustrate the
influence of time t on the velocity field w. the
velocity is increasing with increase the time t.
Fig(8) is provided the graphically illustrations for
effects ofthe non-integer fractional parameter 𝛼 on
the velocity u(r,t) with the non-integer fractional
parameter 𝛼 . the velocity is decreasing with
increase of 𝛼.Fig(9) is prepared to show the effect
of the variations of fractional parameter 𝛽 .the
velocity field w is increasing with
increasefractional parameter 𝛽 .Fig(10) represents
the variation of velocity w for different values of
relaxation 𝜆1 . The velocity is decreasing with
increase 𝜆1.Fig(11 and 12) are prepared to show the
effect of the variation of retardation 𝜆2 and
kinematic viscosity v on the velocity field .The
velocity field w is increasing with increase the
parameters 𝜆2 and v .Fig (13) is established value
of magnetic parameter M .the velocity field is
decreases with increasing of M .Fig(14) illustrate
11. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 68 | P a g e
the influence of time t on the velocity field w. the
velocity is increasing with increase the time t.
Figs(15-20) provide the graphically illustrations
for the effects of (fractional parameters(𝛼 , 𝛽) ,
relaxation 𝜆1 , retardation 𝜆2 , kinematic viscosity v
, magnetic parameter M and time t) on the shear
stress 𝜏1 . The shear stress decreasing with increase
the different parameters.
Figs(21-26) are established to show the behavior
of the parameters (fractional parameters(𝛼 , 𝛽) ,
relaxation 𝜆1 , retardation 𝜆2 , kinematic viscosity v
, magnetic parameter M and time t) on the shear
stress 𝜏2 . The shear stress decreasing with increase
the different parameters.
The velocity w
Fig1:- the velocity w for different value of fractional
parameter {115 , 28 , v=0.165 , M0,=0.6 ,
K13.31114 , t2,R1=1, R2=2, u1=2, u2=2}
Fig2:- the velocity w for different value of fractional
parameter {115 , 28 , v=0.165 , M0.1,=0.4 ,
K13.31114 , t2,R1=1, R2=2, u1=2, u2=2}
Fig3:- the velocity w for different value 1 { 28 , v=0.165 ,
M0.1,=0.4 , K13.31114 , t2,R1=1, R2=2, u1=2, u2=2}
Fig4:- the velocity w for different value 2 { 115 , v=0.165 ,
M0.1, =0.6 , =0.4 , K13.31114 , t2,R1=1, R2=2, u1=2,
u2=2}
Fig5:- the velocity w for different value v {115 , 28 ,
M=0.1 ,=0.6, =0.4, K13.31114 , t2,R1=1, R2=2, u1=2,
u2=2}
1.2 1.4 1.6 1.8 2.0
r
6
4
2
2
1.2 1.4 1.6 1.8 2.0
r
4
3
2
1
1
2
1.2 1.4 1.6 1.8 2.0
r
4
3
2
1
1
2
1.2 1.4 1.6 1.8 2.0
r
3
2
1
1
2
1.2 1.4 1.6 1.8 2.0
r
3
2
1
1
2
……. =0.2
____ =0.4
------ =0.6
……. =0.2
____ 0.8
------ 1.4
……. 15
____ 1
------125
……. 28
____ 2
------224
……..v=0.165
____v=0.265
_ _ _v=0.365
12. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 69 | P a g e
Fig6:- the velocity w for different value M { 115 , 28,
v=0.165,=0.4 , K13.31114 , t2,R1=1, R2=2, u1=2, u2=2}
Fig7:- the velocity w for different value t { 115 ,
28,M0.1, v=0.165,=0.4 , K13.31114 , R1=1, R2=2,
u1=2, u2=2}
The velocity of u
Fig8:- the velocity u for different value of fractional
parameter {115 , 28 , v=0.165 , M0.1,=0.6 ,
K13.31114 , t2,R1=1, R2=2, u1=2, u2=2}
Fig9:- the velocity u for different value {115 , 28 ,
v=0.165 , M0.1, =0.4, K13.31114 , t2,R1=1, R2=2, u1=2,
u2=2}
Fig10:- the velocity u for different value 1 { 28 , v=0.165 ,
M0.1, =0.4,=0.6 , K13.31114 , t2,R1=1, R2=2, u1=2,
u2=2}
Fig11:- the velocity u for different value 2 {115, v=0.165 ,
M0.1, =0.6 ,=0.4, K13.31114 , t2,R1=1, R2=2, u1=2,
u2=2}
1.2 1.4 1.6 1.8 2.0
r
3
2
1
1
2
1.2 1.4 1.6 1.8 2.0
r
6
4
2
2
1.2 1.4 1.6 1.8 2.0
r
8
6
4
2
2
1.2 1.4 1.6 1.8 2.0
r
6
4
2
2
1.2 1.4 1.6 1.8 2.0
r
6
4
2
2
1.2 1.4 1.6 1.8 2.0
r
6
4
2
2
……. M0.1
____ M0.2
------ M0.3
……. t1
____ t2
------ t3
…….=0.2
____ =0.4
------ =0.6
……. =0.2
____ 0.8
------ 1.4
……. 28
____ 2
------224
……. 1
____ 1
------125
13. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 70 | P a g e
Fig12:- the velocity u for different value v {115 , 28 ,
M=0.1 ,=0.6, =0.4, K13.31114 , t1,R1=1, R2=2, u1=2,
u2=2}
Fig13:- the velocity u for different value M {115 , 28 ,
v=0.165 , M0.1,=0.6 , =0.4, K13.31114 , t2,R1=1,
R2=2, u1=2, u2=2}
Fig14:- the velocity u for different value t {115 , 28 ,
v=0.165 , M0.1, =0.6 ,=0.4, K13.31114 , t2,R1=1, R2=2,
u1=2, u2=2}
The shree stress
Fig .15. the shear stress for different value of fractional
parameter {115 , 28 , v=0.165 , M0.1,=0.6 ,
K13.31114 , t2,R1=1, R2=2, u1=1, u2=1,r2, 0.1}
Fig .16. the shear stress for different value of fractional
parameter {115 , 28 , v=0.165 , M0.1,=0.6 ,
K13.31114 , t2,R1=1, R2=2, u1=1, u2=1,r2, 0.1}
Fig .17. the shear stress for different value of 128 ,
v=0.165 , M0.1,=0.6 , =0.6 , K13.31114 , t2,R1=1,
R2=2, u1=1, u2=1,r2, 0.1}
1.2 1.4 1.6 1.8 2.0
r
6
4
2
2
1.2 1.4 1.6 1.8 2.0
r
6
4
2
2
1.2 1.4 1.6 1.8 2.0
r
12
10
8
6
4
2
2
8 10 12 14 16 18 20
t
50
40
30
20
10
10
8 10 12 14 16
t
20
15
10
5
8 10 12 14 16
t
8
6
4
2
……..v=0.165
____v=0.265
_ _ _v=0.365
……. M0.1
____ M0.2
------ M0.3
……. t1
____ t2
------ t3
…….=0.2
____ =0.4
------ =0.6
……. =0.2
____ 0.8
------ 1.4
……. 1
____ 1
------125
14. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 71 | P a g e
Fig .18. the shear stress for different value of 2115 ,
v=0.165 , M0.1,=0.6 , =0.6 , K13.31114 , t2,R1=1,
R2=2, u1=1, u2=1,r2, 0.1}
Fig19:- the the shear stress for different value v {115 ,
28 , M=0.1 ,=0.6, =0.4, K13.31114 , t1,R1=1, R2=2,
u1=2, u2=2,r2,0.1}
Fig .20. the shear stress for different value of M115
28, v=0.165 , M0.1,=0.6 , =0.6 , K13.31114 ,
t2,R1=1, R2=2, u1=1, u2=1,r2, 0.1}
the sheer stress 2
Fig .21 the shear stress for different value of fractional
parameter {115 , 28 , v=0.165 , M0.1,=0.6 ,
K13.31114 , t2,R1=1, R2=2, u1=1, u2=1,r2, 0.1}
Fig .22. the shear stress for different value of fractional
parameter {115 , 28 , v=0.165 , M0.1,=0.6 ,
K13.31114 , t2,R1=1, R2=2, u1=1, u2=1,r2, 0.1}
Fig .23. the shear stress for different value of 1 {28 ,
v=0.165 , M0.1 ,=0.6=0.6 , K13.31114 , t2,R1=1,
R2=2, u1=1, u2=1,r2, 0.1}
8 10 12 14
t
12
10
8
6
2
0
8 10 12 14
t
12
10
8
6
2
0
8 10 12 14
t
4.0
3.0
2.5
2.0
1.5
1.0
8 10 12 14 16 18 20
t
40
30
20
10
10
8 10 12 14
t
15
10
5
8 10 12 14 16
t
6
5
4
3
2
1
1
……. 28
____ 2
------224
……..v=0.165
____v=0.265
_ _ _v=0.365
……. M0.1
____ M0.2
------ M0.3
…….=0.2
____ =0.4
------ =0.6
……. 1
____ 1
------125
……. =0.2
____ 0.8
------ 1.4
15. Sundos Bader et.al. . Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.58-82
www.ijera.com 72 | P a g e
Fig .24. the shear stress for different value of 2 {115 ,
v=0.165 , M0.1 ,=0.6=0.6 , K13.31114 , t2,R1=1,
R2=2, u1=1, u2=1,r2, 0.1}
Fig25:- the the shear stress for different value v {115 ,
28 , M=0.1 ,=0.6, =0.4, K13.31114 , t1,R1=1, R2=2,
u1=2, u2=2,r2,0.1}
Fig .26. the shear stress for different value of M {115 ,
28, v=0.165,=0.6=0.6 , K13.31114 , t2,R1=1, R2=2,
u1=1, u2=1,r2, 0.1}
References
[1] C.Fetecau,C.Fetecau,D.Vieru ,On some helical
flows of Oldroyd –B fluids, Acta Mech . 189
(2007) 53-63.
[2] C.Fetecau,C.Fetecau, The first problem of Stokes
for an Oldroyd –B fluid, Int .J.Non –Linear
Mech.38(2003) 1539-1544.
[3] C.Fetecau , The Rayleigh –Stokes problem for an
edge in an Oldroyd –B fluid .C.R.Acad.
Paris Ser.1335(2003) 979-984.
[4] C.Guilope ,J.c ,Saut ,Global existence and one-
dimensional non –linear stability of shearing
motion of viscoelastic fluids of Oldroyd type
,RAIRO Model ,Math .Anal ,Number
24)1990)369-401.
[5] -Debnath,l.and Bhatta,D.Integral transforms and
Their Application Boca Rato ,London , Newyork
, chapman &Hall.CRC;2007.
[6] Dengke Tong , Xianmin Zhang ,Xinhong Zhang
,Unsteady helical flows of a generalized Oldroyd-
B fluid ,J.Non-Newtonian fluid Mech. 156
(2009) 75-83 .
[7] D.K.Tong,R.H.Wang ,Exact solution for the flow
of non-Newtonian fluid with fractional derivative
in an annular pipe , Sci .Chin .Sec G 48 (2005)
485-495
[8] G.C.Georgiou ,On the stability of the shear flow
of a viscoelasic fluid with slip along the fixed
wall ,Rheol .Acta 35 (1996) 39-47.
[9] H.Stechfest ,Algorithm 368 numerical inversion
of Laplace transform ,Commum ,ACM 13(1)
(1970) 47-49.
[10] H.Stechfest .Remark on Alrorithim 368numerical
inversion of Laplace transform ,Commum ,ACM
13(10) (1970) 624-625.
[11] J.G. Oldroyd , on the formulation of rheological
equation of state ,Proc,R.Soc . Lond .A
200(1950)523-541.
[12] K.R. Rajagopal ,P.K.Bahtngar .Exact solution for
some simple flows of an Oldroyd-B
fluid. Acta Mech 113(1995)233-239.
[13] M.A .Fontelos ,A.Friedman .Stationary non-
Newtonian fluid flows in channel-like and pipe –
like domains. Arch .Rational Mech .Anal
151(2001) 1-43.
[14] N.D.Waters M.J. King ,Unsteady flow of an
elastico –viscous liquid ,Rheol .Acta 93(1970)
345-355.
[15] T.Wenchang ,P.Wenxian .X. Mingyu,Anote on
unsteady flows of a viscoelastic fluids with the
fractional Maxwell model between two parallel
plates ,Int.J.Non –Linear Mech.38(5)(2003) 645-
650.
[16] W.P.Wood , Transient viscoelastic helical flows
in pipe of circular and annular cross –section ,J.
Non -Newtonian fluid Mech.100(2001) 115-126.
[17] X.Mingyu,T.Wenchang ,Theoretical analysis of
the velocity field , stress field and vortex sheet of
generalized second order fluid with fractional
anomalous diffusion ,Sci,Chin (Ser.A)
44(11)(2001) 1387-1399.
8 10 12 14 16
t
10
8
6
4
2
8 10 12 14 16
t
10
8
6
4
2
8 10 12 14 16
t
3.0
2.5
2.0
1.5
1.0
0.5
……. 28
____ 2
------224
……..v=0.165
____v=0.265
_ _ _v=0.365
……. M0.1
____ M0.2
------ M0.3