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no – slip assumption between the wall and the fluid is no longer valid.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.
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.
nternational Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
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.
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.
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.
nternational Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
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.
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.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, 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 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 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.
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.
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.
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
techDynamic characteristics and stability of cylindrical textured journal bea...ijmech
The dynamic characteristics of cylindrical textured hydrodynamic journal bearing are presented in this paper. The Reynolds equation is solved numerically in a finite difference grid in an iterative scheme satisfying the appropriate boundary conditions. Stiffness and damping coefficients of fluid film and stability parameter are found using the first-order perturbation method for different eccentricity ratios and various texture parameters like texture depth and texture density.. From the present analysis, it has been found that stability is enhanced with increase in texture depth, whereas there is an optimum texture density corresponding to the maximum stability of the bearing
While the computed transported turbulence dissipation rate, , works well as part of a differential
equation-based turbulence model in predicting turbulent flows, it doesn’t seem to work well when used to
determine the Kolmogorov length-scale (ℓ퐾표푙) which, like the other Kolmogorov scales, exists within the viscous
sublayer portion of the inner turbulent boundary layer zone. Using may lead to an increase in ℓ퐾표푙 as roughness
increases, the opposite of what should happen. It is proposed here to replace the computed (and its level at the
1
st point off the wall as dictated by wall functions) with the one resulting from basic law-of-the-wall sublayer
relationships which includes the Prandtl-Schlichting (P-S) roughness effect. This approach enables physically
correct prediction of ℓ퐾표푙, particularly a reliable decrease thereof with increasing roughness level.
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.
Effects Of Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
The present analysis is made to investigate the effects of heat source and thermal diffusion on an unsteady free convection flow along a porous vertical plate in a rotating system. The plate is subjected to constant heat and mass flux also. The problem is solved analytically and expressions for velocity. Energy and temperature profiles, skin friction and Nusselt number are obtained. The effects of different parameter entered in the problem are discussed on the primary and secondary velocities, temperature and concentration distributions, primary and secondary skin frictions and Nusselt number with the help of tables and graphs.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Statistical Optimization of process parameters for SiO2-Nickel nanocomposites...IJERA Editor
The optimum combination of process parameters - temperature, time of reduction under nitrogen atmosphere and amount of NiCl2 was delineated to find the maximum yield of nanocrystallite Ni in the synthesized silica gel matrix. A statistically adequate regression equation, within 95% confidence limit was developed by carrying out a set of experiments within the framework of design of experiment. The regression equation is found to indicate the beneficial role of the temperature and time of reduction.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, 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 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 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.
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.
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.
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
techDynamic characteristics and stability of cylindrical textured journal bea...ijmech
The dynamic characteristics of cylindrical textured hydrodynamic journal bearing are presented in this paper. The Reynolds equation is solved numerically in a finite difference grid in an iterative scheme satisfying the appropriate boundary conditions. Stiffness and damping coefficients of fluid film and stability parameter are found using the first-order perturbation method for different eccentricity ratios and various texture parameters like texture depth and texture density.. From the present analysis, it has been found that stability is enhanced with increase in texture depth, whereas there is an optimum texture density corresponding to the maximum stability of the bearing
While the computed transported turbulence dissipation rate, , works well as part of a differential
equation-based turbulence model in predicting turbulent flows, it doesn’t seem to work well when used to
determine the Kolmogorov length-scale (ℓ퐾표푙) which, like the other Kolmogorov scales, exists within the viscous
sublayer portion of the inner turbulent boundary layer zone. Using may lead to an increase in ℓ퐾표푙 as roughness
increases, the opposite of what should happen. It is proposed here to replace the computed (and its level at the
1
st point off the wall as dictated by wall functions) with the one resulting from basic law-of-the-wall sublayer
relationships which includes the Prandtl-Schlichting (P-S) roughness effect. This approach enables physically
correct prediction of ℓ퐾표푙, particularly a reliable decrease thereof with increasing roughness level.
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.
Effects Of Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
The present analysis is made to investigate the effects of heat source and thermal diffusion on an unsteady free convection flow along a porous vertical plate in a rotating system. The plate is subjected to constant heat and mass flux also. The problem is solved analytically and expressions for velocity. Energy and temperature profiles, skin friction and Nusselt number are obtained. The effects of different parameter entered in the problem are discussed on the primary and secondary velocities, temperature and concentration distributions, primary and secondary skin frictions and Nusselt number with the help of tables and graphs.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Statistical Optimization of process parameters for SiO2-Nickel nanocomposites...IJERA Editor
The optimum combination of process parameters - temperature, time of reduction under nitrogen atmosphere and amount of NiCl2 was delineated to find the maximum yield of nanocrystallite Ni in the synthesized silica gel matrix. A statistically adequate regression equation, within 95% confidence limit was developed by carrying out a set of experiments within the framework of design of experiment. The regression equation is found to indicate the beneficial role of the temperature and time of reduction.
Intensify Denoisy Image Using Adaptive Multiscale Product ThresholdingIJERA Editor
This Paper presents a wavelet-based multiscale products thresholding scheme for noise suppression of magnetic resonance images. This paper proposed a method based on image de-noising and edge enhancement of noisy multidimensional imaging data sets. Medical images are generally suffered from signal dependent noises i.e. speckle noise and broken edges. Most of the noises signals appear from machine and environment generally not contribute to the tissue differentiation. But, the noise generated due to above mentioned reason causes a grainy appearance on the image, hence image enhancement is required. For the intent of image denoising, Adaptive Multiscale Product Thresholding based on 2-D wavelet transform is used. In this method, contiguous wavelet sub bands are multiplied to improve edge structure while reducing noise. In multiscale products, boundaries can be successfully distinguished from noise. Adaptive threshold is designed and forced on multiscale products as an alternative of wavelet coefficients or recognize important features. For the edge enhancement. Canny Edge Detection Algorithm is used with scale multiplication technique. Simulation results shows that the planned technique better suppress the Poisson noise among several noises i.e. salt & pepper, speckle noise and random noise. The Performance of Image Intesification can be estimate by means of PSNR, MSE.
Evaluation of Antioxidant Phytochemicals in Different Genotypes of PotatoIJERA Editor
Potato (Solanum tuberosum L.) is considered a natural source of phytochemicals that help lower the risk of chronic diseases. Thus, ascorbic acid, total phenolic content and antioxidant activities of Indian potato genotypes were studied. ‘Kufri Pukhraj’ was the best genotype in terms of nutritional components, since it contained the highest amount of ascorbic acid (19.42 mg/100g), and total phenolic content (68.20 mg GAE/100g). Also ‘Kufri Pukhraj’ had the highest DPPH radical scavenging activity of 67.30%. On the other hand, genotype ‘Kufri Chipsona-1’, most commonly used cultivar for processing, displayed the lowest content of total phenolics (31.30 mg GAE/100g) and total antioxidant activity (28.80%). Correlation analysis between total phenolic content and antioxidant activity (r=0.8502) showed a high degree of correlation indicating that these bioactive components contributes to the total antioxidant activity in potato genotypes.
The Opportunity of Using Wind to Generate Power as a Renewable Energy:"Case o...IJERA Editor
The demand ofsustainable energy is increased daily by expanding our cities and creating new cities and suburbswith huge towers besides increasing in population,moreover the environment and human life is threatening by the pollutions resulted from energy generation. For this reason the researchersattracted todevelop renewable energy and explore its large benefits and unit capacity. Wind power is one of the clean renewable energy resources.Therefore the importance of implementing this resource in Kuwait draws our attention to make this research to emphasis on the technical and economic aspects due to acceptable environmental conditions. Whereas, in some areas of the world, such as in Japan, has some geographical and electrical restrictions such as power fluctuating for land wind generation. And the introductionof large amount of wind power generation tends to be extremely difficult and even impossible in some location. This research is aimed to concentrate on the visibility of utilizing the wind energy as complementary source for the existing steam and gas turbine power stations in Kuwait, furthermore point out the economical perspectives that will guide us to take the right decision. The location of wind farms is very important in this aspect where we cannot build such projects inside the cities between buildings besides meeting the minimum requirements for economic generation. The study prove to us that even at a location which is almost close to the inhabitants buildings ( Kuwait airport ) can get accepted results the historical data was collected from the weather station at internet. The implementation of wind turbine farms is foreseen to be economic in generation for long run and encouragestepping up toward putting the infrastructure design. Furthermoreit is an opportunity for creating new job vacancies.
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.
An Investigative and Concise Review on Evaporation and Condensation Processes...IJERA Editor
The vapour adsorption refrigeration is based on the evaporation and condensation of a refrigerant combined with adsorption or chemical reaction. The towering fossil fuel price and the responsiveness of environmental problems offer many potential applications to thermal powered adsorption cooling. However, the adsorption cooling machines still have some disadvantages that hinder their wide application. The patents surveyed are classified into four main groups: adsorption system development, adsorbent bed innovation, adsorbent/adsorbate material development and novel application of adsorption cooling system. The adsorption refrigeration is based on the evaporation and condensation of a refrigerant combined with adsorption or chemical reaction. Important targets are to reach a high efficiency through optimization measures at various components and the control system. On the other hand measures are to verify to simplify the construction with regard to a low-cost manufacturing, as well as to reach long periods with maintenance-free operation. This review paper gives a comprehensive review on the work carried out on vapour adsorption refrigeration for cryogenic applications.
Mechanical properties of rice husk flour reinforced epoxy bio-compositeIJERA Editor
A bio-composite reinforced with rice husk flour in epoxy resin has been developed. The effect of fibre treatment and weight percentage of rice husk on the mechanical properties was studied and compared with wood dust reinforced epoxy composite. It was observed that addition of rice husk as filler is detrimental to almost all the mechanical properties. About 51% and 26.8% decrease in ultimate strength and Young’s Modulus for 40 wt% untreated rice husk reinforcement was observed. The corresponding decrease in flexural strength and flexural modulus was 51%. Similar trend was also observed in hardness and impact strength. However the mechanical properties of rice husk reinforced biocomposites are found to be superior than wood dust reinforced epoxy composite. SEM microscopy was also done to corroborate the results.
Investigation on the Growth and Physio-Chemical Properties of L-Alanine Mixed...IJERA Editor
Pure and L-alanine an aminoacid mixed bisthiourea cadmium bromide chloride single crystals were grown by slow evaporation technique. A drastic change in morphology was inferred with the concentration of L-alanine. Mixed crystals have better optical transparency as well as NLO efficiency than the pure BTCBC which were imperative for nonlinear applications. Also L-alanine mixing increases the hardness. The AC conductivity of the grown crystals increases with increasing concentration of L-alanine.
Hankel Determinent for Certain Classes of Analytic FunctionsIJERA Editor
Let 1 A denote the class of functions
2
( )
n
n
n f z z a z analytic in the unit disc E {z : |z| <1}.
M denotes the class of functions in 1 A which satisfy the conditions 0
( ). ( )
z
f z f z
and for
0 1, 0
( )
( ( ))
( )
( )
) 1 ( Re
f z
zf z
f z
zf z
. We are interested in determining the sharp upper bound
for the functional
2
2 4 3 a a a for the class M .
Modeling and Analysis of Wholesale Competitive Electricity Markets: A Case Fo...IJERA Editor
Electricity markets all over the world are moving towards greater reliance on competition and this has become a global trend as a method of best practice. However, before competition is introduced in electricity markets it is imperative to model and assess the behavior of the market. The assessment includes calculating the market performance indices to determine the levels of market power exploitation by the Generating Companies (GenCos) that will participate in the market. This paper presents a study on modeling and analysis of wholesale competitive electricity market for a developing country to help regulators assess and predict market behaviour. It involves modeling and simulation of the Zambian power system network in Agent-Based Modeling of Electricity Systems (AMES) using real system data to pick out critical information that enables us to assess the status of the market. The results indicate that market power exploitation is prevalent for the two largest GenCos assessed.
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.
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.
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 Survey on Constellation Based Attribute Selection Method for High Dimension...IJERA Editor
Attribute Selection is an important topic in Data Mining, because it is the effective way for reducing dimensionality, removing irrelevant data, removing redundant data, & increasing accuracy of the data. It is the process of identifying a subset of the most useful attributes that produces compatible results as the original entire set of attribute. Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group called a cluster are more similar in some sense or another to each other than to those in other groups (Clusters). There are various approaches & techniques for attribute subset selection namely Wrapper approach, Filter Approach, Relief Algorithm, Distributional clustering etc. But each of one having some disadvantages like unable to handle large volumes of data, computational complexity, accuracy is not guaranteed, difficult to evaluate and redundancy detection etc. To get the upper hand on some of these issues in attribute selection this paper proposes a technique that aims to design an effective clustering based attribute selection method for high dimensional data. Initially, attributes are divided into clusters by using graph-based clustering method like minimum spanning tree (MST). In the second step, the most representative attribute that is strongly related to target classes is selected from each cluster to form a subset of attributes. The purpose is to increase the level of accuracy, reduce dimensionality; shorter training time and improves generalization by reducing over fitting.
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.
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
Numerical simulation on laminar convection flow and heat transfer over an iso...eSAT Journals
Abstract A numerical algorithm is presented for studying laminar convection flow and heat transfer over an isothermal vertical horizontal plate embedded in a saturated porous medium. By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Velocity and temperature profiles are illustrated graphically. Heat transfer parameters are derived. Keywords
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
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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.
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homotopy analysis method (HAM). The convergence of obtained analytical solutions is explicitly discussed. The
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Numerical simulation on laminar convection flow and heat transfer over a non ...eSAT Journals
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analysis in Matlab environment. Velocity, and temperature profiles for various Prandtl number and n are illustrated graphically.
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For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
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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.
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Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with Slip Condition
1. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 7( Version 2), July 2014, pp.19-33
www.ijera.com 19 | P a g e
Pressure Gradient Influence on MHD Flow for Generalized
Burgers’ Fluid with Slip Condition
Ghada H. Ibraheem, Ahmed M. Abdulhadi
Department of Mathematics, University of Baghdad College of Education, Pure Science, Ibn A-Haitham
Department of Mathematics, University of Baghdad College of Science
Abstract
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.
Keywords: Generalized Burgers’ fluid, Constant pressure gradient, Sinusoidal pressure gradient, Fox H-function.
I. Introduction
In recent years, the flow of non- Newton fluid has received much attention for their increasing industrial
and technological applications, such as extrusion of polymer fluids, exotic lubricants, colloidal and suspension
solutions food stuffs and many others. Because of the complicated behavior, there is no model which can alone
describe the behavior of all non- Newtonian fluids. For this reason, several constitutive equations for all non-
Newtonian fluid models have been proposed. Among them, rate type models have special importance and many
researchers are using equations of motion of Maxwell and Oldroyd fluid flow [5, 9, 13, 16, 17]. Recently, a
thermodynamic framework has been put into place to develop a rate type model known as Burgers’ model [8]
which is used to describe the motion of the earth’s mantle. The Burgers’ model is the preferred model to
describe the response of asphalt and asphalt concrete [5]. Many applications of this type of fluid can be found in
[3, 6, 11, 14, 15, 18]. Fluids exhibiting boundary slip are important in technological applications, such as the
polishing of artificial heart values, polymer melts often exhibit macroscopic wall slip that in general is governed
by a non- linear and non- monotone relation between the slip velocity and the traction [4]. Ebaid [1] and Ali [10]
studied the effect of magnetic field and slip condition on peristaltic transform. Khaleda [2] gives the exact
solution for the slip effect on Stokes and Couette flows due to an oscillating wall. Liancun eta. [19] investigated
effect of slip condition on MHD flow of a generalized Oldroyd- B fluid with fractional derivative. They used the
fractional approach to write down the constitutive equations for a viscoelastic fluid. A closed form of the
velocity distribution and shear stress are obtained in terms of Fox H- function by using the discrete Laplace
transform of the sequential fractional derivative.
No attempt has been made regarding the exact solutions for flows due to constant and sinusoidal pressure
gradient of generalized Burgers’ fluid with fractional derivative and the non- slip condition is no longer valid.
The exact solutions for velocity field and shear stress are obtained by using discrete Laplace transform and they
are written in term of Fox H- function. Many cases are recovered from our solutions.
II. Governing Equations
The constitutive equations for an incompressible fractional Burger’s fluid given by
S I T p , A S ) D ~
+ (1 = ) D ~
D ~
(1+ 3 t
2
t t 1 2
(1)
where T denoted the cauchy stress, pI is the indeterminate spherical stress, S is the extra stress tensor,
T A LL is the first Rivlin- Ericksen tensor with the velocity gradient where L grad V , is the dynamic
viscosity of the fluid, 1
and 3 (< 1
) are the relaxation and retardation times, respectively, 2 is the new
RESEARCH ARTICLE OPEN ACCESS
2. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 7( Version 2), July 2014, pp.19-33
www.ijera.com 20 | P a g e
material parameter of Burger’s fluid, and the fractional calculus parameters such that 0 1 and
p
t D ~
the upper convected fractional derivative define by
T ( . )
~
S S V S LS SL
t t D D (2)
T ( . )
~
A A V ALA AL
t t D D (3)
in which
t D and
t D are the fractional differentiation operators of order and based on the Riemann-
Liouville definition, defined as
,0 1
( )
( )
(1 )
1
[ ( )]
0
d p
t
f
dt
d
p
D f t
t
p
p
t
(4)
here (.) denotes the Gamma function and
( ) 2 S S p
t
p
t
p
t D D D (5)
The model reduced to the generalized Oldroyd- B model when 0 2 and if, in addition to that, 1
the ordinary Oldroyd- B model will be obtained.
We consider the MHD flow of an incompressible generalized Burger’s fluid due to an infinite accelerating plate.
For unidirectional flow, we assume that the velocity field and shear stress of the form
V u(y, t)i ,S S(y, t) (6)
where u is the velocity and i is the unit vector in the x- direction .Substituting equation (6) into (1) and taking
account of the initial condition
S(y,0) 0 , y 0 (7)
we obtain
(1+ D D )S = (1+ D ) ( , ) xy 3 t y
2
1 t 2 t u y t (8)
where S S S S 0 xz yy yz zz , xy yx S S . Furthermore, it assumes that the conducting fluid is permeated by
an imposed magnetic field [0,B ,0] 0 B which acts in the positive y- direction. In the low- magnetic Reynolds
number approximation, the magnetic body force is represented as u 2
0 B , where is the electrical conductivity
of the fluid. Then in the present of a pressure gradient in the x- direction, the equation of motion yields the
following scalar equation:
u
dt x y
du
xy
2
0 S B
p
(9)
where is the constant density of the fluid. Eliminating xy S between Eqs. (8) and (9), we obtain the following
fractional differential equation
u
y
u
v
u
(1+ D ) M(1+ D D )
dx
dp
(1+ D D )
1
t
(1+ D D ) 2
2 1 t 2 t
2
3 t
2
1 t 2 t
2
1 t 2 t
(10)
where
v is the kinematic viscosity and
2
0 B
M is the magnetic dimensionless number.
III. Flow induced by a constant pressure gradient:
Let us consider the flow problem of an incompressible generalized Burgers’ fluid over an infinite plate at
y 0 with fluid occupies the space y 0 and flowing under the action of a constant pressure gradient. Also,
we assumed the existence of slip boundary between the velocity of fluid at the wall u(0, t) and the speed of the
wall, the relative velocity between u(0, t) and the wall is assumed to be proportional to the shear rate at wall.
Initially, the system is at rest and at time t 0 the fluid is suddenly set in motion due to a constant pressure
gradient and by the existence of the slip boundary condition. Referring to Eq. (10), the corresponding fractional
partial differential equation that described such flow takes the form
u
t
u
v
u t t
(1+ D ) M(1+ D D )
(1 ) (1 2 )
A 1+
t
(1+ D D )
2
2 1 t 2 t
2
3 t
2
1 2
2
1 t 2 t
(11)
3. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 7( Version 2), July 2014, pp.19-33
www.ijera.com 21 | P a g e
where
dx
dp
1
A is the constant pressure gradient
The associated initial and boundary conditions are as follow
( ,0) ( ,0) 0 , 0
u y y
t
u y (12)
(0, ) , 0
t
y
u
u t a tb (13)
( , ) , ( , )0 , 0
u y t as y t
y
u y t (14)
where a and b are constants, is the slip strength or slip coefficient. If 0then the general assumed no-slip
boundary condition is obtained. If is finite, fluid slip occurs at the wall but its effect depends upon the length
scale of the flow.
Employing the non- dimensionless quantities
v
av
v
av t
t
v
av y
y
av
u
u
b b b b b b
b b
2 1
2
1
*
1
2 1
2
*
2 1
1
*
2 1
1
* ( )
,
( )
,
( )
,
( )
v
av
av
v
v
av
v
av b b
b b
b b b b
2 1
2
*
2 1
2
*
2 1
2
3
*
3
2
2 1
2
2
*
2
( )
,
( )
M
, M
( )
,
( )
Eqs. (11- 14) in dimensionless form are:
u
y
u
u t t
(1+ D ) M(1+ D D )
(1 ) (1 2 )
A 1+
t
(1+ D D )
2
2 1 t 2 t
2
3 t
2
1 2
2
1 t 2 t
(15)
( ,0) ( ,0) 0 , 0
u y y
t
u y (16)
(0, ) , 0
t
y
u
u t t b (17)
( , ) , ( , )0 , 0
u y t as y t
y
u y t (18)
where the dimensionless mark “ ” has been omitted for simplicity.
Now applying Laplace transform principle [7] to Eq. (15) and taking into account the boundary condition (16),
we find that
(1+ s )
A
(1+ )
( M)(1+ s ) 2
1 2
3
2
1 2
2
2
s
s
u
s
s s
dy
d u
(19)
Subject to boundary conditions
0
1
( 1)
(0, )
y
b dy
du
s
b
u s (20)
y s as y
y
u(y, s) , ( , ) 0 (21)
where u(y, s) is the image function of u(y, t) and s is a transform parameter. Solving Eqs. (19)- (21), we get
( M)
A
( M) (1 )
A
(1 )
( 1)
( , ) 1
c s s
e
c s s
e
s
b
u y s
c y c y
b
(22)
where
2
1
3
2
1 2
(1+ )
( M)(1+ s )
s
s s
c
4. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 7( Version 2), July 2014, pp.19-33
www.ijera.com 22 | P a g e
The shear stress can be calculated from Eq. (8), taking Laplace transform of Eq. (8) and introducing Eq. (22),
we get
( M)
( 1) A
(1+ ) (1 )
(1+ s )
( , ) ( M) 1
2
1
3
2
2 1 2
1
s s s
b
c
e
s
s
y s s b
a y
(23)
where (y, s) is the Laplace transform of
2 1
2
( )
S
( , )
b b
xy
av
y t
.
In order to avoid the burdensome calculations of residues and contour integrals, we will apply the discrete
inverse Laplace transform to get the velocity and the shear stress fields. Now, writing Eq. (22) in series form as
( 1) 2 2
2
0
3
0 0
2
0
1
0
0
2
3
1
1
( 1) 2 2
0 2
3
0 0
2
0
1
1 0
2
3
1
( 1) 2 1
2
0
3
0
2
0
1
0 0
2
3
1
( 1) 2 1 1
2
0
3
0
2
0
1
1 0
2
3
1
1
2
2
2 2
2 2
!
( )
( M)
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
A ( )
2 2 2
2 2 2
!
( )
( M)
!( )!
( ) !
!
( )
!
( M)
A ( ) ( )
1
2 2 2
2 2 2
!
( )
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
( 1) ( )
2 2 2
2 2 2
!
( )
!( )!
( ) !
!
( )
!
( M)
( 1) ( ) ( )
( 1)
( , )
l m n p i
k j
n
n
i
i
m
p
p
m
m
l
l
k
k j
j
j
k
l m n p i
k
n
n
i
i
m
p
p
m
m
l
l
k
k
k
l m n p b
k j
n
m n
p
p
m
m
l
l
k
k j
j
j
k
l m n p b
k
n
m n
p
p
m
m
l
l
k
k
k
b
s
k j
k j
n
k j k j
k j
m
k j
l
p m p n
m
l m
j
y
s
k k k
k
n
k
m
k
l
n
p m p
m
l m
s
k j k j k j
k j
n
k j
m
k j
l
p m p n
m
m
j l
y
b
s
k k k
k
n
k
m
k
l
p m p n
m
l m
b
s
b
u y s
(24)
Applying the discrete inverse Laplace transform to Eq. (24), we get
( 1) 2 1
2 2 2 2
2 2 2
!
( )
!( )!
( ) !
!
( )
!
( M)
( , ) ( 1) ( ) ( )
0
3
2
2
|( 1)
0
2
0
1
1 0
2
3
1
l m p n b
k k k k
k
n
k
m
k
l
n
t
t
p m p
m
l m
u y t t b
n
n
l m p b
m k
p
p
m
m
l
l
k
k
b k
l m p b
m k j
p
p
m
m
l
l
k
k j
j
j
k t
p m p
m
j l m
y
b
2
2
( 1)
0
2
0
1
0 0
2
3
1
1 !( )!
( ) !
!
( )
!
( M)
( )
!
( )
( 1) ( )
( 1) 2 1
2 2 2 2
2 2 2
!
( )
0
3
l m p n b
k j k j k j k j
k j
n
k j
m
k j
l
n
t
n
n
5. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
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( 1) 2 2
2 2 2 2
2 2 2
!
( )
( M)
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
A ( )
( 1) 2 2
2 2 2 2
2 2 2
!
( )
( M)
!( )!
( ) !
!
( )
!
( M)
A ( ) ( )
0
3
2 1
2
( 1)
0 0
2
0
1
0 0
2
3
1
1
0
3
2 1
2
( 1)
0 0
2
0
1
1 0
2
3
1
l m p n i
k j k j k j k j
k j
n
k j
m
k j
l
n
t
t
p m p
m
j l m
y
l m p n i
k k k k
k
n
k
m
k
l
n
t
t
p m p
m
l m
n
n
l m p i
k j
i
i
m
p
p
m
m
l
l
k
k j
j
j
k
n
n
l m p i
k
i
i
m
p
p
m
m
l
l
k
k
k
(25)
In terms of Fox H- function, Eq. (25) takes the simpler from:
,1)
2
,0),(1
2
,0),(1
2
(1
2 1, )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2 1
2
( 1)
0 0
2
0
1
0 0
2
3
1
1
,1)
2
,0),(1
2
,0),(1
2
(1
2 1, )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2 1
2
( 1)
0 0
2
0
1
1 0
2
3
1
,1)
2
,0),(1
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2
2
( 1)
0
2
0
1
0
0
2
3
1
1
,1)
2
,0),(1
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2
2
|( 1)
0
2
0
1
1 0
2
3
1
H
( M)
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
A ( )
( M) H
!( )!
( ) !
!
( )
!
( M)
H A ( ) ( )
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
H ( 1) ( )
!( )!
( ) !
!
( )
!
( M)
( , ) ( 1) ( ) ( )
k j k j
m
k j
l
l m p i
k j k j k j k j
l m p i
k j
i
i
m
p
p
m
m
l
l
k
k j
j
j
k
k k
m
k
l
l m p i
k k k k
l m p i
k
i
i
m
p
p
m
m
l
l
k
k
k
k j k j
m
k j
l
l m p b
k j k j k j k j
l m p b
m k j
p
p
m
m
l
l
k
k j
j
j
k
k k
m
k
l
l m p b
k k k k
l m p b
m k
p
p
m
m
l
l
k
k
b k
t
t
p m p
m
j l m
y
t
t
p m p
m
l m
t
t
p m p
m
l m
j
y
b
t
t
p m p
m
l m
u y t t b
(26)
where the property of the Fox H- function is
(1 , ), ,(1 , )
(0,1),(1 , ), ,(1 , )
1,
, 1
0
1
1 1 1
1 1
H
! ( )
( ) ( )
p p
q q
a A a A
b B b B
p
p q
n
q
j
j j
p
j
j j
n
z
n b B n
z a A n
The solution Eq. (26) should satisfy the boundary condition Eq. (17). To see this, from Eq. (26) it is easy to
obtain
6. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
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www.ijera.com 24 | P a g e
,1)
2
,0),(1
2
,0),(1
2
(1
2 1, )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2 1
2
( 1)
0 0
2
0
1
1 0
2
3
1
,1)
2
,0),(1
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2
2
|( 1)
0
2
0
1
1 0
2
3
1
( M) H
!( )!
( ) !
!
( )
!
( M)
H A ( ) ( )
!( )!
( ) !
!
( )
!
( M)
(0, ) ( 1) ( ) ( )
k k
m
k
l
l m p i
k k k k
l m p i
k
i
i
m
p
p
m
m
l
l
k
k
k
k k
m
k
l
l m p b
k k k k
l m p b
m k
p
p
m
m
l
l
k
k
b k
t
t
p m p
m
l m
t
t
p m p
m
l m
u t t b
and
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 1
2
1
( 1)
0 0
2
0
1
1 0
2
1
3
1
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2
2
1
( 1)
0
2
0
1
0 0
2
1
3
1
0
( M) H
!( )!
( ) !
!
( )
!
( M)
H A ( ) ( )
!( )!
( ) !
!
( )
!
( M)
( 1) ( ) ( )
( , )
k k
m
k
l
l m p i
k k k k
l m p i
k
i
i
m
p
p
m
m
l
l
k
k
k
k k
m
k
l
l m p b
k k k k
l m p b
m k
p
p
m
m
l
l
k
k
k
y
t
t
p m p
m
l m
t
t
p m p
m
l m
b
y
u y t
then
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 1
2
1
( 1)
0 0
2
0
1
1 0
2
1
3
1
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2
2
1
( 1)
0
2
0
1
0 0
2
1
3
1
0
( M) H
!( )!
( ) !
!
( )
!
( M)
H A ( ) ( )
!( )!
( ) !
!
( )
!
( M)
( 1) ( ) ( )
( , )
k k
m
k
l
l m p i
k k k k
l m p i
k
i
i
m
p
p
m
m
l
l
k
k
k
k k
m
k
l
l m p b
k k k k
l m p b
m k
p
p
m
m
l
l
k
k
k
y
t
t
p m p
m
l m
t
t
p m p
m
l m
b
y
u y t
i.e.
y
u y t
u t t b
( , )
(0, )
Adopting the similar procedure in Eq. (23), we obtain the shear stress:
m
p
p
m
m
l
l
k
k j
j
j
k
p m p
m
j l m
y
b
0
2
0
1
0 0
2
1
3
1
0 !( )!
( ) !
!
( )
!
( M)
( )
!
( )
( 1) ( )
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2
2
1
( 1)
0 0
2
0
1
0 0
2
1
3
1
0
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 1
2
1
( 1)
H
( M)
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
A ( )
H
k j k j
m
k j
l
l m p i
k j k j k j k j
l m p i
k j
i
i
m
p
p
m
m
l
l
k
k j
j
j
k
k j k j
m
k j
l
l m p b
k j k j k j k j
l m p b
k j
t
t
p m p
m
j l m
y
t
t
(27)
Special Cases:
1- If 0 then the no- slip condition is obtained. In this special case Eqs. (26) and (27) are simplified
to
7. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
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www.ijera.com 25 | P a g e
,1)
2
,0),(1
2
,0),(1
2
(1
2 1, )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2 1
2
( 1)
0 0
2
0
1
0
2
3
1
1
,1)
2
,0),(1
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2
2
( 1)
0
2
0
1
0
2
3
1
1
( M) H
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
H A
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
( , ) ( 1)
j j
m
j
l
l m p i
j j j j
l m p i
j
i
i
m
p
p
m
m
l
j l
j
j
j j
m
j
l
l m p b
j j j j
l m p b
m j
p
p
m
m
l
j l
j
j
b
t
t
p m p
m
j l m
t y
t
p m p
m
j l m
y
u y t t b
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2
2
1
( 1)
0 0
2
0
1
0
2
1
3
1
0
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 1
2
1
( 1)
0
2
0
1
0
2
1
3
1
0
H
( M)
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
A
H
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
( , ) ( 1)
j j
m
j
l
l m p i
j j j j
l m p i
j
i
i
m
p
p
m
m
l
j l
j
j
j j
m
j
l
l m p b
j j j j
l m p b
j
m
p
p
m
m
l
j l
j
j
t
t
p m p
m
j l m
y
t
t
p m p
m
j l m
y
y t b
2- If 0 and M 0 then Eqs. (26) and (27) reduce to
,1)
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2
2
( 1)
0
2
0
1
0
2
3
1
1
,1)
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2
2
|( 1)
0
2
0
1
1
2
3
1
H
!( )!
( ) !
!
( )
( )
!
( )
H ( 1) ( )
!( )!
( ) !
!
( )
( , ) ( 1) ( ) ( )
k j k j
m
m p b
k j k j k j
m p b
m k j
p
p
m
m
k
k j
j
j
k
k k
m
m p b
k k k
m p b
m k
p
p
m
m
k
k
b k
t
t
p m p
m
m
j
y
b
t
t
p m p
m
m
u y t t b
,1)
2
,0),(1
2
(1
2 1, )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2 1
2
( 1)
0
2
0
1
0
2
3
1
1
,1)
2
,0),(1
2
(1
2 1, )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2 1
2
( 1)
0
2
0
1
1
2
3
1
H
!( )!
( ) !
!
( )
( )
!
( )
H A ( )
!( )!
( ) !
!
( )
A ( ) ( )
k j k j
m
m p
k j k j k j
m p
m k j
p
p
m
m
k
k j
j
j
k
k k
m
m p
k k k
m p
m k
p
p
m
m
k
k
k
t
t
p m p
m
j m
t y
t
p m p
m
m
8. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
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www.ijera.com 26 | P a g e
,1)
2
1
,0),(1
2
1
(1
2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
3 (0,1),(1
1,2
2,4
2
2
1
( 1)
0
2
0
1
0
2
1
3
1
0
,1)
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
3 (0,1),(1
1,2
2,4
2 1
2
1
( 1)
0
2
0
1
0
2
1
3
1
0
H
!( )!
( ) !
!
( )
( )
!
( )
A ( )
H
!( )!
( ) !
!
( )
( )
!
( )
( , ) ( 1) ( )
k j k j
m
m p
k j k j k j
m p
k j
m
p
p
m
m
k
k j
j
j
k
k j k j
m
m p b
k j k j k j
m p b
k j
m
p
p
m
m
k
k j
j
j
k
t
t
p m p
m
j m
y
t
t
p m p
m
j m
y
y t b
which correspond to the flow of a generalized Burgers’ magnetic field effect.
3- Making 0 2 and A0 in Eqs. (26) and (27) the solutions corresponds to slip effects of a
generalized Oldroyd- B fluid in absence of pressure gradient can be recovered, as found by Zheng … etc in
[19].
4- If we set 0 2 , 0 and M 0 in Eqs. (26) and (27) the similar solutions for generalized
Oldroyd- B fluid are recovered, as found by Hyder … etc in [12].
IV. Flow due to a sinusoidal pressure gradient:
Let us consider the flow problem of generalized Burgers’ fluid bounded by an infinite plane wall at y 0 ,
under the action of sinusoidal pressure gradient with the same initial and boundary conditions, Eqs. (16- 18).
In this case the governing equation can be written as
u
y
u
p wt
u
M(1+ D D )
(1+ D D ) cos( ) (1+ D )
t
(1+ D D )
2
1 t 2 t
2
2
3 t
2
0 1 t 2 t
2
1 t 2 t
(28)
where cos( )
1
cos( ) 0 0 p wt
dx
dp
p wt
dx
dp
and 0 p is constant. The associated initial and boundary
condition are as given in Eqs. (16- 18).
Again, by similar procedure as in the previous case the velocity field is found in the form of
0
2
3
1
1
,1)
2
,0),(1
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2
2
|( 1)
0
2
0
1
1 0
2
3
1
( )
!
( )
H ( 1) ( )
!( )!
( ) !
!
( )
!
( M)
( , ) ( 1) ( ) ( )
k
k j
j
j
k
k k
m
k
l
l m p b
k k k k
l m p b
m k
p
p
m
m
l
l
k
k
b k
j
y
b
t
t
p m p
m
l m
u y t t b
l m p b
m k j
p
p
m
m
l
l
t
p m p
m
l m
2
2
( 1)
0
2
0
1
0 !( )!
( ) !
!
( )
!
( M)
2 2 1
2
( 1)
0
2
0 0
2
0
1
0 0
2
3
1
1
0
,1)
2
,0),(1
2
,0),(1
2
(1
2 2 1, )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
2 2 1
2
( 1)
0
2
0 0
2
0
1
1 0
2
3
1
0
,1)
2
,0),(1
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
,0),(1
2
3 (0,1),(1
1,3
3,5
( M) ( )
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
H ( )
( M) ( )
!( )!
( ) !
!
( )
!
( M)
H ( ) ( )
l m p i r
k j
r
r
i
i
m
p
p
m
m
l
l
k
k j
j
j
k
k k
m
k
l
l m p i r
k k k k
l m p i r
k
r
r
i
i
m
p
p
m
m
l
l
k
k
k
k j k j
m
k j
l
l m p b
k j k j k j k j
w t
p m p
m
m
j l
y
p
t
w t
p m p
m
m
l
p
t
(29)
9. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
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www.ijera.com 27 | P a g e
and, by using Eq. (8), the corresponding stress is found in the form of
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 2
2
1
( 1)
0
2
0 0
2
0
1
0 0
2
1
3
1
0
0
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 1
2
1
( 1)
0
2
0
1
0 0
2
1
3
1
0
H
( M) ( )
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
( )
H
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
( , ) ( 1) ( )
k j k j
m
k j
l
l m p i r
k j k j k j k j
l m p i r
k j
r
r
i
i
m
p
p
m
m
l
l
k
k j
j
j
k
k j k j
m
k j
l
l m p b
k j k j k j k j
l m p b
k j
m
p
p
m
m
l
l
k
k j
j
j
k
t
t
w
p m p
m
j l m
y
p
t
t
p m p
m
j l m
y
y t b
(30)
Special Cases:
1- If 0 then the no- slip condition is obtained. In this special case Eqs. (29) and (30) are simplified
into
,1)
2
,0), (1
2
,0), (1
2
(1
2 2 1, )
2
,0), (( 1)
2
,0), (1
2
,0), (1
2
3 (0,1), (1
1,3
3,5
2 2 1
2
( 1)
0
2
0 0
2
0
1
0
2
3
1
1
0
,1)
2
,0), (1
2
,0), (1
2
(1
2 , )
2
,0), (( 1)
2
,0), (1
2
,0), (1
2
3 (0,1), (1
1,3
3,5
2
2
( 1)
0
2
0
1
0
2
3
1
1
( M) ( ) H
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
H
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
( , ) ( 1)
j j
m
j
l
l m p i r
j j j j
l m p i r
j
r
r
i
i
m
p
p
m
m
l
j l
j
j
j j
m
j
l
l m p b
j j j j
l m p b
m j
p
p
m
m
l
j l
j
j
b
t
w t
p m p
m
j l m
y
p
t
t
p m p
m
j l m
y
u y t t b
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 1
2
1
( 1)
0
2
0
1
0
2
1
3
1
0
H
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
( 1)
j j
m
j
l
l m p b
j j j j
l m p b
j
m
p
p
m
m
l
j l
j
j
t
t
p m p
m
j l m
y
b
,1)
2
1
,0),(1
2
1
,0),(1
2
1
(1
2 2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
,0),(1
2
1
3 (0,1),(1
1,3
3,5
2 2
2
1
( 1)
0
2
0 0
2
0
1
0
2
1
3
1
0
0
H
( M) ( )
!( )!
( ) !
!
( )
!
( M)
( )
!
( )
j j
m
j
l
l m p i r
j j j j
l m p i r
j
r
r
i
i
m
p
p
m
m
l
j l
j
j
t
t
w
p m p
m
j l m
y
p
2- If 0 and M 0 then Eqs. (29) and (30) reduce to
10. Ghada H. Ibraheem Int. Journal of Engineering Research and Applications www.ijera.com
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www.ijera.com 28 | P a g e
,1)
2
,0),(1
2
(1
2 2 1, )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2 2 1
2
( 1)
0
2
0
2
0
1
0
2
3
1
1
0
,1)
2
,0),(1
2
(1
2 2 1, )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2 2 1
2
( 1)
0
2
0
2
0
1
1
2
3
1
0
,1)
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2
2
( 1)
0
2
0
1
0
2
3
1
1
,1)
2
,0),(1
2
(1
2 , )
2
,0),(( 1)
2
,0),(1
2
3 (0,1),(1
1,2
2,4
2
2
|( 1)
0
2
0
1
1
2
3
1
H
( )
!( )!
( ) !
!
( )
( )
!
( )
( )
( ) H
!( )!
( ) !
!
( )
H ( ) ( )
!( )!
( ) !
!
( )
( )
!
( )
H ( 1) ( )
!( )!
( ) !
!
( )
( , ) ( 1) ( ) ( )
k j k j
m
m p r
k j k j k j
m p r
k j
r
r
m
p
p
m
m
k
k j
j
j
k
k k
m
m p r
k k k
m p r
k
r
r
m
p
p
m
m
k
k
k
k j k j
m
m p b
k j k j k j
m p b
m k j
p
p
m
m
k
k j
j
j
k
k k
m
m p b
k k k
m p b
m k
p
p
m
m
k
k
b k
t
w t
p m p
m
j m
y
p
t
w t
p m p
m
m
p
t
t
p m p
m
j m
y
b
t
t
p m p
m
m
u y t t b
,1)
2
1
,0),(1
2
1
(1
2 2 , )
2
1
,0),(( 1)
2
1
,0),(1
2
1
3 (0,1),(1
1,2
2,4
2 2
2
1
( 1)
0
2
0
2
0
1
0
2
1
3
1
0
0
,1)
2
1
,0),(1
2
1
(1
2 1, )
2
1
,0),(( 1)
2
1
,0),(1
2
1
3 (0,1),(1
1,2
2,4
2 1
2
1
( 1)
0
2
0
1
0
2
1
3
1
0
H
( )
!( )!
( ) !
!
( )
( )
!
( )
( )
H
!( )!
( ) !
!
( )
( )
!
( )
( , ) ( 1) ( )
k j k j
m
m p r
k j k j k j
m p r
k j
r
r
m
p
p
m
m
k
k j
j
j
k
k j k j
m
m p b
k j k j k j
m p b
k j
m
p
p
m
m
k
k j
j
j
k
t
t
w
p m p
m
j m
y
p
t
t
p m p
m
j m
y
y t b
which correspond to the flow of a generalized Burgers’ magnetic field effect.
V. Numerical results and discussion:
In this work, we have discussed the MHD flow of generalized Burger’s fluid due to accelerating plate with
slip effects. The exact solutions for the velocity field u and the stress in terms of the Fox H-function are
obtained by using the discrete Laplace transform. Moreover, some figures are plotted to show the behavior of
various parameters involved in the expressions of velocity u .
A comparison between non- slip effect (Panel a) and slip effect (Panel b) is also made graphically. Figs. 1-
7 are prepared for flow due to constant pressure gradient where as Figs. 8- 14 for flow due to sinusoidal pressure
gradient.
Fig. 1 shows the variation of the non- integer fractional parameter and the slip coefficient . The velocity
is increasing with the increase of and .
Fig. 2 is depicted to show the changes of the velocity with fractional parameter . In the case of non- slip
condition is fullfield, the influence of is same as that of . However, it is observed that as increasing
there is a variation in velocity value a bout some value of which is greater than 0.4.
Figs. 3 and 4 provide the graphically illustrations for the effects of relaxation and retardation parameters 1
and
3 on the velocity fields. The velocity is increasing with the increased the 1
and 3 for both cases, the slip and
no- slip condition.
The effect of 2 is illustrated in Fig. 5 which shows that 2 has quite the opposite effect to that of 1
and
3 for both cases ( 0& 0 ).
Fig. 6 demonstrates the influence of the magnetic field M. It is noticed for both cases 0& 0 , when
M < 2 there is increasing in the velocity field, however when M > 2 show an opposite effect on the velocity.
In Fig. 7, the variations of the slip coefficient on velocity with the magnetic field parameter. The velocity is
decreasing with increase of the magnetic parameter.
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Figs. 8- 14 provide the graphically illustrations for the velocity for flow due to sinusoidal pressure gradient.
Qualitatively, the observations for sinusoidal pressure gradient flow are similar to that of constant pressure
gradient flow. However, the velocity profile in flow of constant and sinusoidal pressure gradient are not similar
quantitatively.
Fig. 15 demonstrate the velocity changes with time at given points (y=1 and y=2) for Panel (a) 0 and
Panel (b) 0.5 for two types of flows. Comparison shows that the velocity profile in sinusoidal pressure
gradient flow are larger when compared to those of constant pressure gradient flow. The effects of the slip
coefficient and magnetic field are the similar on the both flows.
Fig. 1. The velocity for different value of when keeping other parameters fixed a) 0 b) 0.5 (Constant
p. g.)
Fig. 2. The velocity for different value of when keeping other parameters fixed a) 0 b) 0.5 (Constant
p. g.)
Fig. 3. The velocity for different value of 1 when keeping other parameters fixed a) 0 b) 0.5 (Constant
p. g.)
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Fig. 4. The velocity for different value of 3 when keeping other parameters fixed a) 0 b) 0.5 (Constant
p. g.)
Fig. 5. The velocity for different value of 2 when keeping other parameters fixed a) 0 b) 0.5 (Constant
p. g.)
Fig. 6. The velocity for different value of M when keeping other parameters fixed a) 0 b) 0.5 (Constant
p. g.)
Fig. 7. The velocity for different value of when keeping other parameters fixed a) M 0 b) M 4.5
(Constant p. g.)
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Fig. 8. The velocity for different value of when keeping other parameters fixed a) 0 b) 0.5 (
Sinusoidal p. g.)
Fig. 9. The velocity for different value of when keeping other parameters fixed a) 0 b) 0.5 (
Sinusoidal p. g.)
Fig. 10. The velocity for different value of 1 when keeping other parameters fixed a) 0 b) 0.5 (
Sinusoidal p. g.)
Fig. 11. The velocity for different value of 3 when keeping other parameters fixed a) 0 b) 0.5 (
Sinusoidal p. g.)
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Fig. 12. The velocity for different value of 2 when keeping other parameters fixed a) 0 b) 0.5 (
Sinusoidal p. g.)
Fig. 13. The velocity for different value of M when keeping other parameters fixed a) 0 b) 0.5 (
Sinusoidal p. g.)
Fig. 14. The velocity for different value of when keeping other parameters fixed a) M 0 b) M 4.5 (
Sinusoidal p. g.)
Fig. 15. The velocity for different value of y when keeping other parameters fixed a) 0 b) 0.5 ( Constant
P. & Sinusoidal P.)
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