International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Mixed convection flow of a nanofluid containing gyrotactic microorganisms along a vertical wedge in porous media

A. Mahdy

Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

ABSTRACT

The main objective of the present paper is to examine the case of mixed convection flow due to
gyrotactic microorganisms from an isothermal vertical wedge embedded in a porous medium saturated with a nanofluid. In this pioneering study, the simplest possible boundary conditions are considered, namely those in which the temperature, the nanoparticle fraction and the density of motile microorganisms are constant along the wall of the wedge. The benefits of adding motile microorganisms to the suspension include enhanced mass transfer, microscale mixing, and anticipated improved stability of the nanofluid. Upon the Oberbeckâ€“Boussinesq approximation and non-similarity transformation, the nonlinear model equations are obtained and tackled numerically numerically by using the R.K. Gill and shooting methods to obtain the dimensionless velocity, temperature, nanoparticle concentration and density of motile microorganisms together with the reduced Nusselt, Sherwood and motile microorganism numbers. The bioconvection parameters strongly influence the heat, nanoparticle volume fraction and motile microorganism transport rates. In the absence of bioconvection, the results are compared with the existing data in the open literature and found to be in very good agreement.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Class of invariants for a time dependent linear potential

mounira berrehail , benamira farid

2 Département des Sciences de la matière, Université de Bordj Bou Arreridj Bordj Bou Arreridj 43000, Algeria

Laboratoire de Physique Théorique, Département de Physique, Faculté des Sciences Exactes, Université Constantine 1, Constantine 25000, Algeria

ABSTRACT

General solution of the one-dimensional Schrödinger equation in presence of a time-dependent linear potential is reconsidered in the context of Lewis-Riesenfeld and unitary transformation approaches. Three invariant operators are constructed as limiting cases of a general Hermitian quadratic invariant and their instantaneous eigenfunctions are obtained. Then the corresponding solutions of Schrödinger equation for each invariant operator are derived. These solutions include all known solutions of the system. Furthermore, it is shown how different solutions can be related to each other.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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VISCOELASTIC STRESSES OF NONLINEAR ROTATING FUNCTIONALLY GRADED SOLID AND ANNULAR DISKS WITH GRADUALLY VARYING THICKNESS

M.N.M. Allam , R. Tantawy , A. Yousof , A.M. Zenkour

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Department of Mathematics, Faculty of Science, Damietta University, Damietta 34517, Egypt

Department of Mathematics, Faculty of Science, Damietta University, Damietta 34517, Egypt

Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt

ABSTRACT

Analytical and numerical nonlinear solutions for rotating variable-thickness functionally graded solid and annular disks with viscoelastic orthotropic material properties are presented by using the method of successive approximations. Variable material properties such as Young’s moduli, density and thickness of the disk, are first introduced to obtain the governing equation. As a second step, the method of successive approximations is proposed to get the nonlinear solution of the problem. In the third step, the method effective moduli is deduced to reduce the problem to the corresponding one of a homogeneous but anisotropic material. The results of viscoelastic stresses and radial displacement are obtained for annular and solid disks of different profiles and graphically illustrated. The calculated results are compared and the effects due to many parameters are discussed.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Concomitants of order statistics And record values from general farlie-gumble-morgenestern type bivariate- generalized exponential distribution

H. M. Barakat , E. M. Nigm , A. H. Syam

Departement of mathematics-faculty of science- zagazig university

Departement of mathematics-faculty of science- zagazig university

Higher Technological Institute 10th of Ramadan city

ABSTRACT

We introduce BKb-fgm type bivariate generalized exponential distribution. Some distribution properties of concomitants. Of order statistics as well aas record values for this family are studied. Recurrence. Relations between the momeents of concomitants are obtained. Some of these recurrence relations were not published before even for morgenestern type bivariate distributions. Morever, most of paper results are extended To arbitrary distributions (see remark 3.1)

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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24

Entanglement of thermal states of two-qutrit states
with Long-Range Interaction.

A. R. Mohammed. , T. M. El-Shahat

Math. Dept., Faculty of Science, Al-Azhar University, 71524, Assiut, Egypt.

Math. Dept., Faculty of Science, Al-Azhar University, 71524, Assiut, Egypt.

ABSTRACT

We investigate the effect of the long-range interaction (LRI) with an inverse-square function on
the thermal entanglement in anisotropic two-qutrit Heisenberg XYZ system with Dzyaloshinskii-
Moriya (DM) interaction in the presence of the external magnetic eld, using Negativity and
Measurement-Induced Disturbance (MID) to quantify entanglement. The temperature and mag-
netic eld dependence of the thermal entanglement in this system for this interaction are discussed.
Our results indicate that, when the LRI type interactions exist, there is a rich conduct dependent
between spins on the interaction strength, temperature, DM interaction and magnetic eld.
In addition, we conclude that sudden death is displayed at the critical distance of the entangle-
ment. We nd that for less than a critical distance there are entanglement plateaus dependent
upon the distance between spins. Furthermore it, we will make obvious comparison between the
measurement-induced disturbance (MID) and negativity for this model. we will discover that MID
is more robust than thermal entanglement against temperature T.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Piezoelectric FGPM Rotating Hollow Variable Thickness Disc in Hygrothermal Envairoment

M.N.M. Allam , R. Tantawy , A. Yousof

Mathematics department, Faculty of science, Mansoura University, Mansoura, 35516 Egypt

Mathematics department, Faculty of science, Damietta University, Damietta, 34517 Egypt

Mathematics department, Faculty of science, Damietta University, Damietta, 34517 Egypt

ABSTRACT

a semi-analytical solution for a functionally graded piezoelectric rotating disc with variable thickness is presented. There are pressure on the boundary surface, uniform distribution of hygrothermal effect and electric potentials difference between the inner and outer surface. All material properties of the disc assumed to be a function in the radial direction. Some cases of boundary conditions are presented. At last, numerical results are carried out and discussed. The values used in this study are arbitrary chosen to demonstrate the hygrothermal effect on FGPM with variable thickness.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Mathematical modeling of Hyperthermia in a system of interaction ferromagnetic particles

Ali F. Abu-Bakr , Andery Yu. Zubarev

Mathematics department, Faculty of Science, Menoufia university, Egypt

Theoretical and Mathematical Phyiscs department, Institute of Natural Science and Mathematics, Ural Federal University

ABSTRACT

Hyperthermia is the method of heating materials with the help of micro- or nano- sized magnetic particles embedded in them. The essence of this effect is that the medium with particles is placed in an alternating magnetic field. The speed and degree of heating of the particles can be regulated by the strength and frequency of the applied field. Hyperthermia refers to the heating of organs or tissues to temperatures ranging from 42°C to 46°C where it causes the death of cancer cells [1,2]. In this work, we employ an interparticle interaction of magnetic nanoparticles model based on a system of nonlinear differential equations of rotation equations and stokes equations under the influence of rotation magnetic field. Computer simulation has been used to solve this system numerically and investigate of magnetic hyperthermia in the cell of tumor. The results show that the interparticle interaction under rotating magnetic field can significantly increase the heat production as compared with the calculations in the model of the interparticle interaction in linearly oscillating magnetic field.
References
[1] R.E. Rosensweig. Heating magnetic fluid with alternating magnetic field. Journal of Magnetism and Magnetic Materials, 252, (2002), Pp. 370–374.
[2] A.Yu. Zubarev, L. Iskakova and A.F. Abu-Bakr, Magnetic hyperthermia in solid magnetic colloids, Physica A, 467, (2017), Pp. 59–66.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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MHD electroosmotic micro-pump of non-Newtonian fluids through micro-channel

Mohammed Abdulhameed

Federal Polytechnic Bauchi, Nigeria

ABSTRACT

This problem deals with steady driven pressure flow and heat transfer of electro-magnetohydrodynamic micro-pump of third grade fluids between two micro-parallel plates embedded in a porous medium. The effect of thermal radiation and electro-kinetic have been taken into account. The flow forced by the Lorentz force, produced by the interaction of a vertical magnetic field and an externally horizontal imposed electrical field, is assumed to be unidirectional and one dimensional. Based upon the velocity field, the thermally fully developed heat transfer with radiation effect are analyzed by taking the viscous dissipation, the volumetric heat generation due to Joule heating effect and electromagnetic couple effect into account. Analytical solutions corresponding to the fluid velocity and temperature distribution are obtained in series forms, in the assumption that the non-Newtonian viscoelastic parameter has small values. The effects of permeability of the porous medium K, the dimensionless electrical strength parameter H, Hartmann number Ha, non-Newtonian parameter Lamda, constant pressure P, thermal radiation Nr and non-dimensional parameter Brinkman number gamma1 on the velocity and temperature are investigated graphically and discussed in detail.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Analytical study of chemically reactive Rivlin-Ericksen viscoelastic fluid in a circular tube

Mohammed Salisu Chimo , Mohammed Abdulhameed , Sagir Mahmud Abdullahi

Federal Polytechnic Bauchi, Nigeria

Federal Polytechnic Bauchi

Federal Polytechnic Bauchi

ABSTRACT

This problem deals with the influence of chemically reactive Rivlin-Ericksen viscoelastic fluid in a circular tube with no thermal convection. The fluid starts heat generation because of its reactive nature of chemically viscoelastic fluid which set up free convection currents inside the tube. The governing equations are modelled using the fully developed flow conditions. Analytical algorithm based on the modefied homotopy perturbation method (HPM), incorporating the He's polynomial and combined with the Laplace transform is implemented in time and space with the second grade constitutive model for the viscoelastic liquids. Explicit analytical expressions for the transient state as well as the steady state for velocity field and temperature field have been derived. These solutions are written as the sum between the permanent solutions and the transient solutions. The algorithm is validated against the classical solution of this problem for reactive viscous fluid results. The nature of the wall shear stress and Nusselt number engendered due to the flow are determined. The results also indicate that it takes longer to attain steady-state in the case of molten polymer than water and air.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Effects of the fractional order on nanoparticles in Blood flow through the stenosed artery

Dr. Abdulsalam Ya'u Gital

Abubakar Tafawa Balewa University Bauchi, Nigeria

ABSTRACT

In this paper, based on the applications of nanoparticle, the blood flow along with nanoparticles through stenosed artery is studied. The blood is acted by periodic body acceleration, an oscillating pressure gradient and an external magnetic field. The mathematical formulation is based on Caputo-Fabrizio fractional derivative without singular kernel. The model of ordinary blood, corresponding to time-derivatives of integer order, is obtained as a limiting case. Analytical solutions of the blood velocity and temperature distribution are obtained by means of the Hankel and Laplace transforms. Effects of the order of Caputo-Fabrizio time-fractional derivatives and three different nanoparticles i.e. , and are studied. The results highlights that, models with fractional derivatives bring significant differences compared to the ordinary model. It is observed that the addition of nanoparticle reduced the resistance impedance of the blood flow and temperature distribution through bell shape stenosed arteries as compared to and nanoparticles. On entering in the stenosed area, blood temperature increases slightly, but, increases considerably and reaches its maximum value in the stenosis throat. The shears stress has variation from a constant in the area without stenosis and higher in the layers located far to the longitudinal axis of the artery. This fact can be an important for some clinical applications in therapeutic procedures

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Atomic population inversion of the asymmetric atom-atom interaction and SU(1,1) quantum system

Abdallah A. Nahla , M.M.A. Ahmed

Department of Mathematics, Faculty of Science, Tanta University, Egypt.

Department of Mathematics, Faculty of Science, Al-Azhar University, Egypt.

ABSTRACT

A model of the asymmetric two two-level atoms interacting with the SU(1,1) quantum system is presented. The rotating wave approximation (RWA) and the atom-atom interaction are considered in the Hamiltonian operator. The time-dependent wave function for asymmetric case is obtained analytically via solving the Schrodinger equation. Initially, the SU(1,1) quantum system prepared in the Perelemov coherent state and two atoms are in superposition states. Therefore, the atomic population inversion is obtained and discussed for different values of model parameters such as initial atomic angles, Perelomov coherent parameter, the Bargmann index and the detuning parameters. We note that the quantum system is sensitive to the variation in both the Perelomov coherent parameter and the Bargmann index. Moreover, there are non-classical characteristics of the proposed system in the presence the detuning parameters.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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A theoretical study of variable magnetic field with endoscope effect on peristaltic blood flow of Prandtl nanofluid having TiO2 nanoparticles

Sara Abdelsalam , Muhammed Bhatti

The British University in Egypt

Shanghai University

ABSTRACT

In this article, we have studied the variable magnetic field and endoscope effects on peristaltic blood flow of nanofluid containing TiO2 nanoparticles through a porous annulus. The Prandtl fluid model is taken into account for the present flow. The mathematical modeling comprises the temperature, continuity, nanoparticle concentration, and equations of motion which are further simplified by taking a long peristaltic wave and creeping flow regime. The obtained highly nonlinear partial differential equations are solved using homotopy perturbation scheme. The inclusion of the pertinent parameters is discussed mathematically and graphically for the pressure rise, friction forces, temperature profile, and concentration profile. The trapping phenomenon is also investigated with the help of contours. Results show that the maximum velocity distribution exists near the centre of the annulus, whereas the average time flow boosts the velocity profile.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Exact Solution of the N-dimensional Radial Schrödinger Equation via Laplace Transform Method with the Modified Extended Cornell Potential

M. Abu-Shady , T. A. Abdel-Karim , E. M. Khokha

Department of Mathematics, Faculty of Science, Menoufia University, Shebin El- Kom Egypt

Department of Mathematics, Faculty of Science, Menoufia University, Shebin El- Kom Egypt

Department of Basic Science, Modern Academy for Engineering and Technology, Cairo, Egypt

ABSTRACT

The exact solution of N- dimensional radial Schrödinger equation with the modified extended Cornell potential has been obtained using the Laplace transform (LT) method. The energy eigenvalues and the corresponding wave functions for any state have been determined. The eigenvalues for some special cases of the modified extended Cornell potential have been derived. We have investigated the present results to calculate the mass spectra of heavy quarkonium systems such as charmonium, bottomonium and the meson in comparison with the experimental data and with the other studies. The present results have been improved in comparison with other recent studies and have shown a good agreement with the experimental data.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Quantifying and teleporting Fisher Information for accelerated two-qubit system

Nasser Metwally

Math. Dept., college of Science, P. O. Box 32038 Kingdom of Bahrain, University of Bahrain

ABSTRACT

The quantum Fisher information of an accelerated two qubit system is discussed, where the analytical solution which consists of three different parts is derived. We show that, the Unruh acceleration has a depleting effect on the Fisher information. This depletion depends on the
degree of entanglement of the initial state settings. Fisher information is employed to estimate the states' parameters. Although the precision of estimating the state's parameters decreases as the acceleration increases, it can be maximized at certain values of the Unruh acceleration. Moreover, the contribution of the three different parts on the total Fisher information is investigated.
Additionally, quantum Fisher information is introduced as a measure of estimating the quantity of the teleported information between two users, where only one of them is accelerated. The estimation degree depends on Unruh acceleration, the used single mode approximation (within/beyond), the type of encoded information (classic/quantum) in the teleported state, and the entanglement of the initial communication channel. The estimation degree of the parameters can be maximized if the partners teleport classical information.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Buongiorno's nanofluid model for mixed convection flow over a vertical porous wedge with convective boundary conditions

A. Mahdy

Mathematics Departement, Faculty of Science, South Valley University

ABSTRACT

An exploring of the state of mixed convection flow due to an isothermal vertical wedge submersed in saturated porous medium utilize Buongiorno’s nanofluid paradigm is the master intention of our research. In this pioneering investigation, Buongiorno’s nanoﬂuid model that comprises the effects of both of Brownian motion and thermophoresis is employed. The paradigm takes regard the case when the nanoﬂuid particle fraction on the boundary layer is passively rather than actively controlled. The wall of the wedge is submersed in a uniform porous medium and the convective boundary condition has been employed over the wedge wall. Upon the Oberbeck–Boussinesq approximation and non-similarity transformation, the nonlinear set equations are obtained and tackled numerically by using the R.K. Gill and shooting method. A parametric study of the entire flow regime is procured to clarify the effects of the controlled parameters such as: wedge angle parameter ( ), buoyancy ratio parameter ( ), mixed convection parameter ( ), Biot number ( ), Brownian motion parameter ( ), thermophoresis parameter ( ) and Lewies number ( ); the results are likened with the available data in the open literature and detected to be in very good harmonizing. The eminent lineaments of the achieved outcome have been construed and depicted.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Dynamics of entanglement and non-classicality features of a single-mode nonlinear Jaynes-Cummings model
intensity-dependent coupling

Tarek M. Elshahat

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

ABSTRACT

In this paper, we study the interaction between a Four-level atom and a quantized single-mode
field with "intensity-dependent coupling" involving two-photon processes in a "Kerr medium". The Four-level atom is considered to be in a $\lambda$-type configuration. Using the generalized (nonlinear) Jaynes-Cummings model, the exact analytical solution of the wave function for the considered system under particular condition, has been obtained when the atom is initially in the superposition state and the field is in a coherent
state. By using some particular condition, the $\lambda$-type Four-level reduced to $\Xi$-type three level atom, Then, we study the amount of entanglement of the generated entangled states using the field entropy,purity and Fidelity. Moreover, we evaluate a few of their non-classical properties such as momentum increment, momentum diffusion, Mandel $Q$-parameter, mean photon number, Normal squeezing and establish their non-classicality features.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Electro-osmotic flow of non-Newtonian biofluids through wavy micro-concentric tubes

Kh. S. Mekheimer , R. E. Abo-Elkhair , A. M. A. Moawad

Mathematical Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt

Mathematical Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt

Mathematical Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt

ABSTRACT

Most of the bio-fluids are non-Newtonian fluid with complex flow behavior, for instances the human blood and DNA sample are shear thinning fluids (Jeffery model). The Electrokinetic transport of such fluids by micro-peristaltic pumping has been interested in biomedical engineering and other of medical technology. This kind of fluid transport requires more elegant mathematical models and numerical simulations. Motivated by these developments, the present article analyzed the simultaneous effects of an electric double layer and a transverse magnetic field on peristaltic transport of Jeffrey fluid as blood flow model. We select it as a constitutive relation to describe electroosmotic flow located in the gap between two coaxial horizontal pipes. Under low Reynolds number, long wavelength and Debye linearization approximations, the Poisson-Boltzmann equation together with the governing partial differential equations for mass and momentum are derived with appropriate boundary conditions. The expressions for electric potential, stream function, axial velocity, shear wall stress, and axial pressure gradient have been obtained. Pressure rise and frictional force per wavelength have been evaluated numerically and discussed briefly. Our computational evidence that the electric potential is an increasing function of the thickness of the electrical double layer (EDL). Also, axial flow is accelerated with the adding electric field and decelerated with opposing electric field. Finally, bolus size is reduced as the axial external electric field change its direction from adding to opposing direction.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Peristaltic flow of micropolar fluid with nanoparticles through porous medium under the effects of heat absorption, chemical reaction and wall properties

Shaimaa F. Ramadan

Al-Azhar University

ABSTRACT

In the present analysis we have discussed the effects of heat absorption, chemical reaction and wall properties on peristaltic flow of micropolar nanofluid through a porous medium. The fundamental equations of the motion are first modulated and then simplified under the assumptions of long wavelength and low Reynolds number. The exact solutions have been calculated for the velocity and the microrotation velocity, while the governing equations of energy and nanoparticles equations are solved analytically using homotopy perturbation method. In the end, graphical results are discussed to illustrate the effects of various physical parameters of the problem on these distributions.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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A Theoretical Study Of The Effect Of A Laser Mode With
Different Coherent Angles On The Motion Of Atomic
Vapors

Aly M. Abourabia , Amany Z. Elgarawany

Department of Mathematics-Faculty of Science-Menoufiya University. Shebin-El kom 32511.Egypt

Department of Computer Science-Higher Institute Of Computer and Management Sciences-Integrated Thebes Institutes-1st Maadi Corniche, Cairo 11434.Egypt

ABSTRACT

We follow theoretically the motion of the sodium atoms in vapor state
under the influence of a laser mode in (1 + 1) D, which is achieved via
different optical filters. In the Dirac interaction representation, the
equations of motion are represented via the Bloch form, together with the
Pauli operators to find the elements of the density matrix of the system.
The immergence of the principle of coherence in varying the angles of the
laser mode, permits to evaluate the average force affecting the atoms
acceleration or deceleration, accordingly the corresponding velocities and
temperatures are investigated. The atomic vapor is introduced in a region
occupied by a heat bath presented by the laser field, such that the state of
the atomic vapor is unstable inside the system due to the loss or gain of
its kinetic energy to or from the laser field. This instability is studied
through finding the eigenvalues of the system's entropy. Resorting to the
assumption of Boten, Kazantsev and Pusep, who issued a coupling
between the mean numbers of photons in terms of time, allows the
evaluation of the rate of entropy production of the system under study. A
set of figures illustrating the dynamics of the problem is presented.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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LAGRANGIAN VERSUS HAMILTONIAN FORMALISMS IN CERTAIN PROBLEMS OF DYNAMICS

Hamad M. YEHIA

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

ABSTRACT

It is commonly believed that Lagrangian and Hamiltonian equations for a given mechanical system are equivalent. We show that, even in case of mathematical equivalence, the Lagrangian description has certain advantages. This is related to the fact that in a given set of generalized coordinates, Lagrange’s equations are unique, although the Lagrangian itself is not unique, due to possible gauge transformations. On the other hand, neither the Hamiltonian nor Hamilton’s equations are unique for a given mechanical system. This difference is essential in inverse problems of physical importance, when integrable Hamiltonians are constructed from the solution of an integrability requirement. Physical characteristics of the system are mostly disguised in the Hamiltonian form.
Examples are given from dynamics of a 3-D system with a symmetry, involving a particle or a rigid body about a fixed point.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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The concentration distribution surrounded the growing of gas bubbles in the bio tissues

A. K. Abu-Nab , S. A. Mohammadein

Mathematics Department, Faculty of science, Menoufia University, Shebin El-Koom Egypt

Mathematics Department, Faculty of science, Tanta University, Tanta, Egypt

ABSTRACT

The concentration distribution around growing gas bubble in the blood and bio tissues of divers who ascend to surface too quickly is obtained by Mohammadein and Mohamed model [12] for variant and constant ambient pressure through the decompression process. The mathematical model describing this problem consists of four main equations: mass, convective diffusion, Fick's and Laplace’s equations. The mathematical model is solved analytically to obtain the concentration distribution around a growing gas bubble in biotissues. The growth of gas bubble is affected by initial concentration difference ∆C_0, diffusivity of gas in tissue , the constant K_d at decompression, surface tension , initial void fraction . The relation between the growth of gas bubble and time is obtained from the definition of the concentration distribution around a growing gas bubble in biotissues. The relation between the growth of gas bubble and time is studied under the effect of two different values of initial void fraction , critical bubble radius . The present model is compared with Mohammadein and Mohammed model [12].

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MPH

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Solution to Pauli-Schrodinger Equation in Discrete 1 and Continuum 3 Dimensions
with Application to Valence Electrons of Silicon

Mahmoud-SifedDin A. Taha , Muhammad H. El-Sabaa , Aladin H. Kamel

Faculty of Engineering, Ain Shams University, Cairo, Egypt

Faculty of Engineering, Ain Shams University, Cairo, Egypt

Faculty of Engineering, Ain Shams University, Cairo, Egypt

ABSTRACT

Many-electron solution to Schrodinger Equation is hard to be put in closed form in terms of single-electron solution. Many perturbation theories and methods, including Linked Cluster by Sinanoglu and Coupled Cluster by Cizek, aimed to formulate a means to such a solution. Mostly spinless electrons were addressed by these methods at first to simplify the problem. Using the simplest form of Configuration Interaction and Brillouin-Wigner Perturbation Theory we present a formulated method for constructing the many-electron solution to Pauli-Schrodinger Equation. Spatial and Spin symmetry and asymmetry variation are addressed here. New Physics-relying and Mathematics-simplifying concepts are proposed here to provide a viable finite Energy wavefunction tracing. Silicon valance electron states are presented as an application of our theoretical formulation.

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Numerical Method to Calculate Electron and Phonon Density of States Functions from 3D Brillouin Zone Energy Dispersion and Application to Silicon

Mahmoud-SifedDin A. Taha , Muhammad H. El-Sabaa , Aladin H. Kamel

Faculty of Engineering, Ain Shams University, Cairo, Egypt

Faculty of Engineering, Ain Shams University, Cairo, Egypt

Faculty of Engineering, Ain Shams University, Cairo, Egypt

ABSTRACT

The number of allowed states within the range of energy permitted for the Electrons and the Phonons per unit volume, typically known as Density of States Functions, is an important parameter for electrons out of their Thermal Equilibrium. Usually these functions are needed for current calculation along the energy range but the particle energy are modeled against the inverse space wave vector. Numerical Method depending on contour tracing of particle energy in the 1st Brillouin Zone is presented here. Comparison to approximate formula based on the effective mass of electrons in Silicon is shown. These functions are essential mathematical component to our Full-band Monte Carlo Method for bulk and device conductivity that is presented elsewhere.

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MPH

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International Conference on Mathematics, Trends and Development ICMTD17 ,

Cairo, Egypt, 28 – 30 Dec. 2017 Organized by The Egyptian Mathematical Society,

Web Site: http://www.etms-eg.org

Mathematical Physics.

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Description of Two Three-level Atoms-Field
Interaction via Quantized Caldirola-Kanai Hamiltonian

M. K. Ismail , T. M. El-Shahat , A.-S. F. Obada

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt.

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt.

Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt

ABSTRACT

The system of two Λ-type three-level atoms and a quantized single-mode cavity field was proposed, where the Hamiltonian of the field is performed based on the Caldirola-Kanai damping Hamiltonian. An analytical description of such system was presented. Some of nonclassical features of such system were discussed, such as quantum entanglement, sub-Poissonian statistics and Quadrature fluctuation. Here, the effects of damping parameter, initial atomic states on the evolution of Linear entropy and the nonclassical features were considered for such system. In addition, some of quantum properties was studied via Mandel parameter.
It was found that the damping parameter and initial atomic states play central roles in the evolution of the Linear entropy and nonclassicality evolution of the particle.