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The effect of wall slip conditions, porous media and heat transfer on peristaltic flow of MHD Newtonian fluid in catheterized tube has been studied under the assumptions of longwavelength and low-Reynolds number. The analytical solution has been derived for velocity and temperature.The amplitude ratio (φ), particle concentration (C), catheter size (ε) and the dimensionless flow rate (Q)were used to obtain the pressure gradient.The results for velocity and temperature obtained in the analysis have been evaluated numerically and discussed.Thetube surface was maintained at a constant temperature.The variations of several parameters were discussed by using suitable graphs.The mathematical model was corresponding to the flow in the annular space of two concentric tubes. Military Technical College Kobry Elkobbah, Cairo, Egypt April 19-21,2016 8 International Conference on Mathematics and Engineering Physics (ICMEP-8) 2 Introduction The peristaltic transport through tubes /channels has attracted considerable attention due to their wide applications in medical and engineering sciences, such as urine transport from kidney to bladder, in movement of ovum in the fallopian tubes, in passage of food through esophagus and many others. In industrial applications, these flows occur in blood pumps in heart lung machine, in sanitary fluid transport and transport of corrosive fluids.It is clear theperistaltic transport of two immiscible viscous fluids in circular tube which has been studied by Ramachandra and Usha[1].Takabatakeet al. [2]has investigatedperistaltic pumping incircular tubes withnumerical study of fluid transport and itsefficiency. The theory of fluid suspension is very important in various situations such assedimentation, combustion, powder technology, aerosol filtration, lunar ash flow, fluidization,atmospheric fallout, environmental pollution, etc. Most recently, the interest has developed in applying the theory of particle-fluid suspension to physiological flows including the vasomotion of small blood vessels such as arterioles, venules and capillaries [3-6]. Military Technical College Kobry Elkobbah, Cairo, Egypt April 19-21,2016 8 International Conference on Mathematics and Engineering Physics (ICMEP-8) 3 Since the pioneering works of Latham [7] and Shapiro et al. [8] much has been said about the peristaltic flows.The study of slip flow through porous medium has attracted much attention recently. It is well known that flow through a porous medium has practical applications especially in geophysical fluid dynamics. Examples of natural porous media are beach sand, sandstone, limestone, rye bread, wood, the human lung, bile duct, gall bladder with stones and in small blood vessels. In some pathological situations, the distribution of fatty cholesterol and artery clogging blood clots in the lumen of coronary artery can be considered as equivalent to a porous medium. There are many researches about the porous media such as the influence of compliant wall properties on peristaltic motion in two-dimensional channelwas noticed by Abd Elnaby and Haroun [9].Muthu et al. [10]have analyzed the peristaltic motion of micropolar fluid in circular cylindrical tubes with elastic wall properties.El Shehawey et al.[11] investigated peristaltic transport through a porous medium.Srinivas et al. [12] studied the effect of slip conditions and porous mediaon peristaltic flow.Hayat et al. [13]have investigated the MHD peristaltic channel flow of a Jeffrey fluid with complaint walls and porous medium.Slip-flow was demonstrated experimentally by Derek et al. [14].Hron et al. [15]have presented analytical solutions for the flows of a generalized fluid of complexity two in Military Technical College Kobry Elkobbah, Cairo, Egypt April 19-21,2016 8 International Conference on Mathematics and Engineering Physics (ICMEP-8) 4 special geometries under the assumption that the slip conditions at the boundary affect on the flow. The boundary conditions of Newtonianfluid on peristaltic motion have been considered by Kwang and Fang [16]. The magneto hydrodynamic (MHD) flow of a fluid in a tube with peristalsis has interest in connection with certain problems of the movement of conductive physiological fluids, the blood, blood pump machines and with the need for theoretical research on the operation of a peristaltic MHD compressor. Effect of a moving magnetic field on blood flow was studied by Sud et al. [17], and they observed that the effect of suitable moving magnetic field accelerates the speed of blood.Ebaid [18]studied the effects of magnetic field and wall slip conditions on the peristaltic transport of a Newtonian fluid in an asymmetric channel.The interaction of peristalsis with heat transfer plays a very important role in our life. Radhakrishnamacharya et al. [19] studiedthe effect of heat transfer on the motion of a viscous incompressible Newtonian fluid in a channel with wall effects. Nadeem and Akbar [20]have investigated the MHD peristaltic flow of an incompressible Newtonian fluid in a uniform channel with variable viscosity in the presence of heat transfer analysis.Kothandapani and Srinivas [21]have discussed the influence of elasticity of the flexible walls on the peristaltic motion of a MHD fluid in a porous medium with heat transfer.Radhakrishnamacharya Military Technical College Kobry Elkobbah, Cairo, Egypt April 19-21,2016 8 International Conference on Mathematics and Engineering Physics (ICMEP-8) 5 and Srinisvasulu[22]showed the influence of wallproperties on peristaltic transport with heat transfer.Taneja and Jain [23]have discussed the problem of an MHD free convection flow in the presence of a temperature dependentheat source in a viscous incompressible fluid between a long vertical wavy wall and a parallel flat wall with constant heat flux and slipflow boundary condition. To discover the real reasons of diseases in the human body, we use the endoscope theory. Thus the cathetershave an important role in modifying the pressure gradient and the discovery of these reasons in different situations such as small intestines and stomach. It has become a standard tool for diagnosis and treatment of certain cardiovascular diseasesin modern medicine. Vajravelu et al.[24]noted the peristaltic flow and heat transfer in a vertical porousannulus, with long wave approximation.Mekheimer and Abdelmaboud [25]studied the influence of heattransfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus as an application of anEndoscope.The purpose of the present paper is to provide such an attempt for MHD Newtonian fluid through porous medium. The features of flow characteristics are analyzed by plotting graphs. The paper has been organized as follows. In Section 1, the problem is first modeled and the non-dimensional governing equations are formulated. Section 2includes the solution of the problem under the longMilitary Technical College Kobry Elkobbah, Cairo, Egypt April 19-21,2016 8 International Conference on Mathematics and Engineering Physics (ICMEP-8) 6 wavelength and low-Reynolds number assumptions. Numerical results and discussion are presented in Section 3. The conclusions have been summarized in Section 4. Mathematical Formulation Impose the flow of a Newtonian viscous fluid through a twodimensionalsymmetricalcirculartube of radius . The walls of the tube are considered to be flexible and are taken as a stretched membrane, on which traveling sinusoidal waves of moderate amplitude are imposed. A rigid catheter of radius is symmetrically implemented. The flow is caused by theimposed sinusoidal wave on the compliant wall of tube. The wave shapes can be written as: ( ) 2 ( , ) sin w H z t a b z c t π λ   ′ ′ ′ ′ = + −     (1) Military Technical College Kobry Elkobbah, Cairo, Egypt April 19-21,2016 8 International Conference on Mathematics and Engineering Physics (ICMEP-8) 7 where b (0 ≤ b ≤ ( )) is the amplitude of the wave, ( ≤ L, the length of the tube under consideration) is the wavelength, is the wave propagation speed, t ′ is the time and z ′ is the axial coordinate. The momentum, continuity and energy equationsfor both the fluid and particle phasesare applied on the mathematical model as expressed in [23] and [24], neglecting the inertia force of the particles and can be written as: UAxial momentum (fluid) 2 (1 ) (1 ) o f f f f f f o f f u u u p C u v C u u t z r z K μ ρ σβ ′ ′ ′ ′ ∂ ∂ ∂ ∂   ′ ′ ′ ′ − + + = − − − − +   ′ ′ ′ ′ ∂ ∂ ∂ ∂   1 (1 ) ( ) ( ), f f s p f u u C C r CS u u r r r z z μ ′ ′  ∂ ∂  ∂ ∂     ′ ′ ′ ′ − + + −       ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂       (2) URadial momentum (fluid) Military Technical College Kobry Elkobbah, Cairo, Egypt April 19-21,2016 8 International Conference on Mathematics and Engineering Physics (ICMEP-8) 8 (1 ) (1 ) f f f f f f v v v p C u v C t z r r ρ ′ ′ ′ ′ ∂ ∂ ∂ ∂   ′ ′ − + + = − − +   ′ ′ ′ ′ ∂ ∂ ∂ ∂   2 1 (1 ) ( ) ( ), f f f s p f v v v C C r CS v v r r r r z z μ ′ ′ ′  ∂ ∂  ∂ ∂     ′ ′ ′ ′ − − + + −       ′ ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂       (3) UMass equation (fluid) ( ) ( ) 1 (1 ) (1 ) 0, f f r C v C u r r z ∂ ∂ ′ ′ ′ − + − = ′ ′ ′ ∂ ∂ (4) UAxial momentum (particle) ( ), p p p p p p f p u u u p C u v C CS u u t z r z ρ ′ ′ ′ ∂ ∂ ∂ ′   ∂ ′ ′ ′ ′ ′ + + = − + −   ′ ′ ′ ′ ∂ ∂ ∂ ∂   (5) URadial momentum (particle) ( ), p p p p p p f p v v v p C u v C CS v v t z r r ρ ′ ′ ′ ∂ ∂ ∂ ′   ∂ ′ ′ ′ ′ ′ + + = − + −   ′ ′ ′ ′ ∂ ∂ ∂ ∂   (6) UMass equation (particle) ( ) ( ) 1 0, p p r Cv Cu r r z ∂ ∂ ′ ′ ′ + = ′ ′ ′ ∂ ∂ (7) The energy equation can be expressed as: 2 2 2 2 1 (1 ) p r z T T T K T T T C c v v t r z r r r z ρ   ∂ ∂ ∂ ∂ ∂ ∂   ′ ′ − + + = + + +     ′ ′ ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂ ∂ ∂     2 2 2 2 (1 ) 2 , s r z r r z v v v v v C r z r z r μ ρ     ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂         − + + + +                 ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂             (8) Military Technical College Kobry Elkobbah, Cairo,

the international conference on mathematics and engineering physics/the international conference on mathematics and engineering physics

Peristaltic Flow of Particulate Fluid Suspension through Porous Medium under the effect of Slip Conditions, Heat transfer and Magnetic Field in Catheterized Tube