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A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field: EMF Effec

2021-03-08 来源:九壹网
0 Journal of Physical Science and Application 2(4)(20 1 2)6 1・70 龠Ll HlNG D A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field:EMF Effect on Blood Flow Mona A.E1.Naggar and Medhat A.Messiery Engineering Physics and Mathematics Department,Faculyt ofEngineering,Cairo University,Greater Cairo,Egypt Received:February 02,2012/Accepted:March 05,2012/Published:April 15,2012 Abstract:The present work introduces a mathematical model for ionic luifd that lowsf under the effect of both pulsating pressure and axial electromagnetic field.The fluid is treated as a Newtonian fluid applying Navier—Stokes equ ̄ion.The fluid is considered as a neutral mixture of positive and neg ̄ive ions.The effect of axial electric field is investigated to determine velocity profiles. Hydroelectric eauation of the lfow is deduced under dc and ac external electric ifeld.Hence the effect of applied frequency(O-1 GHz) Yth £ and amplitude f10.350 V/m)is illustrated.The ultimate goal is to approach the problem ofEMF ielfd interaction wiblood flow.The applied Dressure waveform is represented as such to simulate the systolic-diastolic behavior.Simulation was carried out using Maple software using blood plasma parameters;hence velocity profiles under various conditions are reported. Key words:Newtonian fluid,navier・stokes,ionic fluid,blood flow,electric field,hydroelectric flow,simulation,velocity profiles EMF bioeflfects. Nomenclature f: Greek Letters Electrical conductivity Mass density Time Periodic time 尺 C: Radius Viscosiy COCffticient Shear rate PermiRivity of the ionic fluid. Positive ionic concentration Negative ionic concentration Sound velocity The pressure gradient Bodyforce Radia1 force Axia1force Driving force Magnetic permeabilitiy Radial velocity 1.Intr0ductiOn Recently,hemodynamic properties depend greatly on mathematical analysis of fluid dynamics. Hemodynamic variables are known to impact the Axial velociy tAxial velocity when the ionic concentration isp + Axial velocity when the ionic concentration is PP. Capactitance Axial current density Radial current density genesis of atherosclerosis and formation ofthrombosis. It is possible that many cardiovascular diseases can be detected and diagnosed by analyzing the profiles of the Axial electric field Maximum applied electric field Angular magnetic ielfd velocity components.Being an important field of research for quite a while,various mathematical models have been adopted to reproduce blood velocity profiles in different blood vessels.Navier—Stokes Maximum applied magnetic field Reynold’s number Power density equation is one of the most commonly used analytical Corresponding author:Mona A.E1-Naggar,assistant professor,research fields:biophysics,molecular,atomic physics methods to study hydromagnetic and hydroelectric nd aelectromagnetics.E—mail:monaelnaggar@msn.eom. 62 A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field EMF Effect on Blood Flow low.The equatifon is applicable as such to produce velocity components of electrically conducting pulsatile flow phenomena are quasi—steady,while viscous stress and pressure gradient forces experience an instantaneous balance.The plasma layer has a Newtonian lfuids subjected to magnetic,electric ifelds, or both.Blood lfow in large arteries,can be analogous to ionic fluid flow in a cylindrical vesse1. The main issue facing scientists and mathematicians while attempting to simulate the flow in blood vessels is that blood flow rheology is complex.The complexity of which arises from the viscosity property that is not only vessel dependent but shear dependent as wel1.The axial velocity magnitude depends greatly on the wall viscoelastic properties【1],besides the initial distribution of velocity.The wall viscosity affects greatly the shear forces and hence shear rate,whereas the elasticity affects the magnitude of momentum and strain energy relayed to the successive arterial section 【2,3].However,it is obvious that the blood flow models.introduced mainly to deduce hemodynamic variables.mostly neglect the arterial wall viscoelastic properties.Numerical techniques give different models to predict different results of hemodynamic variables. One of the major problems in studying the blood lfow is that its viscosity varies not only with its speed but also with the diameter of the blood vesse1.The current models are far from being comprehensive and lacks consistency hence one can state that the problem lacks general agreement upon a mode1.However, under the general classification of non.Newtonian models,that simultae blood behavior to di fferent degrees of accuracy,there are many variants.The power law.Casson.Carreau and H.B models are the most popular non・Newtonian simulation models[4—9]. The bloodstream is sometimes modeled as a non-Newtonian core with a much 1ess viscous Newtonian walllayer which is known as the two—fluid model¨01.The Casson model has been used to represent the steady flow of blood,in vitro,extending the fluid model with yield stress.shear.dependent viscosity,and a power law of one half,to the pulsatile lfow in arterioles.venules.and capillaries.In the small vessels considered,inertial effects are negligible, surprisingly large lubricating effect during periods when the effective shear viscosiyt is signiifcantly higher than the ultimate high shear viscosity[4]. A mathematical model has been presented for periodic blood flow in a rigid circular tube of thin diameter[5].Blood is presented as a 3一layered lfuid by considering core fluid as a Casson fluid which is covered by a thin layer of Newtonian fluid(plasma). Taking into account the low and high shear rate behavior of blood,the appropritae parameters are empirically determined.Experimental fit results,for each non-Newtonian model,show the dependence of viscosity on the shear rate【6]. The power law,Casson and Carreau models are the most widely used simulation models giving various viscosity shear dependent functions.For a Newtonian lfuid,the shear rate,y,is considered high with constant viscosiyt of blood,77=O.035 Poise. Another two—fluid model with the suspension of all the erythrocytes in the core region as a non—Newtonian lfuid and the plasma in the peripheral layer as a Newtonian fluid one,is represented by Sankar and Lee 【4,8,1 0,1 3].The non—Newtonian fluid in the core region ofthe artery is represented by both Casson fluid and Herschel—Bulkley(H—B)fluid.It is noted that the plug—lfow velocity,velocity distribution,and lfow rate of the two.-lfuid H-・B model are considerably higher than those of the two—fluid Casson model for a given set of parameters[1 0].A velocity magnitude distribution of the grid cells shows that the Newtonian mode1 is close dynamically to the Casson model while the power law model resembles the Carreau model[5]. Since the use of a power—law model to describe velocity profiles does not provide a consistent rheological description,Das et a1.[9]used the Casson model,in vitro,on rat spinotrapezius muscle.Also,it is shown that the two—phase Casson model with a peripheral plasma layer is in quantitative agreement A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field: EMF Effect on BIood Flow 63 with experimentally obtained velocity profiles in venules of rat muscle under low flow rate『81. Chaturani and Samy[1 1]have mentioned that for tube diameter 0.095 mm blood behaves like H.B fluid rather than power—law and Bingham fluids.Iida[1 2] reports that velocity profile in the arterioles having diameter less than 0.1 mm are generally explained fairly by the two models.However,velocity profiles in the arterioles whose diameters are less than 0.065 mm do not conforrn to the Cassort model but can still be explained by H-B fluid mode1.Moreover,H-B fluid model can be reduced to power-law fluid model when the yield stress is zero and Bingham fluid mode1 when its power-law index takes the value one.Thus.the two.fluid H.B model has more suitability than the two.fluid Casson model in the studies of blood flow through narrow arteries.For the two.fluid H.B model and the expressions for the flow quantities obtained by Sankar【1 3】for the two—lfuid Casson model are used to compare these fluid models and bring out the advantage of using the two-fluid H—B model over two.fluid Casson model for blood flow in catheterized arteries.Non—Newtonian behavior also affects the wall shear stress predicting larger values than the Newtonian models【13-15]. Other mathematical models consider blood as a two—phase fluid,an RBC suspension in Newtonian incompressible plasma[16—18].In this type ofanalysis, the RBCs are assumed to be rigid.neutrally buoyant spherical particles,and their specific gravity compared to that of plasma diminishes the effect of gravity on blood. There is experimental evidence of the electric and magnetic fields effect on the haemodynamics of blood. InternationaI institutes have recorded maximum endurable dose of electric, magnetic and electromagnetic fields.The effects of commercial mobile phone signals,GSM,on various physiological processes have been the field of investigation and research ofr quite a while.These signals are proven to affect the cerebral blood flow in healthy humans using positron emission tomography(PET)imaging[1 9,20]. The pulse modulated RF electromagnetic field,emitted by mobile phones is also proven to affect the EEG signals during sleep and wakefulness【2l-23]. Continuous wave RF electromagnetic ifelds also affect cognitive performance[24].Furthermore,EMF effects on microcirculatory system in different tissues in experimental animals are estimated in SAR for static magnetic ifeld ranges of 0.3-l 80 mT[251. For static magnetic fields,acute effects occur when there is movement in the field,such as motion of a person or internal body movement,such as blood flow or heart beat[26].Static magnetic ifelds exert forces on moving charges in the blood,such as ions,generating electrical ifelds and currents around the heart and major blood vessels that can slightly impede the lfow of blood Possible effects range from minor changes in heartbeat to an increase in the abnormal heart rhythms.that might cause ventricular ifbrilltaion[271.These types of acute effects are only likely within fields in excess of 8 T. To date.it is not possible to determine whether any long—term health consequences exist from exposure in the millitesla range because there are no well・-conducted epidemiological or long・-term animal studies.The recommended limits are time.weighted average of 200 mT during the working day for occupational exposure,with a ceiling value of 2T.A continuous exposure Iimit of 40 mT is given for the general public.Static magnetic fields affect implanted metallic devices such as pacemakers present inside the body,and this could have direct adverse health consequences.It is suggested that wearers of cardiac pacemakers,ferromagnetic implants like stainless steel stents and implanted electronic devices should avoid locations where the field exceeds 0.5 mT.EMF exposure.with ranges of 1.5—2 kV/m with frequencies Of 1-l O0 GHz.can interfere with the proper operation ofpacemakers.Also microwave exposure(1 mW/cm ) are recorded to cause reversible nervous system disturbance and blood chemical alterations『28]. In the present study,we adopt a mathematical A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field EMF Effect on Blood Flow simulation mode1 to produce results representing the velocity profiles in axial and radial direction for a purely ionic fluid.The main aim of the modelis to flow is assumed purely ionic.Furthermoreat low shear .rates,that is,y<100 s~,the RBCs aggregate and orfm what is known as rouleaux.Aggregation disperses as study the interaction of incident electric field.both dc and ac,with the ionic fluid.In this study,we consider ionic viscous steady Newtonian fluid that flows in a relatively large rigid cylindrical vessel under incident the shear rate increases,reducing the viscosity of blood However,as the shear rate increases,the shear.thinning characteristics disappear and blood demonstrates a Newtonian behavior f81. ifelds.For the sake of analogy to the blood flow simulation flow parameters follow the actual recorded blood data.If blood is assumed as a whole ionic fluid where the wall shear stress is considered to result in a relatively high shear rate,therein the present work is a close simulation.The flow is through a rigid large arterial section.where rouleaux is not likely to affect the flow.In addition blood flow is pulsatile because of the nature ofthe heart pumping actions which results in lfuctuating pressure gradient.The pressure gradient applied follows the natural systolic—diastolic profile. These assumptions make computational and analytical effort more manageable 2.Model In the present work,analysis of the ionic flow is aiming to be analogous to blood flow in large vessels. To implement this,two main assumptions are adopted. Firstly,the flow is considered to be Newtonian,though blood exhibits an evident non.Newtonian behavior especially in small blood vessels.The blood,being an anamolous fluid,can be represented by the famous two—fluid,Casson or H.B model『4—91.On the other hand,for relatively large cylindrical vessels,the friction with walls becomes less effective.Hence the shear stress is assumed negligible.Also for high shear rates,as expected in a very large vessel,the most commonly used models nearly coincide giving constant viscosity coefifcient value[6]. The choice of the Newtonian model is suitable for the assumption that the shear rate,1,(s-1),in large blood vessels of radii r( 2 mm),is above 1 00 s 【7】.These models show differences within the range of less than 2O%for shear rates higher than l 50 s~.Secondly.the 3.Mathematical Analysis The present work assumes a Newtonian incompressible ionic fluid flowing with velociyt v= r,o, z),in a cylindrical section shown in Fig.1.We start with the Navier-Stokes equation: p Dv=一 +,+ ×[,lfy×v)】+ [ .v](1) where P is the mass densiyt,77 is the viscosiyt coefifcient,Vp the pressure gradient and F is the body force.For incompressible fluids the continuity equation; .v=0is satisfied,hence: .JD Dv=・7p+F+V×【77( ×V)】(2) Adopting the Newtonian Model,the viscosiyt being constant and shear independent,as mentioned above, hence: JD D1J=一7p+F+77V v (3) To produce a time varying velocity distribution along the axis and across the radius,the cylindrical COOrdinates would be suitable to work with zero incident field. A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field: EMF Effect on BIood Flow 65 p =一 Op+F+77[ +{ (4) ∞)=而 (10) Boundary condition to be satisfied is the no slip where、)_《t)and (t)are the axial velocities when the ionic concentration is inserted in Eq.(6)as P or PP. respectively,and£is the permittiviy of the itonic luid.Furthermore,the radialf force component, ,acts boundary condition for the velocity z-component;Vz 0 at =足is applied to the cylindrical flow.Besides, the symmetryabout z-axis imposes the condition; = on the positive and negative ions in opposite directions 0 at r=0.While adopting the Poiseulle parabolic flow creating a time dependent cylindrical capacitor. profile as a radial distribution,Eq.(4)is solved, imposing zero initial condition.The velocity in the z-direction varies with time,,,and radius,r,as follows: vz(rJ£)=Tfo(t)(尺z—rz)(P一 4t一1)(5) The function fo(t)represents the driving force: 厂0(c)= 一 ) (6) 3.,Hydroelectric Flow An incident axial electric field assumed to be applied, 量 (o.o E:).This field essentially induces a magnetic field that circulates in the x-Y plane,Bi d= Bo, .For the hydroelectric flow under investigation, the body force, ,is expressed as electric and magnetic body forces due to the incident and induced ifelds.If the volumetric ionic charge density is±pP, hence; Fz=P +V ) (7) Originally,the current density vector for the flow of positive charges in the axial direction and equals: =1.6 x 10一 x Pe x (8) An axial current densiyt, ,is generated due to the incident electric and induced magnetic fields: = +vr ̄Bo) (9) where o-is the electrical conductivity of the fluid. Hence a radial force component,Fr,is generated;Fr = ・B 3.2 Hydroelectric Capacitance An instantaneous capacitance is created due to the separation of the positive and negative ions that move along the axis with different velocities.The value ofthe cross sectional capacitor created after a time interval, At(s)is: 4.Results For the sake of analogy with the blood flow in a large arterial vessel,the parameters used here are those reported for blood.The concentration of different ions in plasma is given in Ref.【7】.The blood plasma can be considered as a two-fluid ionic solution one ofpositive electric charge concentration Pe and another of negative charge concentration PP.To simulate the blood flow in a large arterial—like section of radius,R= 1 cm,average reported parameters for plasma are as given below[4-7]:P=1,030 kg/m ,77=0.0035 kgm。 s~,Pe一=0.1269 c/m3,Pe+=0.1407 c/m The axial velociyt,Vz(m/s),is represented against time,t(s)for different radii values,r(m).Fig.1 shows the three dimensional representation of the axial velocity versus both time and radius without any external field under a constant pressure gradient of, =.40 Pa/m.It shows a parabolic distributi0n of oz ‘ maximum 0.25 m/s hence delivering a volumetric rate 0f5.2 1,s. 4 1 The Flow under Constant PreSSUFC Gradient The effect of the exposure to an electric field incident along the axial flow is investigated.Firstly,the pressure gradient is assumed constant throughout the arterial section having the same pressure gradient value as above.The axial electric field,Ez,is applied as a steady dc field and varied from 1 00 V/m to 0.35 kV/m. The axial velocity V:is represented against time and radius in Fig.2,for incident electric field 1 00 V/m. While Fig.3 shows the velocity distribution for an A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field EMF Effecton Blood Flow radius under 100 V/m incident ifeld. 350 V/m incident field. incident field of 3 50 kV/m against the cylindrical radius. The axial velocity is represented for both positive and negative ionic distribution.The separation of the lfuid layers is obvious in Fig.2. A force, acts in positive z-direction, on Pe+,produces V of maximum 0.32 m/s,whereas an opposing one acts on PP一,produces V。of maximum 0.1 89 m/s.At a threshold electric field value of 0-3 l kV/m,the negatively charged ionic lfuid layer starts to reverse its direction.As Fig.3 shows the reversing of the negatively charged ionic layer under the effect ofan incident electric ifeld 350 V/m。where V。max--- 02 m/s andv m =0.64m/s. .2 The Flow under Pulsating Pressure Gradient and DC Eiectric Field Secondly,for simplicity the pressure gradient is assumed to be pulsating in sinusoidal manner with periodicity In this case,the pressure gradient takes a mathematical expression to simulate as much as possible the natural biphasic waveform of heart beat. The pressure is assumed a fluctuating function of periodicity,T=750 ms: ap__clz =-40 se孔(。\ 了2丌亭1)/ (、1。 1) Under these conditions three different cases are studied.Namely,the effect of applying dc electric field along the axis。with constant values O 1 0 V/m.1 00 V/m。250 V/m.The velocity waveforms for the positive ionic layer,V z ,are shown in Fig.4.The root mean square velociyt values, m are 0.2009 1 4 m/s.0.22 l 57 m/s and 0.3 l 304 m/s。respectively. .3 The Flow under Pulsating Pressure Gradient and CElectricFie/d The effect of applying AC electric field in the cylindrical section along the axial direction is studied. In this case,the electric field takes the sinusoidal form of =Eosin(e ̄fo. Thus the induced magnetic field is:Bo= cos(2 ̄fO,where£is the permittivity of the ionic lfuid assumed as 0.2656×1 0 加F/m, is its magnetic permeabilitiy,1.2547×1 0。。H/m and f is the frequency ofthe field.For E0=100 V/m and frequency 50 Hz, the simulated velociyt waveform,vz 0,is as shown in Fig.5. It is obvious that the velocity profile exhibits a complex waveform that includes both the pressure wave frequency together with that of the incident electric ifeld besides other harmonics. For different values of the frequency, namely:50 Hz,1 kHz,1 MHz and 1 GHz,the positive ionic layer waveform is represented in Fig.6.Harmonics for the 68 A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field EMF Effecton BIood Flow (a) (d) Fig.6 Axial positive layer velocity, ,under incident ielfd:100 Vim and 50 Hz: (a)f=50 Hz(b)f=l kHz(c)f=1 MHz(d)f =1 GHz. dit_ferent cases are evident while the time axis scale IS enlarged to give a closer view. The root mean square values of the axial velocity component for both positive and negative ionic layers are calculated for dif_ferent Values Of DC and AC field strengths.It is worthwhile to note that beyond the threshold DC electric field strength value given above, namely 330 V/m,the velocity distributions of ionic layers exhibit a 1 80。phase shift.These values are given in Table 1.together with the Reynold’S number. Reynold’S number iS below the turbulent flow value which justiifes the adopted Newtonian approach.This is the case orf different ranges of electric ielfds applied at different frequencies up 1 GHz.Also it applies to both types of ions.For constant ac electric field strength of 1 00 V/m,the Reynold’S number shows consistency though the frequency changes from 1 kHz up tO l GHz.This indicates that the steadiness of low fis not affected by the change in frequency.And consequently the velocities of both positive and negative ionic layers are unaffected.Electric field Furthermore,the power density,Sr,is calculated as the rms value of, Bo.The power densiy,itn mw/m , strength shows a threshold of 350 V/m where Reynold’S number is critical and turbulence can occur at any slight increase of the field. It is noteworthy to point out that the associated is calculated as S0 .It S time variant,dependent on the incident ifeld parameters and dissipated through the radial direction. magnetic field creates forces on the positively charged 5.Discussion Table 1 shows that under the given conditions, particles radial inward while the negatively charged particles experience outward radial forces.This may A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field EMF Effect on Blood Flow 69 Table 1 Flow variables. lead to charge separation along the radial direction. Under a certain value of the magnetic field one can 【3】 M.A.EI-Naggar,An investigation on silicone and silicone rubber stents in comparison with stainless steel stents,in: Proc.of the 5th international workshop on the biological effects of electromagnetic ielfds,Terrasini,Palermo,Italy, 2008. speculate that the positive ions would coincide with the axis while the negative ions stick to the inner wal1. Another time dependent capacitance appears axially along the flow due to the formation of oppositely [4】D.S.Sankar,U.Lee,Pulsatile lfow of two—lfuid nonlinear models for blood flow through catheterized arteries:A comparative study,Journal of Mechanical Science and charged disc-like ionic layers expressed as in Eq.(1 0) and illustrated in Fig.2 for the DC field case. Technology 23(9)(2010)2444-2455. 【5】 R.P.Maurya,H.Khas,Peripheral layer viscosity effects on periodic blood flow through rigid tube ofthin diameter, Journal of Mathematical Analysis and Applications 124 6.Conclusions The ultimate purpose of the present investigation is to study the effect of external electric field on the blood low.Therefore we are keen fto use as much as possible and available,the reported physical data concerning (1987)43-51. 【6】 J.Aroesy,J.tF.Gross,Pulsatile lfow in small blood vessels, Mathematical Problems in Engineering(20 1 0)2 1. 【7】 S.S.Shibeshi,W.E.Collins,The rheology ofblood lfow in a branched arterial system,Appl Rheo1.1 5(6)(2005) 398.405. human blood.There are two major obstacles in building up a model of blood flow namely vessel viscoelasticity and non circular irregular vessel cross section.However this work introduces a simplified approach to the problem.Hence we establish a strong correlation between incident axial electric field and velocity profiles under difie rent conditions. 【8】D.S.Sankar,U.Lee,An open access article distributed under the Creative Commons Attribution License,Clin Hemorheol Microcirc 36(3)(2007)217-233. 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