Role of Inclined Magnetic Field and Copper Nanoparticles on Peristaltic Flow of Nano fluid through Inclined Annulus:Application of the Clot Model

Iqra Shahzadiand S.Nadeem

Department of Mathematics,Quaid-i-Azam University,Islamabad 44000,Pakistan

1 Introduction

Nano fluid dynamics is appeared as the newly developed branch of fluid dynamics,which discovers sundry applications in biology,energetics,and medical science.The nanoparticles particularly metallic nanoparticles like copper have been broadly utilized for diagnosis,treatment,drug delivery,and medical device coating such as in Refs.[1–3].Nanofluids are the suspension of nanoparticles within the considered base fluid and were firstly investigated by Choi.[4]After him Akbar et al.[5]examined the contemporary study of ciliary motion under the impact of metallic nanoparticles.Here they examined the model for mucus layer Nadeem et al.[6]investigated the two phase nano fluid model in a curved channel.Due to the exclusive properties,nano fluids have gained considerable attention from investigators and latest studies explored the indispensable advantages of nano fluids in various biomedical applications as given in Refs.[7–15].

In peristalsis,sinusoidal wave propagation is responsible for flow generation through channel,tube or duct.This phenomenon has great utility in industry and biology.Some physiological examples include transport of bile,urine,chyme and food,cilia transport,male and female breeding cells through their specific tracks.Engineering applications of peristalsis include roller, finger and hose pumps,dialysis,food mixing processes,hear tlung machines,and so forth.Due to such usefulness and immense continuation of peristaltic flows,various investigations are introduced to examine the peristalsis by considering distinct flow con figurations.[16?21]

Flow of electrically conducting fluid under the impact of applied magnetic field is affected by electromagnetic forces apart from other body and surface forces.Mechanics as well as the thermodynamics of the system is influenced by electromagnetic forces.This situation is occurred in case of severe injuries during bleeding reduction.Several disorders in the human body are diagnosis with the help of different diagnostic test like magnetic resonance imaging.MHD has numerous applications in cosmology,astrophysics,sensors,geophysics,magnetic drug engineering and targeting.Sheikholeslami et al.[22]discussed the effects of magnetic field on natural convection of nano fluid through cubic cavity.These facts realize the importance of determining magneto hydrodynamic peristaltic transport of various fluids through different geometries.[23?28]

A genuine neurotic condition is experienced when some blood constituents saved on the wall of the artery get confined from the wall,again join the circulation system,and coagulation occurred.This can prompt to partial or even total blockage of the veins.Mekheimer and Elmaboud[29]examined the mathematical model defined for micropolar fluid of incompressible nature in a stenosed artery having coagulation inside it.In order to discuss such a serious situation and its repercussions regarding vortices,we have considered such an analysis.We intend to model and analyze the peristaltic transport of nano fluid in an inclined annulus under the influence of copper nanoparticles.Inclined magnetic field concept is utilized in the problem formulation.In fact,magnetic nanofluids have both the characteristic of fluid as well as the magnetic field.Such types of fluids have many significant applications such as adjustable filters,modulators,optical switches,gratings,cancer therapy,hyperthermia and drug delivery etc.The influence of magnetic nanoparticles on the tumor cells has been found to more adhesive than invigorating cells.Therefore,we have considered the copper as the nanoparticles.The present analysis is completed with the aid of lubrication approach and solved exactly.Some vital conclusions have been gotten on the premise of the present considered investigation.The impulsion of emerging parameters is presented via graphical illustration,trapping phenomena and tables.The results obtained from the examination have many biomedical engineering applications.

2 Formulation of the Problem

Consider an incompressible,laminar and viscous nano fluid in an annular region of inclined annulus between two coaxial tube in which inner tube have a clot on its walls.An external magnetic field of strengthB0is inclined at an angleαwhereas annulus is inclined at an angleη,respectively.The consequence of induced magnetic field are ignored for the condition of low magnetic Reynolds number.Nano fluid flow phenomena is consist of blood with copper as the nanoparticles.The central tube is maintained at temperatureT0while the outer tube has a sinusoidal wave of amplitudeband wavelengthλthat is traveling down through its wall with constant speedcand having temperatureT1.It is supposed that the thermal equilibrium is maintained between the nanoparticles and the base fluid.The coordinates(R,Z)are elected in such a way thatR-axis is along radial direction andZ-axis lies along length of the tube.Schematic geometry sketch is visualized through Fig.1.

Fig.1 Geometry of the problem.

The two wall surface geometry is described as:

whereR0is the radius of the outer tube,aR0is the radius of inner tube that keeps the clot inside the tube,bis the amplitude of the wave,f(ˉZ,′t)is the arbitrary shape that can be executed by suitable choice,λis the wavelength,cis the wave speed.

In the above equations,is the temperature of the fluid,is the pressure,ηis the inclination angle,B0is the strength of magnetic field,andQ0is the constant heat absorption or generation.For the proposed nanofuid model,ρnfis the density of the nano fluid,μnfis the variable nanofuid viscosity,βnfis the thermal expansion coefficient,σnfis the electrical conductivity of the nano fluid,Knfis the thermal conductivity of the nano fluid and(ρCp)nfis the nano fluid heat capacitance.

The effective viscosity,density and specific heat of nano fluid,are defined as,[14]

Here,ρ,β,cp,andφare the density,thermal expansion coefficient,specific heat and nanoparticle volume fraction,respectively.The subscriptsfandsare used to indicate the fluid and nanoparticle phases,respectively.Numerical values of the physical parameters are given in Table 1.The H–C model(Hamilton&Crosser,1962)for the effective thermal conductivity of nanofluids is used in this analysis.Hence the expression of effective thermal conductivity of nano fluids is given by

Table 1 Thermo physical parameters of blood and copper.

whereKis the thermal conductivity.In this modelndenotes the shape factor of nanoparticles given by 3/ψ,whereψis the sphericity of the nanoparticles and it depends on the shape of the nanoparticles.For spherical nanoparticlesn=3 orψ=1.For cylindrical nanoparticlesψ=0.5 orn=6.For the analysis in this study we assume that the nanoparticles have a spherical shape,i.e.,n=6.

In the fixed frame the no slip boundary conditions are defined as

In the above expressionsGr,?θ,Ren,γ,δ,M,Frrepresent the Grashr of number,dimensionless temperature,Reynolds number,dimensionless heat source parameter,wave number,Hart mann number,and Froude number respectively.After using the lubrication approach,the continuity equation is exactly satisfied and Eqs.(13)–(15)take the form:

The suitable no slip boundary conditions in the wave frame are given as

Dimensionless flow rateFin the laboratory frame is associated to the dimensionless flow rateqin the wave frame is defined as

whereq,q1,andq2are defined as,

The nondimensional form for the suitable choice of the clot model as suggested by Mekheimer and Elmaboud[29]is given as

where maximum height achieved by the clot atz=zd+0.5 is represented byζ,inner tube radius ratio that keeps the clot in position is represented byaand clot axial displacement is represented byzd.

3 Solution of the Problem

Equations(18)and(19)are solved exactly by utilizing the boundary conditions given in Eq.(20).The solutions for temperature and velocity pro file are as follow

whereC1,C2,C3,andC4can be calculated by using Mathematica.

We obtain the pressure gradient as defined below,

wherel1andl2can be calculated by using Mathematica.

4 Graphical Results and Discussion

The assurance of velocity pro file,pressure gradient,pressure rise,and streamlines outcomes in light of included parameters have been exhibited graphically in this section.These graphs are set up by controlling the parameters such asη=0?π/2,Gr=0.5?6,γ=0.1?0.9,φ=0?0.1,Ha=1.5?3.5,α=0?π/3,ζ=0.1?0.25,Re=0.2?2.5,andFr=0.2?1.5.Axial velocitywdescribes the parabolic trajectory against the radial distance for all the involved parameters.Figure 2(a)interprets the impact of inclination angleηon axial velocity.The values ofη=0,π/4,π/2 correspond to the horizontal,inclined and vertical annulus,respectively.The magnitude of the velocity increases in the presence of copper nanoparticles as we move from horizontal to vertical annulus having clot inside it.The amplitude of the velocity pro file increases in the regionr ?[0.1,0.7]and decreases inr ?[0.71,1.2].

The effects ofGronware found increasing from Fig.2(b)in the regionr ?[0.1,0.7]while the opposite trend is observed in the rest of region.This velocity development occurs due to decrease in viscosity.On physical grounds mixed convection is useful in nuclear reactor technology and electronic cooling processes where forced convection is not sufficient to dissipate energy.Figures 3(a)and 3(b)interpret the impact of heat source parameterγand Hartmann numberHaon axail velocity.The variations in heat source parameterγcause increase in the magnitude of velocity whenrlies in the region of 0.1≤r≤0.7 while decreases in the rest of region.The variations of Hartmann numberHaare given in Fig.3(b).It is observed that with an increase in the intensity of magnetic field the velocity pro file increases and then start decreases.It is due to the fact that when magnetic field is applied normal to the fluid,random motion of the particles within the considered base fluid gets slower down and hence flow of blood is retarded.The captured results of Fig.4(a)comprised of increasing impact of velocity pro file upon increasing values of magnetic field inclinationα.As we consider the effects of inclined magnetic field,the magnitude of the velocity increases and inclusion of copper nanoparticles increases the velocity pro file more prominently in comparison to pure blood case.Figure 4(b)is plotted to show the influence of clot heightζon velocity in the presence of copper nanoparticles and concluded that the velocity decreases by increasing the height of the clot which is physically true.It is due to the fact that the increasing values of?causes the resistance to the flow.

Fig.2 Velocity pro file for different values of(a)inclination angle η;(b)Grashroff number Gr.

Fig.3 Velocity pro file for different values of(a)heat source parameter γ;(b)Hartmann number Ha.

Fig.4 Velocity pro file for different values of(a)magnetic field inclination α;(b)Clot height ζ.

From Fig.5(a),it is analyzed thatwincreases with an increase in the values of Reynolds numberRe.Smaller values ofRemakes the fluid flow more laminar therefore the velocity increases by increasingRe.The velocity pro file for different values of Froude numberFris plotted in Fig.5(b)and observed that the significance of velocity increases with an increase inFrin the presence of copper nanoparticles.Change in pressure gradient for inclination angle,Grashoffnumber,heat source parameter,Hartmann number,mgnetic field inclination,clot height,Reynolds number and Froude number is observed through Figs.6–9.The effect of inclination angleηon dp/dzis displayed in Fig.6(a).It is found that upon increasingηthe amplitude of dp/dzincreases.

Fig.5 Velocity pro file for different values of(a)Reynolds number Re;(b)Froude number Fr.

Fig.6 Pressure gradient of distinct values of(a)inclination angle η;(b)Grashroff number Gr.

Fig.7 Pressure gradient of distinct values of(a)heat source parameter γ;(b)Hartman number Ha.

It is observed that the growth in pressure gradient is more prominent when we move from horizontal to vertical annulus.The effect of Grashof numberGron the pressure gradient is discussed in Fig.6(b)and noticed that with the growth of the buoyancy forces the pressure gradient increases in the presence of metallic nanoparticles.Figure 7(a)depicts that the increase in the heat source/sink parameter decreases the amplitude of pressure gradient since more heat is generated inside the considered base fluid.And decrease is more prominent for Cu-blood.Figure 7(b)represents the effect of Hartmann number(Ha)on dp/dz.It is investigated that the pressure gradient in inclined annulus gives higher height for copper nanoparticles than pure blood in the presence inclined magnetic field.

Figure 8(a)reveals that the variation of pressure gradient for an inclined annulus will decrease with the enhancing effect of magnetic field inclinationα.It is revealed that the decrease is more prominent for Cu-blood case in comparison to pure blood.Figure 8(b)represents the behavior of dp/dzversuszfor the different values of clot heightζand revealed that with the increase of clot height the pressure gradient increases under the influence of inclined magnetic field.

Fig.8 Pressure gradient of distinct values of(a)magnetic field inclination α;(b)Clot height ζ.

Fig.9 Pressure gradient of distinct values of(a)Reynolds number Re;(b)Froude number Fr.

Fig.10 Pressure gradient of distinct values of(a)inclination angle eta;(b)Grashroffnumber Gr.

Figure 9 indicates that the behavior of pressure gradient is inversely proportional to the Froude numberFrand directly related to the Reynolds numberReunder the combine effect of copper nanoparticles and inclined magnetic field.The pressure rise per wavelength is important to explained the pumping properly and sketched here from Figs.10–13.

Fig.11 Pressure rise for distinct values of(a)heat source parameter γ;(b)Hartmann number Ha.

Fig.12 Pressure rise for distinct values of(a)magnetic filed inclination α;(b)Clot height ζ.

Fig.13 Pressure rise for distinct values of(a)Reynolds number Re;(b)Froude number Fr.

One regular observation from these figures is that the pressure rise per wavelength reduces with the expansion in flow rate.It is important to note that inclusion of nanoparticles changes the free pumping flux(value ofqfor Δp=0).Figure 10(a)is plotted for the analysis of pressure rise for changing values of inclination angleηand observed that the pressure rise enhances when we move from horizontal to vertical annulus having clot inside it.Pressure rise increases more prominently in the retrograde region as compared to augmented region when the effects of nanoparticles are taken into account.Figure 10(b)shows the impact of Grashof number on the pressure rise against the flow rate.Important observation from this figure is that the increase inGrincreases the pressure rise in the retrograde pumping region(q＜0,Δp＞0)and decreases in augmented pumping region(q＞0,Δp＞0)when the effects of viscous forces are more prominent in the presence of copper nanoparticles and inclined magnetic field.The fluctuation of pressure rise versusqfor different values of heat source parameterγis given in Fig.11(a)and observed that addition of copper nanoparticles increases the retrograde pumping region more significantly than pure blood in the existence of clot.Figure 11(b)shows that the pressure rise increases with an increase in the Hartmann numberHa.It is depicted that by increasing values of Hartmann number pressure rise becomes an increasing function in the region(?1≤q≤ 0)whereas reverse behavior is seen in the rest of the region.Figure 12 describes the effect of magnetic field inclinationαand clot heightζon the pressure rise.It is declared from these figures that the pressure rise per wavelength is directly related toζbut inversely related toα.Furthermore,pressure rise enhsnces in the retrograde pumping region(q＜0,Δp＞0)with the increase of nanoparticle volume fraction.

Fig.14 Streamlines for Copper nanoparticles for distinct values of(a)Gr=4;(b)Gr=5;(c)Gr=6.

Fig.15 Streamlines for Copper nanoparticles for distinct values of(a)η=0;(b)η= π/4;(c)η=2/π.

Figure 13 describes the results obtained for the Δpversus the flow rateqfor increasing values of Reynolds numberReand Froude numberFr.It is seen that the pressure rise is an increasing function ofRewhile decreasing function ofFrand variations are more enhanced in the presence of metallic nanoparticles and under the effect of inclined magnetic field.Trapping,describing an interesting phenomenon for the blood flow pattern in an inclined annulus having clot is discussed in Figs.14–20 by considering the copper as nanoparticles.It is analyzed that the number of trapped bolus increases but the size of the bolus decreases by increasing Grashoffnumber with the inclusion of copper nanoparticles by the closed stream lines as shown in Fig.14.The influence of inclination angleηon streamlines are shown in Fig.15.It is seen that no bolus appears for horizontal annulus but size of bolus increases as we considered the inclined annulus having clot and than decreases for vertical case.From Fig.16,it is inspected that the number of trapping bolus increases when we increase the concentration of nanoparticlesφas contrast with pure blood case.It is important to note that the number of bolus decreases when we further increase the concentraion of copper nanoparticles.

Fig.16 Streamlines for distinct values of nanoparticle volume fraction(a)?=0.00;(b)?=0.05;(c)?=0.1.

Fig.17 Streamlines for Copper nanoparticles for distinct values of(a)Re=0.2;(b)Re=0.4;(c)Re=0.6.

Fig.18 Streamlines for Copper nanoparticles for distinct values of(a)Fr=0.1;(b)Fr=0.15;(c)Fr=0.2.

Figure 17 describes the impact of Reynolds numberReon the trapping phenomena.Increasing the values ofReincreases the number of bolus for Cu-blood.The trapping phenomena for Froude numberFris examined through Fig.18.It is inspected that the streamlines gets closer for increasing values ofFr.The number of the trapped bolus increases with increasing Hartmann numberHain the presence of copper nanoparticles as presented in Fig.19.The trapping phenomena for the heat source parameterγis given in Fig.20.The number of the trapped inner bolus decreases with an increase in the values ofγfor copper nanoparticles.

Fig.19 Streamlines for Copper nanoparticles for distinct values of(a)Ha=0.2;(b)Ha=0.9;(c)Ha=1.2.

Fig.20 Streamlines for Copper nanoparticles for distinct values of(a)γ=0.5;(b)γ=0.7;(c)γ=0.9.

Fig.21 Streamlines for Copper nanoparticles for distinct values of(a)ζ=0.1;(b)ζ=0.21;(c)ζ=0.25.

Fig.22 Streamlines for Copper nanoparticles for distinct values of(a)α=0;(b)α=π/4;(c)α=π/3.

Figures 21 and 22 describe the impact of clot heightζand magnetic field inclinationαon the streamlines pattern.The number of circulating bolus increases by increasing clot heightζwhereas opposite behavior is observed for magnetic field inclinationα.Table 2 shows the variation of temperature pro file for increasing values of heat source parameterγ.It is inspected that the enhancing values of heat source parameterγenhance the temperature of the considered base fluid through metabolic system.

Table 2 Variations of temperature pro file for different values of heat source parameter γ.

5 Conclusions

Impact of inclined magnetic field and copper nanoparticles on the peristaltic flow of nano fluid through inclined annulus having clot inside it is discussed in this analysis.Some crucial observations of the considered analysis are listed below

?Amplitude of velocity pro files exhibit higher results for copper blood than for the pure blood.

?Velocity for viscous fluid flowing through an inclined annulus is less in comparison to that of a nano fluid in the presence of copper nanoparticles.

?Pressure gradient for the flow in a horizontal annulus is lower when compared with that of an inclined and vertical annulus.

?With an increase in Hartmann number the pressure rise enhances in the retrograde pumping region while opposite behavior is observed in the augmented pumping region.

?For the flow through an inclined annulus the pressure rise decreases more for inclined magnetic field when compared with the constant magnetic field.

References

[1]R.U.Haq,Z.H.Khanb,S.T.Hussain,and Z.Hammouch,J.Mol.Liq.221(2016)298.

[2]S.Ijaz and S.Nadeem,Comput.Meth.Prog.Bio.134(2016)43.

[3]S.U.Rahman,R.Ellahi,S.Nadeem,and Q.M.Z.Zia,J.Mol.Liq.218(2016)484.

[4]S.U.S.Choi and J.A.Eastman,ASME Int.Mech.Eng.Cong.Expos.66(1995)99.

[5]N.S.Akbar and A.W.Butt,Comput.Meth.Prog.Bio.134(2016)43.

[6]S.Nadeem and I.Shahzadi,Commun.Theor.Phys.64(2015)547.

[7]N.S.Akbar,Comput.Meth.Prog.Bio.132(2016)45.

[8]S.Nadeem and Iqra shahzadi,Int.J.Heat Mass Trans.97(2016)794.

[9]T.Hayat,S.Farooq,B.Ahmad,and A.Alsaedi,Int.J.Heat Mass Trans.103(2016)1133.

[10]M.Atlas,R.U.Haq,and T.Mekkaoui,J.Mol.Liq.doi:10.1016/j.molliq.2016.08.032.

[11]Y.Xuan,ASME J.Heat Transf.125(2003)151.

[12]H.C.Brinkman,J.Chem.Phys.20(1952)571.

[13]R.K.Tiwari and M.K.Das,Int.J.Heat Mass Transf.50(2007)2002.

[14]R.U.Haq,Z.H.Khanb,S.T.Hussain,and Z.Hammouch,J.Mol.Liq.221(2016)298.

[15]I.Shahzadi,H.Sadaf,S.Nadeem,and A.Saleem,Comput.Meth.Prog.Bio.139(2016)137.

[16]S.Nadeem and N.S.Akbar,Int.J.Numer.Meth.Fluids 66(2011)919.

[17]A.H.Shapiro,M.Y.Jaffrin,and S.Wienberg,J.Fluid Mech.37(1969)799.

[18]T.Hayat,S.Farooq,B.Ahmad,and A.Alsaedi,Int.J.Heat Mass Transf.103(2016)1133.

[19]N.S.Akbar,M.Raza,and R.Ellahi,European Phys.J.Plus 129(2014)155.

[20]Kh.S.Mekheimer,S.Z.A.Husseny,and A.I.Abdellateef,App.Bion.Biomech.8(2011)1.

[21]T.Hayat,S.Farooq,B.Ahmad,and A.Alsaedi,Int.J.Heat Mass Transf.106(2017)244.

[22]M.Sheikholeslami and R.Ellahi,Int.J.Heat Mass Transf.89(2015)799.

[23]S.Nadeem and I.Shahzadi,AIP Advances 6(2015)015110.

[24]M.M.Bhatti,R.Ellahi,and A.Zeeshan,J.Mol.Liq.222(2016)101.

[25]A.Zeeshan,A.Majeed,and R.Ellahi,J.Mol.Liq.215(2016)549.

[26]T.Hayat,S.Farooq,B.Ahmad,and A.Alsaedi,J.Mol.Liq.223(2016)469.

[27]I.Shahzadi and S.Nadeem,J.Mol.Liq.225(2017)365.

[28]T.Hayat,S.Farooq,B.Ahmad,and A.Alsaedi,Results in Physics doi:org/10.1016/j.rinp.2016.12.048.

[29]Kh.S.Mekheimer and Y.Abd Elmaboud,Appl.Bionics Biomech.5(2008)13.