Physiological Flow of Jeffrey Six Constant Fluid Model due to Ciliary Motion

A.Shaheen,S.Hussain,and S.Nadeem

1Department of Mathematics,Capital University of Science and Technology,Islamabad,Pakistan

2Department of Mathematics,Quaid-i-Azam University,Islamabad,Pakistan

1 Introduction

In fluid mechanics,Cilia function in low Reynolds number environment where inertia is negligible.A cilium means an eye lash in Latin,is motile hair like slender that projects from free surface of certain cells.Common in single cell organisms known as eukryotes.These hair like structures wave to move cell or to move something around the cell.Certain tissues like the fallopian tubes in women,the trachea,the ductus efferentes of human males reproductive tract also have a distinct type of cilia that help in the movement of substances along the tissues surfaces.[1]There are two types of Cilia,motile and non-motile cilia.Cilia which lie on the tissues surface are responsible for protecting a person from germs in the lungs are called motile cilia and are found in groups.Whereas,primary cilia are usually found only one at a time on cells.Cilia play an important role in many psychological processes such as locomotion,alimentation,circulation,respiration,and reproduction.Cilia operate in a periodic two-phase movement of effective and recovery stroke.An effective stroke is executed when a cilium extends itself into the fluid and drags the maximum volume of the fluid forward.Whereas,during a recovery stroke a cilium bends towards itself.Ciliated surfaces can have different patterns depending upon the direction of propagation.When the propagative direction of the metachronal wave is the same as the direction of the effective stroke,the beat coordination is called symplectic.If instead both directions oppose each other than the coordination is termed antiplectic.[2]The interaction of cilia and its propulsion has attained much research efforts by physicists and engineers.[3?4]Noreen Sher Akbar et al.[5]represented a detailed study on the functionality of primary cilia,their signaling,cell cycle and also different diseases which developed due to the dysfunctional cilia like tumorigenesis,syndomes etc.Lardner and Shack[6?7]developed a model for movement of viscous fluid due to ciliary activity in the ductus efferentes of male reproductive tract.Velez–Cordero and Lauga[8]explained the envelop model of cilia in a generalized Newtonian fluid by employing a domain perturbation expansion.Ciliary motion its modeling and the dynamics of multi cilia interactions were discussed by Gueron and Liron.Rydholm,et al.[9]represented the mechanical characteristics of primary cilia.They analyzed that a primary cilia in kidney epithelial cells have been observed to generate intercellular calcium in response to fluid flow.They have investigated the dissipative flow and heat transfer of Casson fluids due to metachronal wave propulsion of beating cilia with thermal and velocity slip effects under an oblique magnetic field. Noreen Sher Akbar et al.[10]have discussed the anti-bacterial applications for new thermal conductivity model in arteries with CNT suspended nano fluid.Copper oxide nanoparticles analysis with water as base fluid for peristaltic flow in permeable tube with heat transfer is presented by Noreen Sher Akbar et al.[11]They also investigated the cilia bending and the resulting calcium signal.Movement and locomotion of microorganisms by considering a ciliary motion was explained by John.[12]Chilvers and Callaghan[13]studied the relationship of the power recovery stroke of respiratory cilia using digital hi speed video imaging and then compared the obtained frequency measurement with those attained by photo multiplier and modied photo diode.Makende,[14?16]represented the Heat and mass transfer in a pipe with moving surface:E ff ects of viscosity variation and energy dissipation.Study of heat transfer on physiological driven movement with CNT nano fluids and variable viscosity is presented by Noreen Sher Akbar et al.[17]Motivated by the above work the aim of present endings is to treatise the fluid movement through the ductus differents of the human male reproductive tract.The problem of the two-dimensional motion of non-Newtonian fluid inside a symmetric metachronal wave channel with ciliated walls is discussed.Bio mathematical venture for the metallic nanoparticles due to ciliary motion is presented by Noreen Sher Akbar et al.[18?19]The ciliary system properties are consequented below the impact of low Reynolds number and long wave-length approximation.The modeled equations are solved analytically by homotopy perturbation method First step is that the problem is modelled HPM[20]solution is considered for the resulting equation.The results for velocity pro file,pressure gradient,pressure rise and stream function have been considered for different values of the parameters.The physical features of pertinent parameters are discussed through graphs.The streamlines are sketched for some physical quantity to examine the trapping phenomenon.

熱加工處理后花生中總蛋白含量結(jié)果如圖1所示，采用凱氏定氮法測定水煮前后花生中的總蛋白含量發(fā)現(xiàn)，未加工的鮮花生含蛋白11.16 g/100 g鮮花生，直接水煮和曬干水煮后，蛋白質(zhì)含量分別為11.67 g/100 g鮮花生和11.14 g/100 g鮮花生，三者間蛋白含量沒有顯著差異。Mondoulet L[29]等的研究認(rèn)為，水煮過程中可能會有蛋白損失到水煮液中，本研究也測定了水煮液中的花生蛋白含量。結(jié)果顯示，約200 g鮮花生經(jīng)過直接帶殼水煮或曬干后帶殼水煮，收集的水煮液中花生蛋白量分別為12.00和11.42 mg，可見蛋白溶解進水煮液的數(shù)量很少，故在水煮處理過程中蛋白的損失量可以忽略。

This paper is organized as follows:In Sec.3,we discuss the details of the formulation of the problem.Section 4,describes the proposed solution methodology for the governing systems of partial differential equations.Section 5 presents the numerical results and discussions.In Sec.6,the paper is concluded with a discussion of the results.

2 Model of the Problem

We have considered the ciliary motion phenomenon for the two-dimensional flow of an incompressible in an annulus.The fundamental equations of continuity,momentum,energy and concentration are

The equation of mass transfer and heat transfer,as well as the viscous dissipation effects are given as

where

上海APM線特殊人群專用通道已應(yīng)用掌靜脈識別技術(shù)，通過掌靜脈識別特殊人群，方便特類人群乘坐地鐵外，降低了冒用證件的票務(wù)處理難度。今后，上海地鐵漢中路、諸光路等智慧車站還將實現(xiàn)特類人群使用掌靜脈識別乘地鐵。上海地鐵未來還將探索生物識別技術(shù)，利用生物特征作為虛擬車票的過閘研究。

where velocity field is V,density constant is ρ,material derivative d/dt,Cauchy stress tensor is T,PI is the cylindrical part of the tensor andˉτ is the extra stress tensor of Jeffrey six-constant fluid.

Flow rate in the dimensionless form can be written as

We suppose for small Reynolds number Re?1 and by the Long-wavelength approximation δ? 1 the flow inside the passage is very slow.Therefore,neglecting the non-inertial terms we get

in which

where d,b,c are material constant of a Jeffreys sixconstant fluid model,T1represents the transpose,relaxation time is ?1and delay time is ?2.

3 Problem Formulation

Let us examine a two-dimensional passage of in finite length having ciliated walls.The symplectic metachronal wave generated as a result of ciliary motion is moving in the Z-direction onward the passage c is wave velocity and normal direction is R.The geometry of the problem is characteristics for a Jeffrey six constant fluid as an action of the metachronal wave velocity and cilia.The metachronal wave arrangement canvas proposes that the cilia tips enclosure may be represented mathematically as.

where a is the average breadth,? is the non-dimensional quantity in accordance to the cilia length a,length is λ and velocity of the wave generated by the cilia is c.The cilia tips horizontal position may be expressed as

If the no-slip condition applies on the walls of the channel,then the velocities imparted to the fluid particles are the same as those of the cilia tips.The horizontal velocities of the cilia are

顏曉晨和沈侯雖然在一個學(xué)院，可是專業(yè)不同，顏曉晨是游離在班級之外的人，沈侯也是游離在班級之外的人，兩人完全無交集，就算有學(xué)院必修課，可全院兩百多人，混到大學(xué)畢業(yè)，仍會有很多人叫不出名字。本來，他們的生活應(yīng)該是兩條平行線，可就是因為代寫作業(yè)和論文，顏曉晨進入了沈侯的視線。從那之后，沈侯不想做的作業(yè)，要完成的論文，期末考試前復(fù)印筆記、勾重點……沈侯都會找顏曉晨，顏曉晨從來不拒絕，但只第一次收了他四千塊錢，之后，無論如何，她都不要錢。因為顏曉晨不肯要錢，沈侯也不好意思總找她代寫，只能變得勤快點，借了作業(yè)來抄，一來二去，有意無意地，變成了顏曉晨幫他輔導(dǎo)功課，沈侯也漸漸地不再玩游戲。

模擬計算跨接線纜長度時,不需要考慮車輛偏移的影響，即在直線區(qū)段，設(shè)置車體的中心線與線路中心線重合；在曲線區(qū)段，設(shè)置車輛的心盤點在線路的中心線上。兩相鄰端車輛的車鉤在任何線路狀態(tài)下始終保持一條直線。以跨接線纜SC1為例(該線纜固定點距離車體縱向中心線830 mm，距離車體端墻20 mm，距離軌面630 mm)，采用該線纜來模擬車輛通過曲線時線纜固定點間的距離變化，并對其拉伸和壓縮狀態(tài)進行分析。通過分析,得到以下結(jié)論:

以G2京滬高速公路在鎮(zhèn)江市某互通立交為實例工程。首先對實例工程的新建匝道需求進行計算分析。根據(jù)本文式（1）可知，實例工程應(yīng)設(shè)置4×3=12條匝道。根據(jù)現(xiàn)狀統(tǒng)計（圖5），實例工程已建有11根轉(zhuǎn)向匝道，因此需要再新建一根匝道，根據(jù)現(xiàn)狀分析，k=4，n0=11，i=1，j=4，同時對可行方案數(shù)進行計算，可知可行方案為：

In the above formulation of velocity components,we are able to distinguish between the effective stroke of the cilia and the slow less effective recovery stroke by approximately accounting for the shortening of the cilia.The transformations between the two frames are

Introducing the following non-dimensional variables,

The constitutive equation for a Jeffrey six-constant fluid model is defined as[14]

(ii)Multisinusoidal wave

where

Finally,in simplified form above equation can be written as

The corresponding boundary conditions are defined as

where

4 Solution Methodology

4.1 Perturbation Solution

Since Eqs.(19)to(21)are non-linear equations,therefore we are seeking the analytical solution.For that we employ the regular perturbation method in terms of a variant of Jeffrey six constant fluid parameter α.As perturbation technique,following expansion of w,θ,σ,and p in terms of small parameter α are used

在一定時期，職業(yè)院校意識形態(tài)工作方式方法比較傳統(tǒng)，大多依靠思想政治理論課教師在課堂進行“說教”，“灌輸”政治理論，內(nèi)容不能與時俱進，與國家當(dāng)前的時事政治聯(lián)系不夠緊密，意識形態(tài)工作的知識性邏輯性趣味性不足，不能結(jié)合學(xué)生的生活學(xué)習(xí)、心理狀況、內(nèi)心需求等實際特點，有針對性的開展意識形態(tài)宣教工作，有些學(xué)生甚至看到意識形態(tài)教育就出現(xiàn)反感心理，產(chǎn)生厭學(xué)情緒，效果并不理想。

(3)老空區(qū)積水。經(jīng)本次勘查，區(qū)內(nèi)沿煤層露頭見多處老窯，老窯均已封閉，給調(diào)查帶了諸多不便。本次通過訪問當(dāng)?shù)卮迕窳私饫细G開采及積水情況，再結(jié)合實地調(diào)查，發(fā)現(xiàn)多數(shù)老窯封閉不嚴(yán)，有大量積水，并從裂縫流出地表。由于礦井已生產(chǎn)多年，形成一定采空區(qū)，亦有大量積水，故在煤礦開采要特別注意本區(qū)老空積水，在老空區(qū)附近應(yīng)預(yù)留隔水煤柱，防止老空水引發(fā)突水事故。

我與同伴定在那里，動彈不得。只聽見心口噗地一下，也躥起火苗，隨之一陣痙攣，像一個很久沒有進食的人面對盛宴，有幾乎暈眩的饑餓感，然而又是幸福的。

Flow rate in the dimensionless form can be written as

Using Eqs.(6)and(7)into Eq.(8),we obtain it as

The pressure rise?p can be written as

Velocities in terms of stream functions are defined as

For the flow analysis,we have considered three waveforms,namely,sinusoidal wave,trapezoidal wave,and mulltisinusoidal wave.The dimensionless equations can be written as

(i)Sinusoidal wave

where

(iii)Trapezoidal wave

(iv)Square wave

經(jīng)過護理,對照組的平均住院時間為(87.69±18.97)d,平均住院費用為(9974.67±751.94)元;研究組的平均住院時間是(64.29±15.74)d,平均住院費用是(7128.51±432.81)元,兩組結(jié)果對比存在統(tǒng)計學(xué)差異性(P<0.05)。研究組有20例治愈,4例復(fù)發(fā),對照組有12例治愈,10例復(fù)發(fā),兩組的治愈率和復(fù)發(fā)率存在統(tǒng)計學(xué)差異性 (P<0.05)。

5 Results and Discussions

Fig.1 Temperature graph for different values of α1 when Z=0.75,?=0.22,m=0.2,Br=0.22,α =0.22,β =1.5,α2=0.6.

Fig.2 Temperature graph for different values of α1 when Z=0.75,?=0.22,m=0.2,Br=0.22,α =0.22,β =1.5,α1=0.6.

Fig.3 Temperature graph for different values of Br when Z=0.75,?=0.22,m=0.2,α1=0.22,α =0.22,β =1.5,α2=0.6.

Fig.4 Temperature graph for different values of ? when Z=0.75,Br=0.22,m=0.2,α1=0.22,α =0.25,β =1.5,α2=0.6.

Fig.5 Concentration graph for different values of α1 when Z=0.75,Br=0.22,m=0.2,?=0.22,α =0.25,β =1.5,α2=0.6,Sc=0.7,Sr=0.5.

Fig.6 Concentration graph for different values of α2 when Z=0.75,Br=0.22,m=0.2,?=0.22,α =0.25,β =1.5,α1=0.6,Sc=0.7,Sr=0.5.

Fig.7 Concentration graph for different values of Sc when Z=0.75,Br=0.22,m=0.2,?=0.22,α =0.25,β =1.5,α1=0.6,α2=0.7,Sr=0.5.

Fig.8 Concentration graph for different values of Sr when Z=0.75,Br=0.22,m=0.2,?=0.22,α =0.25,β =1.5,α1=0.6,α2=0.7,Sc=0.5.

Fig.9 Velocity graph for different values of α2when Z=0.75,Br=0.22,m=0.2,?=0.22,α =0.25,β =1.5,α2=0.7.

Fig.10 Velocity graph for different values of ? when Z=0.75,Br=0.22,m=0.2,α1=0.22,α =0.25,β =1.5,α2=0.27.

Fig.11 Pressure rise graph for different values of α1 when Z=0.23,?=0.01,η= π/4,α1=0.22,α2=0.4,β=1.5,B=1.5,Sc=0.7,Sr=0.5.

Fig.12 Frictional forces for different values of α1when Z=0.23,?=0.01,η= π/4,α =0.22,α2=0.4,β =1.5,Br=1.5,Sc=0.7,Sr=0.5.

In this section,we have analyzed the solution for physiological flow of Jeffrey six constant fluid due to ciliary motion through graphs.We have presented the solution attained by Perturbation by framing velocity,pressure rise,pressure gradient,temperature,concentration and streamline graphs for diverse values of the parameters α,Q,ξ,δ,and γ,ST,SHrespectively.Figures 1–2 show that with the increase in α1,α2temperature pro file decreases.Figures 3–4 show that with the increase in Br,? temperature pro file increases.In Figs.5–8,it is depicted that with the increase in α1,α2,ST,SHconcentration pro file increases.Figure 9 shows that increases the value of α2while the velocity pro file in the centre of the tube decreases as well as it gets opposite behaviour nearest of the tube or near the peristaltic wave.In Fig.10 it is depicted that,at the centre of the tube,the velocity pro file is minimum whereas it gets opposite behaviour nearest of the tube or near the peristaltic wave.Pressure rise and frictional forces for diverse values of α1,α2,β and is plotted in Figs.11–16.In these figures,it is depicted that by increasing value of α1,α1,β pressure rise increasing in the region(Q ∈[?2,?1])whereas re flux occur in the last.Three different regions can be recognized from these figures.The retrograde pumping region can also be seen in Figs.11,13,15 when Q<0 and?p>0 and free pumping region can be seen when Q=0 and?p=0.Moreover,augmented pumping region can also be seen in Figs.11,13,15 when Q>0 and?p<0.Figures 12,14,16 show the forces have an opposite behaviour as well as the pressure rise.In Figs.17–18,it is depicted that by increasing value of α1,α2pressure rise decreasing.Figures 19–22 illustrate the streamlines for different wave shapes.

Fig.13 Pressure rise graph for different values ofα2 when Z=0.23,?=0.01,η= π/4,α1=0.22,α =0.4,β=1.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.14 Frictional forces for different values of α2when Z=0.23,?=0.01,η= π/4,α1=0.22,α =0.4,β =1.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.15 Pressure rise graph for different values of β when Z=0.23,?=0.01,η= π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.16 Frictional forces for different values of β when Z=0.23,?=0.01,η = π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.17 Pressure gradient dp/dz for sinusoidal wave when Z=0.23,?=0.01,η= π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.18 Pressure gradient dp/dz for sinusoidal wave when Z=0.23,?=0.01,η= π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.19 Streamlines pattern for sinusoidal wave Z=0.23,?=0.01,η= π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.20 Streamlines pattern for multisinusoidal wave Z=0.23,?=0.01,η = π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

Fig.21 Streamlines pattern for trapizodioal Z=0.23,?=0.01,η = π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

6 Conclusion

In this article,we have analysed the physiological Breakdown of Jeffrey six constant flow due to ciliary mo-tion.The main explanation of the present study is concisely as follows:

?The temperature pro file is enhanced corresponding to increasing values of parameters Brand parameter ?,

?The temperature pro file decreases with increasing the values of α1and α2,

?The nanoparticle concentration field is enhanced corresponding to increasing values of STand SH,

如何運作？林燕玲介紹，黨組織引領(lǐng)就是由全縣駐村第一書記牽頭，跨鄉(xiāng)聯(lián)建就是整合各個鄉(xiāng)鎮(zhèn)的力量，多村捆綁就是將第一書記所在的行政村捆綁在一起，公司化運營就是組建現(xiàn)代企業(yè)化管理的股份制公司，每個村都是股東。在這一思路指引下，三明市省、市、縣第五批派駐清流縣31個貧困村中的12個第一書記所在村聯(lián)合起來作為業(yè)主，整合投入扶貧幫扶資金240萬元，再由清流縣政府配套扶貧資金240萬元，共計480萬元。資金問題迎刃而解。

2.1.5 學(xué)生安全和紀(jì)律狀況總體良好，但安全隱患也應(yīng)重視 由于學(xué)校層面及工程學(xué)院對學(xué)生管理較細(xì)致、要求較嚴(yán)格，對外出組織活動有限制，總體上，學(xué)生的安全狀況和紀(jì)律狀況良好。然而，學(xué)校周邊的安全隱患也不能忽視，如娛樂場所、交通、食品安全等。部分學(xué)生在課余偶爾參與打麻將、玩撲克牌等娛樂，有三成學(xué)生在此娛樂中有一定的經(jīng)濟刺激行為，需要加強思想教育。學(xué)生宿舍無門禁帶來的安全隱患，應(yīng)引起重視。對學(xué)生使用大功率電器的問題，一方面需要繼續(xù)加強安全教育，同時也應(yīng)積極改善學(xué)校的基礎(chǔ)設(shè)施條件，滿足學(xué)生的合理生活需求，治標(biāo)更應(yīng)治本。

? Increases the value of α2while the velocity pro file in the centre of the tube decreases as well as it gets opposite behaviour nearest of the tube or near the peristaltic wave.

?Pressure rise and frictional forces for diverse values of α1,α2,β,it is depicted that by increasing value of α1,α1,β pressure rise increasing in the region(Q ∈ [?2,?1])whereas re flux occur in the last.Three different regions can be recognized from these figures.

?It is clear that frictional forces and pressure rise have an opposite behaviour while compare to each other.

?The pressure gradient increases with the increasing value of φ.

?Stream lines bolus take the form of the shape of the geometry.

Fig.22 Streamlines pattern for square Z = 0.23,Z=0.23,?=0.01,η = π/4,α1=0.22,α =0.4,α2=0.5,Br=1.5,Sc=0.7,Sr=0.5.

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