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

时间：2024-05-22

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.

热加工处理后花生中总蛋白含量结果如图1所示，采用凯氏定氮法测定水煮前后花生中的总蛋白含量发现，未加工的鲜花生含蛋白11.16 g/100 g鲜花生，直接水煮和晒干水煮后，蛋白质含量分别为11.67 g/100 g鲜花生和11.14 g/100 g鲜花生，三者间蛋白含量没有显著差异。Mondoulet L[29]等的研究认为，水煮过程中可能会有蛋白损失到水煮液中，本研究也测定了水煮液中的花生蛋白含量。结果显示，约200 g鲜花生经过直接带壳水煮或晒干后带壳水煮，收集的水煮液中花生蛋白量分别为12.00和11.42 mg，可见蛋白溶解进水煮液的数量很少，故在水煮处理过程中蛋白的损失量可以忽略。

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

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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 and¯τ 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 ǫ1and delay time is ǫ2.

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,ǫ 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

颜晓晨和沈侯虽然在一个学院，可是专业不同，颜晓晨是游离在班级之外的人，沈侯也是游离在班级之外的人，两人完全无交集，就算有学院必修课，可全院两百多人，混到大学毕业，仍会有很多人叫不出名字。本来，他们的生活应该是两条平行线，可就是因为代写作业和论文，颜晓晨进入了沈侯的视线。从那之后，沈侯不想做的作业，要完成的论文，期末考试前复印笔记、勾重点……沈侯都会找颜晓晨，颜晓晨从来不拒绝，但只第一次收了他四千块钱，之后，无论如何，她都不要钱。因为颜晓晨不肯要钱，沈侯也不好意思总找她代写，只能变得勤快点，借了作业来抄，一来二去，有意无意地，变成了颜晓晨帮他辅导功课，沈侯也渐渐地不再玩游戏。

模拟计算跨接线缆长度时,不需要考虑车辆偏移的影响，即在直线区段，设置车体的中心线与线路中心线重合；在曲线区段，设置车辆的心盘点在线路的中心线上。两相邻端车辆的车钩在任何线路状态下始终保持一条直线。以跨接线缆SC1为例(该线缆固定点距离车体纵向中心线830 mm，距离车体端墙20 mm，距离轨面630 mm)，采用该线缆来模拟车辆通过曲线时线缆固定点间的距离变化，并对其拉伸和压缩状态进行分析。通过分析,得到以下结论:

以G2京沪高速公路在镇江市某互通立交为实例工程。首先对实例工程的新建匝道需求进行计算分析。根据本文式（1）可知，实例工程应设置4×3=12条匝道。根据现状统计（图5），实例工程已建有11根转向匝道，因此需要再新建一根匝道，根据现状分析，k=4，n0=11，i=1，j=4，同时对可行方案数进行计算，可知可行方案为：

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

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(3)老空区积水。经本次勘查，区内沿煤层露头见多处老窑，老窑均已封闭，给调查带了诸多不便。本次通过访问当地村民了解老窑开采及积水情况，再结合实地调查，发现多数老窑封闭不严，有大量积水，并从裂缝流出地表。由于矿井已生产多年，形成一定采空区，亦有大量积水，故在煤矿开采要特别注意本区老空积水，在老空区附近应预留隔水煤柱，防止老空水引发突水事故。

我与同伴定在那里，动弹不得。只听见心口噗地一下，也蹿起火苗，随之一阵痉挛，像一个很久没有进食的人面对盛宴，有几乎晕眩的饥饿感，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

经过护理,对照组的平均住院时间为(87.69±18.97)d,平均住院费用为(9974.67±751.94)元;研究组的平均住院时间是(64.29±15.74)d,平均住院费用是(7128.51±432.81)元,两组结果对比存在统计学差异性(P<0.05)。研究组有20例治愈,4例复发,对照组有12例治愈,10例复发,两组的治愈率和复发率存在统计学差异性 (P<0.05)。

5 Results and Discussions

Fig.1 Temperature graph for different values of α1 when Z=0.75,ǫ=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,ǫ=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,ǫ=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 ǫ 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,ǫ=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,ǫ=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,ǫ=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,ǫ=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,ǫ=0.22,α =0.25,β =1.5,α2=0.7.

Fig.10 Velocity graph for different values of ǫ 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,ǫ=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,ǫ=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,ǫ 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,ǫ=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,ǫ=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,ǫ=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,ǫ=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,ǫ=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,ǫ=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,ǫ=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,ǫ=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,ǫ=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 ǫ,

•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,

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2.1.5 学生安全和纪律状况总体良好，但安全隐患也应重视由于学校层面及工程学院对学生管理较细致、要求较严格，对外出组织活动有限制，总体上，学生的安全状况和纪律状况良好。然而，学校周边的安全隐患也不能忽视，如娱乐场所、交通、食品安全等。部分学生在课余偶尔参与打麻将、玩扑克牌等娱乐，有三成学生在此娱乐中有一定的经济刺激行为，需要加强思想教育。学生宿舍无门禁带来的安全隐患，应引起重视。对学生使用大功率电器的问题，一方面需要继续加强安全教育，同时也应积极改善学校的基础设施条件，满足学生的合理生活需求，治标更应治本。

• 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,ǫ=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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