# Effect of Heat and Mass Transfer and Magnetic Field on Peristaltic Flow of a Fractional Maxwell Fluid in a Tube

Authors: F. S. Bayones; A. M. Abd-Alla; Esraa N. Thabet
Journal: Complexity
 (2021)

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## Abstract

Magnetic field and the fractional Maxwell fluids’ impacts on peristaltic flows within a circular cylinder tube with heat and mass transfer were evaluated while assuming that they are preset with a low Reynolds number and a long wavelength. The analytical solution was deduced for temperature, concentration, axial velocity, tangential stress, and coefficient of heat transfer. Many emerging parameters and their effects on the aspects of the flow were illustrated, and the outcomes were expressed via graphs. Finally, some graphical presentations were made to assess the impacts of various parameters in a peristaltic motion of the fractional fluid in a tube of different nature. The present investigation is essential in many medical applications, such as the description of the gastric juice movement of the small intestine in inserting an endoscope.


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## Body

## 1. Introduction

Numerous implementations have drawn interest of physicists, mathematicians, and engineers on magneto-hydrodynamic flow issues. In some applications and geothermal studies, metal alloy substantiation processes are optimized Sources, management of waste fuel, regulation of underground propagation and pollution of chemicals, waste, the construction of energy turbines for MHD, magnetic equipment for wound therapy and cancer tumour treatment, reduction of bleeding during surgery and transport of targeted magnetic particles as medicines. Several extensive works of literature on that fertile field are now available in [1, 2]. Saqib et al. [3] clarified the nonlinear motion of the non-Newtonian fractional model fluid problem. Rashed and Ahmed [4] produced a numerical solution for dusty nanofluids peristaltic motion in a channel using a shooting method. The slip effect’s problem on a peristaltic flow of the fractional fluid of second-grade over a cylindrical tube was examined by Rathod and Tuljappa [5]. Vajravelu et al. [6] obtained the velocity, temperature, and concentration with a magnetic field of a Carreau fluid in a channel with the heat and mass transfer. Ali et al. [7] discussed magnetic field effects on a blood flow that the blood was characterized as the Casson fluid. Zhao et al. [8] explored the motion natural convection temperature of a fraction with a magnetic field of viscoelastic fluid through a porous medium. Abd-Alla et al. [9] were researching the magnetic field’s impact on a peristaltic motion of the fluid through the cylindrical cavity. Afzal et al. [10] analyzed the effect of the diffusivity convection and magnetic field in nanofluids on the peristaltic motion through the nonuniform channel. Heat and mass transfer’s effects and magnetic field of the peristaltic motion in a planar channel were examined by Hayat and Hina [11]. The impact of the temperature and the magnetic field of peristaltic motion through a porous medium was debated by Srinivas and Kothandapani [12]. Ramzan et al. [13] discussed the heat flux and magnetic field’s influences in Maxwell fluid flow through a two-way strained surface. Rachid [14] calculated the movement of viscoelastic fluid peristaltic transport under the Maxwell fractional model. The impact of a viscosity and a magnetic field of the peristaltic motion of synovial nanofluid in an asymmetric channel was reconnoitered by Ibrahim et al. [15]. Aly and Ebaid [16] inspected the slip conditions' effects of a peristaltic motion of nanofluids. Carrera et al. [17] checked the extension of a fractional Maxwell fluid and viscosity to the peristaltic motion. Zhao [18] exhibited the convection flow, the magnetic field, and velocity slip of a peristaltic motion of a fractional fluid. Abd-Alla et al. [19] obtained the solution to the peristaltic motion problem in an endoscope tube. The analytical solution of the transport of viscoelastic fluid through a channel in the fractional peristalsis movement model was presented by Tripathi et al. [20]. The magnetic field effect on peristaltic movement in a vertical annulus was exposed by Nadeem and Akbar [21]. Srinivas et al. [22] were determining the effects on Newtonian fluid’s peristaltic movement into porous channels of wall slip conditions, magnetic field, and heat transfer. Recent research expansions on the subject beginning from [23–33].This paper aims to inspect the impacts of magnetic fields, heat and mass transfer, and fractional Maxwell fluids on the peristaltic flow of Jeffrey fluids. Both two-dimensional equations of motion and heat and mass transfer are generalized under the presence of low Reynolds numbers and a long wavelength. The temperature, concentration, axial velocity, tangential stress, and coefficient of heat transfer are empirical solutions, and the wave shape is found. In the problem, the relevant parameters are specified pictorially. The findings obtained are displayed and discussed graphically. For physicists, engineers, and individuals interested in developing fluid mechanics, the outcomes described in this paper are essential. The different potential fluid mechanical flow parameters for the Jeffrey peristaltic fluid are also supposed to serve as equally good theoretical estimates. Indeed, the current investigation is firmly believed to receive considerable attention from the researchers towards further peristaltic development with a variety of applications in physiological, modern technology, and engineering.


## 2. Formulation of the Problem

Take the MHD peristaltic flow through uniform coaxial tubes of a viscoelastic fluid through the fractional Maxwell fluid model. If the flow is transversely subject to a consistent magnetic field, electrical conductivity exists (Figure1). Furthermore, it is supposed the inner and outer tube temperatures are T0 and T1, and concentrations are C0 and C1, respectively. We picked a cylindrical coordinate R¯ and Z¯. The equations for the tube walls are given by(1)R1¯−a1=0,R2¯−a2=+bsin2πλZ¯−ct¯.Figure 1
The geometry of the problem.The equation of the fractional Maxwell fluid is given by(2)1+λ1¯α1D˜t¯α1S¯=μγ.,where 0≤α1≤1.D˜t¯α1 is defined as follows:(3)D˜t¯α1S¯=Dt¯α1S¯+V¯.∇S¯−L¯S¯−S¯L¯T,where(4)γ.=∇V¯+∇V¯T.Also, note thatDt¯α1=∂t¯α1, of order α1 concerning t and defined as follows:(5)Dt¯α1ft=1Γ1−α1ddt∫0tfξt−ξα1dξ,0≤α1≤1.The equation of motion can be written in the fixed frame which are derived [32, 33] as(6)ρ∂∂t¯+U¯∂∂R¯+W¯∂∂Z¯U¯+∂p¯∂R¯=1R¯∂∂R¯R¯S¯R¯R¯+∂∂Z¯S¯R¯Z¯−S¯θ¯θ¯R¯,ρ∂∂t¯+U¯∂∂R¯+W¯∂∂Z¯W¯+∂p¯∂Z¯=1R¯∂∂R¯R¯S¯R¯Z¯+∂∂Z¯S¯Z¯Z¯+ρgαtT¯−ToρgαcC¯−Co−σBo2W¯,ρCp∂∂t¯+U¯∂∂R¯+W¯∂∂Z¯T¯=K∂2∂R¯2+1R¯∂∂R¯+∂2∂Z¯2T¯+Qo,∂∂t¯+U¯∂∂R¯+W¯∂∂Z¯C¯=Dm∂2∂R¯2+1R¯∂∂R¯+∂2∂Z¯2C¯+DmKTTm∂2∂R¯2+1R¯∂∂R¯+∂2∂Z¯2T¯,∂U¯∂R¯+U¯R¯+∂W¯∂Z¯=0.The transformation between these two frames can be written as follows:(7)r¯−R¯=0,z¯−Z¯=−ct¯,u¯−U¯=0,w¯−W¯=−c.The relevant governed boundary conditions for the considered flow analysis can be listed as(8)w¯+c=0,u¯=0atr¯=r1¯,w¯+c=0atr¯=r2¯+bsin2πz¯λ,T¯−T1¯=0,C¯−C1¯=0,atr¯=r1¯,T¯−T0¯=0,C¯−C0¯=0atr¯=r2¯.The leading motion equations of the flow for fluid in the wave frame are given by(9)ρu¯∂∂r¯+w¯+c∂∂z¯u¯+∂p¯∂r¯=1r¯∂∂rr¯S¯r¯r¯+∂∂z¯Sr¯z¯−S¯θ¯θ¯r¯,ρu¯∂∂r¯+w¯+c∂∂z¯w¯+∂p¯∂z¯=1r¯∂∂r¯r¯S¯r¯z¯+∂∂z¯S¯Z¯Z¯+ρgαtT¯−To+ρgαcT¯−To−σBo2w¯+c,ρCpu¯∂∂r¯+w¯+c∂∂z¯T¯=K∂2∂r¯2+1r∂∂r¯+∂2∂z¯2T¯+Qo,u¯∂∂r¯+w¯+c∂∂z¯C¯=Dm∂2∂r¯2+1r∂∂r¯+∂2∂z¯2C¯+DmKTTm∂2∂r¯2+1r∂∂r¯+∂2∂z¯2T¯,∂u¯∂r¯+u¯r¯+∂w¯∂z¯=0,where S¯ depends only on r and t. After using the initial condition S¯t¯=0, we find S¯r¯r¯=S¯θ¯θ¯=S¯z¯z¯=S¯r¯θ¯=0, and(10)1+λ1¯α1∂α1∂t¯α1S¯r¯z¯=μ∂w¯∂r¯.We present the following dimensionless parameters for further analysis:(11)r=r¯a2,z=z¯λ,t=ct¯λ,u=u¯cδ,w=w¯c,λ1=cλ1¯λ,p=a22p¯cλμ,δ=a2λ,θ=T¯−T0T1−T0,Pr=μCPK,Re=ρca2μ,Gr=ρgαtT1−Toa22μc,M=σμB0a2,Sc=μDmρ,Sr=ρDmKTT1−ToμTmC1−C0,Br=ρgαcC1−Coa22μc,Θ=C¯−C0C1−C0,β=a22QoKT1−To,S=a2S¯μc,r1=r1¯a2=ε<1,r2=r2¯a2=1+φsin2πz.wherever φ=b/a2<1 is the wave amplitude.


## 3. Solution of the Problem

For the abovementioned modifications and nondimensional variables listed earlier, the preceding equations are reduced to(12)Reδ3u∂∂r+w+1∂∂zu+∂p∂r=δr∂∂rrSrr+δ2∂∂zSrz−δSθθr,(13)Reδu∂∂r+w+1∂∂zw+∂p∂z=1r∂∂rrSrz+δ∂∂zSzz+Grθ+BrΘ−M2w+1,(14)RePrδu∂∂r+w+1∂∂zθ=∂2∂r2+1r∂∂r+δ2∂2∂z2θ+β,(15)Reδu∂∂r+w+1∂∂zΘ=1Sc∂2∂r2+1r∂∂r+δ2∂2∂z2Θ+Sr∂2∂r2+1r∂∂r+δ2∂2∂z2θ,(16)∂u∂r+ur+∂w∂z=0.With boundary conditions(17)w+1=0,u=0atr=r1=ε,w+1=0atr=r2=1+φsin2πz,θ−1=0,Θ−1=0atr=r1,θ=0,Θ=0atr=r2.


## 4. The Analytical Solution

Furthermore, the hypothesis of the long wavelength approach is also supposed. Now,δ is very small so that it can be tended to zero. Thus, the δ≪1 dimensionless governing equations (12)–(15) by using this hypothesis may be written as(18)∂p∂r=0,(19)fdpdz−Grθ−BrΘ+M2w+1=∂2w∂r2+1r∂w∂r,∂2θ∂r2+1r∂θ∂r+β=0,1Sc1rddrr∂Θ∂r+Sr1rddrr∂θ∂r=0,equation (18) specifies that p is only a function of z.Temperature, concentration, and axial velocity solutions can be described as follows:(20)θ=logr/r2logr1/r2+β4r12logr/r2−r22logr/r1logr1/r2−β4r2,Θ=SrScβ4r2+c1logr+c2,w=4c2+4c1logr+f×dp/dzr2−Grr2logr/r2−1/logr1/r2+β/4r12r2logr/r2−1−r22r2logr/r1−1/logr1/r2−β/16r4−BrSrScβ×r4/16+c1r2logr−1+c2r2+M2r24−fM2r2,where(21)f=1+λ1α1Dtα1,A=−4−f×dpdzr12+Grr12logr1/r2−1logr1/r2+β4r14logr1/r2−1+r12r22logr1/r2−β16r14,B=−4−f×dpdzr22+Gr−r22logr1/r2+β4−r12r22−r24logr1/r2−1logr1/r2−β16r24,C1=4+SrScβr22−r124logr1/r2,C2=SrScβr12logr2−r22logr1−4logr24logr1/r2,C3=A−B4logr1/r2,C4=Blogr1−Alogr24logr1/r2.The heat transfer coefficient is indicated as follows:(22)Zr=∂θ∂r×∂r2∂z.So, the solution of heat transfer is given by(23)Zr=−rβ2+1rlogr1/r2+r12/r−r22/rβ4logr1/r2×2φπcos2πz.Using the definition of the fractional differential operator (5) we find the expression of f as follows:(24)f=ft=1+λ1α1t−α1Γ1−α1.


## 5. Results and Discussion

In this section, the effect of different parameters is shown graphically in Figures2–7 such as fractional parameter α1, heat source/sink parameter β, wave amplitude φ, radius ratio ε, Hatman number M, Grashof number Gr, relaxation time λ1, the Soret number Sr, and the Schmidt number Sc on the temperature θ, the concentration Θ, axial velocity w, tangential stress srz, and heat transfer coefficient Zr. MATLAB software is used to identify the quantitative influences of various physical parameters implicated in the our study. Approximate analytical results are numerically evaluated for temperature, concentration, axial velocity, tangential stress, and the heat transfer coefficient for various values of parameters. For this object, Figures 2–7 are displayed.Figure 2
Discrepancies of the temperatureθ against the r−axis for various values of β and φ..
(a)(b)Figure 3
Discrepancies of the concentrationΘ against the r−axis for various values of ε,φ,Sc, and Sr.
(a)(b)(c)(d)Figure 4
Discrepancies of the axial velocityw against the r− and the z−axes for various values of λ1λ1,φ,α1,M, and Sc.
(a)(b)(c)(d)(e)(f)Figure 5
Discrepancies of the axial tangential stresssrz against the z−axis for various values of α1,M,β, and Sc.
(a)(b)(c)(d)Figure 6
Discrepancies of the heat transfer coefficientZr against the z−axis for various values of ε and φ.
(a)(b)Figure 7
Discrepancies of the axial velocity,w, the concentration, Θ, the temperature, θ, and the heat transfer coefficient Zr in 3D against r- and z-axis under the influence of α1,Sr, ε, and φ..
(a)(b)(c)(d)Figure2 has been plotted to clarify the variations of β and φ on the temperature distribution θ. Figure 2 shows that θ decreases when β increases in the range 0≤r≤0.32, while θ increases when β increases in the range 0.32≤r≤1.2. Moreover, θ decreases when φ increases in the range 0≤r≤0.32, while θ increases when φ increases in the range 0.32≤r≤1.4. In addition, the temperature decreases with the radial increase and the boundary conditions are fulfilled.Figure3 displays the discrepancy of the concentration with the radial for various values of ε,φ,Sc and Sr. It is indicated that the concentration increases with increasing ε and φ. However, Θ decreases with increasing Sr and Sc. In addition, the concentration decreases with the radial increase and the boundary conditions are fulfilled.The impacts ofGr,λ1,φ,α1,M, and Sc on the axial velocity w are illustrated in Figure 4. It is indicated that the axial velocity profiles decreases with increasing Gr,λ1, and φ in the range 0≤r≤0.32, while it increases in the range 0.32≤r≤0.45, In addition to this, the axial velocity profile decreases with increasing α1 in the whole range 0≤z≤1, while it increases with increasing M in the whole range 0≤z≤1, the axial velocity profiles decreases with increasing Sc in the range 0≤z≤53 as well, and it increases in the range 0.53≤r≤0.88 and then decreases again in the range 0.88≤z≤1. Also, it is observed that the velocity has oscillatory behavior due to peristaltic motion concerned.The effect ofα1,M,β and Sc can be observed from Figure 5, in which the tangential stress is illustrated for the various values of α1,M,β, and Sc. With the increase of α1 and Sc, the tangential stress decreases. Moreover, tangential stress increases with increasing M and β. It is noticed that one can observe the tangential stress is in oscillatory behavior, which may be due to peristalsis.Figure6 explains the influence of ε and φ on the heat transfer coefficient Zh. Obviously, the increase in ε and φ increases the amplitude of the heat transfer coefficient in the whole range z. From Figure 6, one can observe that heat transfer coefficient is an oscillatory behavior in the whole range, which may be due to peristalsis.Figure7 is plotted in 3D schematics concern the axial velocity w, the concentration Θ, the temperature θ, and the heat transfer coefficient Zh concerning r and z axes in the presence α1,Sr, ε, and φ. It is indicated that the axial velocity decreases by increasing α1, Also, the concentration decreases by increasing Sr, the temperature increases with increasing of ε as well, otherwise the heat transfer coefficient increases by increasing φ. For all physical quantities, we obtain the peristaltic flow in 3D overlapping and damping when the state of particle equilibrium is reached and increased. The vertical distance of the curves is greater, with most physical fields moving in peristaltic flow.


## 6. Conclusions

The concluding remarks are listed as follows:(1)
The axial velocity decreases and increases with the increase ofα1,φ,Gr,λ1, and Sc due to the increase in the Lorentz force.(1)
The temperature increases with the increase of the wave amplitude and radius ratio.(2)
The concentration decreases with the increase of bothSr, Sc and it increases with the increase of both ε and φ.(3)
The tangential stress decreases and increases with the increase of bothα1, Sc, and it increases with the increase both M and β.(4)
The study of the phenomenon under effect ofα1, β, φ, ε, M, Gr, λ1, Sr, and Sc was performed.(5)
This study has indeed been widely applied in many fields of science, such as medicine and the medical industry. Thus, in the field of fluid mechanics, it is considered as extremely essential. When inserting an endoscope through the small intestine, this study describes the movement of the gastric juice.



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*Source: 9911820-2021-06-17.xml*