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Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube
Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube
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Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube M.A. Imran, A. Shaheen, El-Sayed M. Sherif, M. Rahimi-Gorji, Asiful H. Seikh PII: DOI: Reference:
S0577-9073(19)31010-X https://doi.org/10.1016/j.cjph.2019.11.029 CJPH 1032
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
10 January 2019 22 November 2019 28 November 2019
Please cite this article as: M.A. Imran, A. Shaheen, El-Sayed M. Sherif, M. Rahimi-Gorji, Asiful H. Seikh, Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.11.029
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Highlights • Peristaltic flow of a Jeffrey six constants nanofluid has been modeled. • The governing equations are solved analytically by HPM. • Direct relationship between nanoparticles concentration and value of thermophoresis parameter.
• Velocity, temperature and frictional forces are decreasing functions of thermophoresis parameter.
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Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube M. A. Imran∗, A. Shaheen†, El-Sayed M Sherif ‡, M. Rahimi-Gorji §, Asiful H. Seikh
Abstract: Peristaltic flow of non-Newtonian nano fluid through a non-uniform surface has been investigated in this paper. The fluid motion along the wall of the surface is caused by the sinusoidal wave traveling with constant speed. The governing equations are converted into cylindrical coordinate system and assuming low Reynolds number and long wave length partial differential equations are simplified. Analytically solutions of the problem are obtained by utilizing the homotopy perturbation method (HPM). In order to insight the impact of embedded parameters on temperature, concentration and velocity some graphs are plotted for different peristaltic waves. At the end, some observations were made from the graphical presentation that velocity, pressure rise and nano particle concentration are increasing function of thermophoresis parameter Nt while temperature and frictional forces show opposite trend.
Keywords: Peristaltic flow; nanoparticles; Jeffrey six constant fluid; Homotopy perturbation method.
∗
Department of Mathematics, University of Management and Technology, Lahore, Pakistan.
Email:[email protected] † Department of Mathematics, Comsats University Islamabad Lahore Campus,Pakistan.Email: [email protected] ‡ Center of Excellence for Research in Engineering Materials, King Saud University, P.O. Box-800, Riyadh 11421, Saudi Arabia,2Electrochemistry and Corrosion Laboratory, Department of Physical Chemistry, National Research Centre, El-Behoth St. 33, Dokki, Cairo 12622, Egypt Corresponding e-mail address: [email protected] § Faculty of Medicine and Health Science, Ghent University, Gent 9000, Belgium. Email: [email protected] , [email protected] ¶ Center of Excellence for Research in Engineering Materials, King Saud University, P.O. Box-800, Riyadh 11421, Saudi Arabia. Email: [email protected]
2
¶
3
Nomenclature U,¯W C¯
fluid velocity components
T¯
temperature of fluid
nanopartical concentration
DB
Brownian diffusion cofficient
DT
thermospheric diffusion coefficient
τ
ratio of heat capacity of the nanoparticle
a0
radius of the inlet
Re
Reynolds number
Nt
thermophoresis number
t
is the time
δ
long wavelength
Gr
thermophoresis number
Nb
Brownian motion number
µ
dynamic viscosity
ρp
density of the particle
λ
wave length
c
wave speed
b
wave amplitude
1
relaxation time
2
delay time
d, b, c
Jeffrey six constant
σ
dimensionless concentration
θ
dimensionless temperature
w
dimensionless velocity
Pr
Prandtl number
Ec
Eckert number
1
Introduction
Peristalsis is a critical system for moving toward fluid in a channel or tube by methods for a usage contractile ring over the tube, which drives the material forward. Peristalsis is a radially symmetrical contraction of muscles which propagates in a way down a muscular tube. This process is quite important for fluid transport in living organisms and industry. Blood pumps in dialysis and heart lung machine works because of the peristaltic phenomena. Most of the fluids in nature possess non-Newtonian characteristics. Non-Newtonian fluids change their viscosity or flow behavior under stress. Not all non-Newtonian fluids behave in the same way when stress is applied; some become more solid while others more fluid. Some non-Newtonian fluids react as a result of the amount of stress applied, while others react as a result of the length of time that stress is applied. In a few technique of physiology and designing, peristaltic stream concerns are comprehensively found in channel/tube. Such streams rise in gulping of sustenance amid throat, pee transportation from kidney to bladder, evaluation chyme in gastrointestinal tract, ovum development in the female fallopian tube, spermatozoa move in the channels efferent of male conceptive tract, vasomotion of limited veins, water development from ground to above branches of adult
4 trees referred to as [1–5]. Pumping of blood in lungs, transportation of mordant fluids, heart and hygienic fluid is due to the service of peristalsis. Srivastava [6] was the first one who investigated the peristaltic during male reproductive system experimentally and numerically. Peristaltic flow has been discussed in the vas deferens by Gupta [7] , Guha [8] and Batra [9]. Hussain and Malik [10] studied the effects of Joule heating and viscous dissipation on MHD Sisko nanofluid over a stretching cylinder. Khalil et al. [11–16] have discussed numerical solutions of non-Newtonian fluids. Hydromagnetic flow of fluid in a uniform pipe with variable thickness was investigated by Hakeem et al. [17]. Mair et al. [18] explained the boundary layer flow of MHD tangent hyperbolic nanofluid over a stretching sheet. Bilal et al. [19] discussed double diffusion on MHD Prandtl nano fluid. Peristaltic transportation of channel flow for non-Newtonian fluid with some inclination has been studied by Vajravelu et al. [20]. Study of peristaltic flow in the presence of magnetic field has also achieved a lot of importance in daily life and engineering sciences. Effects of MHD on the peristaltic flows for different modes of heat transfer like conduction, convection and radiations are reported in the Refs. [21–23]. The rate of heat transfer is dependent on the temperatures of the systems and the properties of the prevailing medium through which the heat is transferred. Different authors have discussed [24, 25] the effect of force on the heat convection and mass transfer. Study of peristaltic flow in the presence of magnetic field has also achieved a lot of importance in daily life and engineering sciences. A study dealing MHD nanofluid flow with Stagnation mixed convection and slip condition for stretching sheet has been discussed [26]. Effects of MHD on the peristaltic flows for different modes of heat transfer like conduction, convection and radiations are reported in the Refs. [27]. Abbasi et al. [28,29] discussed Peristaltic motion of nanofluid with MHD mixed convective and Joule heating effects. Since the first investigation done by the Choi [30], the study of nanofluids have attracted the attention of many researchers due to its tremendous applications in various fields of life such as biomedical devices, treatment of tumor, nuclear reactor, microchips, cooling, radiators and nonmedicinal etc. Different authors have discussed [31, 32] heat transfer analysis for peristaltic flow of Carreau-Yasuda fluid through a curved channel with radial magnetic field. In [33, 34] discussed the calculations for the estimation of transient surface warmth motion amid ultra-quick surface cooling.The demonstrated conditions are settled
5 scientifically by homotopy bother technique. The outcomes for velocity profile, pressure gradient, pressure rise and stream function have been considered for various estimations of the parameters. The physical highlights of applicable parameters are examined through diagrams. The streamlines are outlined for some physical amount to analyze the catching trapping phenomenon. Present study deals with the peristaltic flow of non-Newtonian nano fluid through a non-uniform tube. The fluid motion along the wall of the surface is caused by the sinusoidal wave traveling with constant speed. The governing equations are converted into cylindrical coordinate system and assuming low Reynolds number and long wave length partial differential equations are simplified. Analytically solutions of the problem are obtained by utilizing the homotopy perturbation method (HPM). In order to insight the impact of embedded parameters on temperature, concentration and velocity some graphs are plotted for different peristaltic waves. At the end, some streamlines have also been plotted.
2
Problem Formulation
Let us consider incompressible Jeffrey’s six constant nanofluid bearing the peristaltic flow in a non-uniform vertical tube. The flow is generated by a sinusoidal wave train conduction along the walls of the tube with constant speed c. The geometry of the problem cylindrical Coordinates system (R, Z) is shown in Fig. ( 1) ¯ = a(Z) ¯ + b sin 2π (Z¯ − ct), h λ ¯ = a0 + K Z, ¯ a0 is the radius of the inlet, K is the constant whose magnitude where a(Z) depend on the length of the tube , λ is the wavelength, b is wave amplitude, c is the propagation velocity and t is the time. We consider the cylindrical coordinate system (R, Z), in which Z-axis lies along the center line of the tube and R is transverse to it. The
6
Figure 1: Geometry of the problem.
fundamental equations of continuity, momentum, nanoparticle concentration are [35] U¯ (1) U¯R + ¯ + W¯Z = 0, R ∂ ∂ P¯ ∂ ∂ 1 ∂ ¯ ∂ τ¯¯¯ ¯ ¯ ρ (2) + U ¯ + W ¯ U¯ = − ¯ + ¯ ¯ (R¯ τR¯ R¯ ) + ¯ (¯ τR¯ Z¯ ) + ¯θθ , ¯ ∂t ∂R ∂Z ∂R R ∂R ∂Z R ¯ ∂ ∂ ∂ ¯ ¯ ¯ = − ∂ P + 1 ∂ (R¯ ¯ τR¯ Z¯ ) + ∂ (¯ ρ +U ¯ +W ¯ W τZ¯Z¯ ) ¯ ¯ ¯ ¯ ∂t ∂R ∂Z ∂Z R ∂R ∂ Z¯ + ρgα(T¯ − T¯0 ) + ρgα(C¯ − C¯0 ), (3)
∂ ¯ ∂ U¯ ¯ + W ∂R ∂ Z¯
∂ ¯ ∂ U¯ ¯ + W ∂R ∂ Z¯
2¯ ¯ ∂ 2 T¯ ∂ T 1 ∂ T T¯ = α ¯2 + R ¯ ∂R ¯ + ∂ Z¯ 2 ∂R " " ¯ ¯ 2 ¯ 2 ## ∂C ∂T ∂ C¯ ∂ T¯ DT¯ ∂ T¯ ∂T + τ DB + + + , (4) ¯ ∂R ¯ ∂ Z¯ ∂ Z¯ ¯ ∂R T¯0 ∂R ∂ Z¯
C¯ = DB
∂ 2 C¯ 1 ∂ C¯ ∂ 2 C¯ ¯2 + R ¯ ∂R ¯ + ∂ Z¯ 2 ∂R DT¯ ∂ 2 T¯ 1 ∂ T¯ ∂ 2 T¯ + ¯ ¯2 + R ¯ ∂R ¯ + ∂ Z¯ 2 . T0 ∂ R
(5)
7 The constitutional equation for a Jeffrey fluid form is defined as [36] d¯ τ τ¯ + 1 − W.¯ τ + τ¯.W + d(¯ τ .D + D.¯ τ ) + b¯ τ : DI + cDtr¯ τ = dt dD 2µ D + 2 ( − W.D + D.W + 2dD.D + bD : DI) . dt
(6)
in which ∇V¯ + (∇V¯ )t1 , 2 ∇V¯ − (∇V¯ )t1 W(antisymmetric measurement of velocity gradient) = , 2 D(symmetric measurement of velocity gradient) =
The transformations takes the form ¯ r¯ = R,
z¯ = Z¯ − c1 t¯,
u¯ = U¯ ,
¯ − c1 , w¯ = W
The resultant boundary conditions are: ¯ ∂W = 0, ∂¯ r ¯ = 0, W
∂ T¯ = 0, ∂¯ r T¯ = T¯0 ,
∂ C¯ = 0, ∂¯ r
C¯ = C¯0 ,
at
at
r¯ = 0,
¯ = a(Z) ¯ + b sin r¯ = h
(7) 2π ¯ (Z − c1 t). λ
(8)
Introducing the following non-dimensional variables, R = U = Re = Pr = Nb =
¯ ¯ R r¯ Z¯ z¯ W w¯ a¯ τ , r= , Z= , z= , W = , w= , τ= , a a λ λ c1 c1 c1 µ 0 λU¯ λ¯ u c1 t¯ a2 p¯ 1 c1 2 c1 , u= , t= , p= , λ1 = , λ2 = , ac1 ac1 λ c1 λµ0 a a ¯ ac1 ρ c2 a h λkz qw a , Ec = , δ = , h= =1+ + φ sin 2πz, Nu = µ0 cp T0 λ a a0 kT0 2 ¯ ¯ µ 0 cp µ ¯(θ) Q0 a T − T0 k , µ(θ) = , β1 = , θ= , α= , ¯ k µ0 kT0 (ρc)f T0 (ρc)p DB C¯0 gαa3 C¯0 (ρc)p DT C¯0 C¯ − C¯0 , Br = , N = , σ = . t (ρc)f υ2 (ρc)f α C¯0
keeping in mind the assumptions of δ << 1 and low Reynolds number approximation we
8 have in simplified form as ∂p = 0, ∂r 3 ! ∂p 1 ∂ ∂w 1 ∂ ∂w = r + αα1 r + ∂z r ∂r ∂r r ∂r ∂r 5 ! 1 ∂ ∂w α 2 α2 r + Gr θ + Br σ, r ∂r ∂r 2 1 ∂ ∂θ ∂θ ∂σ ∂θ r + Nb + Nt , 0 = r ∂r ∂r ∂r ∂r ∂r 1 ∂ ∂σ Nt 1 ∂ ∂θ 0 = r + r . r ∂r ∂r Nb r ∂r ∂r
(9)
(10) (11) (12)
The corresponding boundary conditions are defined as ∂w = 0, ∂t
∂θ = 0, ∂r
w = −1,
3
θ = 0,
∂σ = 0, ∂r
at
r = 0,
σ=0
at
r =h=1+
(13) λkz + φ sin(2πz). a0
(14)
Solution of the problem
Analytical technique called homotopy perturbation method (HPM) is used to eliminates fixed perturbation method and apply to Eqs. (10)-(12), "
∂θ ∂σ H(q, θ) = L(θ) − L(θ10 ) + qL(θ10 ) + q Nb + Nt ∂r ∂r
Nt H(q, σ) = L(σ) − L(σ10 ) + qL(σ10 ) + q Nb
1 ∂ r ∂r
∂θ ∂r
∂θ r ∂r
2 #
,
,
3 ! ∂w 1 ∂ αα1 r , H(q, w) = L(w) − L(w10 ) + qL(w10 ) + q[ r ∂r ∂r 5 ! 1 ∂ ∂w ∂p + α 2 α2 r + Gr θ + Br σ − ]. r ∂r ∂r ∂z
(15)
(16)
(17)
9 Taking L =
1 ∂ ∂ (r ∂r ) r ∂r
and we can define initial guesses satisfying boundary conditions r2 − h2 ), 4 r2 − h2 Nt σ10 (r, z) = ( ) , 4 Nb 2 2 r − h ∂p0 w10 (r, z) = ( ) − 1, 4 ∂z θ10 (r, z) = (
(18) (19) (20)
Let us define θ(r, q) = θ0 + qθ1 + q 2 θ2 + q 3 θ3 + . . .
(21)
σ(r, q) = σ0 + qσ1 + q 2 σ2 + q 3 σ3 + . . .
(22)
w(r, q) = w0 + qw1 + q 2 w2 + q 3 w3 + . . .
(23)
The solution for velocity, temperature and nanoparticle phenomena can be written as for q→1 Nt 4 Nt 4 (Nt )2 6 r2 − h2 )+ (r − h4 ) + (r − h4 ) + (r − h6 ), 4 32 64 576 Nt (Nt )2 6 + (r4 − h4 ) + (r − h6 ), 16 288
θ(r, z) = (
σ(r, z) = (
r2 − h2 Nt r2 − h2 Nt r2 − h2 Nt r4 − h4 (Nt )2 ) ) ) ) −( +( +( , 4 Nb 2 Nb 4 Nb 32 Nb
r2 − h2 dp ) + a1 (r2 − h2 ) + a2 (r4 − h4 ) + a3 (r6 − h6 ) 4 dz +a12 (r4 − h4 ) + a13 (r6 − h6 ) + a14 (r8 − h8 ) + a15 (r10 − h10 ),
(24) (25)
w(r, z) = −1 + (
(26)
dp 16 [F + a16 ] . (27) = dz −h4 φ2 F = 2Q − − 1. (28) 2 The pressure rise ∆p and friction force F are defined as follows Z 1 dp ∆p = dz, (29) dz 0 Z 1 dp 2 F = h − dz. (30) dz 0 In order to investigate the flow analysis, we have taken four waveforms, namely, sinusoidal, multi sinusoidal square wave and trapezoidal waves which are given below
10 1. Sinusoidal wave h(z) = 1 +
λkz + φ sin(2πz) a0
2. Multi Sinusoidal wave h(z) = 1 +
λkz + φ sin(2mπz) a0
3. Square wave h(z) = 1 +
∞ λkz 4 X (−1)n+1 + φ{ cos(2π(2n − 1)z)} a0 π n=1 (2n − 1)
4. Trapezoidal wave ∞ λkz 32 4 X sin π8 (2n − 1) h(z) = 1 + sin(2π(2n − 1)z)} + φ{ 2 a0 π π n=1 (2n − 1)2
4
Results and discussion
This section deals with the solution for nanofluid flow of a non Newtonian fluid model with a non-uniform vertical tube. The expression for temperature,concentration, velocity, pressure rise, pressure gradient and streamlines are calculated numerically and analyzed for different flow parameters. Figs. 2-3 show the temperature profile for different values of Nt and Nb . It is depicted that with the increase in Nt and Nb temperature profile decreases. Fig. 4 shows that with the increase in Nb (brownian motion parameter) concentration profile increases. Fig. 5 is presented to see the impact of Nt (thermophoresis parameter) on concentration profile and observed that by increasing the values of Nt parameter concentration profile decreases. Figs. 6-9 show the velocity profile for different values of α, α1 (fluid parameters) Nt (thermophoresis parameter) and Nb (brownian motion parameter). Fig. 6-7 show that velocity profile gets decreasing function in the region (−0.6 ≤ r ≤ 0.6) whereas it get opposite behavior in the rest of the region. Fig. 8-9, it is depicted that velocity profile gets
decreasing function by increasing the value of Nt and Nb . Figures 10, 13 show the pressure rise (versus flow rate) for diverse value of Nt , α3 . In these figures, it is depicted that by increasing value of Nt pressure rise increasing. From figure 12 by increasing value of α3 pressure rise decreasing. Figs. 11, 13 show the friction force for diverse values of α3 , Nt . Figs. 14-17 are arranged to analyze the behavior of pressure gradient for different waves
11 shape. It is noticed from these figures that for z ∈ [0, 0.5] and [1, 1.5], the pressure gradient
remains lower where as higher values of pressure gradient are observed for z ∈ [0.51, 1].
Moreover, increasing the values of φ results in an increase in pressure gradient. Figs. 18-21 illustrate the streamlines for different wave shapes. In these figures, it is depicted that by increasing value of α1 , trapped bolus is increasing.
12
Figure 2: Variation of temperature graph for Nb when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22.
Figure 3: Variation of temperature graph for Nt when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22.
13
Figure 4: Variation of concentration graph for Nb when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22.
Figure 5: Variation of concentration graph for Nt when Nb = 8, Z = 0.2, k = 0.1 ,a0 = 0.01 and λ = 0.1,φ = 0.22.
14
Figure 6: Variation of velocity graph for α when Nt = 8, Z = 0.2, k = 0.1 ,a0 = 0.01 and λ = 0.1,φ = 0.22, Nb = 8, α1 = 0.9.
Figure 7: Variation of velocity graph for α1 when Nt = 8, Z = 0.2, k = 0.1 ,a0 = 0.01 and λ = 0.1,φ = 0.22, Nb = 8, α = 0.08, α1 = 0.9.
15
Figure 8: Variation of velocity graph for Nt when = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, Nb = 8, α1 = 8, α = 0.03.
Figure 9: Variation of velocity graph for Nb when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
16
Figure 10: Variation of pressure rise for Nt when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8.
Figure 11: Variation of frictional force for Nt when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, Nb = 0.03.
17
Figure 12: Variation of pressure rise for α3 when Nt = 8, Z = 0.2, Nb = 7, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α2 = 0.03.
Figure 13: Variation of frictional force for α3 when Nt = 8, Z = 0.2, Nb = 7, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α2 = 0.03.
18
Figure 14: Pressure gradient
dp dz
for sinusoidal wave when Nt = 8, Z = 0.2, k = 0.1,
a0 = 0.01 and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
Figure 15: Pressure gradient
dp dz
for multisinusoidal wave when Nt = 8, Z = 0.2, k = 0.1,
a0 = 0.01 and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
19
Figure 16: Pressure gradient
dp dz
for Trapizodioal wave when Nt = 8, Z = 0.2, k = 0.1,
a0 = 0.01 and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
Figure 17: Pressure gradient
dp dz
for square wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01
and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
20
Figure 18: Streamlines pattern for sinusoidal wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
Figure 19: Streamlines pattern for multisinusoidal wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
21
Figure 20: Streamlines pattern for Trapizodioal wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
Figure 21: Streamlines pattern for square wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
22
5
Conclusion
In this article, we have analyse the nano particle analysis for peristaltic flow of a NonNewtonian fluid due to non uniform vertical tube. The main explanation of the present study is concisely as follows: • The pressure rise decreases with an increase in the value of α while an opposite behavior is observed for thermopherses parameters Nt .
• An increase in the values of thermophoresis parameter Nt leads to a reduction in the velocity and temperature profiles.
• The concentrated nanoparticle field is enhanced by increasing values of thermophoresis parameter Nt while it reduces for larger Brownian motion parameter Nb .
• The velocity field is increases by increasing values of thermophoresis parameter Nt while it reduces for larger Brownian motion parameter Nb .
• The pressure gradient is an increasing function of φ. • The frictional forces are increased when we increase the parameter α where as an opposite effect is observed for Nt .
Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
6
Acknowledgement
The author extends his appreciation to the Deanship of Scientific Research at KSU for funding the work through the research group project No. RGP-160.
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Journal Pre-proof
Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube M.A. Imran, A. Shaheen, El-Sayed M. Sherif, M. Rahimi-Gorji, Asiful H. Seikh PII: DOI: Reference:
S0577-9073(19)31010-X https://doi.org/10.1016/j.cjph.2019.11.029 CJPH 1032
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
10 January 2019 22 November 2019 28 November 2019
Please cite this article as: M.A. Imran, A. Shaheen, El-Sayed M. Sherif, M. Rahimi-Gorji, Asiful H. Seikh, Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.11.029
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Highlights • Peristaltic flow of a Jeffrey six constants nanofluid has been modeled. • The governing equations are solved analytically by HPM. • Direct relationship between nanoparticles concentration and value of thermophoresis parameter.
• Velocity, temperature and frictional forces are decreasing functions of thermophoresis parameter.
1
Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube M. A. Imran∗, A. Shaheen†, El-Sayed M Sherif ‡, M. Rahimi-Gorji §, Asiful H. Seikh
Abstract: Peristaltic flow of non-Newtonian nano fluid through a non-uniform surface has been investigated in this paper. The fluid motion along the wall of the surface is caused by the sinusoidal wave traveling with constant speed. The governing equations are converted into cylindrical coordinate system and assuming low Reynolds number and long wave length partial differential equations are simplified. Analytically solutions of the problem are obtained by utilizing the homotopy perturbation method (HPM). In order to insight the impact of embedded parameters on temperature, concentration and velocity some graphs are plotted for different peristaltic waves. At the end, some observations were made from the graphical presentation that velocity, pressure rise and nano particle concentration are increasing function of thermophoresis parameter Nt while temperature and frictional forces show opposite trend.
Keywords: Peristaltic flow; nanoparticles; Jeffrey six constant fluid; Homotopy perturbation method.
∗
Department of Mathematics, University of Management and Technology, Lahore, Pakistan.
Email:[email protected] † Department of Mathematics, Comsats University Islamabad Lahore Campus,Pakistan.Email: [email protected] ‡ Center of Excellence for Research in Engineering Materials, King Saud University, P.O. Box-800, Riyadh 11421, Saudi Arabia,2Electrochemistry and Corrosion Laboratory, Department of Physical Chemistry, National Research Centre, El-Behoth St. 33, Dokki, Cairo 12622, Egypt Corresponding e-mail address: [email protected] § Faculty of Medicine and Health Science, Ghent University, Gent 9000, Belgium. Email: [email protected] , [email protected] ¶ Center of Excellence for Research in Engineering Materials, King Saud University, P.O. Box-800, Riyadh 11421, Saudi Arabia. Email: [email protected]
2
¶
3
Nomenclature U,¯W C¯
fluid velocity components
T¯
temperature of fluid
nanopartical concentration
DB
Brownian diffusion cofficient
DT
thermospheric diffusion coefficient
τ
ratio of heat capacity of the nanoparticle
a0
radius of the inlet
Re
Reynolds number
Nt
thermophoresis number
t
is the time
δ
long wavelength
Gr
thermophoresis number
Nb
Brownian motion number
µ
dynamic viscosity
ρp
density of the particle
λ
wave length
c
wave speed
b
wave amplitude
1
relaxation time
2
delay time
d, b, c
Jeffrey six constant
σ
dimensionless concentration
θ
dimensionless temperature
w
dimensionless velocity
Pr
Prandtl number
Ec
Eckert number
1
Introduction
Peristalsis is a critical system for moving toward fluid in a channel or tube by methods for a usage contractile ring over the tube, which drives the material forward. Peristalsis is a radially symmetrical contraction of muscles which propagates in a way down a muscular tube. This process is quite important for fluid transport in living organisms and industry. Blood pumps in dialysis and heart lung machine works because of the peristaltic phenomena. Most of the fluids in nature possess non-Newtonian characteristics. Non-Newtonian fluids change their viscosity or flow behavior under stress. Not all non-Newtonian fluids behave in the same way when stress is applied; some become more solid while others more fluid. Some non-Newtonian fluids react as a result of the amount of stress applied, while others react as a result of the length of time that stress is applied. In a few technique of physiology and designing, peristaltic stream concerns are comprehensively found in channel/tube. Such streams rise in gulping of sustenance amid throat, pee transportation from kidney to bladder, evaluation chyme in gastrointestinal tract, ovum development in the female fallopian tube, spermatozoa move in the channels efferent of male conceptive tract, vasomotion of limited veins, water development from ground to above branches of adult
4 trees referred to as [1–5]. Pumping of blood in lungs, transportation of mordant fluids, heart and hygienic fluid is due to the service of peristalsis. Srivastava [6] was the first one who investigated the peristaltic during male reproductive system experimentally and numerically. Peristaltic flow has been discussed in the vas deferens by Gupta [7] , Guha [8] and Batra [9]. Hussain and Malik [10] studied the effects of Joule heating and viscous dissipation on MHD Sisko nanofluid over a stretching cylinder. Khalil et al. [11–16] have discussed numerical solutions of non-Newtonian fluids. Hydromagnetic flow of fluid in a uniform pipe with variable thickness was investigated by Hakeem et al. [17]. Mair et al. [18] explained the boundary layer flow of MHD tangent hyperbolic nanofluid over a stretching sheet. Bilal et al. [19] discussed double diffusion on MHD Prandtl nano fluid. Peristaltic transportation of channel flow for non-Newtonian fluid with some inclination has been studied by Vajravelu et al. [20]. Study of peristaltic flow in the presence of magnetic field has also achieved a lot of importance in daily life and engineering sciences. Effects of MHD on the peristaltic flows for different modes of heat transfer like conduction, convection and radiations are reported in the Refs. [21–23]. The rate of heat transfer is dependent on the temperatures of the systems and the properties of the prevailing medium through which the heat is transferred. Different authors have discussed [24, 25] the effect of force on the heat convection and mass transfer. Study of peristaltic flow in the presence of magnetic field has also achieved a lot of importance in daily life and engineering sciences. A study dealing MHD nanofluid flow with Stagnation mixed convection and slip condition for stretching sheet has been discussed [26]. Effects of MHD on the peristaltic flows for different modes of heat transfer like conduction, convection and radiations are reported in the Refs. [27]. Abbasi et al. [28,29] discussed Peristaltic motion of nanofluid with MHD mixed convective and Joule heating effects. Since the first investigation done by the Choi [30], the study of nanofluids have attracted the attention of many researchers due to its tremendous applications in various fields of life such as biomedical devices, treatment of tumor, nuclear reactor, microchips, cooling, radiators and nonmedicinal etc. Different authors have discussed [31, 32] heat transfer analysis for peristaltic flow of Carreau-Yasuda fluid through a curved channel with radial magnetic field. In [33, 34] discussed the calculations for the estimation of transient surface warmth motion amid ultra-quick surface cooling.The demonstrated conditions are settled
5 scientifically by homotopy bother technique. The outcomes for velocity profile, pressure gradient, pressure rise and stream function have been considered for various estimations of the parameters. The physical highlights of applicable parameters are examined through diagrams. The streamlines are outlined for some physical amount to analyze the catching trapping phenomenon. Present study deals with the peristaltic flow of non-Newtonian nano fluid through a non-uniform tube. The fluid motion along the wall of the surface is caused by the sinusoidal wave traveling with constant speed. The governing equations are converted into cylindrical coordinate system and assuming low Reynolds number and long wave length partial differential equations are simplified. Analytically solutions of the problem are obtained by utilizing the homotopy perturbation method (HPM). In order to insight the impact of embedded parameters on temperature, concentration and velocity some graphs are plotted for different peristaltic waves. At the end, some streamlines have also been plotted.
2
Problem Formulation
Let us consider incompressible Jeffrey’s six constant nanofluid bearing the peristaltic flow in a non-uniform vertical tube. The flow is generated by a sinusoidal wave train conduction along the walls of the tube with constant speed c. The geometry of the problem cylindrical Coordinates system (R, Z) is shown in Fig. ( 1) ¯ = a(Z) ¯ + b sin 2π (Z¯ − ct), h λ ¯ = a0 + K Z, ¯ a0 is the radius of the inlet, K is the constant whose magnitude where a(Z) depend on the length of the tube , λ is the wavelength, b is wave amplitude, c is the propagation velocity and t is the time. We consider the cylindrical coordinate system (R, Z), in which Z-axis lies along the center line of the tube and R is transverse to it. The
6
Figure 1: Geometry of the problem.
fundamental equations of continuity, momentum, nanoparticle concentration are [35] U¯ (1) U¯R + ¯ + W¯Z = 0, R ∂ ∂ P¯ ∂ ∂ 1 ∂ ¯ ∂ τ¯¯¯ ¯ ¯ ρ (2) + U ¯ + W ¯ U¯ = − ¯ + ¯ ¯ (R¯ τR¯ R¯ ) + ¯ (¯ τR¯ Z¯ ) + ¯θθ , ¯ ∂t ∂R ∂Z ∂R R ∂R ∂Z R ¯ ∂ ∂ ∂ ¯ ¯ ¯ = − ∂ P + 1 ∂ (R¯ ¯ τR¯ Z¯ ) + ∂ (¯ ρ +U ¯ +W ¯ W τZ¯Z¯ ) ¯ ¯ ¯ ¯ ∂t ∂R ∂Z ∂Z R ∂R ∂ Z¯ + ρgα(T¯ − T¯0 ) + ρgα(C¯ − C¯0 ), (3)
∂ ¯ ∂ U¯ ¯ + W ∂R ∂ Z¯
∂ ¯ ∂ U¯ ¯ + W ∂R ∂ Z¯
2¯ ¯ ∂ 2 T¯ ∂ T 1 ∂ T T¯ = α ¯2 + R ¯ ∂R ¯ + ∂ Z¯ 2 ∂R " " ¯ ¯ 2 ¯ 2 ## ∂C ∂T ∂ C¯ ∂ T¯ DT¯ ∂ T¯ ∂T + τ DB + + + , (4) ¯ ∂R ¯ ∂ Z¯ ∂ Z¯ ¯ ∂R T¯0 ∂R ∂ Z¯
C¯ = DB
∂ 2 C¯ 1 ∂ C¯ ∂ 2 C¯ ¯2 + R ¯ ∂R ¯ + ∂ Z¯ 2 ∂R DT¯ ∂ 2 T¯ 1 ∂ T¯ ∂ 2 T¯ + ¯ ¯2 + R ¯ ∂R ¯ + ∂ Z¯ 2 . T0 ∂ R
(5)
7 The constitutional equation for a Jeffrey fluid form is defined as [36] d¯ τ τ¯ + 1 − W.¯ τ + τ¯.W + d(¯ τ .D + D.¯ τ ) + b¯ τ : DI + cDtr¯ τ = dt dD 2µ D + 2 ( − W.D + D.W + 2dD.D + bD : DI) . dt
(6)
in which ∇V¯ + (∇V¯ )t1 , 2 ∇V¯ − (∇V¯ )t1 W(antisymmetric measurement of velocity gradient) = , 2 D(symmetric measurement of velocity gradient) =
The transformations takes the form ¯ r¯ = R,
z¯ = Z¯ − c1 t¯,
u¯ = U¯ ,
¯ − c1 , w¯ = W
The resultant boundary conditions are: ¯ ∂W = 0, ∂¯ r ¯ = 0, W
∂ T¯ = 0, ∂¯ r T¯ = T¯0 ,
∂ C¯ = 0, ∂¯ r
C¯ = C¯0 ,
at
at
r¯ = 0,
¯ = a(Z) ¯ + b sin r¯ = h
(7) 2π ¯ (Z − c1 t). λ
(8)
Introducing the following non-dimensional variables, R = U = Re = Pr = Nb =
¯ ¯ R r¯ Z¯ z¯ W w¯ a¯ τ , r= , Z= , z= , W = , w= , τ= , a a λ λ c1 c1 c1 µ 0 λU¯ λ¯ u c1 t¯ a2 p¯ 1 c1 2 c1 , u= , t= , p= , λ1 = , λ2 = , ac1 ac1 λ c1 λµ0 a a ¯ ac1 ρ c2 a h λkz qw a , Ec = , δ = , h= =1+ + φ sin 2πz, Nu = µ0 cp T0 λ a a0 kT0 2 ¯ ¯ µ 0 cp µ ¯(θ) Q0 a T − T0 k , µ(θ) = , β1 = , θ= , α= , ¯ k µ0 kT0 (ρc)f T0 (ρc)p DB C¯0 gαa3 C¯0 (ρc)p DT C¯0 C¯ − C¯0 , Br = , N = , σ = . t (ρc)f υ2 (ρc)f α C¯0
keeping in mind the assumptions of δ << 1 and low Reynolds number approximation we
8 have in simplified form as ∂p = 0, ∂r 3 ! ∂p 1 ∂ ∂w 1 ∂ ∂w = r + αα1 r + ∂z r ∂r ∂r r ∂r ∂r 5 ! 1 ∂ ∂w α 2 α2 r + Gr θ + Br σ, r ∂r ∂r 2 1 ∂ ∂θ ∂θ ∂σ ∂θ r + Nb + Nt , 0 = r ∂r ∂r ∂r ∂r ∂r 1 ∂ ∂σ Nt 1 ∂ ∂θ 0 = r + r . r ∂r ∂r Nb r ∂r ∂r
(9)
(10) (11) (12)
The corresponding boundary conditions are defined as ∂w = 0, ∂t
∂θ = 0, ∂r
w = −1,
3
θ = 0,
∂σ = 0, ∂r
at
r = 0,
σ=0
at
r =h=1+
(13) λkz + φ sin(2πz). a0
(14)
Solution of the problem
Analytical technique called homotopy perturbation method (HPM) is used to eliminates fixed perturbation method and apply to Eqs. (10)-(12), "
∂θ ∂σ H(q, θ) = L(θ) − L(θ10 ) + qL(θ10 ) + q Nb + Nt ∂r ∂r
Nt H(q, σ) = L(σ) − L(σ10 ) + qL(σ10 ) + q Nb
1 ∂ r ∂r
∂θ ∂r
∂θ r ∂r
2 #
,
,
3 ! ∂w 1 ∂ αα1 r , H(q, w) = L(w) − L(w10 ) + qL(w10 ) + q[ r ∂r ∂r 5 ! 1 ∂ ∂w ∂p + α 2 α2 r + Gr θ + Br σ − ]. r ∂r ∂r ∂z
(15)
(16)
(17)
9 Taking L =
1 ∂ ∂ (r ∂r ) r ∂r
and we can define initial guesses satisfying boundary conditions r2 − h2 ), 4 r2 − h2 Nt σ10 (r, z) = ( ) , 4 Nb 2 2 r − h ∂p0 w10 (r, z) = ( ) − 1, 4 ∂z θ10 (r, z) = (
(18) (19) (20)
Let us define θ(r, q) = θ0 + qθ1 + q 2 θ2 + q 3 θ3 + . . .
(21)
σ(r, q) = σ0 + qσ1 + q 2 σ2 + q 3 σ3 + . . .
(22)
w(r, q) = w0 + qw1 + q 2 w2 + q 3 w3 + . . .
(23)
The solution for velocity, temperature and nanoparticle phenomena can be written as for q→1 Nt 4 Nt 4 (Nt )2 6 r2 − h2 )+ (r − h4 ) + (r − h4 ) + (r − h6 ), 4 32 64 576 Nt (Nt )2 6 + (r4 − h4 ) + (r − h6 ), 16 288
θ(r, z) = (
σ(r, z) = (
r2 − h2 Nt r2 − h2 Nt r2 − h2 Nt r4 − h4 (Nt )2 ) ) ) ) −( +( +( , 4 Nb 2 Nb 4 Nb 32 Nb
r2 − h2 dp ) + a1 (r2 − h2 ) + a2 (r4 − h4 ) + a3 (r6 − h6 ) 4 dz +a12 (r4 − h4 ) + a13 (r6 − h6 ) + a14 (r8 − h8 ) + a15 (r10 − h10 ),
(24) (25)
w(r, z) = −1 + (
(26)
dp 16 [F + a16 ] . (27) = dz −h4 φ2 F = 2Q − − 1. (28) 2 The pressure rise ∆p and friction force F are defined as follows Z 1 dp ∆p = dz, (29) dz 0 Z 1 dp 2 F = h − dz. (30) dz 0 In order to investigate the flow analysis, we have taken four waveforms, namely, sinusoidal, multi sinusoidal square wave and trapezoidal waves which are given below
10 1. Sinusoidal wave h(z) = 1 +
λkz + φ sin(2πz) a0
2. Multi Sinusoidal wave h(z) = 1 +
λkz + φ sin(2mπz) a0
3. Square wave h(z) = 1 +
∞ λkz 4 X (−1)n+1 + φ{ cos(2π(2n − 1)z)} a0 π n=1 (2n − 1)
4. Trapezoidal wave ∞ λkz 32 4 X sin π8 (2n − 1) h(z) = 1 + sin(2π(2n − 1)z)} + φ{ 2 a0 π π n=1 (2n − 1)2
4
Results and discussion
This section deals with the solution for nanofluid flow of a non Newtonian fluid model with a non-uniform vertical tube. The expression for temperature,concentration, velocity, pressure rise, pressure gradient and streamlines are calculated numerically and analyzed for different flow parameters. Figs. 2-3 show the temperature profile for different values of Nt and Nb . It is depicted that with the increase in Nt and Nb temperature profile decreases. Fig. 4 shows that with the increase in Nb (brownian motion parameter) concentration profile increases. Fig. 5 is presented to see the impact of Nt (thermophoresis parameter) on concentration profile and observed that by increasing the values of Nt parameter concentration profile decreases. Figs. 6-9 show the velocity profile for different values of α, α1 (fluid parameters) Nt (thermophoresis parameter) and Nb (brownian motion parameter). Fig. 6-7 show that velocity profile gets decreasing function in the region (−0.6 ≤ r ≤ 0.6) whereas it get opposite behavior in the rest of the region. Fig. 8-9, it is depicted that velocity profile gets
decreasing function by increasing the value of Nt and Nb . Figures 10, 13 show the pressure rise (versus flow rate) for diverse value of Nt , α3 . In these figures, it is depicted that by increasing value of Nt pressure rise increasing. From figure 12 by increasing value of α3 pressure rise decreasing. Figs. 11, 13 show the friction force for diverse values of α3 , Nt . Figs. 14-17 are arranged to analyze the behavior of pressure gradient for different waves
11 shape. It is noticed from these figures that for z ∈ [0, 0.5] and [1, 1.5], the pressure gradient
remains lower where as higher values of pressure gradient are observed for z ∈ [0.51, 1].
Moreover, increasing the values of φ results in an increase in pressure gradient. Figs. 18-21 illustrate the streamlines for different wave shapes. In these figures, it is depicted that by increasing value of α1 , trapped bolus is increasing.
12
Figure 2: Variation of temperature graph for Nb when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22.
Figure 3: Variation of temperature graph for Nt when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22.
13
Figure 4: Variation of concentration graph for Nb when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22.
Figure 5: Variation of concentration graph for Nt when Nb = 8, Z = 0.2, k = 0.1 ,a0 = 0.01 and λ = 0.1,φ = 0.22.
14
Figure 6: Variation of velocity graph for α when Nt = 8, Z = 0.2, k = 0.1 ,a0 = 0.01 and λ = 0.1,φ = 0.22, Nb = 8, α1 = 0.9.
Figure 7: Variation of velocity graph for α1 when Nt = 8, Z = 0.2, k = 0.1 ,a0 = 0.01 and λ = 0.1,φ = 0.22, Nb = 8, α = 0.08, α1 = 0.9.
15
Figure 8: Variation of velocity graph for Nt when = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, Nb = 8, α1 = 8, α = 0.03.
Figure 9: Variation of velocity graph for Nb when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
16
Figure 10: Variation of pressure rise for Nt when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8.
Figure 11: Variation of frictional force for Nt when Nb = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, Nb = 0.03.
17
Figure 12: Variation of pressure rise for α3 when Nt = 8, Z = 0.2, Nb = 7, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α2 = 0.03.
Figure 13: Variation of frictional force for α3 when Nt = 8, Z = 0.2, Nb = 7, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α2 = 0.03.
18
Figure 14: Pressure gradient
dp dz
for sinusoidal wave when Nt = 8, Z = 0.2, k = 0.1,
a0 = 0.01 and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
Figure 15: Pressure gradient
dp dz
for multisinusoidal wave when Nt = 8, Z = 0.2, k = 0.1,
a0 = 0.01 and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
19
Figure 16: Pressure gradient
dp dz
for Trapizodioal wave when Nt = 8, Z = 0.2, k = 0.1,
a0 = 0.01 and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
Figure 17: Pressure gradient
dp dz
for square wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01
and λ = 0.1, Nb = 7, α1 = 8, α = 0.03.
20
Figure 18: Streamlines pattern for sinusoidal wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
Figure 19: Streamlines pattern for multisinusoidal wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
21
Figure 20: Streamlines pattern for Trapizodioal wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
Figure 21: Streamlines pattern for square wave when Nt = 8, Z = 0.2, k = 0.1, a0 = 0.01 and λ = 0.1, φ = 0.22, α1 = 8, α = 0.03.
22
5
Conclusion
In this article, we have analyse the nano particle analysis for peristaltic flow of a NonNewtonian fluid due to non uniform vertical tube. The main explanation of the present study is concisely as follows: • The pressure rise decreases with an increase in the value of α while an opposite behavior is observed for thermopherses parameters Nt .
• An increase in the values of thermophoresis parameter Nt leads to a reduction in the velocity and temperature profiles.
• The concentrated nanoparticle field is enhanced by increasing values of thermophoresis parameter Nt while it reduces for larger Brownian motion parameter Nb .
• The velocity field is increases by increasing values of thermophoresis parameter Nt while it reduces for larger Brownian motion parameter Nb .
• The pressure gradient is an increasing function of φ. • The frictional forces are increased when we increase the parameter α where as an opposite effect is observed for Nt .
Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
6
Acknowledgement
The author extends his appreciation to the Deanship of Scientific Research at KSU for funding the work through the research group project No. RGP-160.
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