/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 3, 2016, 266-276 ISSN 1307-5543 – www.ejpam.com Peristaltic Flow of Newtonian Nanofluid through an Inclined Annulus Cylinder A. I. Abdellateef, Syed Zahoor ul Haque∗ Department of Mathematics and Statistics, Caledonian college of engineering Oman, University of Glasgow, Scotland, UK Abstract. This investigation is concerned with the peristaltic transport of Nanofluid in an inclined an- nulus tube. Transport equations involve the combined effects of Brownian motion and thermophoretic diffusion of nanoparticles. Mathematical modeling is carried out by utilizing long wavelength and low Reynolds number assumptions. Attention has been focused on the behaviors of Brownian motion parameter (Nb), thermophoresis parameter (Nt) and inclination of the annulus. The result indicates an appreciable increase in the temperature and nanoparticles concentration with the increase in the strength of Brownian motion effects, and the inclination angle increases the pressure rise. Key Words and Phrases: peristaltic flow, heat transfer, Nanofluid, frictional force, Viscous fluid, In- clined Endoscopic; 1. Introduction In the peristaltic flow, wave propagates along the length of the annulus tube, Nano fluid present in the tube get transport in the same direction where the wave propagates. peristal- sis pumping technique is used by many organs such as urethra, male reproductive system and gastro-intestinal tract. Shapiro, Jaffrine and Weinberg [12] investigated the linearized problem of the peristaltic transport of the fluid in two-dimensional channel and tube contains incompressible fluid. They even considered that the wave length to be infinite. In continua- tion to that Jaffrine [5], investigated the effects of Reynolds number and wave number, when these are very small but not infinitesimal. Furthermore experiments of Weinberg, Eckstein, and Shapiro [19] Established that the theory of Jaffrine [5] is valid up to Reynolds number of about 10. This research was continued by Takabatake and Ayukawa [15] in which they investigated the problem of peristaltic in case of two-dimensional tube numerically and it was concluded that validity of the perturbation solutions by Jaffrine [5] is restricted within a lim- ited range than that which he had predicted earlier. Moreover they found the reflux near the ∗Corresponding author. Email address: syed.zahoor@cce.edu.om (S. ul Haque) http://www.ejpam.com 266 c© 2016 EJPAM All rights reserved. A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. Math, 9 (2016), 266-276 267 central axis with a large Reynolds number. Takabatake, Ayukawa, and Mori [16] studied nu- merically the peristaltic transport of the fluid and its efficiency in cylindrical tubes and since nano fluid has many potential applications so the new researchers paying intensive attention for references [1–4, 7, 8, 10, 17] and [13], this lead towards effective thermal capacity, for many production of heat transfer fluids in heat exchange, in plants and in automotive cooling significations, due to their extensive thermal properties, a plenty of literature available which deals with the study of nano fluid and its applications (Yoo et al. [20], Menca et al. [11], Wang and Mujumdar [18]) of the non-linear peristaltic transport of a newtonian fluid in an inclined symmetric channel considered by Kothandapani [9]. In the present paper, we investigated the peristaltic transport of Nano viscous incompress- ible Newtonian fluid in an inclined annulus tube under the assumptions of long wavelength and lower Reynolds number. Attention has been focused on the behaviors of Brownian mo- tion parameter (Nb), thermophoresis parameter (Nt) and inclination of the annulus cylinder. The result indicates an appreciable increase in the temperature and Nanoparticles concentra- tion with the increase in the strength of Brownian motion effects, and the inclination angle increases the pressure and all these also discussed through graphs. 1.1. Equations Consider incompressible Newtonian fluid through coaxial inclined tubes such that the inner tube is rigid (endoscope or catheter) and moving with constant velocity where the outer have a sinusoidal wave traveling down its wall. Figure 1: Simplified model geometry of the problem. A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. Math, 9 (2016), 266-276 268 The geometry of the wall surfaces described in Figure 1 are r ′1 =a1, r ′2 =a′ + b′ sin[ 2π λ (z′ − c t ′)], (1) where a1, a′ are the radius of the inner and the outer tubes at any axial distance z′, b′ are the amplitude of the wave, λ′ is the wavelength, c′ is the propagation velocity and t ′ is the time. We choose a cylindrical coordinate system (r ′, z′), where z′-axis lies along the centerline of the inner and outer tubes and r ′ is the distance measured radially. The equations governing the problem are: ∇̄ · q̄=0, (2) ρ f Dq̄ Dt =− ∇̄p̄+µ∇̄2q̄+ρg[sin(ψ)ẑ − cos(ψ)R̂], (3) (ρc) f DT̄ Dt =k∇̄2 T̄ + (ρc)p � DB(∇̄C̄) · (∇̄T̄) + DT (T0) (∇̄T̄) · (∇̄T̄) � , (4) DC̄ Dt =DB∇̄ 2C̄ + DT (T0) ∇̄2 T̄ , (5) where ρ f is the density of the fluid, ρp is the density of the particle, C is the volumetric volume expansion coefficient, q is the velocity vector, Ts is the fluid temperature in static condition, d/dt represents the material time derivative, p is the pressure, C is the nanoparticle phenomena, DB is the Brownian diffusion coefficient and DT is the thermophoretic diffusion coefficient. The problem has been studied in cylindrical coordinate system (r ′, z′), radial, and axial coordinates respectively, introducing a wave frame (r ′, z′) moving with velocity C away from the fixed frame (R′, Z ′) by the transformations. ∂ u′ ∂ z′ + ∂ v′ ∂ r ′ + v′ r ′ =0 (6) ρ � u′ ∂ u′ ∂ z′ + v′ ∂ u′ ∂ r ′ � =− ∂ p′ ∂ z′ +µ �∂ 2u′ ∂ z′2 + ∂ 2u′ ∂ r ′2 + 1 r ′ ∂ u′ ∂ r ′ � +ρgα(T ′ − T ′0) +ρgα(C ′ − C ′0) +ρg sin(ψ) (7) ρ � u′ ∂ v′ ∂ z′ + v′ ∂ v′ ∂ r ′ � =− ∂ p′ ∂ r ′ +µ �∂ 2v′ ∂ z′2 + ∂ 2v′ ∂ r ′2 + 1 r ′ ∂ v′ ∂ r ′ − v′ r ′2 � −ρg cos(ψ) (8) The energy equation [ū ∂ T̄ ∂ z̄ + v̄ ∂ T̄ ∂ r̄ ] =α( ∂ 2 T̄ ∂ r̄2 + 1 r ∂ T̄ ∂ r̄ + ∂ 2 T̄ ∂ z̄2 ) +τ � DB ¦ ( ∂ C̄ ∂ r̄ ∂ T̄ ∂ r̄ ) + ( ∂ C̄ ∂ z̄ ∂ T̄ ∂ z̄ ) © + DT T0 ¦ ( ∂ T̄ ∂ r̄ )2 + ( ∂ T̄ ∂ z̄ )2 ©� (9) A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. Math, 9 (2016), 266-276 269 [ū ∂ C̄ ∂ z̄ + v̄ ∂ C̄ ∂ r̄ ] =DB( ∂ 2C̄ ∂ r̄2 + 1 r ∂ C̄ ∂ r̄ + ∂ 2C̄ ∂ z̄2 ) + DT T0 ( ∂ 2 T̄ ∂ r̄2 + 1 r ∂ T̄ ∂ r̄ + ∂ 2 T̄ ∂ z̄2 ) (10) where v′,u′ are the velocity components in r ′ and z′-directions respectively, ρ is the density, p′ is the pressure, τ= (ρc)p (ρc) f is the ratio between the effective heat capacity of the nano particle material and heat capacity of the fluid and µ is the viscosity. The boundary conditions are: u′ =− 1, T = T 2, C = C0 at r ′ = r ′2, u′ =− 1, T = T 1 , C = C0 at r ′ = r ′1, (11) where V ′ is the velocity of the inner tube, v′ and u′ are the velocity components in the r ′-and z′ direction, respectively. We introduce the following nondimensional variables and parameters p = a2 µcλ p′, u= u′ c , v = v′ cδ0 , t = c λ t ′, Re = ρca µ , r1 = r ′1 a = ε φ = b a , δ0 = a λ , z = z′ λ , r2 = r ′2 a = 1+ b′ a sin(2πz), θ = T − Ts T1 − Ts , Nt = (ρc)pDt(T1 − Ts) (ρc) f αT0 , σ = (C − C0) C0 , Nb = (ρc)pDBC0 (ρc) f α , α= k (ρc) f , Gr = gαd2 1 (T1 − Ts) υc , Br = gαd2 1 C0 υc (12) where φ is the amplitude ratio, Re is the Reynolds number, δ0 is the dimensionless wave number and Fr = c2 ga2 the Froude number. After using the above assumption, the equations of motion in the dimensionless form reduce to ∂ v ∂ r + v r + ∂ u ∂ z =0, (13) Reδ0 � v ∂ u ∂ r + u ∂ u ∂ z � =− ∂ p ∂ z + � δ2 0 ∂ 2u ∂ z2 + ∂ 2u ∂ r2 + 1 r ∂ u ∂ r � + Grθ + Brσ+ Re Fr sin(ψ), (14) Reδ3 0 � v ∂ v ∂ r + u ∂ v ∂ z � =− ∂ p ∂ r +δ0 � δ2 0 ∂ 2v ∂ z2 + ∂ 2v ∂ r2 + 1 r ∂ v ∂ r − v r2 � − Re Fr δ0 cos(ψ), (15) δ0prRe � v ∂ θ ∂ r + u ∂ θ ∂ z � = ∂ 2θ ∂ r2 + 1 r ∂ θ ∂ r +δ2 0 ∂ 2θ ∂ z2 + Nb � ∂ σ ∂ r ∂ θ ∂ r +δ2 0 ∂ σ ∂ z ∂ θ ∂ z � + Nt � ( ∂ θ ∂ r )2 +δ2 0( ∂ θ ∂ z )2 � (16) δ0Repr C0τ � v ∂ σ ∂ r + u ∂ σ ∂ z � = ∂ 2σ ∂ r2 + 1 r ∂ σ ∂ r +δ2 0 ∂ 2σ ∂ z2 + Nt Nb � ∂ 2θ ∂ r2 + 1 r ∂ θ ∂ r +δ2 0 ∂ 2θ ∂ z2 � (17) A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. Math, 9 (2016), 266-276 270 Equation (15) shows that p is not a function of r. Using the long wavelength approximation and dropping terms of order δ0 and higher, it follows from Equations (13) to (17) that the appropriate equation describing the flow is dp dz − Grθ − Brσ = � ∂ 2u ∂ r2 + 1 r ∂ u ∂ r � + Re Fr sin(ψ). (18) 0= ∂ 2θ ∂ r2 + 1 r ∂ θ ∂ r + Nb � ∂ σ ∂ r ∂ θ ∂ r � + Nt � ( ∂ θ ∂ r )2 � (19) 0= ∂ 2σ ∂ r2 + 1 r ∂ σ ∂ r + Nt Nb � ∂ 2θ ∂ r2 + 1 r ∂ θ ∂ r � (20) The boundary conditions are obtained from the manipulation of the viscous no-slip conditions at the boundary of the wall of the outer and inner tubes, u= 0,θ = 0, σ = 0 at r2 = 1+φ sin(2πz), u= 0,θ = 1, σ = 1 at r1 = ε, (21) where r1 = ε, r2 = 1+φ sin(2πz) and the wall temperature ratio. The solution of Equations (19) and (20), valid in r1 ≤ r ≤ r2 satisfying the corresponding boundary conditions (21) is given by σ =− Nt Nb � rA− rA 2 rA 1 − rA 2 � +   1+ Nt Nb Ln[ r1 r2 ]   Ln[ r r2 ] (22) θ = rA− rA 2 rA 1 − rA 2 (23) where A = � Nb + Nt � /Ln[ r2 r1 ] on substitution from the equations (22), (23) in equation (18) and solving it analytically as a second order ordinary differential equation, the solution will be of the form u=c1 + c2Ln[r] + r2 4 p+   r(A+2) (A+2)2 − rA 2 4 r2 � rA 1 − rA 2 �   � Nt Nb Br−Gr � − Br   1+ Nt Nb Ln � r1 r2 �   � −r2 8 + r2 4 Ln � r r2 � − r2 8 � − Re Fr r2 4 Sin[ψ] (24) The instantaneous volume flow rate Q(z, t) is given by Q(z, t) = 2π ∫ r2 r1 r u dr (25) A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. Math, 9 (2016), 266-276 271 Following the analysis given by Shapiro et al. [6], the mean volume flow, Q over a period is obtained as Q(z, t) = Q π − φ2 2 + 2φSin[2πz] +φ2Sin[2πz]2, (26) where Q is the time-average of the flow over one period of the wave. Make combination between Eqs. (25) and (26), we get ∂ p ∂ z . The pressure rise ∆p(t) and the friction force (at the wall) on the outer and inner tubes are F2 and F1 respectively, in a cylinder of length L, in their non-dimensional forms, are given by ∆p = ∫ 1 0 ∂ p ∂ z dz, (27) F1 = ∫ 1 0 r1 2 � − ∂ p ∂ z � dz (28) F2 = ∫ 1 0 r2 2 � − ∂ p ∂ z � dz (29) 1.2. Graphical Results and Discussion To discuss the results obtained above quantitatively, we shall assume the form of the flow rate Q, in period (Z − t) as in Srivastava and Srivastava [14]: Q(z, t) = Q π − φ2 2 + 2φSin[2πz] +φ2Sin[2πz]2. Where Q is the time -average of the flow over one period of the wave. This form of Q has been assumed in view of the fact that the constant values of Q gives always∆P negative, and hence there will be no pumping action. We shall now compute the dimensionless pressure rise ∆p, the inner / outer frictional forces F1, F2(on the inner and outer surface) for various values of dimensionless flow average Q, radius ratio ε and the velocity of the inner tube v. Following Srivastava [14], we use the values of the various parameters in equations (27) to (29) as: a = 1.25cm and L = λ = 8.01cm. Figure 2 represent the variation of pressure rise with Q for fixed thermophoresis and Brow- nian motion parameters Nt , Nb, the pressure rise gradually increases as the area of flow be- tween the annulus decrease(inner radius ε increases). Also the pressure rise for endoscope increases as (Inclined angle of the cylinder annulus ψ increase) and the increase in both of the amplitude ratio and Reynolds number gives rise in pressure rise. The friction force have been plotted in Figure 3, which show an opposite character in comparison to pressure rise. Effects of temperature profile θ have been shown through Figures 4a and 4c, the behaviour of temperature mainly depends on whether Nt or Nb is less than one, to illustrate it, a sud- den decrease in the temperature is seen in Figures 4a and 4c when both Nt and Nb are in the range of 0.1−0.8. Figures 4b and 4d here the nanopartical phenomenon decrease when there is an increase in the values of thermophoresis parameter Nt and increases gradually with an increase in Brownian motion parameter Nb. A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. Math, 9 (2016), 266-276 272 (a) (b) (c) (d) Figure 2: Variation of pressure rise over the length versus Q̄. A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. Math, 9 (2016), 266-276 273 (a) (b) (c) (d) (e) (f) Figure 3: Variation of inner and outer frictional forces F1, F2 respectively. A. Abdellateef, S. Zahoor / Eur. J. Pure Appl. 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