The Mathematical Analysis for Peristaltic Flow of Hyperbolic Tangent Fluid in a Curved Channel

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1 Commun. Theor. Phys Vol. 59, No. 6, June 15, 213 The Mathematical Analysis for Peristaltic Flow of Hyperbolic Tangent Fluid in a Curved Channel S. Nadeem and E.N. Maraj Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Received December 17, 212; revised manuscript received February 19, 213 Abstract In the present paper, we have investigated the peristaltic flow of hyperbolic tangent fluid in a curved channel. The governing equations of hyperbolic tangent fluid model for curved channel are derived including the effects of curvature. The highly nonlinear partial differential equations are simplified by using the wave frame transformation, long wave length and low Reynolds number assumptions. The reduced nonlinear partial differential equation is solved analytically with the help of homotopy perturbation method HPM. The physical features of pertinent parameters have been discussed by plotting the graphs of pressure rise and stream functions. PACS numbers: d Key words: tangent hyperbolic fluid, curved channel, peristaltic flow, HPM 1 Introduction The peristaltic flows of viscous and non-newtonain fluid have been discussed theoretically and experimentally by many researchers because of its application in physiology and industry. [1 1] Physiologically this mechanism occur in the study of urine transport from kidney to bladder, swallowing food through the esophagus, chyme motion in the gastrointestinal tract, vasomotion of small blood vessels and movement of spermatozoa in the human reproductive tract. There are many industrial and engineering applications in which peristaltic pumps are used to handle a wide range of fluids particularly in chemical pharmaceutical industries. It is also used in sanitary fluid transport, blood pumps in heart lungs machine and transport of corrosive fluids where the contact of the fluid with machinery parts are prohibited. Recently, Nadeem and Akram [11] have considered the peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. In another study, Nadeem and Akbar [12] have discussed the numerical analysis of peristaltic transport of a tangent hyperbolic fluid in an endoscope and considered a cylindrical geometry. However, only few papers are available in literature which discussed the peristaltic flow in a curved channel. Sato et al. [13] have initiated the peristaltic flow in a curved channel. Very recently, Hayat et al. [14] have examined the peristaltic flow of viscous fluid in a curved channel with complaint walls. To our knowledge the peristaltic flow of hyperbolic tangent fluid model in curved channel is still unexplored. Therefore, in the present investigation we have highlighted the peristaltic flow of tangent hyperbolic fluid in a curved symmetric channel. The governing equation of two dimensional hyperbolic tangent fluid model in a curved channel including the effects of curvature are modelled and simplified by using the wave transformation from moving to fixed frame. After the transformation and using assumptions of the long wave length and low Reynolds number approximation the reduced highly nonlinear partial differential equation is solved analytically with the help of homotopy perturbation method. The homotopy perturbation method was first introduced by He in [15 17] In this method, a homotopy with an imbedding parameter p [, 1] is constructed, and the imbedding parameter is considered as a small parameter, and the series solution for highly non linear differential equation is computed by taking into account p 1. The expression for the pressure rise is computed by using numerical integration. At the end the physical phenomena is discussed by plotting graphs. 2 Mathematical Model For an incompressible fluid the conservation of mass and momentum are given by div V =, 1 ρ d V dt = div S+ρ f, 2 where ρ is the density, V is the velocity vector, S is the Cauchy stress tensor, f represents the specific body force and d/ dt represents the material time derivative. The constitutive equation for hyperbolic tangent fluid is given by [12] S = p Ī + τ, 3 τ = [η + η + η tanhγ γ n ] γ, 4 in which pī is the sperical part of the stress due to constraint of incompressibility, τ is the extra stress tensor, Corresponding author, ehnber@gmail.com c 213 Chinese Physical Society and IOP Publishing Ltd

2 73 Communications in Theoretical Physics Vol. 59 η is the infinity shear rate viscosity, η is the zero shear rate viscosity, Γ is the time constant, n is the power law index and γ is defined as 1 γ = γ 2 ij γ ji = i j 1 Π, 5 2 where Π = trgrad V + grad V T 2 in which Π is the second invariant strain tensor. We consider the constitutive Eq. 4, the case for which η = and Γ γ < 1. The component of extra stress tensor therefore, can be written as τ = [η Γ γ n γ] = [η 1 + Γ γ 1 n γ] = η [1 + nγ γ 1] γ. 6 3 Mathematical Formulation Consider a curved channel of half width a coiled in a circle with centre O and radius R is filled with an incompressible hyperbolic tangent fluid. The flow in the channel is induced by sinusoidal waves of small amplitude b traveling along the flexible walls of the channel. The equation of wall surfaces are H X, 2π t = a + b sin[ λ X ] c t, upper wall, 7 H X, 2π t = a b sin[ λ X ] c t, lower wall. 8 In above equations c is the speed and λ denotes the wave length. With the help of Eqs. 5 and 6, Eqs. 1 and 2 in component form for two dimensional flow take the form R [ V ] + ρ t Ū ρ + V t R + R Ū Ū X + R Ū X =, + V V R + R Ū V X Ū2 = P R + 1 Ū V = R P X + 1 R [ τ R R] + R R [ τ R Θ] + X τ R Θ R 9 τ Θ Θ, 1 X [τ Θ Θ]. 11 In the above equations, P is the pressure, V and Ū are the velocity components in radial R and axial X directions respectively, R is constant radius and τ s represent the stresses which are defined as τ R R = 2η [1 + nγ γ 1] V, 12 R τ R Θ = τ Θ R = η [1 + nγ γ 1] Ū R + R V X Ū, 13 τ Θ Θ = 2η [1 + nγ γ 1] R Ū X + V. 14 The flow phenomena is unsteady in the fixed frame. To carry out a steady analysis we switch from fixed frame to wave frame r, x moving with the wave speed c. The transformation between the two frames is given by x = X c t, r = R, ū = Ū c, v = V, 15 where v and ū are the velocity components along r and x-directions in the wave frame. With the help of these transformations Eqs. 9 to 11 take the form r [ r + R v] + ρ c v x + v v r + R ū + c v r + R x R ū r + R x =, 16 ū + c2 r + R = p r + 1 r + R r [ r + R τ r r ] + R r + R ρ c ū x + vū x [τ r θ] r + R ū + c ū r + R x τ θ θ r + R, 17 ū + c v + r + R = R r + R p x + 1 r + R r [ r + R τ r θ] + R x [τ θ θ]. 18 r + R Introducing the following non dimensional variables and velocity stream function relation x = 2π x λ, r = r d, u = ū c, v = v c, ψ = ψ cd, δ = 2πd λ, k = R d, ρcd Re =, η p = 2πd2 p cλη, γ = d c γ, We = Γc d, t = 2πc t λ, u = ψ r, v = δ k ψ r + k x, 19 where Re is the Reynolds number, δ is the wave number, k is the curvature parameter and We is the Weissenberg number. Equation 16 is identically satisfied and Eqs. 17 and 18 under long wavelength and low Reynolds number approximations after eliminating pressure can be casted in the following dimensionless from

3 No. 6 Communications in Theoretical Physics 731 [ 1 nr + k 4 ϕ r ϕ { n3 r 3 + nwe 2 2 ϕ 3 ϕ 3 2 r 2 r 3 + r + k ϕ 2 ϕ 1 k + r r 3 r ϕ 2 4 r + k 3 r r + k 2 ϕ r 2 ϕ r ϕ r + k r ϕ r 3 ϕ r 3 4 ϕ }] r 4 =. 2 The dimensional volume flow rate in fixed frame is defined as Q = H H Ū d R. 21 in which H is a function of X and t. The above expression in wave frame becomes F = H H ūd r, 22 where H is function of x alone. From Eqs. 15, 21, and 22 we can write Q = F + 2c H. 23 The time average flow over a period T at a fixed position X is Q = 1 T T Qdt. 24 Invoking Eq. 23 into Eq. 24 and then integrating, we get Q = q + 2cd. 25 We define the dimensionless mean flows Q in the fixed frame and q in the wave frame, according to Q= Q cd, q = F cd. 26 Equation 22 reduces to in which h q = h Q = q ψ dr = ψh ψ h. 28 r Selecting ψh = q/2, we have ψ h = q/2 and the appropriate boundary conditions in the wave frame are defined as ψ = q 2, ψ r = 1, at r = h = 1 + ǫ sinx, ψ = q 2, ψ = 1, r at r = h = 1 ǫ sinx, 29 where ǫ = b/a is the amplitude ratio. 4 Solution of the Problem In order to solve Eq. 2 using HPM, we construct the following equation [15 17] Using [ 4 ϕ Hϕ, p = 1 p r 4 4 ψ { + nwe r 4 ] 2 ϕ r + k 3 r [ + p 1 nr + k 4 ϕ r ϕ n3 r ϕ 3 ϕ 3 2 r 2 r 3 r + k ϕ r + k 2 ϕ r 2 ϕ r r + k 2 ϕ 1 k + r r 3 r 2 ϕ r ϕ r 3 ϕ r 3 4 ϕ }] r 4 =. 3 φ = φ + pφ 1 + p 2 φ With the help of Eq. 31, equating the like powers of p, we obtain the following systems i Zeroth order system 4 ϕ r 4 = 4 ψ r 4, 32 subject to the boundary conditions ψ = q 2, ψ r = 1, at r = h = 1 + ǫ sin x, ψ = q 2, ψ = 1, r at r = h = 1 ǫ sin x. 33 ii First Order System 4 ϕ 1 r 4 4 ϕ r ψ r nr + ϕ k4 r ϕ { n3 r 3 + nwe 2 ϕ 2 4 ϕ 2 ϕ r + k 3 r r + k 2 r r ϕ 2 ϕ 3 ϕ 2 ϕ 3 ϕ 3 + r + k r 2 r r 3 2 r 2 r 3 r + k ϕ 2 ϕ 4 ϕ } 1 k + r r 3 r r 4 =, 34 ψ =, ψ r =, at r = h = 1 + ǫ sin x, ψ =, ψ =, at r = h = 1 ǫ sin x. 35 r

4 732 Communications in Theoretical Physics Vol. 59 iii Second Order System 4 ϕ 2 r 4 4 ϕ 1 r nr + ϕ 1 k4 r ϕ { 1 n3 r 3 + nwe 4 ϕ r + k 3 r ϕ 2 ϕ 1 r + k r 2 r 2 + ϕ 3 ϕ 1 r r 3 + ϕ 1 3 ϕ 2 ϕ r r 3 2 2r + k 3 ϕ 3 ϕ 1 r 3 r 3 1 k + r ϕ r ψ ψ =, r =, at r = h = 1 + ǫ sinx, ψ =, ψ r ϕ 1 r 4 ϕ 2 ϕ 1 r + k 2 r r 2 + ϕ 1 2 ϕ r r 2 3 ϕ 1 r 2 r ϕ 1 3 ϕ r 2 r 3 4 ϕ 1 r 4 + ϕ 1 4 ϕ } r r 4 =, 36 =, at r = h = 1 ǫ sinx. 37 The solutions of the above systems satisfying the boundary conditions are directly written as ϕ = ψ = a 3r 4h + a 4r 3 4h 3, 38 ϕ 1 = b + b 1 r + b 2 r 2 + b 3 r 3 + b 4 r 4 + b 5 logk + r + b 6 r logk + r, 39 ϕ 2 = c + c 1 r + c 2 r 2 + c 3 r 3 + c 4 r 4 + c 5 r 5 + c 6 r 6 + c 7 logk + r + c 8 r logk + r + c 9 r 2 logk + r + c 1 r 3 logk + r. 4 Invoking Eqs. 38 to Eq. 4 into Eq. 31, and substituting p 1, we obtain the following solution ψr, x = b + c + b 1 + c 1 a 3 r + b 2 + c 2 r 2 + b 3 + c 3 + a 4 4h 4h 3 r 3 + b 4 + c 4 r 4 + c 5 r 5 + c 6 r 6 + b 5 + c 7 logk + r + b 6 + c 8 r logk + r + c 9 r 2 logk + r + c 1 r 3 logk + r. 41 The expression for the pressure rise is computed as where P = 2π p dx, 42 x p x = 1 [ ψr r + k 1 + n We k r r + k ψ ψr ] rr 1 r + k ψ rr. 43 The expression for pressure gradient p/x can be computed by substituting the value of ψ from Eq. 41 into Eq Results and Discussion Figures 1 3 are plotted to show the pressure rise against the flow rate in order to see the effect of pertinent parameters i.e. curvature parameter k, Weissenberg number We and power law index n. Figure 1 shows the effect of curvature parameter k on pressure rise. Fig. 2 Explains that the behavior for different values of We where k =.1, =.2, n =.45, and r = 1. Fig. 1 Shows the effect of curvature parameter k on pressure rise. Plots are shown for different values of k where We =.1, =.2, n =.85, and r = 1. It is observed that pressure rise decreases with increase in curvature parameter k in the region [.5,.2] and increases in the region [.2,.5] respectively. From Fig. 2 we note that in augmented pumping region [.5,.2], the pressure rise decreases. However, it increases in the peristaltic pumping region i.e. [.2,.5], with increase in We. Figure 3 describes the effects off various values of power law index n. Here pressure rise decreases for increasing values of n in augmented pumping region [.5, ], while it increases in the peristaltic pumping region [,.5]. Figures 4 6 are the graphical results of fluid

5 No. 6 Communications in Theoretical Physics 733 velocity for different values of curvature k, Wessenberg number We and n. We observe that in these graphs the velocity profile is slighted tilted towards right which is mainly because we are considering flow in a curved channel. where as it increases in the region [, 1]. Figures 5 and 6 are plotted for different values of We and n, in both cases velocity profile behavior is opposite to that for k i.e. velocity increases in region [ 1, ] and decreases in region [, 1] for increasing values of We and n respectively. Fig. 3 Describe the effect on n on pressure rise, where We =.65, =.2; k =.5, and r = 1. Fig. 5 Shows the effect of We where k =.1, =.1, n =.1, q =.9, and x =.1. Fig. 4 Is a plot for velocity component ur, x against r. It shows the effect of curvature parameter k where We =.1, =.1, n =.1, q =.5, and x =.1. Figure 4 shows the effect of curvature parameter k. In the region [ 1, ] velocity decreases with increase in k, Fig. 6 Describe the behavior of fluid velocity for different values of n where We =.1, k =.1, =.1, q =.5, and x =.1. Fig. 7 Here the streamlines are plotted for against different values of curvature parameter k, i.e. k =.1,.2,.3 while We =.1, =.65, n =.85, and q =.25.

6 734 Communications in Theoretical Physics Vol. 59 Figures 7 9 are the stream lines for different values of pertinent parameters. In Fig. 7 streamlines are plotted for various values of k. We note that as the curvature parameter increases, the size of the bolus starts to decrease in the lower half of the channel where as, it remains invariant on the upper half of the channel. However, for large values of We the bolus size increases rapidly and bolus disappears for k = 1. Figure 8 shows the effect of We. Here the effect of increase in We is quite opposite to that of k, the bolus appearing in the upper half decreases and bolus in the lower half varies gradually with increase in W e. However, for large values of We, size of the bolus grows rapidly and vanishes for We = 1. Figure 9 displays the streamlines for different values of power law index n. For n =, we get streamlines for the case of viscous fluid. In this case the number and size of bolus remain same in upper and lower half of the channel, however, with small change in n the size of the bolus starts to decrease in upper half and gradually disappears. Fig. 8 Here the streamlines are plotted for against different values of We, i.e. We =.1,.5,.1 while k =.5, =.95, n =.99, and q =.99. Fig. 9 Here the streamlines are plotted for against different values of n, i.e. n =,.15,.2,.25 while k = 1, =.85, We =.1, and q =.9.

7 No. 6 Communications in Theoretical Physics 735 Appendix a = h 2 k 2, = logh + k, a 2 = log h + k, a 3 = 2h + q, a 4 = 2h + 3q, a 5 = 1 + n, a 6 = 3 + 2k 2, a 7 = h 2 3k 2, a 8 = 1 + k 2, a 9 = h 2 + 3k 2, = 1 4nWe, 1 = 1 + 2nWe, 2 = 1 + 4nWe, 3 = 1 + 6nWe, 4 = 1 + 6nWe, 5 = 1 + 1nWe, 6 = q 2, 7 = q 2, 8 = 6k 5q, 9 = 4k + q, a 2 = q 2, a 21 = q 2, a 22 = 1 49nWe, a 23 = 5 13nWe, a 24 = 2 5nWe, a 25 = 2 + 5nWe, a 26 = nWe, a 27 = 1 1nWe, a 28 = 1 2nWe, a 29 = 2 + n8 + 3q 2 We, a 3 = 4 2n29 + 3q 2 We, a 31 = n 2 We 2, a 32 = n 2 We 2, a 33 = 11 + q 2, a 34 = 5 + q 2, a 35 = k 2 + 9kq, a 36 = 24 + q 2, a 37 = 45 + q 2, a 38 = 5 26nWe, a 39 = 5 14nWe, a 4 = 12nWe + q nWe, a 41 = 36nWe + q nWe, a 42 = nWe, a 43 = nWe, a 44 = nWe, a 45 = nWe, a 46 = 8 + q 2, a 47 = 44 + q 2, a 48 = 18 + q 2, a 49 = q 2, a 5 = 4 + n16 + 3q 2 We, a 51 = 2 + n4 + 9q 2 We, a 52 = n 2 We 2, a 53 = 9 + 5a 4, a 54 = a 2, a 55 = 2 + a 4, a 56 = 3 + a 4, a 57 = a 54 14k 2, a 58 = a 54 14k 2, a 59 = 2a 5 k 2 + a 56 nwe, a 6 = 2a 5 k 2 + a 56 nwe, b = 16h 9 a 5h 1 3 h 7 knwe + 9 a 54 h 4 k 2 nwe a 2 h 3 k 3 nwe, b 1 = 16h a 2 h + 27 a 54 kh 2 k 2 nwe, b 2 = 16h 92a 5h h 5 knwe + 9 a 54 h 2 k 2 nwe, b 3 = 16h 918hk 2 9 a 54 k 3 nwe, b 4 = 16h 9 a 5h 6 3 h 3 knwe, b 5 = 9a2 1 k3 nwe 4h 6, b 6 = 9a2 1 k2 nwe 4h 6, c = 32h 15 2a a 2 a 5kh 16 + a 5 2a a k 2 nweh a 53 kn 2 We 2 h a 5 a k 4 nweh a 54 a 5 k 2 27a 5 k nwe 3a 4 k 2 nwe 54 k 2 nwenwekh a 5 k a 2 1 a 54nWenWek 2 h a a 54 a 5 a a a 54 a 5 k 2 54a 2 1 k2 nwe 45 a 54 a 5 k 4 knweh a + a 2 a 5 a a 2 1 a 54k 2 nwe 9a a 54 a 59 nwek 2 h a a a 2 1 a 55 18a a 2 a a2 1 k2 k 3 n 2 We 2 h a a 3 1 a 8 54a a 2 1 a 2a 8 27a 2 1 a 54k 4 k 2 n 2 We 2 h 7 9a a 54 a 2 a 58 k 3 n 2 We 2 h a a a a a 54a a a a 2k 4 n 2 We 2 h a a a a a a 54a 2 k 5 n 2 We 2 h a a a a 2 1 a2 2 k6 n 2 We 2 h 3, c 1 = 32h 15 8a a 2 5 a h a 5 knweh 15 + a 5 36a 24k 2 knweh a 2 1k 2 n 2 We 2 h 12 + a 5 12a 18k 2 nweknweh a a 54 a a 2 1nWe + 18 a 4 nwe 18a 2 1k 2 nwek 2 nweh 1 + a 5 12a + a 2 a 7 36a k a + a 2 k k 4 knweh a a 2 1a 59 18a a 2 a 6 18a a 54 a 5 a 7 9a a 54 a 5 k a a a a 2 144a 2 1k 2 18 a 4 k 2 k 2 nweh 8 + a a a 55 k 3 n 2 We 2 h 7 + a a + a 2 a 8 54k k 4 k 2 n 2 We 2 h 6 + a a a 54 18a a 57 18a a 2 a 58 63a a 54 k a 54 k 2 k 3 n 2 We 2 h 5 + a a a a 3 + a a a a a a k 2 k 4 n 2 We 2 h 4 + a a a a 4 + a a a a a 54 k 2 k 5 n 2 We 2 h 3 + a a a a 2 324a a2 54 k6 n 2 We 2 h 2, c 2 = 32h 15 4a a 2 5 a kh14 + a 5 14a 1a a 4 2a a k 2 nweh a 4 6a 53 kn 2 We 2 h 12 + a k 2 k 3 n 2 We 2 h 6 + a 5 12a + 27k a 6 k 2 + 6k 4 nweh a 5 a 54 k k 2 + 6a 4 k 2 kn 2 We 2 h a + a 2 a 5 36a 5 k 2 9 a 54 nwek 2 nweh 9 + 6a a 54 a 5 a a a 54 a 5 k a 5 a 54 k a 2 1k 2 nweknweh 8

8 736 Communications in Theoretical Physics Vol a a 2 1 a a a 2 a 6 18a + a 2 a 5 k a a 3 1nWe 36a a 2 1a 2 45a 2 1a 54 k 2 k 2 n 2 We 2 h a a 54 a a 2 1a 54 k 4 k 2 n 2 We 2 h a a a 2 9a a 3 1a a a 2 1a 2 a a a 2 1a 54 k 2 k 3 n 2 We 2 h 4 + a a a a a a a 54 k 4 n 2 We 2 h 3 + a a a a a a 54 k 5 n 2 We 2 h 2 162a a a a 2 1 a a a 3 1 a a 2 a 54 k 6 n 2 We 2 h, c 3 = 32h 15 16a a 2 5 a h14 48 a 5 knweh a 2 1 k2 n 2 We 2 h 1 + a 5 72a + 48k 2 knweh a a 2 1a 2 54k 6 n 2 We 2 + a 5 12a + 18k 2 knweh 9 + a 5 36a 18k 2 k 3 nweh 7 + a a a 55 k 3 n 2 We 2 h 5 + 9a a 5 a a 4 nwe k 2 nwek 2 nweh 8 + 6a a 5 a 7 a 54 9a a 5 a 54 k a a a a a 4 k 2 k 2 nweh 6 + a 2 1 a 5418a 63a k 2 18k 2 k 3 n 2 We 2 h 3 + a a a k2 k 4 n 2 We 2 h 2 + a k2 k 4 n 2 We 2 h 4 + a 2 1 a a + 162k 2 k 5 n 2 We 2 h, c 4 = 32h 15 2a a 2 5 a kh a 2 1 a 54k 4 n 2 We 2 h a 2 1 k5 n 2 We 2 h 4 27a 2 1 a 54k 6 n 2 We 2 h a 4 kn 2 We 2 h 1 + 1a a 5 + 1a a 4 a a 5 k 2 nweh a 5 2a a 6 k 2 nweh a 5 k 2 + a 54 nwek 2 nweh a 5 a 54 k 2 72 nwek 3 nweh a 5 a 54 k nwe 3 k 2 nwe 3a 4 k 2 nweknweh 8, c 5 = 32h 15 8a a 2 5 a h a 5 knweh a 2 1 k2 n 2 We 2 h 8 18a 2 1 k4 n 2 We 2 h a 24k 2 knwe a 5 h 9, c 6 = 32h 15 4a a 5 nweh a 2 a 1kn 2 We 2 h 8 12a 2 1k 3 n 2 We 2 h 6, c 7 = 32h 15 36a a 55 k 3 n 2 We 2 h a a 2 1 a a 54k 4 n 2 We 2 h 5 648a a 2 1 k5 n 2 We 2 h a a 2 1a 54 k 6 n 2 We 2 h a a 5 a 7 36a a 5 k 2 k 2 nweh a a a a 2 1k 2 k 3 n 2 We 2 h 6, c 8 = 32h 15 36a a 55 k 3 n 2 We 2 h a a 2 a 1a 54 k 4 n 2 We 2 h 5 648a a 2 1k 5 n 2 We 2 h a a 2 1a 54 k 6 n 2 We 2 h a 7 36k 2 a a 5 k 2 nweh k 2 a a 2 1k 3 n 2 We 2, c 9 = 9a2 1a 5 nwek 2 8h 6, c 1 = 9a3 1n 2 We 2 k 2 8h 9. References [1] S. Nadeem and M. Hameed, J. Math. Analy. Appl [2] S. Nadeem and Noreen Sher Akbar, Appl. Math. Comput [3] S. Nadeem and Noreen Sher Akbar, Commun. Nonl. Sci. Numer. Simul [4] S. Nadeem, Noreen Sher Akbar, Naveeda Bibi, and Sadaf Ashiq, Commun. Nonl. Sci. Numer. Simul [5] S. Nadeem and Safia Akram, Commun. Nonlinear Sci. Numer. Simul [6] Kh.S. Mekheimer, Phys. Lett. A [7] Kh.S. Mekheimer and Y. Abd Elmabound, Phys. Lett. A [8] Kh.S. Mekheimer and Y. Abd Elmabound, Appl. Math. Mod [9] D. Tripathi, K.S. Pendey, and S. Das, Appl. Math. Comput [1] A. Tripathi and R.P. Chhabra, Chem. Eng. Commun [11] S. Nadeem and Safia Akram, Commun. Nonl. Sci. Numer. Simul [12] S. Nadeem, Noreen Sher Akbar, T. Hayat, and A. Hendi, International Journal of Heat and Mass Transfer [13] H. Sato, T. Kawai, T. Fujita, and M. Okabe, Trans. Jpn. Soc. Mech. Eng. B [14] T. Hayat, Maryiam Javed, and Awatif.A. Hendi, International Journal of Heat and Mass Transfer [15] Ji-Hun He, Appl. Math Comp [16] Ji-Huan He, Asymptotic Methods for Solitary Solutions and Compactons, Abstract and Applied Analysis [17] Ji-Huan He, International Journal of Modern Physics B