Peristaltic Transport of a Hyperbolic Tangent Fluid Model in an Asymmetric Channel

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1 Peristaltic Transport of a Hyperbolic Tangent Fluid Model in an Asymmetric Channel Sohail Nadeem and Safia Akram Department of Mathematics Quaid-i-Azam University 4530 Islamabad Pakistan Reprint requests to S. N; snqau@hotmail.com Z. Naturforsch. 64a ; received August / revised December In the present analysis we have modeled the governing equations of a two dimensional hyperbolic tangent fluid model. Using the assumption of long wavelength and low Reynolds number the governing equations of hyperbolic tangent fluid for an asymmetric channel have been solved using the regular perturbation method. The expression for pressure rise has been calculated using numerical integrations. At the end various physical parameters have been shown pictorially. It is found that the narrow part of the channel requires a large pressure gradient also in the narrow part the pressure gradient decreases with the increase in Weissenberg number We and channel width d. Key words: Modeling of Hyperbolic Tangent Fluid Model; Asymmetric Channel; Analytical Solutions.. Introduction Peristaltic transport is a well known process of a fluid transport which is induced by a progressive wave of area contraction or expansion along the length of distensible tube containing the fluid. It is used by many systems in the living body to propel or to mix the contents of a tube. The peristalsis mechanism usually occur in 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 engineering processes as well in which peristaltic pumps are used to handle a wide range of fluids particularly in chemical and pharmaceutical industries. It is also used in sanitary fluid transport blood pumps in heart lung machine and transport of corrosive fluids where the contact of the fluid with the machinery parts is prohibited. Because most of the physiological fluids behave like a non-newtonian fluid therefore some interesting studies dealing with the flows of non-newtonian fluids are given in 5]. Motivated by possible applications in industry and physiology and previous studies regarding the peristaltic flows of Non-Newtonian fluid models we discussed the tangent hyperbolic fluid model. The governing equations of hyperbolic tangent fluid model for peristaltic fluid flow in a two dimensional asymmet- ric channel has been modeled in the present paper. To the best of the authors knowledge no attempt has been made to study the hyperbolic tangent fluid model for peristaltic problems. The governing equations are reduced using long wave length approximation and then the reduced problem has been solved by the regular perturbation method. The expression for pressure rise is computed numerically using mathematics software Mathematica. At the end the graphical results are presented to discuss the physical behaviour of various parameters of interest.. Fluid Model For an incompressible fluid the balance of mass and momentum are given by divv = 0 ρ dv = divs + ρf dt 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 0 ] S = PI + τ 3 τ = η +η 0 + η tanhγ γ n] γ / 09 / $ c 009 Verlag der Zeitschrift für Naturforschung Tübingen

2 560 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel in which PI is the spherical part of the stress due V ρ to constraint of incompressibility τ is the extra stress t +Ū V X + V V = P τ XȲ Ȳ Ȳ X τȳȳ Ȳ. 0 tensor η is the infinite shear rate viscosity η 0 is the We introduce a wave frame xȳ moving with velocity c away from the fixed frame XȲ by the trans- zero shear rate viscosity Γ is the time constant n is the power law index and γ is defined as formation γ = i γ ij γ ji = j Π 5 where Π = trac gradv +gradv T. Here Π is the second invariant strain tensor. We consider constitution 4 the case for which η = 0and Γ γ <. The component of extra stress tensor therefore can be written as τ = η 0 Γ γ n ] γ = η 0 + Γ γ n ] γ = η 0 + nγ γ ] γ. 3. Mathematical Formulation 6 Let us consider the peristaltic transport of an incompressible hyperbolic tangent fluid in a two dimensional channel of width d + d. The flow is generated by sinusoidal wave trains propagating with constant speed c along the channel walls. The geometry of the wall surface is defined as ] Y = H = d π + ā cos λ X c t ] 7 Y = H = d π b cos λ X c t+φ where ā and b are the amplitudes of the waves λ is the wave length d + d is the width of the channel c is the velocity of propagation t is the time and X is the direction of wave propagation. The phase difference φ varies in the range 0 φ π in which φ = 0 corresponds to a symmetric channel with waves out of phase and φ = π the waves are in phase further ā b d d andφ satisfies the condition ā + b + ā b cosφ d + d. The equations governing the flow of a tangent hyperbolic fluid are given by Ū X + V Ȳ Ū Ū ρ +Ū t X + V Ū Ȳ = 0 8 = P X τ X X τ XȲ X Ȳ 9 x = X c t ȳ = Ȳ ū = Ū c v = V and px= PXt. Further we define x = x λ y = ȳ d u = ū c t = c λ t h = h d h = h d τ xx = λ η 0 c τ xx d τ xy = η 0 c τ xȳ d τ yy = η 0 c τ d ȳȳ δ = λ Re = ρc d We = Γ c d P = p η 0 d cλη 0 γ = γ d c. Using the above non-dimensional quantities and the resulting equations in terms of stream function Ψ u = Ψ v = δ Ψ 9 and 0 can be written as y x Ψ δre y x Ψ ] Ψ x y y = P x δ τ xx x τ 3 xy y where δ 3 Re Ψ y x Ψ x y = P y δ τ xy x δ τ yy y ] Ψ x τ xx = + nwe γ ] Ψ x y Ψ τ xy = + nwe γ ] y δ Ψ x τ yy = δ + nwe γ ] Ψ x y γ = δ Ψ Ψ + x y y δ Ψ x + δ Ψ ] / x y 4

3 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel 56 in which δ ReWe represent the wave Reynolds and Weissenberg numbers respectively. Under the assumptions of long wavelength δ and low Reynolds number and neglecting the terms of order δ and higher 3 and 4 take the form P x = y + n We Ψ y ] Ψ y 5 P = 0. 6 y Elimination of pressure from 5 and 6 yield y + n We Ψ y ] Ψ = 0. 7 y The dimensionless mean flow Θ is defined by ] Θ = F + + d 8 in which F = h x h x Ψ y dy = Ψh x Ψh x 9 h x= + acosπx 0 h x=dbcosπx + φ. The boundary conditions in terms of stream functionψ are defined as: Ψ = F Ψ = F Ψ y = fory = h x Ψ y = fory = h x where a b φ andd satisfy the following relation: a + b + abcosφ + d. 4. Perturbation Solution For the perturbation solution we expand Ψ F and P as Ψ = Ψ 0 +WeΨ + OWe F = F 0 +WeF + OWe 3 P = P 0 +WeP + OWe. 4 Substituting above expressions in 5 7 and and collecting the powers of We we obtain the following systems: 4.. System of Order We 0 4 Ψ 0 = 0 5 y4 P 0 x = n 3 Ψ 0 y 3 6 Ψ 0 = F 0 Ψ 0 y = ony = h x 7 Ψ 0 = F 0 Ψ 0 y = ony = h x System of Order We 4 Ψ y 4 = n n y Ψ 0 y P x = Ψ n 3 y 3 + n y 9 Ψ 0 y 30 Ψ = F Ψ y = 0ony = h x 3 Ψ = F Ψ y = 0ony = h x Solution for System of Order We 0 Solution of 5 satisfying the boundary conditions 7 and 8 can be written as: Ψ 0 = F 0 + h h h h 3 y3 3h + h y + 6h h y F0 y + h h 3 + h h 3 3h h h F ] 0 h 3 3h h. The axial pressure gradient at this order is 33 dp 0 dx = nf 0 + h h h h For one wavelength the integration of 34 yields P λ0 = 0 dp 0 dx. 35 dx

4 56 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel 4.4. Solution for System of Order We Substituting the zeroth-order solution 33 into 9 the solution of the resulting problem satisfying the boundary conditions take the following form: Ψ = C 0 +C y +C y + 88n n y 3! +C 3 3! F0 + h h h h 3 ] y 4 where 6 C 0 = h h 3 F A h3 h h h 3 h h h h ] ] h3 6 3! A h h + h h 3 h4 4 F 36 C = 6h h F h h 3 + Ah h h + h 3 h h 3 h 3 h h h h h ] C = 6F h + h h + h h h 3 + A 4h h 3 h 3 h h h 3 h h h + h h + h ] C 3 = 3! h h 3 F A h 3 h h h 3 h h ] A = n ] n 88 F0 + h h h h 3. The axial pressure gradient at this order is ] dp dx =nc F0 + h h 344nh +h h h For one wavelength the integration of 37 yields P λ = 0 dp dx. 38 dx Summarizing the perturbation results for small parameter We the expression for stream functions and pressure gradient can be written as: Ψ = F + h h h h 3 y 3 3h + h y + 6h h y F y + h h 3 + h h 3 3h h h F ] 39 h 3 3h h + We B +Cy + D y! + E y3 3! + A y 4 ] dp dx = nf + h h h h 3 n A + We h h 3 h 3 h h 40 h 3 h h F + h h ] 44nh + h h h 3 where 6 A ] B = h h 3 h 3 h h h 3 h h h h h3 A h h 6 3! + h h 3 h4 4 C = A h h h + h 3 h 3 h h ] h h 3 h 3 h h h + h D = A 4h h 3 h 3 h h h 3 h h h + h h + h ] h h 3 A 3! h 3 h h h 3 h h ] E = A = n ] F + n 88 h h h h 3. In the above solution when γ 0thenµ µ 0 or n = 0 the solutions of Mishra and Rao 6] are a special case of our problem. The non-dimensional pressure rise over one wavelength P λ for the axial velocity are dp P λ = dx 4 0 dx where dp/dx is defined in 40.

5 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel 563 Fig.. Variation of P λ with Θ for different values of We at a = 0.5 b = 0.5 d = 0.4 n = 0.04 and φ = π 6. Fig. 4. Variation of P λ with Θ for different values of d at a = 0.5 b = 0.5 We = 0.03 n = 0.04 and φ = π 4. Fig.. Variation of P λ with Θ for different values of φ at a = 0.5 d = 0.5 We = 0.03 n = 0.04 and b = 0.7. Fig. 5. Variation of P λ with Θ for different values of a at b = 0.7 d = 0.7 We = 0.03 n = 0.04 and φ = π 4. Fig. 3. Variation of P λ with Θ for different values of n at a = 0.5 d = 0.5 We = 0.03 b = 0.7 and φ = π 6. Fig. 6. Variation of P λ with Θ for different values of b at a = 0.5 d = 0.5 We = 0.03 n = 0.04 and φ = π 6.

Variation of dp dx with x for different values of d at a = 0.5 b = 0.5 We = 0.03 n = 0.04 Θ = 0.4 and φ = π. a b c Fig. 9.

6 564 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel Fig. 7. Variation of dp dx with x for different values of We at a = 0.5 b = 0.5 d = 0. n = 0.04 Θ = 0.4 and φ = π. Fig. 8. Variation of dp dx with x for different values of d at a = 0.5 b = 0.5 We = 0.03 n = 0.04 Θ = 0.4 and φ = π. a b c Fig. 9. Stream lines for three different values of Q. afor Q = 0.4 b for Q = 0.5 c for Q = 0.6. The other parameters are chosen as a = 0.5 b = 0.5 d =.0 n = 0.09 We = 0.04 and φ = 0.0.

S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel 565 a b Fig. 0. Stream lines for two different values of We. a for We = 0.4 b for We = 0.04.

7 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel 565 a b Fig. 0. Stream lines for two different values of We. a for We = 0.4 b for We = The other parameter are chosen as a = 0.54 b = 0.5 d =.0 n = 0.09 Q = 0.5 and φ = 0.0. a b c Fig.. Stream lines for three different values of a. afor a = 0.5 b for a = 0.54 c for a = The other parameters are chosen as b = 0.5 d =.0 Q = 0.3 φ = 0.0 We = 0.04 and n = 0.09.

8 566 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel 5. Results and Discussion The analytical solution of the hyperbolic tangent model is presented. The expression for pressure rise P λ is calculated numerically using mathematics software. The effects of various parameters on the pressure rise P λ are shown in Figures 6 for various values of Weissenberg number We amplitude ratio φ tangent hyperbolic power law index n channel width d and wave amplitudes a b. It is observed from Figure that pressure rise decreases for small values of Θ 0 Θ.45 with the increase in We and for large Θ.45 Θ the pressure rise increases. We also observe that for different values of We there is a linear relation between P λ and Θ i. e. the pressure rise decreases for small Θ and increases for large Θ. The pressure rise P λ for different values of φ are illustrated in Figure. It is shown that P λ decreases with the increase in φ for Θ 0.9] and after that P λ increases. The graphs of P λ for different values of power law index n are presented in Figure 3. It is seen that with the increase in n P λ decreases for Θ 0.6] and for Θ.6] it is increasing. It is observed that the pressure rise decreases with the increase in d and increases with the increase in a and b for small Θ and for large Θ the results are opposite see Figs Figures 7 and 8 represent that for x 00.] and 0.6] the pressure gradient is small we say that the flow can easily pass without imposition of large pressure gradient while in the narrow part of the channel x 0.0.6] to retain same flux large pressure gradient is required. Moreover in the narrow part of the channel the pressure gradient decreases with the increase in We and d. 5.. Trapping Phenomena Another interesting phenomena in peristaltic motion is trapping. It is basically the formation of an internally circulating bolus of fluid by closed stream lines. This trapped bolus pushed a head along peristaltic waves. Figures 9 illustrate the stream lines for different values of Q We and a. The stream lines for different values of volume flow rate Q are shown in Figure 9. It is found that with the increase in volume flow rate Q the size and the number of trapping bolus increases. In Figure 0 the stream lines are prepared for different value of Weissenberg number We. It is depicted that the size of the trapped bolus increases with the increase in We. It is observed from Figure that the size and the number of the trapping bolus increases with the increases in amplitude of the wave a. 6. Conclusion In the present paper we have investigated the peristaltic flow of tangent hyperbolic fluid in an asymmetric channel. The governing two dimensional equations have been modeled and then simplified using long wave length approximation. The simplified equations are solved analytically using regular perturbation method. The results are discussed through graphs. We conclude the following observations:. It is observed that in the peristaltic pumping region the pressure rise decreases with the increases in We φ n andd and increases with the increases in a and b.. The pressure gradient decreases with the increases in both We and d. 3. The size of the trapping bolus increases with the increases in Q We and decreases with the increases in a. ] T. Hayat F. M. Mahomed and S. Asghar Nonlinear Dynamics ] T. Hayat and N. Ali Appl. Math. Computation ] D. Srinivasacharya M. Mishra and A. R. Rao Acta Mechanica ] T. Hayat Y. Wang A. M. Siddiqui and K. Hutter Math. Probl. Eng ] T. Hayat and N. Ali Physica A ] M. Ealshahed and M. H. Haroun Probl. Eng ] T. Hayat N. Ahmad and N. Ali Commun. Nonlinear Science and Numerical Simulation ] M. H. Haroun Computation Material Science ] T. Hayat and N. Ali Physica A ] T. Hayat N. Ali and Z. Abbas Phys. Lett. A ] L. Ai and K. Vafai Numer. Heat Transfer part A ] I. Pop and D. B. Ingham Convective heat transfer: Mathematical and computational modeling of viscous

9 S. Nadeem and S. Akram Hyperbolic Tangent Fluid Model in an Asymmetric Channel 567 fluids and porous media Pergamon Amsterdam New York 00. 3] A. H. Shapiro M. Y. Jaffrin and S. L. Weinberg J. Fluid Mech ] M. H. Haroun Commun. Nonlin. Sci. Numer. Simul ] Y. Wang T. Hayat and K. Hutter Intern. J. Nonlinear Mech ] M. Mishra and A. R. Rao Z. angew. Math. Phys

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

Commun. Theor. Phys. 59 213 729 736 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,