Effects of Heat and Mass Transfer on Peristaltic Flow of Carreau Fluid in a Vertical Annulus

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1 Effects of Heat and Mass Transfer on Peristaltic Flow of Carreau Fluid in a Vertical Annulus Sohail Nadeem and Noreen Sher Akbar Department of Mathematics Quaid-i-Azam University 530 Islamabad 000 Pakistan Reprint requests to S. N.; snqau@hotmail.com Z. Naturforsch. 5a ; received June / revised November 009 This article is devoted to the study of peristaltic transport of a Carreau fluid in a vertical annulus under the consideration of long wavelength. The flow is investigated in a wave frame of reference moving with the velocity of the wave. Exact solutions have been evaluated for temperature and concentration field while approximated analytical and numerical solutions are found for the velocity field using i the perturbation method and ii the shooting method. The effects of various emerging parameters are investigated graphically. Key words: Peristaltic Flow; Carreau Fluid; Annulus; Perturbation Solution; Numerical Solutions.. Introduction Peristalsis is an important mechanism for transporting fluids. The word peristalsis comes from the Greek word peristalikos which means clasping and compressing. Peristaltic transport is a form of fluid transport induced by a progressive wave of area contraction or expansion along the length of a distensible tube containing fluids. In physiology peristalsis used by the body to propel or mix the contents of a tube as in ureter gastro-intestinal tract bile duct and other glandular ducts. Some worms use peristalsis as a means of locomotion. Peristalsis mechanism has attracted the attention of many researcher s since the first investigation of Latham []. A number of exact analytical and numerical studies of peristaltic transport for Newtonian and non- Newtonian fluids have been investigated in references [ 9]. In the literature numerous studies regarding peristaltic motion have been done for Newtonian fluids. But such approach fails to give proper understanding of peristalsis in blood vessels chyme movement in intestine semen transport in ductus afferents of male reproductive tract and transport of spermatozoa in cervical canal. In these body organs the fluid viscosity varies across the thickness of the duct. Most of the physiological fluid do not behave like a Newtonian fluid in reality. Limited studies [0 ] are available in literature for peristalsis involving non-newtonian fluids. To the best of our knowledge only one article [7] exist in peristaltic literature which studies the peristaltic flow in connection with mixed convection heat and mass transfer. Mixed convection is the combination of free convection and forced convection. Free convection is occurred due to temperature difference in a fluid at different locations while forced convection is the flow of heat due to some externally applied forces. Keeping in mind the importance of peristaltic mechanism in connection with mixed convection heat and mass transfer we present the effects of mixed convection heat and mass transfer on peristaltic flow of a Carreau fluid in a vertical annulus. Exact solutions have been calculated for energy and concentration while analytical and numerical solutions have been presented for the velocity profile. At the end of the article graphical results have been presented for various parameter of interest.. Mathematical Model For an incompressible fluid the balance of mass and momentum are given by divv = 0 ρ dv = PI + div τ ij + ρf dt where ρ is the density V is the velocity vector PI is the spherical part of the stress due to constraint of incompressibility τ ij is the Cauchy stress tensor f re / 0 / $ 0.00 c 00 Verlag der Zeitschrift für Naturforschung Tübingen Download Date 7/5/ :5 PM

2 7 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid presents the specific body force and d/dt represents the material time derivative. The constitutive equation for a Carreau fluid is given by [] η η η 0 η = [ n +Γ γ ] 3 τ ij = η 0 [ + ] n Γ γ γ ij in which τ ij is the extra stress tensor η is the infinite shear rate viscosity η 0 is the zero shear rate viscosity Γ is the time constant n is the power law index and γ is defined as γ = γ ij γ ij = Π. 5 i j Here Π is the second invariant strain tensor. 3. Mathematical Formulation Let us consider the peristaltic flow of an incompressible Carreau fluid in a vertical annulus. The inner tube is rigid and maintained at temperature T the outer tube has a sinusoidal wave travelling down its walls and maintained at temperature T 0. The geometry of the wall surface is defined as R = a R = a + bsin π λ Z c t 7 where a is the radius of the inner tube a is the radius of the outer tube at inlet b is the wave amplitude λ is the wavelength c thewavespeedand t the time see Fig.. The governing equations of Carreau fluid along with heat and mass transfer are Ū R + Ū R + W = 0 Z ρ t +Ū R + W Z Ū = p R R + R R τ + Z τ 3 τ R ρ t +Ū R + W W Z = p Z R + R R τ 3 + Z τ 33 + ρgα T T 0 +ρgα C C Fig.. Geometry of the problem. ρc p t +Ū R + W T Z T = k R + R T R + T Z t +Ū R + W C Z C = D R + R C R + C Z + DK T T m T R + R T R + T Z. In the above equations P is the pressure Ū W are the velocity components in the radial and axial directions respectively T is the temperature C is the concentration of the fluid T 0 is the constant temperature of the inner tube C 0 is the constant concentration of inner tube ρ is the density k denotes the thermal conductivity c p is the specific heat at constant pressure T m is the temperature of the medium D is the coefficients of mass diffusivity K T is the thermal-diffusion ratio. In the fixed coordinates R Z the flow between the two tubes is unsteady. It becomes steady in a wave frame r z moving with the same speed as the wave moves in the Z-direction. The transformations between the two frames are r = R z = Z c t 3 ū = Ū w = W c Download Date 7/5/ :5 PM

3 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid 73 where ū and w are the velocities in the wave frame. Reδ u The appropriate boundary conditions in the wave r + w w = z frame are of the following form: P w = at r = r a z + r r rτ 3 +δ z τ 33 +Grθ + Brσ ReδP r u r + w θ = z w = at r = r b T = T 0 at r = r c T = T at r = r d C = C 0 at r = r e C = C at r = r f We introduce the dimensionless variables R = R a r = r a Z = Z λ z = z λ W = W c w = w c γ a = c γ u = λ ū a c U = λū a c P = a P cλη 0 θ = T T T 0 T t = c t λ δ = a λ Re = ρca η 0 τ = λ τ cη 0 τ = λ τ cη 0 τ 33 = λ τ 33 cη 0 r = r a = ε r = r = + φ sinπz τ 3 = τ 3 = a τ 3 a cη 0 We = cγ β = Q 0a a k T T 0 Gr = gαa3 T T 0 v Br = αga3 C C 0 v 5 Sr = ρdk T T 0 T η 0 T m C 0 C Sc = η 0 Dρ σ = C C C 0 C in which R and Z are the dimensionless form of radial and transverse components respectively Sr is the Soret number Sc Schmidt number Br is the local concentration Grashof number Gr is the local temperature Grashof number and We is the Weissenberg number. Making use of 3 and 5 9 to take the form u r + u r + w = 0 z Reδ 3 u = P r + δ u r + w z r r rτ + z τ 3 τ r 7 where τ ij = θ r + θ r r + δ θ z + β Reδ u r + w σ z = r σ + δ σ Sc r r r z + Sr r θ + δ θ r r r z [ + ] n Re γ γ ij γ = u r γ 33 = w z u γ 3 = γ 3 = z δ + w r γ = δ γ ij γ ij i = j i j γ = γ ij γ ij i j i j [ n w τ 3 = τ 3 = + Re r γ = u r ] w r. 9 0 Here δ Re represent the wave and the Reynolds numbers respectively. Under the assumptions of long wavelength δ and low Reynolds number i. e. neglecting the terms of order δ and higher 7 to 0 take the form P r = 0 P z = r r rτ 3 +Grθ + Brσ 3 0 = θ r + θ r r + β 0 = r σ +Sr r θ. 5 Sc r r r r r r Download Date 7/5/ :5 PM

4 7 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid The corresponding boundary conditions are dp = Q + r r a 7 + We a a 77 a 77 w = at r = r = ε a The pressure rise P and the friction force F on inner and outer tubes F 0 F i aregivenby w = at r = r = + φ sinπz θ = at r = r θ = 0 at r = r σ = at r = r σ = 0 at r = r. Solution of the Problem b c d e f Solving and 5 subject to the boundary conditions c to f we obtain the expression for temperature and concentration field as follows: θrz= a a lnr+a 3 r + a. 7 σrz= SrSc a a lnr+a 3 r + a + a 7 lnr + a... Perturbation Solution To get the solution of 3 we employ the regular perturbation. For the perturbation solution we expand w QandP as w = w 0 + We w + OWe 9a Q = Q 0 + We Q + OWe 9b P = P 0 + We P + OWe. 9c The perturbation results for small parameter We satisfying the conditions a and b for velocity fields and pressure gradient can be written as r dp w = + + a 5 lnr + a 7 + a 9r + a 30 r lnr + a 3 r + a lnr + a + We [ a 5 r 0 + a 59 r + a 0 r + a r + a r + a 3 r + a 5r lnr + a r lnr + a 7 r lnr + a r lnr + a 9 lnr + a 70 r lnr + a 7 r lnr + a 7 r lnr 3 + a 73 r lnr + a 7 lnr + a 77 ]. 30 P = 0 F 0 = r 0 F i = r 0 dp 3 dp dp 33 3 where dp is defined in 3. The dimensionless expressions for the five considered wave forms are given by the following equations:. Sinusoidal wave: r z= + φ sinπz.. Triangular wave: { } r z=+φ π n+ 3 sinπn z. n= n 3. Square wave: { } r z=+φ π n+ cosπn z. n n=. Trapezoidal wave: { 3 r z=+φ π n= 5. Multi sinusoidal wave: hz= + φ sinmπz... Numerical Solution sin π n n sinπn z The present problem consisting of 3 and is also solved numerically by employing the shooting method. The numerical results are compared with the perturbation results see Fig. and Table. 5. Numerical Results and Discussion In this section we present the solution for the Carreau fluid graphically. The expression for pressure rise P is calculated numerically using mathematics }. Download Date 7/5/ :5 PM

5 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid 75 Table.. Shows numerical values of comparison of axial velocity for perturbation and numerical solutions when ε = 0. We = 0. z = 0. dp/ = 0.3 φ = 0.3 Sr = 0.5 Sc = 0.3 β = 0.0 Gr = 0.0 Br = 0.0 n = 0.. r Numerical solution Perturbation solution Error Fig. 3. Pressure rise versus flow rate for ε = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 φ = 0. n = 0.. Fig.. Comparison of velocity field for ε = 0. We = 0. z = 0. dp/ = 0.3 φ = 0.3 Sr = 0.5 Sc = 0.3 β = 0.0 Gr = 0.0 Br = 0.0 n = 0.. software. The effects of various parameters on the pressure rise P are shown in Figures 3 to for various values of Weissenberg number We amplitude ratio φ radius ratio ε and power law index n. It is observed from these figures that the pressure rise increases with the increase in We φ ε while the pressure rise decreases with increase in n. The peristaltic pumping region is Q.5 for Figure 3 Q 0. for Figure Q 0.3 for Figure 5 and Q for Figure otherwise there is augmented pumping. Figures 7 to represent the behaviour of the frictional forces. It is depicted that these forces have an opposite behaviour as compared with the pressure rise. Figure 5 is prepared to see the behaviour of different wave forms on pressure rise. It is analyzed that a square wave has the best pumping characteristics while a triangular wave has the worst pumping characteristics. Effects of temperature profile are shown through Fig.. Pressure rise versus flow rate for ε = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 We = 0. n = 0.. Fig. 5. Pressure rise versus flow rate for We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 n = 0.. Download Date 7/5/ :5 PM

6 7 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid Fig.. Pressure gradient versus flow rate for We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = ε = 0.. Fig. 9. Frictional force versus flow rate for inner tube for We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 n = 0.. Fig. 7. Frictional force versus flow rate for inner tube for ε = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 φ = 0. n = 0.. Fig. 0. Frictional force versus flow rate for inner tube for We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = ε = 0.. Fig.. Frictional force versus flow rate for inner tube for ε = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 We = 0. n = 0.. Fig.. Frictional force versus flow rate for outer tube for ε = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 φ = 0. n = 0.. Download Date 7/5/ :5 PM

7 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid 77 Fig.. Frictional force versus flow rate for outer tube for ε = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 We = 0. n = 0.. Fig. 5. Pressure rise versus flow rate for We = 0. ε = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 φ = 0. n = 0.. Fig. 3. Frictional force versus flow rate for outer tube for We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 n = 0.. Fig.. Temperature profile for φ = 0. ε = 0. z = 0.. Fig.. Frictional force versus flow rate for inner tube for We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = ε = 0.. Fig. 7. Concentration field for φ = 0. ε = 0. z = 0. Sr = 3 Sc = 0.5. Download Date 7/5/ :5 PM

8 7 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid Fig.. Concentration field for φ = 0. ε = 0. z = 0. β = 3 Sc = 0.5. Figure. It is seen that with the increase in β the temperature profile increases. The concentration field σ for different values of β SrandScareshowninFigures 7 to 9. We observed that the concentration field decreases with the increase in β heat source parameter Sr Soret number and Sc Schmidth number. Figures 0 to are prepared to see the behaviour of pressure gradient for different wave shapes. It is observed from the figures that for z [00.5] and [..5] the pressure gradient is small and large pressure gradient occurs for z [0.5] moreover it is seen that with the increase in φ the pressure gradient increases. Figure 5 shows the streamlines for different wave forms. It is observed that the size of the trapped bolus in a triangular wave is small as compared with the other wave forms. Fig. 9. Concentration field for φ = 0. ε = 0. z = 0. Sr = 3 β = 0.5. Fig. 0. Pressure gradient versus z sinusoidal wave for ε = 0. Q =.5 We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0. Acknowledgement The authors are thankful to Higher Education Commission for the financial support of this work.. Appendix a = lnr lnr a = + β r r a 3 = β lnr lnr a = β r lnr r lnr lnr a 5 = SrSc a lnr +a 3 r a + a a = SrSc a a lnr +a 3 r + a Fig.. Pressure gradient versus z square wave for ε = 0. Q =.5 We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0. Download Date 7/5/ :5 PM

9 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid 79 Fig.. Pressure gradient versus z trapezoidal wave for ε = 0. Q =.5 We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0. Fig.. Pressure gradient versus z multisinusoidal wave for ε = 0. Q =.5 We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0. a 5 = r r lnr lnr Fig. 3. Pressure gradient versus z triangular wave for ε = 0. Q =.5 We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0. a 7 = a 5 a lnr lnr a = a 5 a 7 lnr a 9 = Gra 3 SrScBra 3 a a SrScBra a 0 = Gra a a + Bra 7 a = Gra SrScBra + Bra a a a = a 9 r + a 30r lnr + a 3 r a 3 = a 9 r + a 30 r lnr + a 3 r a = a 9 r + a 30r lnr a = a a 3 a 7 = r + r a5lnr + lnr a = + a + a 3 + a lnr + lnr a 9 = a 9 a 30 = a 0 a 3 = a 0 a a 3 = a 30 + a 3 a 33 = a 9 a 3 = dp 0 a 9 + a 3 a 9 a 35 = dp 0 a 3 + dp0 + + a 9 a + a 3 a 3 dp0 dp0 a 3 = a 5 + dp0 + dp0 a 37 = a a 5 a 3 + a a 3 a 30 a 5 + a 30 a dp0 a 5 a 9 Download Date 7/5/ :5 PM

10 790 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid a b c d e dp0 a 3 = a 30 + a 30 a 3 a 39 = a 9 a 30 a 0 = a 30 a = dp0 a 5 dp0 + a a 5 +a Fig. 5. Streamlines for a sinusoidal wave b multisinusoidal wave c square wave d trapezoidal wave e triangular wave for ε = 0. Q =.5 We = 0. Sr = 5 Sc = 0. Gr = Br = 3 β = 0.0 φ = 0.. a = a 9 a 33 a 3 = dp 0 a = dp 0 a 33 +a 30 + a 3 a 33 + a 9 a 3 a 3 + dp 0 a 5a 33 +a 30 + a 3 a 3 + a 9 a 35 + a a 33 Download Date 7/5/ :5 PM

11 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid 79 a 5 = dp 0 a = dp 0 a 7 = dp 0 a 35 + dp 0 a 5a 3 +a 30 + a 3 a 35 + a 9 a 3 + a a 3 a 3 + dp 0 a 5a 35 +a 30 + a 3 a 3 + a 9 a + a a 35 a + dp 0 a 5a 3 +a 30 + a 3 a + a a 3 a = dp 0 a 5a + a a a 9 = dp 0 a 50 = dp 0 a 5 = dp 0 a 37 + dp 0 a 5a 3 +a 30 + a 3 a 37 + a a 3 + a 30 a 3 a 3 + dp 0 a 5a 39 +a 30 + a 3 a 3 + a a 39 + a 30 a 35 + a 9 a 37 a 39 +a 30 + a 3 a 39 + a 30 a 3 + a 9 a 3 a 5 = a 30 a 33 + a 9 a 39 a 53 = dp 0 a 5a 37 + a 30 a + a a 37 a 5 = dp 0 a 5a 0 + a 30 a 37 + a a 0 a 55 = a 39 +a 30 + a 3 a 0 + a 30 a 3 a 5 = a 30 a 39 + a 9 a 0 a 57 = a 30 a 0 n a 5 = a 0 n n a 59 = a 3 + a 5 a 0 = n a + n n a 5 7 a 5 n n a = a 5 + a 50 3 n 3n a 55 + a 57 5 n n a = a + a 9 a 7 = n a 5 a 3 = a = a 5 = n a n a 7 n a 9 + n a = a 50 n + a 55 a 7 = n a 5 + n a = a 5 n a 9 = a 53 n a 70 = a 5 a 7 = a 7 = n a 55 + n a 57 n a 5 3n a 57 n a 5 3 3n a 57 3 n a 73 = a 5 a 7 = a 5 r 0 + a 59r + a 0r + a r + a r + a 3 r + a lnr + a 5 r lnr + a r lnr + a 7 r lnr + a r lnr + a 9 lnr + a 70 r lnr + a 7 r lnr + a 7 r lnr 3 + a 73 r lnr a 75 = a 5 r 0 + a 59r + a 0r + a r + a r + a 3 r + a lnr + a 5 r lnr + a r lnr + a 7 r lnr + a r lnr + a 9 lnr + a 70 r lnr + a 7 r lnr + a 7 r lnr 3 + a 73 r lnr a 7 = a 7 a 75 + a lnr lnr a 77 = a 7 a 5 lnr r r + a 5 r lnr r r lnr a r r 5 + a r 7 Download Date 7/5/ :5 PM

12 79 S. Nadeem and N. Sher Akbar Peristaltic Flow of Carreau Fluid a 79 = a r r 9 + a lnr r lnr 30 r r 3 + a 3 r a 0 = a 5 r + a r + a r lnr r lnr r r + a a lnr r lnr 3 + a 0 lnr r 0 lnr 5 r0 + a r 0 5 a 3 r r r + a lnr r lnr 5 r + a r + a r 3 r r + a lnr r lnr 7 r0 r a 9 r lnr r lnr r lnr r lnr + r r + a 70 lnr r lnr + a 7 lnr r lnr 3 + a 7 lnr 3 r lnr a 73 lnr r lnr + a 7 r lnr r lnr r r r lnr +r lnr r lnr +r lnr 9 + r r + r r 5 r r r r 3 r lnr + r lnr + r lnr r lnr + r lnr +r lnr r r + a 77 r r. r r 0 [] T. W. Latham Fluid motion in a peristaltic pump M. Sc. Thesis Massachusetts Institute of Technology Cambridge 9. [] A.El. Hakeem A.E.I Naby A.E.M.El Misery and I. I. El Shamy Physica A [3] S. Nadeem and N. S. Akbar Commun. Nonlinear Sci. Numer. Simul [] S. Srinivas and M. Kothandapani Int. Commun. Heat Mass Transf [5] S. Nadeem T. Hayat N. S. Akbar and M. Y. Malik Int. Commun. Heat Mass Transf [] S. Nadeem and N. S. Akbar Commun. Nonlinear Sci. Numer. Simul [7] S. Srinivas and R. Gayathri Appl. Math. Comput [] S. Srinivas and M. Kothandapani Appl. Math. Comput [9] S. Srinivas R. Gayathri and M. Kothandapani Comput. Phys. Commun [0] A. El. Hakeem A. E. I. Naby A. E. M. El Misery and I. I. El Shamy Appl. Math. Comput [] M. Ealshahed and M. H. Haroun Math. Probl. Eng [] T. Hayat N. Ali and S. Asghar Appl. Math. Comput [3] Y. Wang T. Hayat N. Ali and M. Oberlack Physica A [] M. Kothandapani and S. Srinivas Int. J. Nonlinear Mech [5] A. El. Hakeem A. E. I. Naby A. E. M El. Misery and I. I. El Shamy J. Phys. A [] S. Nadeem and S. Akram Commun. Nonlinear Sci. Numer. Simul [7] N. T. M. Eldabe M. F. El-Sayed A. Y. Ghaly and H. M. Sayed Archive Appl. Mech Download Date 7/5/ :5 PM