Heat transfer analysis of bi-viscous ciliary motion fluid

Transcription

1 International Journal of Biomathematics Vol. 8, No. (5) 556 (3 pages) c World Scientific Publishing Compan DOI: /S Noreen Sher Akbar DBS&H CEME National Universit of Sciences and Technolog Islamabad, Pakistan noreensher@gmail.com Z. H. Khan Department of Mathematics, Universit of Malakand Chakdara, Dir. (Lower), Khber Pakhtunkhwa, Pakistan Received 3 March 4 Accepted 9 December 4 Published 5 Februar 5 The impulsion sstem of cilia motion is deliberated b biviscosit fluid model. The problem of two-dimensional motion of biviscosit fluid privileged in a smmetric channel with ciliated walls is considered. The features of ciliar structures are resolute b the supremac of viscous effects above inertial possessions b the long-wavelength and low Renolds approimation. Closed-form solutions for the longitudinal pressure gradient, temperature and velocities are obtained. The pressure gradient and volume flow rate for different values of the biviscosit are also premeditated. The flow possessions for the biviscosit fluid resolute as a function of the cilia and metachronal wave velocit. Kewords: Ciliar motion; smmetric channel; biviscosit fluid; heat transfer; eact solutions. Mathematics Subject Classification : 76Z Nomenclature U, V :Velocitcomponents M : Hartmann number X, Y :Coordinates a : Wave s amplitudes β : Apparent viscosit coefficient µ β : Plastic viscosit µ : Viscosit of the fluid Θ,T : Temperature ε : Cilia length β : Slenderness parameter 556-

2 N. S. Akbar & Z. H. Khan ν : Kinematic viscosit of the fluid t :Time c : Wave speed φ : Phase difference F :Flowrate p : Yield stress α : Eccentricit of the elliptical motion B r : Brinkman number Re : Modified Renolds number. Introduction Ciliar movement is produced in the aoneme b the unidirectional descending of the eternal doublets of microtubules. Cilia are hair-like configurations that obtrude from the eteriors of certain creatures and collapse in a wave-like stle to transport the fluids. The are eistent in nearl all clusters of the phsical kingdom because their motilit shows a critical part in certain biological progressions such as breathing, imitation, movement and rotation. These hair-like adjuncts shatter or transfer in a whip-like lopsided stle containing an actual thump and retrieval stroke. Furthermore, a collection of cilia function beat with a continuous phase-lag with their neighbors. This indicates to the construction of metachronal waves which are recognized to improve the fluid flow due to cilia. However, dissimilar designs of metachronal breakers are recommended rendering to the association between their subtleties and that of the actual strokes of establishing cilia [7, 8, 6, 7, 9,, ]. When the metachronal breakers are transported in the similar track as the actual thump, the ciliar shattered coordination is called smplectic. If instead both commands oppose each other then the organization is designated as antiplectic. An outstanding assessment and illustration about cilia-driven flow are originated in the papers [,,, 4, 8,, 3, 4]. In humanoid phsiques, the stream twist due to the episodic measure of cilia is complicated in the impulsion of numerous organic liquids, including the flow of epididmal fluid in the ductus efferentes of a male reproductive tract, transport of ovulator mucus and the ovum in the oviduct and the removal of tracheobronchial mucus in the respirator track. Ciliar flaws can lead to numerous human diseases. More specificall, failure of transport functionalit of cilia can affect the successful mammalian reproduction. The main theme of our work is associated with consideration of fluid transport due to ciliar motion through ductus efferentes of the male reproductive tracts. A close inspection of cilia literature has been studied b man researchers [3 6, 9, 3, 5]. The problem of two-dimensional motion of biviscosit fluid and heat transfer in a smmetric channel with ciliated walls is considered. The features of ciliar structures are resolute b the supremac of viscous effects above inertial possessions b the long-wavelength and low Renolds approimation. This is the first paper 556-

3 in literature to discuss heat transfer for ciliar motion. Closed-form solutions for the longitudinal pressure gradient, temperature and velocities are obtained. The pressure gradient and volume flow rate for different values of the biviscosit are also premeditated. Streamlines and isotherms are displaed for understanding the proposed model.. Fluid Model The constitutive equation for incompressible biviscosit fluids [, 3] is defined as follows: { (µβ + p / π)e ij, π > π c, S ij = (µ β + p / () π c )e ij, π < π c, where π = e ij ; e ij is the (i, j) component of deformation rate, which is: e ij = ( vi + v ) j. () j i 3. Development of the Problem We consider an incompressible MHD biviscosit fluid in a smmetric channel with channel width. The channel is ciliated with metachronal wave pattern which propagates along the wall of the channel. The -coordinate is measured along the channel, where -coordinate is transverse to it. The magnetic field B is imposed on the flow in the -direction normal to the flow. It is assumed that the wall of the channel is heated uniforml at a constant temperature T with smmetr at center. The proposed problem has been modeled considering the following assumptions: Flow is incompressible, two-directional and two-dimensional. Wall of the channel is fleible. Metachronal wave pattern propagates along the walls of the channel. The effect of magnetic field induced b the Renolds number is sufficientl small to be neglected as compared to the eternal magnetic field. Due to long-wavelength and low Renolds number approimation, the velocit component in the -direction is more important than the one in the -direction. ThephsicalmodelandcoordinatesstemareshowninFig.. The geometr of the metachronal wave form proposes that the covering of the cilia advices can be stated precisel as: [ ( )] π Y = f(x, t) =± a + aε cos (X ct) = ±L = ±H. (3) λ Sleigh [9, ] observed that cilia tips move in elliptical paths; therefore, the horizontal position of the cilia tips can be written as: ( ) π X = g(x, t) =X + εαa sin (X ct). (4) λ 556-3

4 N. S. Akbar & Z. H. Khan Fig.. A phsical sketch of ciliated channel with metachronal wave pattern. The horizontal and vertical velocities of the cilia are given as [9, ]: U = ( ) ( π λ εαac cos π λ (X ct)) ( ) ( π λ εαac cos π (5) λ (X ct)), V = ( ) ( π λ εαac sin π λ (X ct)) ( ) ( π λ εαac sin π (6) λ (X ct)). The epression for fied and wave frames are related b the following relations: = X ct, ȳ = Ȳ, ū = Ū c, v = V, p( ) =P ( X,t). (7) We introduce the following non-dimensional quantities: = π λ, h = h a, = ȳ a, u = ū c, v = v c, t = π t λ, β = πa λ, P = πa P µcλ, ρca Re = µ, S = Sa µc, θ = T T, β = µ β πc /p, (8) T B r = P r E c, E c = c c p T, P r = µc p k. The stream function and velocit field are related b the epressions: u = Ψ, Ψ v = δ. (9) Under the long-wavelength and low Renolds number assumption, the dimensionless governing equations take the following form: ( + ) 4 Ψ β 4 M Ψ =, () 556-4

5 dp d = [( + ) ] Ψ β M (Ψ+) [ θ = B r β, () ( ) ] Ψ, () subject to the boundar conditions: ψ =, ψ, at =, u = πφαβ cos(π), at = h =+φcos(), πφαβ (π) θ =, at =, θ =, at = h. (4) The flow rates in fied and wave frame are related b [4]: (3) Q = F +. (5) 4. Solution of the Problem The fourth-order momentum equation () has an eact solution of the form: M β Ψ(, ) =C + C + C 3 e β + + C 4 e M β β +, (6) where β and M are the flow governing parameters for the proposed model and the constants of integration C i,i =,, 3, 4 are obtained using boundar conditions (3) through Mathematica 9. A closed-form epression for the pressure gradient is obtained b substituting (6) into (): dp d = M C, (7) where C (β,β,φ,m,α,d,,q). The dimensionless pressure rise P is obtained b substituting (7) into the following equation: ( ) dp P = d. (8) d The energ equation () has also a closed-form solution: θ(, ) =Ω +Ω β B r M 4 (β +) [ 4 (β +) M β { C3e M β β + } ] + C4e M β β + + C 3 C 4, (9) where Ω and Ω are constants of integration, which are obtained using boundar conditions (4)

6 N. S. Akbar & Z. H. Khan 5. Results and Discussion In this section, the effects of the different flow controlling parameters, such as the upper limit apparent viscosit coefficient β, slenderness parameter β, Hartmann number M, eccentricit of the elliptic motion α, flow rate and cilia length φ on the pressure rise, pressure gradient, velocit, temperature, streamlines and isotherms have been discussed and phsicall interpreted. 5.. Flow characteristics The velocit field is illustrated in Figs. (a) (f), for different values of upper limit apparent viscosit coefficient β, slenderness parameter β, Hartmann number M, eccentricit of the elliptic motion α, flowrateq and cilia length φ. It is seen that the behavior of velocit near the channel walls and at center is not similar in view of the Hartmann number M and upper limit apparent viscosit coefficient β. The velocit field increases due to increase in M and β near the channel walls while velocit field decreases at the center of the channel. However it is seen that u(, ).8.6. M =.5,,, 3, 4, 5 =, Q =.5, =.5 =, =, = u(, ).8.6. =.5,,, 3, 4, 5 M =, Q =.5, =.5 =, =, = (a) (b) (c) u(, ) =,.,,.6,.8, M =, Q =.5, =.5 =, =, = u(, ). =,.,,.6,.8,.8 u(, ). Q =,.,,.6,.8,.8.6 u(, ).6. =,.,,.6,.8, =, Q =.5, M = =, =, = M =, =, =.5 =, =, = M =, Q =, =.5 - =, =.5, = (d) (e) (f) Fig.. Effect of the different flow parameter on velocit profile along -aes

7 .8 M =.5,,, 3, 4, 5.3. (, ) (, ).. Q =, =., = = =, B r =.5, M = (, ) =.3 Q =, =., =, = =, B r =.5, = =.5,,, 3, 4, 5. Q =, =., = M = =, B r =.5, = Fig (a) (b) (c) Effect of the different flow parameters on temperature profile along -ais. the velocit field increases uniforml with the increase in slenderness parameter β, eccentricit of the elliptic motion α and cilia length φ but decreases graduall with the flow rate Q. Temperature profiles are displaed in Figs. 3(a) 3(c). It is observed that with the increasing values of Hartmann number M and upper limit apparent viscosit coefficient β, temperature profile decreases, however, when we increase slenderness parameter β, temperature profile increases. 5.. Pumping characteristics Numerical integration is performed for the pressure rise per wavelength. It is noticed that the pressure rise and volume flow rate have opposite behaviors. From Fig. 4, it is found that in pumping region ( P >), the pressure rise decreases with the increase of upper limit apparent viscosit coefficient β and slenderness parameter β, while pressure rise increase with the increases in Hartmann number M and cilia length φ. Figures 4(a) and 4(f) also show that in the augmented pumping region for P <, pressure rise gives the opposite results for all the parameters as compared to the pumping region ( P > ). Free pumping region holds for P =. The pressure gradient for different values of the governing parameters is plotted in Fig. 5. Magnitude of pressure gradient increases with the increase in slenderness parameter β, Hartmann number M, eccentricit of the elliptic motion α, flowrate Q and cilia length φ, while it decreases with the increase in upper limit apparent viscosit coefficient β. It is also observed that the maimum pressure gradient occurs when =8and near the channel walls the pressure gradient is small. This leads to the fact that flow can easil pass in the middle of the channel Trapping phenomena The trapping for different values of β,m and φ are shown in Figs Figure 6 shows that with an increase in upper limit apparent viscosit coefficient β, the fluid becomes thicker and encloses slowl that decreases size and number of trapping 556-7

8 N. S. Akbar & Z. H. Khan 75 6 =, =., =, =. 6 M =, =.5, =, = P 3 P P M =.5,,, 3, 4, Q (a) 75 =, M =, =, = =,.,,.6,.8, Q (c) P =.5,,, 3, 4, Q (b) M =, =.5, =.5, = =,.,,.6,.8, Q (d) Fig. 4. Variations in pressure rise vs. flow rate for different flow governing parameters. bolus. With the increase in the value of the M, mean electromagnetic forces are higher than the viscous forces, then the size of the trapping bolus and the number of bolus decreases see Fig. 7. Figure 8 illustrate that the size and number of enclosed streamlines increase when cilia length increases. A line connecting points of equal temperature is called an isotherm. From Figs. 9 and, the small orange numbers are contour labels, which identif the value of an isotherm (75 F, 85 F). So it is analzed that when we increase upper limit apparent viscosit coefficient β then temperature is 75 F, 85 F, but for increasing flow rate Q temperature decreases graduall (less than 75 F, 85 F)

9 3 Q =, =., =.5, =, = =.3,.6,,, 3, 5 =,.,,.6,.8, 8 dp/d dp/d 4 dp/d -4 - dp/d M =.5,,, 3, 4, Q =, =., =.5, =, M = (a) (b) (c) Q =.5, =., M =, =, = =,.5,.6,.65,.7, Fig dp/d 6 Q = =, =., =.5, =, M = (d) (e) (f) dp/d Q =, =, =., =, M = =,.5,,, 3, 4 Q =, =, =.5, =., M = The influence of the governing parameters on the pressure gradient along -ais..5.5 =.5 = = (a) (b) (c) Fig. 6. Influence of upper limit apparent viscosit coefficient β on streamlines

10 nd Reading Februar 3, 5 5:8 WSPC S IJB 556 N. S. Akbar & Z. H. Khan M= Q= Q = The effects of the flow rate Q on the streamlines = = Fig. 9. = (a) (c) (b) Fig (a).75 Q = Variations in streamlines with the Hartmann number M (c) (b) Fig. 7. (a) M=5 M = (b).75 (c) The effects of the upper limit apparent viscosit coefficient β on the isotherms. 556-

11 Q =.5 Q = Q = (a) (b) (c) Fig.. The effects of the flow rate Q on the isotherms. 6. Conclusion In this work, heat transfer analsis of biviscosit fluid b ciliar motion is investigated. The ke finding of the current analsis is as follows: The velocit field increases due to increase in M and β near the channel walls while velocit field decreases at the center of the channel. However, it is seen that the velocit field increases uniforml with the increase in slenderness parameter β, eccentricit of the elliptic motion α and cilia length φ but decreases graduall with the flow rate Q. It is analzed that with the increasing values of Hartmann number M and upper limit apparent viscosit coefficient β temperature profile decreases, however, when we increase slenderness parameter β, temperature profile increases. It is noticed that the pressure rise and volume flow rate have opposite behaviors. Pressure rise decreases with the increase in upper limit apparent viscosit coefficient β and slenderness parameter β, while pressure rise increase with the increase in Hartmann number M and cilia length φ. When ( P = ), the free pumping region holds. Magnitude of pressure gradient increases with the increase in slenderness parameter β, Hartmann number M, eccentricit of the elliptic motion α, flowrateq and cilia length φ, while decreases with the increase in upper limit apparent viscosit coefficient β. With the increase in the value of the M mean electromagnetic forces are higher than the viscous forces, then the size of the trapping bolus and the number of bolus decreases. It is analzed that when we increase upper limit apparent viscosit coefficient β then temperature is 75 F, 85 F, but for increasing flow rate Q temperature decreases graduall than (75 F, 85 F). 556-

12 N. S. Akbar & Z. H. Khan References [] N. S. Akbar, Blood flow analsis of Prandtl fluid model in tapered stenosed arteries, Ain Shams Engrg. J. 5 (4) [] N. S. Akbar, Nanofluid analsis for the intestinal flow in a smmetric channel, IEEE Trans. Nanobiosci. 3(4) (4) 5. [3] N. S. Akbar, Influence of magnetic field on peristaltic flow of a Casson fluid in an asmmetric channel: Application in crude oil refinement, J. Magn. Magn. Mater. 378 (5) [4] N. S. Akbar, Heat transfer and carbon nanotubes analsis for the peristaltic flow in adivergingtube, Meccanica 5 (5) [5] N. S. Akbar and A. W. Butt, CNT suspended nanofluid analsis in a fleible tube with ciliated walls, European Phs. J. Plus 9 (4) 74. [6] N. S. Akbar and A. W. Butt, Heat transfer analsis of viscoelastic fluid flow due to metachronal wave of cilia, Int. J. Biomath. 7(6) (4) 4566, 4 pp. [7] N. S. Akbar, A. W. Butt and N. F. M. Noor, Heat transfer analsis on transport of copper nanofluids due to metachronal waves of cilia, Curr. Nanosci. (6) (4) [8] N. S. Akbar and Z. H. Khan, Heat transfer stud of an individual multiwalled carbon nanotube due to metachronal beating of cilia, Int. Commun. Heat Mass Transfer 59 (4) 4 9. [9] N. S. Akbar and Z. H. Khan, Metachronal beating of cilia under the influence of Casson fluid and magnetic field, J. Magn. Magn. Mater. 378 (5) [] N. S. Akbar, Z. H. Khan and S. Nadeem, Metachronal beating of cilia under influence of Hartmann laer and heat transfer, European Phs. J. Plus 9 (4) 76. [] N. S. Akbar, Z. H. Khan and S. Nadeem, Peristaltic impulsion of MHD biviscosit fluid in a lopsided channel: Closed form solution, European Phs. J. Plus 9 (4) 3. [] N. S. Akbar and S. Nadeem, Blood flow analsis in tapered stenosed arteries with pseudoplastic characteristics, Int. J. Biomath. 7(6) (4) 4565, 8 pp. [3] N. S. Akbar and S. Nadeem, Eact solution of peristaltic flow of biviscosit fluid in an annulus: A note, Aleandria Engrg. J. 53 (4) [4] N. S. Akbar, S. U. Rahman, R. Ellahi and S. Nadeem, Blood flow stud of Williamson fluid through stenosed arteries with permeable walls, European Phs. J. Plus 9 (4) 4. [5] N. S. Akbar, S. U. Rahman, R. Ellahi and S. Nadeem, Nanofluid flow in tapering stenosed arteries with permeable walls, Int. J. Thermal Sci. 85 (4) [6] J. R. Blake, A spherical envelope approach to ciliar propulsion, J. Fluid Mech. 46 (97) [7] J. R. Blake, Flow in tubules due to ciliar activit, Bull. Math. Biol. 35 (973) [8] S. N. Khaderi and P. R. Onck, Fluid-structure interaction of three-dimensional magnetic artificial cilia, J. Fluid Mech. 78 () [9] T. J. Lardner and W. J. Shack, Cilia transport, Bull. Math. Biophs. 34 (97) [] C. E. Miller, An investigation of the movement of Newtonian liquids initiated and sustained b the oscillation of mechanical cilia, in Aspen Emphsema Conf., B. V. Amsterdam (967), pp [] M. A. Sleigh, The Biolog of Cilia and Flagella (MacMillian, New York, 96). 556-

13 [] M. A. Sleigh, Patterns of ciliar beating, in Aspects of Cell Motilit, Soc. Epl. Biol. Sump, Vol. XXII (Academic Press, New York, 968). [3] D. J. Smith, E. A. Gaffne and J. R. Blake, A viscoelastic traction laer model of muco-ciliar transport, Bull. Math. Biol. 69 (7) [4] R. Velez-Cardero and E. Lauga, Waving transport and propulsion in a generalized Newtonian fluid, J. Non-Newtonian Fluid Mech. 99 (3)