- Email: [email protected]
Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion
ARTICLE IN PRESS Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
Contents lists available at ScienceDirect
Engineering Science and Technology, an International Journal j o u r n a l h o m e p a g e : h t t p : / / w w w. e l s e v i e r. c o m / l o c a t e / j e s t c h
Press: Karabuk University, Press Unit ISSN (Printed) : 1302-0056 ISSN (Online) : 2215-0986 ISSN (E-Mail) : 1308-2043
H O S T E D BY
Available online at www.sciencedirect.com
ScienceDirect
Full Length Article
Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion M.M. Bhatti a,*, A. Zeeshan b, M.M. Rashidi c,d a
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China Department of Mathematics, International Islamic University Islamabad, Pakistan Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Shanghai 201804, China d ENN-Tongji Clean Energy Institute of advanced studies, Shanghai 200072, China b c
A R T I C L E
I N F O
Article history: Received 30 December 2015 Received in revised form 11 March 2016 Accepted 14 March 2016 Available online Keywords: Cilia motion Magnetohydrodynamics Particle fluid Metachronal wave
A B S T R A C T
In this article, the influence of magnetohydrodynamics (MHD) on cilia motion of particle–fluid suspension through a porous planar channel has been investigated. The governing equations of Casson fluid model for fluid phase and particulate phase are solved by taking the assumption of long wavelength and neglecting the inertial forces due to laminar flow. The solutions for the resulting differential equations have been obtained analytically and a close form of solutions is presented. The expression for pressure rise along the whole length of the channel is evaluated numerically. The influences of all the physical parameters are demonstrated graphically. Trapping mechanism has also been discussed with the help of streamlines. It is observed that due to the influence of magnetohydrodynamics and particle volume fraction, velocity of the fluid decreases. It is also found that pressure rise shows similar behaviour for particle volume fraction and Casson fluid parameter. Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction The term cilia is often used for “eukaryotic cells,” and it is derived from the word “eyelashes,” The term cilia is used when different types of cilia elements are appendages on a single cell. The inner layer of these cells is described by a cylindrical core which is known as axoneme. The axoneme contains cylindrical arrangements of molecular motors/dynein and elastic elements (known as microtubules). All the cilia elements on the surface of microorganism describe the same beating pattern. The length range of each cilium element is about 2 μm to mm and its diameter is about 0.2 μm. The shape of the cilia elements is very much similar to hair like motile appendages which can be observed in the nervous system, digestive system, male and female reproductive system. Cilia motion plays a vital role in various physiological processes such as reproduction, alimentation, locomotion and respiration. Metachronal wave that is generated due to the waves of ciliary elements propagates along out of phase direction in a distensible tube/channel. The main benefit of metachronal wave is to control the continuity of the flow and also to help us increase the amount of fluid propelled. Ciliary moments are found in different shapes depending on the ciliary system, such as oscillatory, excitable, helical, beating and planar. When the * Corresponding author. Tel.: +8613162146836. E-mail address: [email protected] (M.M. Bhatti). Peer review under responsibility of Karabuk University.
metachronal wave propagates along the same path as the effective stroke, then this phenomenon is known as symplectic metachronism, whereas when they propagate along the opposite direction, then this phenomenon is known as antiplectic. Several authors investigated analytically and numerically the motion of cilia in different situations with various biological fluids [1–5]. On the other hand, magnetohydrodynamics is very helpful to control the flow of fluid. Magnetohydrodynamics (MHD) deals with the study of electrically conducting fluids such as salt water or electrolytes, plasmas and liquid metals. The application of magnetic field can be found in various engineering process and geophysical studies. Magnetohydrodynamics is also applicable in magnetic drug targeting for different types of cancer diseases. Magnetohydrodynamics is useful and applicable in various microchannel design for creating continues and non-pulsating flow. In biomedical engineering, magnetohydrodynamics helps in the regulation of hyperthermia, MRI and magneto-fluid rotary blood pumps, etc. Magnetohydrodynamics oscillation and waves are very famous tools for astrophysical plasmas and remote diagnostic. Magnetohydrodynamics sensors are often used to measure the angular velocity. Akbar and Khan [6–8] analysed the effects of magnetohydrodynamics on metachronal beating of cilia for Casson fluid model. Ellahi et al. [9] studied the effects of magnetohydrodynamics on peristaltic flow of Jeffery fluid in a rectangular duct through a porous medium. Mekheimer et al. [10] examined the particle fluid suspension of peristaltic flow in a non-uniform annulus. Akbar and Butt [11] analysed
http://dx.doi.org/10.1016/j.jestch.2016.03.001 2215-0986/Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NCND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
2
the effects of heat transfer on metachronal of wave cilia for Rabinowitsch fluid model. Later, Akbar and Khan [6–8] characterize the effects of heat transfer for bi-viscous ciliary motion fluid. Some relevant studies on the present analysis can be found from the list of references [6–8,12–29]. With the above analysis in mind, the purpose of this present investigation is to analyse the effects of magnetohydrodynamics on metachronal wave of particle–fluid suspension due to cilia motion in the uniform porous planar channel. The governing equations of particle–fluid suspension have been solved for Casson fluid model under the assumption of long wavelength and creeping flow regime. The solution for the resulting equation has been obtained analytically and a closed-form solution is presented for fluid phase and particulate phase. The impact of all the physical parameters is discussed and plotted. This paper is summarized as follows; Sec. (2) describes the mathematical formulation of the problem, Sec. (3) is devoted to methodology and solution of the problem and finally, Sec. (4) characterizes the numerical results and discussion.
The governing equation of motion, continuity for fluid phase and particulate phase can be written as
2.1. Fluid phase
∂U f ∂V f + = 0, ∂X ∂Y ⎛ ∂U
(5)
∂U
∂U ⎞
∂ ⎛ ∂ ⎞ + (1 − C ) ⎜ τ XX τ XY + ⎟ ⎝ ∂X ⎠ ∂Y μ + CS U p − U f + J x × B − s U f , (6) k
(
⎛ ∂V
∂V
2π , Y = F X , t = a + a ε cos X − ct λ
(
(
)
)
(1)
2π . X = G X , t = X 0 + a ε α cos X − ct λ
(
(
)
)
(2)
The vertical and horizontal velocities for cilia motion can be written as [26])
2π 2π aε α c cos X − ct λ λ U= , 2π 2π 1− aε α c cos X − ct λ λ
( (
−
) )
2π 2π aε α c sin λ λ V = 2π 2π 1− aε α c sin λ λ −
( X − ct ) . ( X − ct )
)
∂V ⎞
∂P
(1 − C ) ρ f ⎜ f + U f f + V f f ⎟ = − (1 − C ) ⎝ ∂t ∂X ∂Y ⎠ ∂Y
∂ ⎛ ∂ ⎞ + (1 − C ) ⎜ τ YX τ YY + ⎟ ⎝ ∂X ⎠ ∂Y μ + CS Vp − V f + J y × B − s V f , k
2. Mathematical formulation Let us consider the unsteady irrotational, hydromagnetic particle– fluid suspension model, which is incompressible, and electrically conducting by an external magnetic field is applied through a twodimensional porous planar channel. A metachronal wave is travelling with a constant velocity c that is generated due to collective beating of cilia along the walls of the channel whose inner surfaces are ciliated. We have selected a Cartesian coordinate system for the channel in such a way that is X − axis taken along the axial direction and Y − axis is taken along the transverse direction (See Fig. 1). The envelop for cilia tips is assumed to be written as [6–8]
∂P
(1 − C ) ρ f ⎜ f + U f f + V f f ⎟ = − (1 − C ) ⎝ ∂t ∂X ∂Y ⎠ ∂X
(
)
(7)
2.2. Particulate phase
∂U p ∂Vp + = 0, ∂X ∂Y
(8)
⎛ ∂U p ∂U p ∂U p ⎞ ∂P Cρ p ⎜ + U p + Vp ⎟ = −C + CS U f − U p , ⎝ ∂t ∂X ∂Y ⎠ ∂X
(
)
⎛ ∂Vp ∂Vp ∂Vp ⎞ ∂P Cρ p ⎜ + U p + Vp ⎟ = −C + CS V f − Vp . ⎝ ∂t ∂X ∂Y ⎠ ∂Y
(
)
(9)
(10)
The mathematical expression for the drag coefficient and the empirical relation for the viscosity of the suspension can be described as
S= (3)
9μ 0 4 + 3 8C − 3C 2 + 3C μ0 , μs = , λ (C ) , λ (C ) = 2 ˇ 1 − χC 2a (2 − 3C )2 ⎡2.49C + 1107 e −1.69 C ⎤ ⎢ T ⎦⎥
χ = 0.07e ⎣
.
(11)
The stress tensor of Casson fluid is defined as (4)
τ
1n
= τ 01 n + μ sγ 1 n,
(
(12)
)
τ i , j = 2Ei , j μb + 2π D Py .
(13)
In the above equation, π = Ei , j we have consider Py = 0 Now, it is convenient to define the transformation variable from fixed frame to wave frame
, x = X − ct
y = Y , u f , p = U f , p − c , v f , p = V f , p, p = P .
(14)
Introducing the following non-dimensional quantities
u f ,p v f ,p y a 2 ρa c p , Re = , u f , p = , v f , p = , p= , a c cδ λcμ s μs μ √ 2π D Sa 2 k a B02a 2σ , M= , k= , ζ = b , β= . N= (15) μs μs μs λ Py
x =
Fig. 1. Geometry of the problem.
x , λ
y =
Using Eq. (14) and Eq. (15) and taking the approximation of long wavelength and neglecting the inertial forces, then Eq. (5) to Eq. (10)
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
3
reduces to the following form. The resulting equations for fluid phase and particulate phase can be written as
The expression for dimensionless stream function satisfying equation of continuity is defined as
1 ⎞ ∂ 2u f 1 dp ⎛ NC = ⎜1+ ⎟ − (u f + 1) − M 2 (u f + 1) + ( up − u f ) , dx ⎝ ζ ⎠ ∂y 2 k (1 − C )
(16)
u f ,p =
dp = N ( u f − up ) , dx
(17)
∂Ψ f , p ∂Ψ f , p , v f ,p = − , ∂y ∂x
where N1 =
(27)
2πφαβ cos 2π ( x ) ζ + km2ζ . , N2 = 1 − 2πφαβ cos 2π ( x ) k (1 + ζ )
and their corresponding non-dimensional boundary conditions are
u′f = 0 at
y = 0 and u f = −1 −
2πφαβ cos 2π ( x ) , 1 − 2πφαβ cos 2π ( x )
4. Numerical results and discussion
y = h = 1 + φ co os 2π x.
at
(18)
3. Solution of the problem The exact solution of Eq. (15) and Eq. (16) can be written as uf = −
(1 − C ) (1 + kM 2 ) + kdp dx + ((1 − C ) (1 + kM 2 ) N1 − kdp dx ) cosh N 2 y sech N 2h , (1 − C ) (1 + kM 2 )
(19)
In this section, the graphical results of different pertinent parameters are sketched for velocity profile, pressure rise and stream lines. For this purpose, Figs. 2 to 12 have been plotted against a , b , C , k , M and z . It is depicted from Fig. 2 that when the parameters a and b increase, the velocity of fluid shows the opposite behaviour near the walls. It can be noticed from Fig. 3a that when the particle volume fraction (C ) increases, the magnitude of the velocity decreases. It can be observed from Fig. 3b that when the porosity parameter (k ) increases, velocity of the fluid increases. From
(1 − C ) (1 + kM 2 ) + kdp dx + ⎛⎜⎝ (1 − C ) (1 + kM 2 ) N1 − k ⎞⎟⎠ cosh N 2 y secch N 2h − (1 N ) (1 − C ) (1 + kM 2 ) dp dx dx , up = − (1 − C ) (1 + kM 2 ) dp
The volume flow rate is given by h
Q f = (1 − C ) ∫ u f dy,
(21)
0
h
Q p = C ∫ updy,
(22)
0
where
Q = Q f + Q p,
Q=
(23)
Fig. 4a, we can observe that Hartmann number ( M ) is very helpful to control the flow because when the Hartmann number increases, the velocity of the fluid decreases, whereas the opposite behaviour has been observed for Casson fluid parameter (z ) as shown in Fig. 4b. Figs. 5 and 6 are sketched for pressure rise versus average volume flow rate (Q ) It can be noticed from Fig. 5a that when the particle volume fraction increases, the pressure rise decreases in retrograde pumping region ( ΔP > 0, Q < 0) , but its attitude is opposite in co-pumping region ( ΔP < 0, Q > 0) . It can be scrutinized from Fig. 5b that the pressure rise is decreasing in retrograde pumping region and its behaviour seems opposite in free pumping (ΔP < 0, Q < 0) region and co-pumping region. It can be observed
h ((1 − C ) (1 + kM 2 ) + (k (1 − C )C (1 + km2 ) (1 N )) dp dx ) ζ + km2ζ + ((1 − C ) (1 + kM 2 ) N1 − kdp dx ) k (1 + ζ ) tanh N 2h
(−1 + C ) (1 + kM 2 ) ζ + kM 2ζ
The pressure gradient dp dx is obtained after solving above Eq. (24), we get
(
)
(1 − C ) (1 + kM 2 ) (h + Q ) ζ + kM 2ζ + N1 k (1 + ζ ) tanh N 2h dp = . dx h ( −k + (1 − C )C (1 + kM 2 ) (1 N )) ζ + kM 2ζ + k k (1 + ζ ) tanh N 2h (25) The non-dimensional pressure rise ( ΔP ) is evaluated numerically by using the following expression 1
dp dx. dx 0
ΔP = ∫
(26)
(20)
.
(24)
from Fig. 6a that when the Hartmann number ( M ) increases, the pressure rise decreases in co-pumping and free pumping regions while it increases in retrograde pumping region. It can be seen from Fig. 6b that due to Casson fluid parameter, pressure rise decreases in retrograde pumping region but its behaviour starts changing in the same region and become opposite in free pumping and copumping regions. Another most interesting part of this section is trapping, which is taken into account by drawing stream against all the physical parameters. Generally, it is the formulation of internally circulating bolus that is enclosed by various streamlines. From Fig. 7, we can observe that when the parameter a increases, the size of trapped
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS 4
M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
Fig. 2. Velocity profile for various values of a and b when C = 0.6, M = 1, z = 3, Q = 2, φ = 0.6 .
Fig. 3. Velocity profile for various values of C and k when M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
bolus slowly reduces. In Fig. 8, same behaviour can be observed but the size of the trapping bolus slowly reduces as compared to Fig. 7. It can be observed from Fig. 9 that when the particle volume fraction (C ) increases, the number of trapped bolus increases. It
can be noticed from Fig. 10 that when the porosity parameter (k ) increases, the magnitude of the bolus slowly reduces, whereas the number bolus also increases. It can be visualized from Fig. 11 and Fig. 12 that when the Hartmann number ( M ) and Casson
Fig. 4. (a) Velocity profile for various values of M and z when M = 1, C = 0.6, k = 0.5, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Fig. 5. Pressure rise vs volume flow rate for various values of C and k when M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
5
Fig. 6. Pressure rise vs volume flow volume flow rate for various values of M and z when k = 0.5, C = 0.6, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
fluid parameter (z ) increase then the number of trapping bolus reduces. 5. Conclusion In this article, the influence of magnetohydrodynamics on cilia motion of particle–fluid suspension through a porous planar channel has been investigated. The governing equation of motion and continuity for fluid phase and particulate phase has been obtained by neglecting the inertial forces and taking the long wavelength assumption. The solution for the resulting differential equation has been obtained analytically and the closed-form solution is
presented. The major outcomes for the flow problem are: Velocity of the fluid decreases due to the magnetic field while its behaviour is opposite for porosity parameter. Velocity of the fluid behaves as decreasing function due to particle volume fraction.
• • •
The behaviour of pressure rise is same for Casson fluid parameter and particle volume fraction. The present analysis can also be reduced to Newtonian fluid by taking z → ∞ as a special case of our study. The present investigation depicts various interesting behaviour that warrants further study on cilia motion with particle volume fraction for various biological fluids.
Fig. 7. Stream lines for various values of a , (a ) a = 0.25, (b ) a = 0.35, (c ) a = 0.4 when C = 0.6, M = 1, z = 3, k = 0.5, b = 0.5, Q = 2, φ = 0.6 .
Fig. 8. Stream lines for various values of b , (a ) b = 0.2, (b ) b = 0.4, (c ) b = 0.6 when C = 0.6, M = 1, z = 3, a = 0.2, k = 0.5, Q = 2, φ = 0.6 .
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
6
Fig. 9. Stream lines for various values of C , (a ) C = 0, (b ) C = 0.1, (c ) C = 0.2 when k = 0.5, M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Nomenclature
a B0 c Py C M P Q Re k
Mean radius of the channel Magnetic field Wave velocity Yield stress Volume fraction density Hartmann number Pressure in fixed frame Volume flow rate Reynolds number Porosity parameter
S t U , V X0 X , Y
Drag force Time Velocity components in fixed frame Reference position of the cilia Cartesian coordinate axis in fixed frame
Greek symbols σ Electric conductivity of the fluid λ Wavelength μs Viscosity of the fluid Amplitude ratio φ ρ Fluid density
Fig. 10. Stream lines for various values of k , (a ) k = 0.2, (b ) k = 0.3, (c ) k = 0.5 when C = 0.6, M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6.
Fig. 11. Stream lines for various values of M , (a ) M = 0.1, (b ) M = 1, (c ) M = 2 when C = 0.6, k = 0.5, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
7
Fig. 12. Stream lines for various values of z , (a ) z = 0.5, (b ) z = 1, (c ) z = 3 when C = 0.6, M = 1, k = 0.5, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
τ β ε μb γ ζ α
Stress tensor Wave number Ratio of the cilia length Plastic viscosity Shear rate Casson fluid parameter Eccentricity of the elliptic path
Subscripts Fluid phase f p Particulate phase References [1] C. Barton, S. Raynor, Analytical investigation of cilia induced mucous flow, Bull. Math. Biol. 29 (1967) 419–428. [2] S. Peter, Studies on cilia III. Further studies on the cilium tip and a “sliding filament” model of ciliary motility, J. Cell Biol. 39 (1968) 77–94. [3] J.R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46 (1971) 199–208. [4] S.K. Guha, H. Kaur, A.M. Ahmed, Mechanics of spermatic fluid transport in the vas deferens, Med. Biol. Eng. 13 (1975) 518–522. [5] H.L. Agarwal, Anawaruddin, Cilia transport of bio-fluid with variable viscosity, Indian J. Pure Ap. Mat. 15 (1984) 1128–1139. [6] S.N. Akbar, H.Z. Khan, Metachronal beating of cilia under the influence of Casson fluid and magnetic field, J. Magn. Magn. Mater. 378 (2015) 320–326. [7] N.S. Akbar, H.Z. Khan, Heat transfer analysis of bi-viscous ciliary motion fluid, Int. J. Biomath. 8 (2015) 1550026. [8] N.S. Akbar, Z.H. Khan, Influence of magnetic field for metachronal beating of cilia for nanofluid with Newtonian heating, J. Magn. Magn. Mater. 381 (1) (2015) 235–242. [9] R. Ellahi, M.M. Bhatti, A. Riaz, M. Sheikholeslami, Effects of magnetohydrodynamics on peristaltic flow of Jeffrey fluid in a rectangular duct through a porous medium, J. Porous Media 17 (2014) 143–157. [10] K.S. Mekheimer, Y. Abd Elmaboud, Peristaltic transport of a particle–fluid suspension through a uniform and non-uniform annulus, Appl. Bionics Biomech. 5 (2008) 47–57. [11] N.S. Akbar, A.W. Butt, Heat transfer analysis of Rabinowitsch fluid flow due to metachronal wave of cilia, Results Phys. 5 (2015) 92–98.
[12] M. Hameed, A.A. Khan, R. Ellahi, M. Raza, Study of magnetic and heat transfer on the peristaltic transport of a fractional second grade fluid in a vertical tube, Eng. Sci. Technol. 37 (2015) 469–502. [13] T.J. Lardner, W.J. Shack, Cilia transport, Bull. Math. Biophys. 34 (1972) 325–335. [14] C.S. Bagewadi, S. Bhagya, Solutions in variably inclined MHD flows, J. Appl. Fluid Mech. 4 (2011) 77–83. [15] C.J. Misra, S.K. Pandey, Peristaltic transport of a particle-fluid suspension in a cylindrical tube, Comput. Math. Appl. 28 (1994) 131–145. [16] M. Saxena, V.P. Srivastava, Particulate suspension flow induced by sinusoidal peristaltic waves, Jpn J. Appl. Phys. 36 (1997) 385. [17] M.R. Krishnamurthy, B.C. Prasannakumara, B.J. Gireesha, R.S.R. Gorla, Effect of chemical reaction on MHD boundary layer flow and melting heat transfer of Williamson nanofluid in porous medium, Eng. Sci. Technol. 19 (2015) 53–61. [18] K.S. Mekheimer, E.F. El Shehawey, A.M. Elaw, Peristaltic motion of a particle-fluid suspension in a planar channel, Int. J. Theor. Phys. 37 (1998) 2895. [19] L.M. Srivastava, V.P. Srivastava, Peristaltic transport of a particle-fluid suspension, J. Biomech. Eng. 111 (1989) 157–165. [20] N. Freidoonimehr, M.M. Rashidi, Dual solutions for MHD Jeffery–Hamel nano-fluid flow in non-parallel walls using predictor homotopy analysis method, J. Appl. Fluid Mech. 9 (2016). [21] C.S.K. Raju, N. Sandeep, V. Sugunamma, M.J. Babu, J.R. Reddy, Heat and mass transfer in magnetohydrodynamics Casson fluid over an exponentially permeable stretching surface, Eng. Sci. Technol. 19 (2015) 45–52. [22] G.M. Pavithra, B.J. Gireesha, Unsteady flow and heat transfer of a fluid-particle suspension over an exponentially stretching sheet, Ain Shams Eng. J. 5 (2014) 613–624. [23] A. Khalid, I. Khan, A. Khan, S. Shafie, Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium, Eng. Sci. Technol. 52 (2015) 309–317. [24] N.S. Akbar, A.W. Butt, Heat transfer analysis of viscoelastic fluid flow due to metachronal wave of cilia, Int. J. Biomath. 07 (2014) 1450066, doi:10.1142/ S1793524514500661 [14 pages]. [25] N.S. Akbar, A.W. Butt, CNT suspended nanofluid analysis in a flexible tube with ciliated walls, Eur. Phys. J. Plus 129 (2014) 174. [26] N.S. Akbar, Biomathematical analysis of carbon nanotubes due to ciliary motion, Int. J. Biomath. 08 (2015) 1550023, doi:10.1142/S1793524515500230 [11 pages]. [27] N.S. Akbar, Z.H. Khan, Heat transfer study of an individual multiwalled carbon nanotube due to metachronal beating of cilia, Int. Commun. Heat Mass Transf. 59 (2014) 114–119. [28] S. Nadeem, H. Sadaf, Trapping study of nano fluids in an annulus with cilia, AIP Adv. 5 (2015) 127204. [29] N.S. Akbar, Z.H. Khan, S. Nadeem, Metachronal beating of cilia under influence of Hartmann layer and heat transfer, Eur. Phys. J. Plus 129 (2014) 176.
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
Contents lists available at ScienceDirect
Engineering Science and Technology, an International Journal j o u r n a l h o m e p a g e : h t t p : / / w w w. e l s e v i e r. c o m / l o c a t e / j e s t c h
Press: Karabuk University, Press Unit ISSN (Printed) : 1302-0056 ISSN (Online) : 2215-0986 ISSN (E-Mail) : 1308-2043
H O S T E D BY
Available online at www.sciencedirect.com
ScienceDirect
Full Length Article
Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion M.M. Bhatti a,*, A. Zeeshan b, M.M. Rashidi c,d a
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China Department of Mathematics, International Islamic University Islamabad, Pakistan Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Shanghai 201804, China d ENN-Tongji Clean Energy Institute of advanced studies, Shanghai 200072, China b c
A R T I C L E
I N F O
Article history: Received 30 December 2015 Received in revised form 11 March 2016 Accepted 14 March 2016 Available online Keywords: Cilia motion Magnetohydrodynamics Particle fluid Metachronal wave
A B S T R A C T
In this article, the influence of magnetohydrodynamics (MHD) on cilia motion of particle–fluid suspension through a porous planar channel has been investigated. The governing equations of Casson fluid model for fluid phase and particulate phase are solved by taking the assumption of long wavelength and neglecting the inertial forces due to laminar flow. The solutions for the resulting differential equations have been obtained analytically and a close form of solutions is presented. The expression for pressure rise along the whole length of the channel is evaluated numerically. The influences of all the physical parameters are demonstrated graphically. Trapping mechanism has also been discussed with the help of streamlines. It is observed that due to the influence of magnetohydrodynamics and particle volume fraction, velocity of the fluid decreases. It is also found that pressure rise shows similar behaviour for particle volume fraction and Casson fluid parameter. Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction The term cilia is often used for “eukaryotic cells,” and it is derived from the word “eyelashes,” The term cilia is used when different types of cilia elements are appendages on a single cell. The inner layer of these cells is described by a cylindrical core which is known as axoneme. The axoneme contains cylindrical arrangements of molecular motors/dynein and elastic elements (known as microtubules). All the cilia elements on the surface of microorganism describe the same beating pattern. The length range of each cilium element is about 2 μm to mm and its diameter is about 0.2 μm. The shape of the cilia elements is very much similar to hair like motile appendages which can be observed in the nervous system, digestive system, male and female reproductive system. Cilia motion plays a vital role in various physiological processes such as reproduction, alimentation, locomotion and respiration. Metachronal wave that is generated due to the waves of ciliary elements propagates along out of phase direction in a distensible tube/channel. The main benefit of metachronal wave is to control the continuity of the flow and also to help us increase the amount of fluid propelled. Ciliary moments are found in different shapes depending on the ciliary system, such as oscillatory, excitable, helical, beating and planar. When the * Corresponding author. Tel.: +8613162146836. E-mail address: [email protected] (M.M. Bhatti). Peer review under responsibility of Karabuk University.
metachronal wave propagates along the same path as the effective stroke, then this phenomenon is known as symplectic metachronism, whereas when they propagate along the opposite direction, then this phenomenon is known as antiplectic. Several authors investigated analytically and numerically the motion of cilia in different situations with various biological fluids [1–5]. On the other hand, magnetohydrodynamics is very helpful to control the flow of fluid. Magnetohydrodynamics (MHD) deals with the study of electrically conducting fluids such as salt water or electrolytes, plasmas and liquid metals. The application of magnetic field can be found in various engineering process and geophysical studies. Magnetohydrodynamics is also applicable in magnetic drug targeting for different types of cancer diseases. Magnetohydrodynamics is useful and applicable in various microchannel design for creating continues and non-pulsating flow. In biomedical engineering, magnetohydrodynamics helps in the regulation of hyperthermia, MRI and magneto-fluid rotary blood pumps, etc. Magnetohydrodynamics oscillation and waves are very famous tools for astrophysical plasmas and remote diagnostic. Magnetohydrodynamics sensors are often used to measure the angular velocity. Akbar and Khan [6–8] analysed the effects of magnetohydrodynamics on metachronal beating of cilia for Casson fluid model. Ellahi et al. [9] studied the effects of magnetohydrodynamics on peristaltic flow of Jeffery fluid in a rectangular duct through a porous medium. Mekheimer et al. [10] examined the particle fluid suspension of peristaltic flow in a non-uniform annulus. Akbar and Butt [11] analysed
http://dx.doi.org/10.1016/j.jestch.2016.03.001 2215-0986/Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NCND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
2
the effects of heat transfer on metachronal of wave cilia for Rabinowitsch fluid model. Later, Akbar and Khan [6–8] characterize the effects of heat transfer for bi-viscous ciliary motion fluid. Some relevant studies on the present analysis can be found from the list of references [6–8,12–29]. With the above analysis in mind, the purpose of this present investigation is to analyse the effects of magnetohydrodynamics on metachronal wave of particle–fluid suspension due to cilia motion in the uniform porous planar channel. The governing equations of particle–fluid suspension have been solved for Casson fluid model under the assumption of long wavelength and creeping flow regime. The solution for the resulting equation has been obtained analytically and a closed-form solution is presented for fluid phase and particulate phase. The impact of all the physical parameters is discussed and plotted. This paper is summarized as follows; Sec. (2) describes the mathematical formulation of the problem, Sec. (3) is devoted to methodology and solution of the problem and finally, Sec. (4) characterizes the numerical results and discussion.
The governing equation of motion, continuity for fluid phase and particulate phase can be written as
2.1. Fluid phase
∂U f ∂V f + = 0, ∂X ∂Y ⎛ ∂U
(5)
∂U
∂U ⎞
∂ ⎛ ∂ ⎞ + (1 − C ) ⎜ τ XX τ XY + ⎟ ⎝ ∂X ⎠ ∂Y μ + CS U p − U f + J x × B − s U f , (6) k
(
⎛ ∂V
∂V
2π , Y = F X , t = a + a ε cos X − ct λ
(
(
)
)
(1)
2π . X = G X , t = X 0 + a ε α cos X − ct λ
(
(
)
)
(2)
The vertical and horizontal velocities for cilia motion can be written as [26])
2π 2π aε α c cos X − ct λ λ U= , 2π 2π 1− aε α c cos X − ct λ λ
( (
−
) )
2π 2π aε α c sin λ λ V = 2π 2π 1− aε α c sin λ λ −
( X − ct ) . ( X − ct )
)
∂V ⎞
∂P
(1 − C ) ρ f ⎜ f + U f f + V f f ⎟ = − (1 − C ) ⎝ ∂t ∂X ∂Y ⎠ ∂Y
∂ ⎛ ∂ ⎞ + (1 − C ) ⎜ τ YX τ YY + ⎟ ⎝ ∂X ⎠ ∂Y μ + CS Vp − V f + J y × B − s V f , k
2. Mathematical formulation Let us consider the unsteady irrotational, hydromagnetic particle– fluid suspension model, which is incompressible, and electrically conducting by an external magnetic field is applied through a twodimensional porous planar channel. A metachronal wave is travelling with a constant velocity c that is generated due to collective beating of cilia along the walls of the channel whose inner surfaces are ciliated. We have selected a Cartesian coordinate system for the channel in such a way that is X − axis taken along the axial direction and Y − axis is taken along the transverse direction (See Fig. 1). The envelop for cilia tips is assumed to be written as [6–8]
∂P
(1 − C ) ρ f ⎜ f + U f f + V f f ⎟ = − (1 − C ) ⎝ ∂t ∂X ∂Y ⎠ ∂X
(
)
(7)
2.2. Particulate phase
∂U p ∂Vp + = 0, ∂X ∂Y
(8)
⎛ ∂U p ∂U p ∂U p ⎞ ∂P Cρ p ⎜ + U p + Vp ⎟ = −C + CS U f − U p , ⎝ ∂t ∂X ∂Y ⎠ ∂X
(
)
⎛ ∂Vp ∂Vp ∂Vp ⎞ ∂P Cρ p ⎜ + U p + Vp ⎟ = −C + CS V f − Vp . ⎝ ∂t ∂X ∂Y ⎠ ∂Y
(
)
(9)
(10)
The mathematical expression for the drag coefficient and the empirical relation for the viscosity of the suspension can be described as
S= (3)
9μ 0 4 + 3 8C − 3C 2 + 3C μ0 , μs = , λ (C ) , λ (C ) = 2 ˇ 1 − χC 2a (2 − 3C )2 ⎡2.49C + 1107 e −1.69 C ⎤ ⎢ T ⎦⎥
χ = 0.07e ⎣
.
(11)
The stress tensor of Casson fluid is defined as (4)
τ
1n
= τ 01 n + μ sγ 1 n,
(
(12)
)
τ i , j = 2Ei , j μb + 2π D Py .
(13)
In the above equation, π = Ei , j we have consider Py = 0 Now, it is convenient to define the transformation variable from fixed frame to wave frame
, x = X − ct
y = Y , u f , p = U f , p − c , v f , p = V f , p, p = P .
(14)
Introducing the following non-dimensional quantities
u f ,p v f ,p y a 2 ρa c p , Re = , u f , p = , v f , p = , p= , a c cδ λcμ s μs μ √ 2π D Sa 2 k a B02a 2σ , M= , k= , ζ = b , β= . N= (15) μs μs μs λ Py
x =
Fig. 1. Geometry of the problem.
x , λ
y =
Using Eq. (14) and Eq. (15) and taking the approximation of long wavelength and neglecting the inertial forces, then Eq. (5) to Eq. (10)
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
3
reduces to the following form. The resulting equations for fluid phase and particulate phase can be written as
The expression for dimensionless stream function satisfying equation of continuity is defined as
1 ⎞ ∂ 2u f 1 dp ⎛ NC = ⎜1+ ⎟ − (u f + 1) − M 2 (u f + 1) + ( up − u f ) , dx ⎝ ζ ⎠ ∂y 2 k (1 − C )
(16)
u f ,p =
dp = N ( u f − up ) , dx
(17)
∂Ψ f , p ∂Ψ f , p , v f ,p = − , ∂y ∂x
where N1 =
(27)
2πφαβ cos 2π ( x ) ζ + km2ζ . , N2 = 1 − 2πφαβ cos 2π ( x ) k (1 + ζ )
and their corresponding non-dimensional boundary conditions are
u′f = 0 at
y = 0 and u f = −1 −
2πφαβ cos 2π ( x ) , 1 − 2πφαβ cos 2π ( x )
4. Numerical results and discussion
y = h = 1 + φ co os 2π x.
at
(18)
3. Solution of the problem The exact solution of Eq. (15) and Eq. (16) can be written as uf = −
(1 − C ) (1 + kM 2 ) + kdp dx + ((1 − C ) (1 + kM 2 ) N1 − kdp dx ) cosh N 2 y sech N 2h , (1 − C ) (1 + kM 2 )
(19)
In this section, the graphical results of different pertinent parameters are sketched for velocity profile, pressure rise and stream lines. For this purpose, Figs. 2 to 12 have been plotted against a , b , C , k , M and z . It is depicted from Fig. 2 that when the parameters a and b increase, the velocity of fluid shows the opposite behaviour near the walls. It can be noticed from Fig. 3a that when the particle volume fraction (C ) increases, the magnitude of the velocity decreases. It can be observed from Fig. 3b that when the porosity parameter (k ) increases, velocity of the fluid increases. From
(1 − C ) (1 + kM 2 ) + kdp dx + ⎛⎜⎝ (1 − C ) (1 + kM 2 ) N1 − k ⎞⎟⎠ cosh N 2 y secch N 2h − (1 N ) (1 − C ) (1 + kM 2 ) dp dx dx , up = − (1 − C ) (1 + kM 2 ) dp
The volume flow rate is given by h
Q f = (1 − C ) ∫ u f dy,
(21)
0
h
Q p = C ∫ updy,
(22)
0
where
Q = Q f + Q p,
Q=
(23)
Fig. 4a, we can observe that Hartmann number ( M ) is very helpful to control the flow because when the Hartmann number increases, the velocity of the fluid decreases, whereas the opposite behaviour has been observed for Casson fluid parameter (z ) as shown in Fig. 4b. Figs. 5 and 6 are sketched for pressure rise versus average volume flow rate (Q ) It can be noticed from Fig. 5a that when the particle volume fraction increases, the pressure rise decreases in retrograde pumping region ( ΔP > 0, Q < 0) , but its attitude is opposite in co-pumping region ( ΔP < 0, Q > 0) . It can be scrutinized from Fig. 5b that the pressure rise is decreasing in retrograde pumping region and its behaviour seems opposite in free pumping (ΔP < 0, Q < 0) region and co-pumping region. It can be observed
h ((1 − C ) (1 + kM 2 ) + (k (1 − C )C (1 + km2 ) (1 N )) dp dx ) ζ + km2ζ + ((1 − C ) (1 + kM 2 ) N1 − kdp dx ) k (1 + ζ ) tanh N 2h
(−1 + C ) (1 + kM 2 ) ζ + kM 2ζ
The pressure gradient dp dx is obtained after solving above Eq. (24), we get
(
)
(1 − C ) (1 + kM 2 ) (h + Q ) ζ + kM 2ζ + N1 k (1 + ζ ) tanh N 2h dp = . dx h ( −k + (1 − C )C (1 + kM 2 ) (1 N )) ζ + kM 2ζ + k k (1 + ζ ) tanh N 2h (25) The non-dimensional pressure rise ( ΔP ) is evaluated numerically by using the following expression 1
dp dx. dx 0
ΔP = ∫
(26)
(20)
.
(24)
from Fig. 6a that when the Hartmann number ( M ) increases, the pressure rise decreases in co-pumping and free pumping regions while it increases in retrograde pumping region. It can be seen from Fig. 6b that due to Casson fluid parameter, pressure rise decreases in retrograde pumping region but its behaviour starts changing in the same region and become opposite in free pumping and copumping regions. Another most interesting part of this section is trapping, which is taken into account by drawing stream against all the physical parameters. Generally, it is the formulation of internally circulating bolus that is enclosed by various streamlines. From Fig. 7, we can observe that when the parameter a increases, the size of trapped
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS 4
M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
Fig. 2. Velocity profile for various values of a and b when C = 0.6, M = 1, z = 3, Q = 2, φ = 0.6 .
Fig. 3. Velocity profile for various values of C and k when M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
bolus slowly reduces. In Fig. 8, same behaviour can be observed but the size of the trapping bolus slowly reduces as compared to Fig. 7. It can be observed from Fig. 9 that when the particle volume fraction (C ) increases, the number of trapped bolus increases. It
can be noticed from Fig. 10 that when the porosity parameter (k ) increases, the magnitude of the bolus slowly reduces, whereas the number bolus also increases. It can be visualized from Fig. 11 and Fig. 12 that when the Hartmann number ( M ) and Casson
Fig. 4. (a) Velocity profile for various values of M and z when M = 1, C = 0.6, k = 0.5, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Fig. 5. Pressure rise vs volume flow rate for various values of C and k when M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
5
Fig. 6. Pressure rise vs volume flow volume flow rate for various values of M and z when k = 0.5, C = 0.6, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
fluid parameter (z ) increase then the number of trapping bolus reduces. 5. Conclusion In this article, the influence of magnetohydrodynamics on cilia motion of particle–fluid suspension through a porous planar channel has been investigated. The governing equation of motion and continuity for fluid phase and particulate phase has been obtained by neglecting the inertial forces and taking the long wavelength assumption. The solution for the resulting differential equation has been obtained analytically and the closed-form solution is
presented. The major outcomes for the flow problem are: Velocity of the fluid decreases due to the magnetic field while its behaviour is opposite for porosity parameter. Velocity of the fluid behaves as decreasing function due to particle volume fraction.
• • •
The behaviour of pressure rise is same for Casson fluid parameter and particle volume fraction. The present analysis can also be reduced to Newtonian fluid by taking z → ∞ as a special case of our study. The present investigation depicts various interesting behaviour that warrants further study on cilia motion with particle volume fraction for various biological fluids.
Fig. 7. Stream lines for various values of a , (a ) a = 0.25, (b ) a = 0.35, (c ) a = 0.4 when C = 0.6, M = 1, z = 3, k = 0.5, b = 0.5, Q = 2, φ = 0.6 .
Fig. 8. Stream lines for various values of b , (a ) b = 0.2, (b ) b = 0.4, (c ) b = 0.6 when C = 0.6, M = 1, z = 3, a = 0.2, k = 0.5, Q = 2, φ = 0.6 .
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
6
Fig. 9. Stream lines for various values of C , (a ) C = 0, (b ) C = 0.1, (c ) C = 0.2 when k = 0.5, M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Nomenclature
a B0 c Py C M P Q Re k
Mean radius of the channel Magnetic field Wave velocity Yield stress Volume fraction density Hartmann number Pressure in fixed frame Volume flow rate Reynolds number Porosity parameter
S t U , V X0 X , Y
Drag force Time Velocity components in fixed frame Reference position of the cilia Cartesian coordinate axis in fixed frame
Greek symbols σ Electric conductivity of the fluid λ Wavelength μs Viscosity of the fluid Amplitude ratio φ ρ Fluid density
Fig. 10. Stream lines for various values of k , (a ) k = 0.2, (b ) k = 0.3, (c ) k = 0.5 when C = 0.6, M = 1, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6.
Fig. 11. Stream lines for various values of M , (a ) M = 0.1, (b ) M = 1, (c ) M = 2 when C = 0.6, k = 0.5, z = 3, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001
ARTICLE IN PRESS M.M. Bhatti et al. / Engineering Science and Technology, an International Journal ■■ (2016) ■■–■■
7
Fig. 12. Stream lines for various values of z , (a ) z = 0.5, (b ) z = 1, (c ) z = 3 when C = 0.6, M = 1, k = 0.5, a = 0.2, b = 0.5, Q = 2, φ = 0.6 .
τ β ε μb γ ζ α
Stress tensor Wave number Ratio of the cilia length Plastic viscosity Shear rate Casson fluid parameter Eccentricity of the elliptic path
Subscripts Fluid phase f p Particulate phase References [1] C. Barton, S. Raynor, Analytical investigation of cilia induced mucous flow, Bull. Math. Biol. 29 (1967) 419–428. [2] S. Peter, Studies on cilia III. Further studies on the cilium tip and a “sliding filament” model of ciliary motility, J. Cell Biol. 39 (1968) 77–94. [3] J.R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46 (1971) 199–208. [4] S.K. Guha, H. Kaur, A.M. Ahmed, Mechanics of spermatic fluid transport in the vas deferens, Med. Biol. Eng. 13 (1975) 518–522. [5] H.L. Agarwal, Anawaruddin, Cilia transport of bio-fluid with variable viscosity, Indian J. Pure Ap. Mat. 15 (1984) 1128–1139. [6] S.N. Akbar, H.Z. Khan, Metachronal beating of cilia under the influence of Casson fluid and magnetic field, J. Magn. Magn. Mater. 378 (2015) 320–326. [7] N.S. Akbar, H.Z. Khan, Heat transfer analysis of bi-viscous ciliary motion fluid, Int. J. Biomath. 8 (2015) 1550026. [8] N.S. Akbar, Z.H. Khan, Influence of magnetic field for metachronal beating of cilia for nanofluid with Newtonian heating, J. Magn. Magn. Mater. 381 (1) (2015) 235–242. [9] R. Ellahi, M.M. Bhatti, A. Riaz, M. Sheikholeslami, Effects of magnetohydrodynamics on peristaltic flow of Jeffrey fluid in a rectangular duct through a porous medium, J. Porous Media 17 (2014) 143–157. [10] K.S. Mekheimer, Y. Abd Elmaboud, Peristaltic transport of a particle–fluid suspension through a uniform and non-uniform annulus, Appl. Bionics Biomech. 5 (2008) 47–57. [11] N.S. Akbar, A.W. Butt, Heat transfer analysis of Rabinowitsch fluid flow due to metachronal wave of cilia, Results Phys. 5 (2015) 92–98.
[12] M. Hameed, A.A. Khan, R. Ellahi, M. Raza, Study of magnetic and heat transfer on the peristaltic transport of a fractional second grade fluid in a vertical tube, Eng. Sci. Technol. 37 (2015) 469–502. [13] T.J. Lardner, W.J. Shack, Cilia transport, Bull. Math. Biophys. 34 (1972) 325–335. [14] C.S. Bagewadi, S. Bhagya, Solutions in variably inclined MHD flows, J. Appl. Fluid Mech. 4 (2011) 77–83. [15] C.J. Misra, S.K. Pandey, Peristaltic transport of a particle-fluid suspension in a cylindrical tube, Comput. Math. Appl. 28 (1994) 131–145. [16] M. Saxena, V.P. Srivastava, Particulate suspension flow induced by sinusoidal peristaltic waves, Jpn J. Appl. Phys. 36 (1997) 385. [17] M.R. Krishnamurthy, B.C. Prasannakumara, B.J. Gireesha, R.S.R. Gorla, Effect of chemical reaction on MHD boundary layer flow and melting heat transfer of Williamson nanofluid in porous medium, Eng. Sci. Technol. 19 (2015) 53–61. [18] K.S. Mekheimer, E.F. El Shehawey, A.M. Elaw, Peristaltic motion of a particle-fluid suspension in a planar channel, Int. J. Theor. Phys. 37 (1998) 2895. [19] L.M. Srivastava, V.P. Srivastava, Peristaltic transport of a particle-fluid suspension, J. Biomech. Eng. 111 (1989) 157–165. [20] N. Freidoonimehr, M.M. Rashidi, Dual solutions for MHD Jeffery–Hamel nano-fluid flow in non-parallel walls using predictor homotopy analysis method, J. Appl. Fluid Mech. 9 (2016). [21] C.S.K. Raju, N. Sandeep, V. Sugunamma, M.J. Babu, J.R. Reddy, Heat and mass transfer in magnetohydrodynamics Casson fluid over an exponentially permeable stretching surface, Eng. Sci. Technol. 19 (2015) 45–52. [22] G.M. Pavithra, B.J. Gireesha, Unsteady flow and heat transfer of a fluid-particle suspension over an exponentially stretching sheet, Ain Shams Eng. J. 5 (2014) 613–624. [23] A. Khalid, I. Khan, A. Khan, S. Shafie, Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium, Eng. Sci. Technol. 52 (2015) 309–317. [24] N.S. Akbar, A.W. Butt, Heat transfer analysis of viscoelastic fluid flow due to metachronal wave of cilia, Int. J. Biomath. 07 (2014) 1450066, doi:10.1142/ S1793524514500661 [14 pages]. [25] N.S. Akbar, A.W. Butt, CNT suspended nanofluid analysis in a flexible tube with ciliated walls, Eur. Phys. J. Plus 129 (2014) 174. [26] N.S. Akbar, Biomathematical analysis of carbon nanotubes due to ciliary motion, Int. J. Biomath. 08 (2015) 1550023, doi:10.1142/S1793524515500230 [11 pages]. [27] N.S. Akbar, Z.H. Khan, Heat transfer study of an individual multiwalled carbon nanotube due to metachronal beating of cilia, Int. Commun. Heat Mass Transf. 59 (2014) 114–119. [28] S. Nadeem, H. Sadaf, Trapping study of nano fluids in an annulus with cilia, AIP Adv. 5 (2015) 127204. [29] N.S. Akbar, Z.H. Khan, S. Nadeem, Metachronal beating of cilia under influence of Hartmann layer and heat transfer, Eur. Phys. J. Plus 129 (2014) 176.
Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion, Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001