Numerical treatment for investigation of squeezing unsteady nanofluid flow between two parallel plates A.K. Gupta, S. Saha Ray PII: DOI: Reference:
S0032-5910(15)00292-2 doi: 10.1016/j.powtec.2015.04.018 PTEC 10928
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Powder Technology
Received date: Revised date: Accepted date:
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Please cite this article as: A.K. Gupta, S. Saha Ray, Numerical treatment for investigation of squeezing unsteady nanofluid flow between two parallel plates, Powder Technology (2015), doi: 10.1016/j.powtec.2015.04.018
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ACCEPTED MANUSCRIPT Numerical treatment for investigation of squeezing unsteady nanofluid flow between two parallel plates
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A. K. Gupta National Institute of Technology Department of Mathematics Rourkela-769008, India
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S. Saha Ray National Institute of Technology Department of Mathematics Rourkela-769008, India Email:
[email protected]
Abstract
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In this paper, a new method based on the Chebyshev wavelet expansion is proposed for solving a coupled
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system of nonlinear ordinary differential equations to model the unsteady flow of a nanofluid squeezing between two parallel plates. Chebyshev wavelet method is applied to compute the numerical solution of
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coupled system of nonlinear ordinary differential equations in order to model squeezing unsteady nanofluid flow. The approximate solutions of nonlinear ordinary differential equations thus obtained by Chebyshev wavelet method are compared with those of obtained by Adomian decomposition method (ADM), fourth order Runge-Kutta method and homotopy analysis method (HAM). The results obtained by the above methods are illustrated graphically and are discussed in details. The present scheme is very simple, effective and appropriate for obtaining numerical solution of squeezing unsteady nanofluid flow between parallel plates. Key words: Nanofluids; squeeze number; Prandtl number; Eckert number; Chebyshev wavelet method. 1.
Introduction
Squeezing flow between parallel plates is a fascinating area of research as it occurs in numerous applications in science and engineering which include hydro dynamical machines, polymer processing,
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ACCEPTED MANUSCRIPT chemical processing equipment, formation and dispersion of fog, damage of crops due to freezing, compression, transient loading of mechanical components, injection modeling and the squeezed films in
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power transmission.
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The seminal work on the squeezing flow under the impulsive deviation of lubrication was reported by Stefan [1] in 1874. Domairry and Aziz [2] analysed the magneto hydrodynamic squeezing flow of a
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viscous fluid between parallel disks. The hydrodynamic squeezing flow of a viscous fluid by using
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homotopy analysis method (HAM) was studied by Rashidi et al. [3]. Mahmood et al. [4] investigated the heat transfer characteristics in the squeezed flow over a porous surface. Muhamin et al. [5] examined the
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effect of suction on the flow past a shrinking surface along with heat and mass transfer phenomena. AbdEl Aziz [6] considered the outcome of time-dependent chemical reaction on the flow of a viscous fluid
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past an unsteady stretching sheet. Influence of thermal radiation and time-dependent chemical reaction on
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the flow past an unsteady shrinking sheet was studied by Hayat et al. [7]. Mustafa et al. [8] studied heat and mass transfer characteristics in a viscous fluid which is squeezed between parallel plates. They found
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that the magnitude of local Nusselt number is an increasing function of Pr and Ec. In contemporary years, nanofluids technology is proposed and studied by some researchers experimentally or numerically to regulate heat transfer in a process [9, 12]. Nanofluids are fluids with suspensions of metals, oxides, carbides or carbon nanotubes in a base fluid. Choi [13] was the first to introduce the term nanofluid that represents the fluid in which nano-scale particles are suspended in the base fluid with low thermal conductivity such as water, ethylene glycol, and oil. Pantzali et al. [14] studied the effect of the use of a nanofluid in a miniature plate heat exchanger (PHE) with modulated surface both experimentally and numerically. They found that the considered nanofluids (CuO–water) can be a promising solution towards designing efficient heat exchanging systems, particularly when the entire volume of the equipment is the main issue. Nanofluids are also significant for the production of nanostructured materials (sizes below 100 nm) for the engineering of complex fluids, and also for cleaning oil from surfaces owing to their excellent wetting and spreading behavior. 2
ACCEPTED MANUSCRIPT Many of the publications on nanofluids about to understand their behavior so that they can be utilized where straight heat transfer enhancement is paramount as in many industrial applications, transportation,
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electronics, nuclear reactors as well as biomedicine and food. These fluids enhance thermal conductivity of the base fluid enormously. Due to the tiny size of nanoelements these fluids are very stable and have no
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additional problems, such as erosion, additional pressure drop, sedimentation and non-Newtonian
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behavior. The low volume fraction of nanoelements is required for conductivity enhancement. Enhancement of heat transfer performance in many industrial fields such as power, manufacturing and
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transportation, is an essential topic from an energy saving perspective. The low thermal conductivity of
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conventional heat transfer fluids such as water and oils is a primary limitation in enhancing the performance and the compactness of such systems.
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Khanafer et al. [15] was first to conduct a numerical investigation on the heat transfer enhancement by
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adding nano-particles in a differentially heated enclosure. They established that the suspended nanoparticles substantially increase the heat transfer rate at any given Grashof number. Sheikholeslami et
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al. [16] investigated the heat transfer characteristics in squeezed flow by using the Adomian Decomposition Method (ADM). They used a Maxwell-Garnett (MG) model and a Brinkman model respectively. They found that for the case in which two plates are moving together, the Nusselt number increases with the increase of nanoparticle volume fraction ( ) and Eckert number (Ec), while it decreases with the growth of the squeeze number (S). It is assumed that the plate moves in the same or opposite direction to the free stream. Here Cu nanofluids are taken to investigate the effect of the volume fraction parameter of the nanofluid with the Prandtl number Pr 6.2 on the flow and heat transfer characteristics ( ). Physical interpretation to several embedding parameters is assigned through graphs for temperature ( ) and tables for skin friction coefficient f (1) and local Nusselt number (1) . Sheikholeslami et al. [17] examined the difficulty of free convection between a circular enclosure and a sinusoidal cylinder. They concluded that isotherms, streamlines, the quantity, size and formation of the
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ACCEPTED MANUSCRIPT cells inside the enclosure depend strongly upon the Rayleigh number and the values of amplitude. Free convection of ferrofluid in a cavity heated from below within the presence of magnetic field used to be
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studied by Sheikholeslami et al. [18, 19]. They determined that particles with a smaller size have higher capacity to dissipate heat, and a larger volume fraction would deliver a driving force which results in
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increase in temperature profile. Lattice Boltzmann method was used to examine magnetohydrodynamic
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flow employing Cu-water nanofluid in a concentric annulus by Sheikholeslami et al. [20, 21].
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The literature survey reveals that no investigation regarding the combined effect of heat and mass transfer in the squeezing flow between parallel plates has been presented. Thus it seems a worthwhile attempt to
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compute a numerical solution for such a problem by using Chebyshev wavelet method. Graphical results representing the noticeable features of various physical parameters are sketched and are discussed in
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details. In this study, the Chebyshev wavelet method is applied to find the numerical solutions of
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nonlinear differential equations governing the problem of unsteady squeezing nanofluid flow and heat transfer. The effects of the squeeze number (S), the nanofluid volume fraction ( ), Prandtl number Pr
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and Eckert number (Ec) on Nusselt number (1) and skin fraction coefficient f (1) are investigated. Macroscopic models for nanofluid flow and heat transfer can be classified as single-phase and two-phase models [22]. Single-phase models consider nanoparticles and base fluid as a single homogeneous fluid with respect to its effective properties. Two-phase models control momentum, continuity and energy equations for particles and base fluid utilizing three distinct approaches such as Eulerian-Mixture model (EMM), Eulerian-Eulerian model (EEM) and thermal dispersion effect. However, the two-phase modeling results exhibit higher heat transfer enhancement in comparison to the homogeneous singlephase model. Also, the heat transfer enhancement increases with increase in Reynolds number and nanoparticle volume concentration as well as with decrease in the nanoparticle diameter.
There are numerous numerical and experimental studies related to nanofluid heat and fluid flow on the macro and microscale. From the numerical aspect, most of the studies have been completed making use of 4
ACCEPTED MANUSCRIPT the homogeneous (single-phase) modeling for the nanofluid. In this process, the nanofluid is considered a homogeneous mixture of nanoparticles and the base liquid. Single-phase models with and without
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Eulerian-Eulerian, thermal dispersion effect and Eulerian-Mixture two-phase models are estimated by comparing expected convective heat transfer coefficients and friction factors with experimental results
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from literature. Dispersion model that uses velocity gradient to define dispersion conductivity is located
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to be more effective at entry region in comparison with other single-phase models. However, two-phase models forecast convective heat transfer coefficient and friction factor more accurately at the entry
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region.
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Although two-phase models provide a better understanding of both phases, single-phase models are computationally more efficient, however produce less detail about each phase. In despite of the single-
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phase modeling, in the two-phase modeling, the nanoparticle and the base fluid are considered as two
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different phases with different velocities and temperatures. In this method, the interfaces between the phases are taken into account in the governing equations. There are a few studies that used two-phase
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approach to study nanofluids [23]. The heat transfer enhancement results for two-phase modeling exhibit higher magnitudes in comparison to the single-phase modeling results. This paper has been organized as follows: in section 2, the problem is formulated and a coupled system of nonlinear ordinary differential equations for unsteady squeezing nanofluid flow and heat transfer is derived. The mathematical preliminaries of Chebyshev wavelet is presented in section 3. The approximation of function using Chebyshev wavelet is presented in section 4. In section 5, the Chebyshev wavelet method is applied to solve the resulting governing system of nonlinear equations. The numerical results and discussions are discussed in section 6 and section 7 concludes the paper.
2.
Mathematical Formulation:
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ACCEPTED MANUSCRIPT The unsteady flow and heat transfer in two-dimensional squeezing nanofluid between two infinite parallel plates is considered in this paper (Fig. 1). The hypothesis of the problem can be found in more details in
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Ref. [16, 24]. The governing equations are as follows:
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u v 0 x y
2u 2u u p u u u v nf 2 2 x y x y t x
nf 2T 2T x 2 y 2 C p
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k nf T T T u v t x y Cp
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2u 2u u p u u u v nf 2 2 x y y y t x
nf
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nf
nf
nf
u 2 u u 2 4 x y x
(1)
(2)
(3)
(4)
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where u and v are the velocities in x and y directions respectively. The effective density nf , the effective
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dynamic viscosity nf , the effective heat capacity C p
nf
and the effective thermal conductivity k nf of
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the nanofluid are defined below in eq. (5) to eq. (8):
C 1 C p nf
Cp
p f
(5)
s
nf 1 f s
nf K nf Kf
(6)
f
1 2.5
2 K
K
K s 2 K f 2 K f K s K s 2K f
f
(Brinkman)
(7)
(Maxwell-Garnett)
(8)
s
subject to the following boundary conditions: v vw
v
dh , dt
u T 0 y y
T TH
at y h(t ),
at y 0.
(9) 6
ACCEPTED MANUSCRIPT with the following parameters:
l (1 t )
v
0.5
,
l f ( ) , 2(1 t )
u
x f ( ), 2(1 t )
T , TH
A1 (1 )
s f
2.5
f 3 f
f f f f 0,
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From eqs. (10), (3) and (4) we have
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Substituting eq. (10) in eqs. (2) and (3), we have f iv SA1 1
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y
A2 Pr Ec f f 2 4 2 f 2 0 2.5 A3 (1 ) A3
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Pr S
p s
p f
In eq. (12), S
K nf kf
,
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A3
,
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C C
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where A2 and A3 are dimensionless constants given by A2 (1 )
l is the squeeze number, 2v f
Pr
Ec
f C p f f Kf f
C
p f
is the Prandtl number,
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x is the Eckert number. 2(1 t )
The boundary conditions are given by f (0) 0, f (0) 0, f (1) 1, f (1) 0,
(0) 0, (1) 1, Skin friction coefficient and Nusselt number are the physical quantities defined by 7
(10)
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(11)
(12)
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3.
u y y h ( t ) 2 nf vw
Nu
,
lK nf
T y y h ( t )
KTH
Chebyshev wavelets
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Cf
nf
useful tools in the numerical simulations of physical problems.
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The Chebyshev wavelets are constructed from their corresponding polynomials. These wavelets are
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The second kind Chebyshev wavelets n,m t k , n, m, t have four arguments; defined on interval
n 1 n t k 1 , k 1 2 2 elsewhere
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k/2 2 U m 2k t 2n 1 ,
n, m t
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[0, 1) by [25]
0,
(13)
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where n 1,2,3,...,2 k 1 , k is assumed to be any positive integer, m is the degree of the second kind
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U m (t ) , U m (t ), m 0, 1, 2, . . . , M are the second kind Chebyshev polynomials of
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Here U m (t )
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Chebyshev polynomials and t is the normalized time.
degree m defined on the interval [-1, 1] and satisfy the following formulae U 0 t 1 ,
U1 t 2t ,
U m1 t 2tU m t U m1 t , m 1,2,3,...
4.
Function approximation
A function f (t ) defined over [0,1) may be expanded in terms of Chebyshev wavelets as [26] f t
c
n, m n, m
t
(14)
n 1 m 0
If the infinite series in eq. (14) is truncated, then eq. (14) can be written as 8
ACCEPTED MANUSCRIPT f t
2 k 1 M 1
c
n, m n, m
t C T t
(15)
n 1 m 0
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where (t ) is 2 k 1 M 1 matrix, given by
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t 1,0 t , 1,1 t ,..., 1, M 1 t , 2,0 t ,..., 2, M 1 t ,..., 2k 1 ,0 t ,..., 2k 1 , M 1 t T
c n, m
f t n,m t dt
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1
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Also, C is a 2 k 1 M 1 matrix whose elements can be calculated from the formula
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Application of Chebyshev wavelet method for solving squeezing unsteady nanofluid flow
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5.
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with n 1,...,2k 1, m 0,...,M 1.
f
iv
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Consider the general unsteady fluid flow eqs. (11) and (12) of the form [24] S f 3 f f f f f 0,
Pr S f Pr Ec f 2 4 2 f 2 0
(16) (17)
with following boundary conditions
f 0 0,
f 0 0,
f 1 1,
f 1 0,
0 0,
1 1,
(18)
The Chebyshev wavelet solutions of f and are sought by assuming that f iv and can be expanded in terms of Chebyshev wavelets as
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ACCEPTED MANUSCRIPT 2 k 1 M 1
f
iv
a n,m n,m
(19)
n 1 m 0
2 k 1 M 1
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bn,m n,m
(20)
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n 1 m 0
n 1 m 0
0
n
a n,m
n 1 m 0
f
a n,m
0 0 0
a n,m
n 1 m 0
(22)
2
2 2
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n,m d d d C1
2k 1 M 1
1
n 1 m 0
f
n,m
0 0
2k 1 M 1
d d C C
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2k 1 M 1
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f
(21)
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a n,m n,m d C1
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f
2k 1 M 1
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Integrating eq. (19) four times with respect to from 0 to , the following equations are obtained
n,m d d dd C1 0 0 0 0
C 2 C3
3 6
C2
2 2
(23)
C3 C 4
(24)
Similarly integrating eq. (20) twice with respect to from 0 to , the following equations are obtained 2k 1 M 1
bn,m n,m d C5 n 1 m 0
(25)
0
2k 1 M 1
n 1 m 0
0 0
bn,m
d d C C n,m
5
(26)
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By using the boundary conditions given in eq. (18), we can get the values of C1 , C 2 , …, C 6 as follows 2 k 1 M 1 2 k 1 M 1 C1 3 an, m n, m 1 d d dd an, m n, m 1 d d d 1 , n 1 m 0 0 0 0 0 0 0 0 n 1 m 0
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ACCEPTED MANUSCRIPT C2 0 ,
2 k 1 M 1
2 n 1 m 0
0 0 0
1 n,m 1 d d dd an,m n,m 1 d d d , 0 0 0 0
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C4 0 ,
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3 3 2 M 1 an, m 2 2 n 1 m 0 k 1
C3
2 k 1 M 1
and C6 1
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C5 0 ,
bn, m
n 1 m 0
1 d d. n, m
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0 0
2 k 1 M 1
n 1 m 0
2 k 1 M 1
an, m
n 1 m 0
2
k 1
M 1
n 1 m 0
n 1 m 0
1 d d dd n, m
0 0 0 0
a n,m
k 1 3 3 2 M 1 an, m n, m 1 d d dd 6 2 2 n 1 m 0 0 0 0 0
3
n, m 1 d d d 1 0 0 0
2
0 0 0 0
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2 k 1 M 1
n,m d d dd 3 an,m
an, m
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f
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Substituting the above values of C1 ,…, C4 in eq. (24), we have
n,m 1 d d d
(27)
0 0 0
Similarly, putting the values of C5 and C6 in eq. (26), we can obtain 2 k 1 M 1
bn, m n 1 m 0
0 0
Now substituting the collocation points l
2 k 1 M 1
n,m d d 1 bn,m
n 1 m 0
1 d d n, m
0 0
(28)
l 0.5 for l 1, 2, .. . ,2 k 1 M in eqs. (27) and (28), we have 2k 1 M
k 1 2 2 k 1 M equations in 2 2 k 1 M unknowns in a n ,m and bn , m , n 1,...,2 , m 0,...,M 1. By solving
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ACCEPTED MANUSCRIPT this system of equations using mathematical software, the Chebyshev wavelet coefficients a n,m and bn,m can be obtained. Numerical Results and Discussion
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6.
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In this study, Chebyshev wavelet method (CWM) is used to solve the problem of unsteady squeezing
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nanofluid flow. The effects of dynamic parameters such as the squeeze number, Prandtl number, nanofluid volume fraction, and Eckert number on flow and heat transfer characteristics are investigated.
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The present numerical method is validated by comparing the obtained results with those of other methods
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available in literature. Comparison between the results obtained by the Chebyshev wavelet method, fourth order Runge–Kutta method, Homotopy analysis method (HAM) and Adomian Decomposition Method (ADM) for different values of active parameters is shown in Tables 1-3 and exhibited in Figs. 2-7. All
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present problem.
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these comparisons demonstrate that Chebyshev wavelet method offers a highly accurate solution for the
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The comparison of the absolute errors for velocity f ( ) and temperature ( ) have been presented in Table 1 which is constructed using the results obtained by Chebyshev wavelet method and fourth order Runge–Kutta method at different values of when S 1, Pr 6.2, Ec 0.01, 0.02 and 0.01 respectively. Similarly, Table 2 shows the comparison of approximate solutions obtained by Chebyshev wavelet method, homotopy analysis method (HAM) and Adomian decomposition method (ADM) for
(1) at different values of Pr and Ec, when S 0.5, and 0.1. Agreement between present numerical results and solutions obtained by ADM and HAM appear very satisfactory through illustrations in Table 2. Again Table 3 shows the comparison of approximate solutions of Chebyshev wavelet method and homotopy analysis method (HAM) for f (1) and (1) at different values of S, when Ec Pr 1 and 0.1. In the present analysis, to examine the accuracy and reliability of the
Chebyshev wavelets, we compare the approximate solutions of Chebyshev wavelet with the solutions obtained by HAM and ADM as given in Refs. [8] and [24] respectively. 12
ACCEPTED MANUSCRIPT To show the influence of embedding physical parameters on the velocity and temperature, Figs. 2-7 have been proposed. Figure 3 shows the combined effects of positive and negative squeeze number (S) on the
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temperature θ. For fixed values of parameters, ( Pr Ec 1.0 and 0.1 ) the temperature θ decreases from 0 to 1. Considerable reduction in the temperature field is observed for large values of S. It is
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quite obvious that the temperature is relatively high when the plates are moving towards each other. The
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effects of Prandtl number and Eckert number on the temperature θ are depicted in Figs. 3 and 4. The small values of Pr (< 1) characterize liquid materials, which have high thermal diffusivity but low
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viscosity. The existence of viscous dissipation effects significantly increases the temperature θ. The
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thermal boundary layer thickness is found to decrease upon increasing Pr and Ec. It is apparent that an increase in the values of Pr largely decreases the thermal diffusivity which therefore decays the thermal
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boundary layer thickness.
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The positive and negative squeeze numbers have different effects on the velocity profile. Fig. 5 demonstrates the effect of the squeeze number on the temperature profile taking Pr 6.2, Ec 0.5,
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0.06 and 0.1. It is important to remind that the squeeze number (S) describes the movement of the plates ( S 0 corresponds to the plates moving apart, while S 0 corresponds to the plates moving together). Fig. 6 shows that the volume fraction has a significant effect on the temperature profile. Fig. 7 demonstrates the effect of Eckert number, squeeze number and volume fraction on temperature.
7.
Conclusion
In this paper, the squeezing unsteady nanofluid flow equations have been solved by using Chebyshev wavelet method. The results thus obtained are compared with fourth order Runge–Kutta method, Adomian decomposition method (ADM) and also with homotopy analysis method (HAM). The obtained results demonstrate the accuracy and efficiency of the proposed method based on Chebyshev wavelet method and its applicability to coupled system of nonlinear ordinary differential equations. Agreement between present numerical results obtained by Chebyshev wavelet method and those of ADM and HAM appears very satisfactory through illustrative results in Tables 1-3. The effects of the squeeze number, 13
ACCEPTED MANUSCRIPT Prandtl number, nanofluid volume fraction and Eckert number on Nusselt number and skin friction coefficient are studied. The results show that when the two plates move toward together; the Nusselt
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number has a direct relationship with nanoparticle volume fraction and Eckert number while it has a reverse relationship with the squeeze number. Large values of Pr and Ec increase the temperature field
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rapidly. The magnitude of local Nusselt number is an increasing function of Pr and Ec. The application of
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the proposed method for the solutions of unsteady squeezing nanofluid flow equation satisfactorily
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justifies its simplicity, efficiency and applicability.
Acknowledgements
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This research work was financially supported by DST, Government of India under Grant No. SR/S4/MS.:722/11. Again the authors would like convey their sincere thanks to learned anonymous
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reviewers for their useful comments for the improvement and betterment of the manuscript.
M.J. Stefan, Versuch über die scheinbare adhesion sitzungsber Sächs Akad Wiss Wein, Math-Nat
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[1]
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Table 1
Comparison of absolute errors obtained by Chebyshev wavelet method and numerical
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method given in ref. [24] for f ( ) , ( ) at various points of when S 1, Pr 6.2, Ec 0.01, 0.02
0 0.14135866 0.28066605 0.41578075 0.54437882 0.66385692 0.77122923 0.86301562 0.93511971 0.98269524 1
2.896167E-3 2.909093E-3 2.917100E-3 2.901735E-3 2.845957E-3 2.721995E-3 2.498219E-3 2.145984E-3 1.635576E-3 9.271310E-4 0
Table 2
1.03496254 1.03497316 1.03494912 1.03479437 1.03435407 1.03338623 1.03153805 1.02829803 1.02289474 1.01411321 1
MA
0 0.14229173 0.28242726 0.41817098 0.54712047 0.66662248 0.77368163 0.86486385 0.93619281 0.98303883 1
D
1.03206637 1.03206407 1.03203202 1.03189263 1.03150811 1.03066423 1.02903983 1.02615205 1.02125916 1.01318608 1
NU
NM ( ) CWM ( )
TE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fourth order Runge-Kutta Chebyshev wavelet Absolute Errors method [16] Method (Present method) f NM ( ) f CWM ( ) f ( ) ( ) f ( ) ( ) 0 9.330730E-4 1.761213E-3 2.390230E-3 2.741652E-3 2.765553E-3 2.452402E-3 1.848234E-3 1.073101E-3 3.435894E-4 0
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and 0.01 taking k 1 and M 6.
Comparison of approximate solutions obtained by Chebyshev wavelet method, homotopy
analysis method (HAM) and Adomian decomposition method (ADM) for (1) at different values of Pr and Ec when S 0.5 and 0.1 taking k 2, M 8 and k 1, M 4. Pr
Ec
0.5 1 2 5 1 1 1 1
1 1 1 1 0.5 1.2 2 5
Table 3
Mustafa et al. [8] Sheikholeslami et al. Present method (CWM) (HAM) [16] (ADM) M 8, k 2 M 4, k 1 1.522368 3.026324 5.98053 14.43941 1.513162 3.631588 6.052647 15.13162
1.52236749518 3.02632355855 5.98053039715 14.4394132325 1.51316180648 3.63158826816 6.05264710721 15.1316178324
1.52236745096 3.02632345997 5.98052980356 14.4394056608 1.51316172998 3.63158815197 6.05264691995 15.1316172998
1.51530555134 3.01037588947 5.94157660731 14.2932191862 1.50518794473 3.61245106737 6.02075177895 15.0518794473
Comparison of approximate solutions obtained by Chebyshev wavelet method and
homotopy analysis method (HAM) for f (1) and (1) at different values of S when Ec Pr 1 and
0.1. 17
ACCEPTED MANUSCRIPT S
Mustafa et al. (HAM) [8] f (1) (1)
Present method (CWM) f (1) (1)
1.0 0.5 0.01 0.5 2.0
2.170090 2.614038 3.007134 3.336449 4.167389
2.17009087 2.61740384 3.00713375 3.33644946 4.16738921
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3.31989925 3.12949109 3.04709193 3.02632345 3.11854353
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3.319899 3.129491 3.047092 3.026324 3.118551
Fig. 2
Influence of S on
Fig. 1
Geometry of problem.
for Pr Ec 1.0 and 0.1.
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Influence of Pr on for S Ec 1.0 and 0.1.
Fig. 4
Influence of Ec on for S Pr 1.0 and 0.1.
Fig. 5
Influence of S on for Pr 6.2, Ec 0.5, 0.06 and 0.1.
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Fig. 3
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Influence of on for Pr 6.2, Ec 0.5, S 1 and 0.1.
Fig. 7
Influence of different values of Ec, S and on for Pr 6.2 and 0.1.
Fig. 8
Comparison of approximate solutions obtained by Chebyshev wavelet method taking k 2, M 8 and
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Fig. 6
k 1, M 4.
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ACCEPTED MANUSCRIPT
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Squeezing unsteady nanofluid flow between two parallel plates
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Analytical Methods (ADM, HAM)
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Numerical Results
Graphical abstract
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Numerical Approach using CWM
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Chebyshev wavelet method is applied. Compared the solutions with ADM, fourth order RK method and homotopy analysis method (HAM). To the best information of the authors, this model has not been solved ever before.
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Highlights
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