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Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates
Accepted Manuscript
Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates Muhammad Danial Hisham , Abdul Rauf , Dumitru Vieru , Aziz Ullah Awan PII: DOI: Reference:
S0577-9073(18)30854-2 https://doi.org/10.1016/j.cjph.2018.10.009 CJPH 666
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
20 June 2018 8 September 2018 17 October 2018
Please cite this article as: Muhammad Danial Hisham , Abdul Rauf , Dumitru Vieru , Aziz Ullah Awan , Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.10.009
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ACCEPTED MANUSCRIPT Highlights
Unsteady flows of two immiscible Maxwell fluids are studied. New analytical solutions for velocity and shear stress are determined. Semi-analytical solutions are obtained using Talbot algorithm. The developed method can be generalized for n-Maxwell fluids.
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Corresponding author: [email protected]
ACCEPTED MANUSCRIPT Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates Muhammad Danial Hisham1, Abdul Rauf2, Dumitru Vieru3, Aziz Ullah Awan1 1 University of the Punjab, Lahore, Pakistan 2 Air University Multan, Pakistan 3 Technical University of Iasi, Romania
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Abstract. Unsteady flows of two immiscible Maxwell fluids in a rectangular channel bounded by two moving parallel plates are studied. The fluid motion is generated by a time-dependent pressure gradient and by the translational motions of the channel walls in their planes. Analytical solutions for velocity and shear stress fields have been obtained by using the Laplace transform coupled with the finite sine-Fourier transform. These analytical solutions are new in the literature and the method developed in this paper can be generalized to unsteady flows of n-layers of immiscible fluids. By using the Laplace transform and classical method for ordinary differential equations, the second form of the Laplace transforms of velocity and shear stress are determined. For the numerical Laplace inversion, two accuracy numerical algorithms, namely the Talbot algorithm and the improved Talbot algorithm are used.
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1. Introduction
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Key words. Immiscible Maxwell fluids; analytical solutions; Talbot algorithm; integral transforms.
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One of the states of matter is called phase and it can be a gas, a liquid or a solid. Flows of several phases are known as multiphase flows. Two-phase flows are the simplest multiphase flows and can be solid-liquid flow, gas-solid flow, gas-liquid flow and liquid-liquid flow. Flows of immiscible liquids have important practical applications. For example, the flows of oil and water are important in oil recovery processes or in reducing of resistance to flow in pipelines. Dispersive flows, liquid extraction processes and co-extrusion flows in polymer processing industry are other applications of the immiscible liquidliquid flows. Interesting problems of multiphase flows are reported by Bird et al. [1]. The pressure-driven motion of two immiscible fluids in an inclined channel has been investigated by Redapangu et al. [2] under consideration of the viscosity and density gradients. The authors used the multiphase lattice Boltzmann approach. They found that the increasing of viscosity ration between fluids the displacement rate decreases. Hasnain and Alba [3] studied the buoyant displacement flows of two immiscible viscous fluids in an inclined twodimensional channel. The regularized lattice Boltzmann model for immiscible two-phase flows of power-law fluids have been studied by Ba et al. [4]. The closed forms solutions for the magneto-hydrodynamic Couette steady flows with heat transfer of two-immiscible fluids through a horizontal channel with isothermal boundaries under the influence of electric and magnetic fields have
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been obtained by Nikodijevic et al. [5] and Zivojin et al. [6]. Satpathi et al. [7] have investigated the unsteady laminar flow of two immiscible fluids in a channel due to a time-dependent pressure gradient by considering a Maxwell fluid and a Newtonian fluid. Using the Laplace transform, they obtained approximate solutions for fluid velocities. Also, a numerical scheme is developed in order to obtain numerical solutions. Raju and Rao [8] have studied the unsteady two-layered fluid flows through a rectangular channel with porous walls in a rotating system. The flow is generated by a constant pressure gradient and an oscillating channel wall. Heat transfer in the peristaltic transport of an incompressible magneto-hydrodynamic second grade fluid through a porous medium in an inclined asymmetric channel has been analyzed by Ramesh and Devakar [9]. Using the regular perturbation method, authors obtained solutions for the stream function, temperature and pressure gradient. Shen et al. [10] studied the unsteady natural convection flow and heat transfer of fractional Maxwell viscoelastic nanofluid in magnetic field over a vertical plate by considering the time-fractional generalized Cattaneo heat flux. Unsteady flow of two immiscible Newtonian fluids through a porous medium between two parallel plates has been investigated by Agarwal and Kumar [11]. One of the plates is moving with constant velocity and, after the fluids reach the stationary state, the plate is suddenly stopped. Roch-Carrier et al. [12] studied the flows of three immiscible fluids in a microchannel consisting of two parallel plates under the influence of an electric field. Interesting topics of fluids flows are analyzed in the references [13-21]. Unsteady electro-osmotic flows of second grade and Maxwell fluids with fractional constitutive equations have been investigated in the references [2224]. In the present paper, unsteady flows of two immiscible Maxwell fluids between two moving parallel plates are studied. The fluid motion is influenced by a time-dependent pressure gradient applied in the flow direction. The wall motions are translational motions with time-dependent velocities in their planes. Analytical solutions for fluid velocity and shear stress have been obtained by employing the Laplace transform coupled with the finite sineFourier transform. The obtained analytical solutions are new in the literature and the method developed in this paper can be generalized to unsteady flows of n-layers of immiscible Maxwell fluids. Using the Laplace transform and classical method for ordinary differential equations, second forms of the Laplace transforms of velocity and shear stress are determined and the semianalytical solutions are obtained by using two accuracy numerical algorithms, namely the Talbot algorithm and the improved Talbot algorithm for the numerical inversion of the Laplace transforms. A numerical comparison between analytical and semi-analytical solutions is carried out. Also, the influence of the system parameters on the fluids flow has been numerically and graphically analyzed. 2. Mathematical formulation of the problem. The flow domain is D {( x, y, z) (, ) [0, h2 ] (, )} ,with the channel walls situated in planes y 0 and y h2 0 . The channel flat wall
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ACCEPTED MANUSCRIPT situated in the plane y 0 is moving along the x -axis with the velocity U 0 f1 (t ) , while the plane wall y h2 is moving along the x -axis with the u10 U 0 g1 (t ) (Fig.1). Functions f1 (t ), g1 (t ) are differentiable functions velocity u20 of exponential order at infinity and f1 (0) g1 (0) 0 . In this paper the velocity field is considered to be of the form V u( y, t ),0,0 . In the region
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y [0, h1 ], h1 h2 flows a Maxwell fluid with the density 1 ,viscosity 1 , relaxation time 1 1 / G1 , G1 the elastic modulus, velocity u1( y, t ) and the shear stress 1 ( y, t ) . In the region y [h1 , h2 ], flows a Maxwell fluid with the density 2 ,viscosity 2 , relaxation time 2 2 / G2 , velocity u2 ( y, t ) and the
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shear stress 2 ( y, t ) .
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Fig. 1. Flow geometry
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The fluids are assumed to be incompressible and immiscible and the flow is unsteady, one-dimensional and fully developed. Motions of the fluids are generated by the movements of the two walls of the channel and by the timedependent pressure gradient in the flow direction. Based on the assumptions about velocity fields, the continuity equation is identically satisfied and the governing equations of motion are [5, 7, 11]
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For the Maxwell fluid in region y [0, h1 ], h1 h2
1
u1 p 1 , t x y
1 1
2
(1)
1 u 1 1 , t y For the Maxwell fluid in region y [h1 , h2 ],
u2 p 2 , t x y
(2)
(3)
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2 2
2 u 2 2 . t y
(4)
Along with the above equations, we consider the following initial, boundary and the interface fluid-fluid conditions: (5)
u1(0, t ) U 0 f1 (t ), u2 (h2 , t ) U 0 g1 (t ),
(6)
u1(h1 , t ) u2 (h1 , t ), 1 (h1 , t ) 2 (h1 , t ) .
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uk ( y,0) 0, k ( y,0) 0, k 1, 2 ,
(7)
Introducing the non-dimensional variables
u h t h p x y , y , t 12 , uk k , k 2 k , k k , k 1, 2, p 2 , h2 h2 h2 U0 1U 0 k 1U 0 2 h t h 2t h 1 1 2 1 , 2 2 2 1 , a0 2 , b0 2 , f(t)= f1 2 , g(t)= g1 2 , h 1 , h2 h2 1 1 h2 1 1
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x
(8)
into Eqs. (1)-(6), we obtain the dimensionless governing equations
u2 p 2, t x y
(10)
2 u b0 2 , t y
(12)
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2 2
(9)
(11)
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a0
1 u1 , t y
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1 1
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u1 p 1, t x y
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with the following non-dimensional conditions:
uk ( y,0) 0, k ( y,0) 0, k 1, 2 ,
(13)
u1 (0, t ) f (t ), u2 (1, t ) g (t ),
(14)
u1 (h, t ) u2 (h, t ), 1 (h, t ) 2 (h, t ) .
(15)
3. Analytical solution In order to determine analytical solutions of the Eqs. (9)-(12) with conditions (13)-(15) we use the finite sine-Fourier transform with respect to
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p (t ) 1.5 0.5cos(5t 3 / 4) cos(5t / 3) x
(16)
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The diagram of the function (t ) is given in figure 2. This function is a periodic function with the period t0 6 / 5 .
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Fig. 2. Diagram of the pressure gradient
p (t ) x
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Applying the Laplace transform to Eqs. (9)-(12) and using the initial conditions (13), we obtain the problem in transformed domain
1 ( y, s) , y
(17)
1 u1 ( y, s) , 1 1s y
(18)
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su1 ( y, s) ( s)
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1 ( y, s)
a0 su2 ( y, s) ( s)
2 ( y, s) , y
(19)
b0 u2 ( y, s) , 1 2 s y with the boundary and interface transformed conditions 2 ( y, s)
(20)
u1 (0,s) f (s), u2 (1,s) g (s),
(21)
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(22)
In the above relation ( y, s) ( y, t ) exp( st ) dt denotes the Laplace 0
transform of function ( y, t ) [25]. The finite sine-Fourier transform of the function ( y, s), y [a,b] is defined as [26]
a
n( y a) , b a, n 1, 2,... , ba
and the inverse Fourier transform is given by
2 (y, s) sin(n y)n (s) . b a n1
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b
n ( s) ( y, s)sin( n y) dy, n
(23)
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(24)
Eliminating 1 between Eqs. (17) and (18), respectively 2 between (19) and (20), we obtain equations for transformed velocities
2u1 ( y, s) , y [0, h] , y 2
(25)
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s(1 1s)u1 ( y, s) (1 1s)( s)
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2 u2 ( y , s ) (26) a0 s(1 2 s)u2 ( y, s) (1 2 s) ( s) b0 , y [h,1] . y 2 Applying the finite sine-Fourier transform to Eqs. (25), (26) and using conditions (21) and (22), we obtain the following expressions of the Fourier transform of fluid transformed velocities: (1) n n n f ( s) 1 1s 1 (1) n ( s ) u (h, s) , 2 2 1 n s(1 1 s) n s(1 1 s) n s(1 1 s) 2n n n , n 1, 2,..., h
(27)
b0nu2 (h, s) (1) n b0n g ( s) 1 (1) n (1 2 s) ( s) , n a0 s(1 2 s) b0n2 a0 s (1 2 s ) b0n2 a0 s (1 2 s ) b0n2 n n , n 1, 2,.... 1 h
(28)
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u1n ( s)
u2 n ( s )
Function (27) can be written in the suitable form s(1 1s) 1 (1) n 1 u1n ( s ) f (s) f ( s ) u1 (h, s ) n n [ s(1 1 s) n2 ] n (1) n s(1 1 s) 1 1s 1 (1) n u ( h , s ) ( s ), 1 2 n [ s(1 1 s) n ] n s(1 1 s) n2
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(29)
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and, by using the pairs of functions
h y 1 , y [0, h], 1n , h n y (1) n 1 2 ( y ) , y [0, h], 2 n , h n and applying the inverse Fourier transform to Eq. (29) we obtain 1 ( y )
s(1 1s) sin( n y ) h y y 2 f ( s) u1 (h, s) f ( s ) 2 h h h n 1 n [ s (1 1 s) n ] (1) n s(1 1 s)sin( n y) 2 2 1 (1) n (1 1s) sin( n y) u1 (h, s) ( s ) . h n [ s (1 1 s) n2 ] h n s(1 1 s) n2 n 1 n 1
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u1 (y, s)
Now, from (18) and (31), we obtain
(31)
1 u1 ( y, s) T11 (y, s )u1 (h, s ) T12 (y, s ) f ( s ) T13 ( y, s) ( s), 1 1s y
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1 (y, s )
(30)
T11 ( y, s )
(1) n s(1 1 s) cos( n y ) 1 1 2 , h(1 1 s) s(1 1 s) 2n n 1
T12 ( y, s )
s(1 1 s) cos( n y ) 1 1 2 , h(1 1 s) s(1 1 s) 2n n 1
(32)
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2 [1 (1) n ]cos( n y ) T13 (y, s ) . h n 1 s (1 1 s) 2n
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A similar manner leads to
(1) n a0 s(1 2 s) (1) n 1 1 g (s) g ( s ) u2 (h, s) 2 n n [a 0 s(1 2 s) b0n ] n n a0 s(1 2 s) 1 2 s 1 (1) u2 (h, s) ( s), 2 n [a 0 s(1 2 s) b0n ] n a 0 s(1 2 s) b0n2
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u2 n ( s )
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By using the pairs function-sine Fourier transform 1 y 1 3 ( y ) , y [h,1], 3n , 1 h n yh (1) n 1 4 ( y ) , y [h,1], 4 n , 1 h n and applying the inverse Fourier transform to Eq. (33), we obtain
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(33)
(34)
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u2 (y, s)
2u (h, s) a0 s (1 2 s) sin(n ( y - h)) 1 y yh u2 (h, s) g ( s) 2 1 h 1 h 1 h n 1 n [a0 s(1 2 s) b0n2 ]
(35) 2 g(s) (1) n a0 s(1 2 s) sin(n ( y - h)) 2( s) [1 ( 1) n ](1 2 s) sin(n ( y - h)) [a s(1 s) b 2 ] 1 h 1 h n 1 n [a0 s(1 2 s) b02n ] n 1 n 0 2 0 n
From (20) and (28), we have
T21 ( y, s)
u2 ( y, s ) T21 ( y, s)u2 (h, s) T22 ( y, s) g(s)+T23 ( y, s) ( s) y
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2 (y, s ) b0
b0 a0 s(1 2 s) cos(n ( y - h)) 1 2 , (1 h)(1 2 s) a0 s(1 2 s) b02n n 1
T23 ( y, s)
(36)
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b0 (1) n a0 s(1 2 s) cos(n ( y - h)) T22 ( y, s) 1 2 , (1 h)(1 2 s) a0 s(1 2 s ) b02n n 1
2b0 [1 (1) n ]cos(n ( y - h)) . 1 h n 1 a0 s (1 2 s ) b02n
u1 (h, s ) u2 (h, s )
A1 ( s ) , A0 ( s )
(37)
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A0 ( s ) T11 (h, s) T21 (h, s),
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Using the interface liquid-liquid conditions (22), we obtain the following expression of the velocity on the interface y h
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A1 ( s ) T12 (h, s ) f(s)+T22 (h, s ) g(s)+ T13 (h, s ) T23 (h, s ) ( s ).
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In order to obtain the inverse Laplace transforms of the functions u1 ( y,s), u2 ( y,s), u1 (h,s) u2 (h,s) we recall the following inverse Laplace transforms:
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1 L1 21 (t ; m, n), s (1 ms) n ms L1 2 (t ; m, n) 1 (t ; m, n), s (1 ms) n 1 ms L1 1 (t ; m, n) 2 (t ; m, n), s (1 ms) n s (1 ms ) L1 (t ) 2n1 (t ; m, n), s (1 ms) n
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(38)
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t exp 2m sinh t 1 4mn , 1 (t ; m, n) 2m 1 4mn
1 lim 1 (t; m, n) lim 2 (t; m, n) exp( nt) . m 0 m 0 2
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t 1 4mn t 2 (t ; m, n) exp . cosh 2m 2m Functions given by Eq. (39) satisfy the property
(39)
(40)
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Using Eqs. (31), (35) and (38), we obtain for velocities u1 ( y, t ), u2 ( y, t ) the following expressions: (h y ) f (t ) y u1 ( y, t ) u1 (h, t ) h h
2 (1) n sin( n y) u1 (h, t ) 2 2nu1 (h, t) 1 (t ; 1 , 2n ) h n 1 n
(41)
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2 sin( n y) f (t ) 2 2n f (t ) 1 (t; 1 , 2n ) h n 1 n
( y h) g (t ) 1 y u2 (h, t ) 1 h 1 h
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u2 ( y , t )
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2 [1 (1) n ]sin( n y) (t ) 1 (t ; 1 , 2n ) 2 (t; 1 , 2n ) , h n 1 n
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2b0 2 b 2 (1) n sin(n ( y - h)) nu2 (h, t) 1 (t ; 2 , 0 2n ) u2 (h, t ) 1 h n 1 n a0 a0
(42)
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2b0 2 b 2 (1) n sin(n ( y - h)) n g (t ) 1 (t ; 2 , 0 2n ) g (t ) 1 h n 1 n a0 a0 [1 (1) n ]sin(n ( y - h)) b b 2 (t ) 1 (t ; 2 , 0 2n ) 2 (t; 2 , 0 2n ) , a0 (1 h) n 1 n a0 a0
t
where f1 (t ) f 2 (t ) f1 (t ) f 2 () d is the convolution product of the functions 0
f1 (t ) , f 2 (t ) and u1 (h, t ) u2 (h, t ) is given by
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u1 (h, t ) u2 (h, t )
A1 (t ) A0 (t )
T12 (h, t ) * f (t ) T22 (h, t ) * g (t ) [ T13 (h, t ) T23 (h, t )]* (t ) , T11 (h, t ) T21 (h, t )
(43)
where
t 2 1 2 (t; 1 , 2n ) 1 (t; 1 , 2n ) , exp h1 h 1 1 n 1
T12 (h, t )
t 2 1 exp (1) n 2 (t; 1 , 2n ) 1 (t; 1 , 2n ) , h1 h 1 1 n 1
T13 (h, t )
4 [(1) n 1]1 (t; 1 , 2n ), h n 1
T21 (h, t )
t b0 2b0 exp (1 h) 2 2 (1 h) 2
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T11 (h, t )
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t b0 2b0 T22 (h, t ) exp (1 h) 2 2 (1 h) 2
n 1
2
(t; 2 ,
(1) n 1
n
b0 2 b n ) 1 (t; 2 , 0 2n ) , a0 a0
b0 2 b0 2 2 (t; 2 , n ) 1 (t; 2 , n ) ,(45) a0 a0
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4b0 b [1 (1) n ]1 (t; 2 , 0 2n ). a0 (1 h) n 1 a0
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T23 (h, t )
(44)
4. Semi-analytical solution.
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In this section, we determine a new form of solutions to the problem given by Eqs. (9)-(15) using the Laplace transform coupled with the classical method for the ordinary differential equations. The general solutions of equations (25) and (26) are
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1 u1 ( y, s) C1 ( s) exp y w1 ( s) C2 ( s) exp y w1 ( s) ( s), s
1 u2 ( y, s) D1 ( s) exp y w2 ( s) D2 ( s) exp y w2 ( s) ( s), a0 s where, a w1 ( s) s(1 1s), w2 ( s) 0 s(1 2 s) . b0
(46)
(47)
Now, Eqs. (18), (20) and (46) lead to the following form of the Laplace transforms of the shear stresses i ( y, t ), i 1, 2 :
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1 ( y, s)
1 C1 ( s) w1 ( s) exp( y w1 ( s)) C2 ( s) w1 ( s) exp(y w1 ( s)) , 1 1s
(48)
b0 2 ( y, s) D1 ( s) w2 ( s) exp( y w2 ( s)) D2 ( s) w2 ( s) exp(y w2 ( s)) . 1 2 s Using the boundary conditions (21) and the interface conditions (22), we have
Q4 ( s) R1 ( s ) Q2 ( s ) R2 ( s ) 1 , C2 ( s ) f ( s ) ( s ) C1 ( s ), Q1 ( s)Q4 ( s ) Q2 ( s)Q3 ( s ) s
D1 ( s )
Q1 ( s ) R2 ( s ) Q3 ( s ) R1 ( s ) 1 , D2 ( s) g ( s) ( s) exp w2 ( s) Q1 ( s )Q4 ( s ) Q2 ( s )Q3 ( s ) a0 s
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C1 ( s )
D1 ( s ) exp 2 w2 ( s) , where,
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(49)
Q1 ( s ) 2sinh h w1 ( s) , Q2 ( s) exp (h 2) w2 ( s) exp h w2 ( s) , Q3 ( s )
2 cosh h w1 ( s ) 1 1s
, Q ( s)
a0b0
1 2s
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exp (h 2)
w2 ( s) exp h w2 ( s) ,
( s) (1 a0 ) ( s) ( s ) R1 ( s ) f ( s ) exp h w1 ( s ) g ( s ) , (50) exp (h 1) w2 ( s) a0 s a0 s s
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a0b0 1 1 exp h w1 ( s ) R2 ( s ) ( s) f ( s) ( s) exp ( h 1) w2 ( s) . g ( s) a0 s 1 1s 1 2s s
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The Laplace transforms given by Eqs. (46), (48)-(50) are complicated, therefore it is difficult to obtain the inverse Laplace transforms of these functions by the analytical methods. In order to obtain the inverse Laplace transforms of the above functions, we use two accuracy numerical algorithms, namely the fixed Talbot algorithm [27] and the improved Talbot algorithm [28] for numerical Laplace transform inversion. Let F ( y, s) be the Laplace transform of function f ( y, t ) . Using the Talbot idea for the Laplace transform inversion, the function f ( y, t ) is approximated by [27] M 1 r exp(rt ) f ( y, t ) F ( y , r ) Re exp(tz (k )) F ( y, z (k )) 1 i(k ) , (51) M 2 k 1 where 2M r , z () r(cot i ), (, ), 5t (52) k () ( cot 1) cot , k . M
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Using the improved Talbot algorithm, the inverse Laplace transform is given by [28] f ( y, t )
1M exp(tz1 (k )) F ( y, z1 (k )) i 1 (k ) , t k 1
(53)
where
M i cot() , [, ], t
(54)
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z1 ()
(2k 1) 1 () ( cot() 1) cot(), k . M
In the above relations, M , , , , are constants to be specified by the user.
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5. Numerical results and discussions
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In this paper we have studied the unsteady laminar one-dimensional flow of two immiscible Maxwell fluids in a rectangular channel bounded by two parallel plates. The bottom plate and the upper plate have translational motions with time-dependent velocities in their planes. The fluid motion is influenced by a time-oscillating pressure gradient applied in the flow direction. Analytical solutions for the velocity and shear stress fields have been obtained by using the Laplace transform coupled with the finite sine-Fourier transform. These analytical solutions are new in the literature and the method developed in this paper can be generalized to the flow of n-layers of immiscible Maxwell fluids. Analytical solutions obtained in Section 3 can be easily customized for the Newtonian fluids. If the fluid situated in the domain y [0, h] is a Newtonian fluid, then the relaxation time 1 becomes zero. By using Eq. (30), Eq. (32) is written for Newtonian fluids as 2 (1) n 1 n2 T11N ( y, s ) cos( n y ), h n 1 s n2
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T12N ( y, s )
2 n2 cos( n y ), h n 1 s n2
(55)
2 1 (1) n n2 cos( n y ) h n 1 s n2 The inverse Laplace transforms of the above relations, for y h are given by T13N ( y, s )
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(56)
Using Eq. (43), the velocity on the interface is given by
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2 n f (t ) exp( 2nt ) sin( n y) h n 1
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T12N (h, t )* f (t ) T22 (h, t )* g (t ) [ T13N (h, t ) T23 (h, t )]* (t ) u (h, t ) u2 (h, t ) (57) T11N (h, t ) T21 (h, t ) Now, using properties (40), the velocity for the Newtonian fluid has the following analytical expression: 2 u1N ( y, t ) (1) n 1 n u1N (h, t) exp( 2nt ) sin( n y) h n 1 N 1
(58)
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2 [1 (1) n ]sin( n y) (t ) exp( 2nt ) , h n 1 n In a similar manner, velocity of the second Maxwell fluid can be customized for the Newtonian fluid by making the relaxation time 2 equal to zero into Eqs. (36) and (42). Solutions for the following problems are determined: flows of two-layered Maxwell fluids, one fluid is Newtonian fluid and other one is Maxwell fluid, respectively both fluids are Newtonian fluids. In order to have the validation of the analytical solutions, we have determined semi-analytical solutions by using the Laplace transform coupled with the classical method for the ordinary differential equations. The Laplace transform u ( y, s) of the velocity u ( y, t) and the Laplace transform ( y, s) of the shear stress ( y, t ) have complicated analytical expressions, therefore, the inverse Laplace transform can not be easily obtained through analytical methods. In order to obtain the inverse Laplace transforms we have used two accuracy numerical algorithms, namely the fixed Talbot algorithm and the improved Talbot algorithm for numerical inversion of the Laplace transforms. In the non-dimensional forms of the governing equations appear four characteristic parameters, namely, the non-dimensional relaxation times 1 and 2 , respectively a0 2 the ratio of fluids density and b0 2 the ratio of 1 1 fluids viscosity. In order to study the fluids behavior, the numerical computations are made by taking [7] Kg Ns N 1 , 2 [0.05,1] 2 , G1 , G2 1 2 , 1 1000 3 , a0 0.45, b0 0.35 . m m m To numerical simulations, we used for the bottom wall velocity f (t ) sin(t ) , respectively for the upper plate velocity g (t ) sin(2t ) .
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In Fig. 3 are plotted curves corresponding to the velocities and shear stresses of both fluids, versus variable y for different values of the time t for 1 0.15 and 2 0.25 . The first Maxwell fluid with velocity u1 ( y, t ) flows in the layer y [0,0.4] and the second Maxwell fluid with velocity u2 ( y, t ) flows in the layer y [0.4,1] . It is observed from Fig. 3 that in neighbor of the position y 0.62 the shear stress changes its sign, therefore it becomes braking force. As consequence, the fluid velocity increases till to the maximum value attains in the position where the shears stress changes its sign. a The influence of the ratio of fluids kinematic viscosities 0 1 is b0 2 analyzed by graphs given in Fig. 4. It is clear that, for the increasing values of 1 the ratio , the viscous force in the first fluid increases, therefore the 2 maximum value of the fluid velocity is decreasing. The influence of the ratio h of the widths of two regions of flow has been analyzed by graphs presented in Fig. 5 at different values of the time t. It is observed that, if the first layer is thinner, the fluid located in the second layer has higher velocity. This is due to the interface velocity which is bigger than the velocity from the plate y 0 . For small values of the spatial coordinate y, the velocity of the first fluid has ery small variations wihle, the velocity of the second fluid has large variations and attains a maximum close to the wall y 1. In order to make a comparison between numerical results obtained with the analytical solution (41) and with the semi-analytical solutions (46) coupled with the numerical algorithms (51) and (53), we use the following notations: Let u1a (y,0.75) be the numerical values of the velocity u1 (y,0.75) given by Eq. (46) using the formula (51), u1b (y, 0.75) -the numerical values of the velocity u1 (y,0.75) given by Eq. (46) using the formula (53), respectively u1c (y,0.75) the numerical values of the velocity given by analytical solution (41). The absolute errors
u1a (y,0.75) u1b (y,0.75) , u1a (y,0.75) u1c (y,0.75) ,
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u1b (y,0.75) u1c (y,0.75) are given in Table 1 for yk 0.2k , k 0,1,...,15. For numerical calculations we used for Eqs. (51)-(54) following values of the parameters: M 25, 0.9, 0.4, 0.55, 0.01.
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u1a (y,0.75) u1b (y,0.75)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30
1.357.10 -4 1.341.10 -4 2.678.10 -4 3.723.10 -4 4.348.10 -4 4.480.10 -4 4.112.10 -4 3.301.10 -4 2.149.10 -5 7.970.10 -5 6.001.10 -4 1.888.10 -4 2.932.10 -4 3.637.10 -4 3.958.10 -4 3.904.10
u1a (y,0.75) u1c (y,0.75)
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u1b (y,0.75) u1c (y,0.75)
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1.357.10 -4 7.737.10 -3 1.475.10 -3 2.044.10 -3 2.436.10 -3 2.620.10 -3 2.575.10 -3 2.302.10 -3 1.814.10 -3 1.140.10 -4 3.216.10 -4 5.920.10 -3 1.549.10 -3 2.501.10 -3 3.407.10 -3 4.237.10
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1.455.10 -4 6.396.10 -3 1.208.10 -3 1.672.10 -3 2.002.10 -3 2.172.10 -3 2.164.10 -3 1.972.10 -3 1.599.10 -3 1.060.10 -4 3.816.10 -4 4.032.10 -3 1.256.10 -3 2.138.10 -3 3.011.10 -3 3.847.10
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Fig. 3. Velocity and shear stress profiles for a0 0.35, b0 0.45 and different values of the time t. 16
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Fig. 4 Velocity profiles for different values of ratio a0 / b0 , 1 0.15, 2 0.25
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Fig. 5 Velocity profiles for different values of the ratio h of widths of two regions for a0 0.525, b0 0.35 , 1 0.15, 2 0.25 and different values of the time t .
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6. Conclusions Unsteady laminar one-dimensional flows of two immiscible Maxwell fluids in a rectangular channel bounded by two moving parallel plates have been studied. The fluid motion is generated by the both plates and by a timedependent pressure gradient which acts in the flow direction. Analytical solutions for the velocity and shear stress fields have been obtained by using the Laplace transform coupled with the finite sine-Fourier transform. These analytical solutions are new in the literature and the method developed in this paper can be generalized to the flow of n-layers of immiscible Maxwell fluids. Solutions for the flow of Maxwell fluids have been particularized for Newtonian fluids be making the relaxation times to be equal to zero. Thus, are determined solutions for the flow of two immiscible liquids, both fluids being Maxwel fluids, one fluid is Newtonian fluid and another one is Maxwell fluid or, both fluids are Newtonian fluids. It was found that the fluid velocity attains a maximum value near the wall y 1. The maximum value of the velocity is decreasing if the ratio of the kinematic viscosities increases. For small values of the ratio of the widths of two regions, the second fluid flows faster than the fluid situated near the wall y 0 .
References
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[1] R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, John Wiley and Sons, In., New York, 1960. [2] P. R. Redapangu, K. C. Sahu, S. P. Vanka, A study of pressure-driven displacement flow of two immiscible liquids using a multiphase lattice Boltzmann approach, Physics of Fluids, 24, 102110, 2012, doi:10.1063/1.4760257. [3] A.Hasnain, K. Alba, Buoyant displacement flow of immiscible fluids in inclined ducts: A theoretical approach, Physics of Fluids 29, 052102, 2017, doi: 10.1063/1.4982896 [4] Y. Ba, N. Wang, H. Liu, Q. Li, G. He, Regularized lattice Boltzmann model for immiscible two-phase flows with power-law rheology, Phys. Rev. E 97, 033307 – Published 13 March 2018. [5] D. Nikodijevic, Z. Stamenkovic, D. Milenkovic, B. Blagojevic, J. Nikodijevic, Flow and heat transfer of two immiscible fluids in the presence of uniform inclined magnetic field, Math. Problems Eng., Volume 2011, Article ID 132302, doi: 10.1155/2011/132302. [6] S. M. Zivojin, D. D. Nikodijevic, B. D. Blagojevic, S. R. Savic, MHD flow and heat transfer of two immiscible fluids between moving plates, Transactions of the Canadian Society for Mechanical Engineering, 34 (3-4), (2010), 351-372. [7] D. K. Satpathi, B. V. Rathish Kumar, P. Chandra, Unsteady-state laminar flow of viscoelastic gel and air in a channel: Application to mucus transport in a cough machine simulating trachea, Math. Comput. Modelling, 38 (2003) 6375.
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[8] T. L. Raju, B. N. Rao, Unsteady two-layered fluid flow of conducting fluids in a channel between parallel porous plates under transverse magnetic field in a rotating system, Int. J. Appl. Mech. Eng., 21 (2) (2016) 423-446, doi: 10.1515/ijame-2016-0026. [9] K. Ramesh, M. Devakar, Effect of heat transfer on the peristaltic transport of MHD second grade fluid through porous medium in an inclined asymmetric channel, Chinese Journal of Physics 55(3) (2017) 825-844, doi:10.1016/j.cjph.2016.10.028. [10] M, Shen, S, Chen, F. Liu, Unsteady MHD flow and heat transfer of fractional Maxwell viscoelastic nanofluid with Cattaneo heat flux and different particle shapes, Chinese Journal of Physics 56(3) (2018), 1199-1211, doi:/10.1016/j.cjph.2018.04.024. [11] M. Agarwal, D. Kumar, An unsteady flow of two immiscible viscous fluids in porous medium between two impermeable parallel plates, impulsively stopped from relatively motion, Int. J. Math. Appl., 5 (1-A) (2017) 103-114. [12] N. Roche-Carrier, G. D. Ngoma, F. Erchiqui, I. Hamani, Investigation of there immiscible fluids in a microchannel accounting for the pressure gradient and the electroosmotic flow, In Proceedings of the 7th International Conference on Simulation and Modelling Metodologies, Technologies and Applications (SIMULTECH 2017), pages 428-433, ISBN: 978-989-758-265-3, doi:10.5220/0006481004280433, SCITEPRESS-Science and Technology Publications. [13] M. S. Malashetty, J. C. Umavathi, J. P. Kumar, Magnetoconvection of two-immiscible fluids in vertical enclosure, Heat MassTransfer, 42 (2006) 977933, doi: 10.1007/s00231-005-0062-x. [14] J. D. Petrovic, Z. M. Stamenkovic, M. M. Kocic, M. D. Nicodijevic, Porous medium magnetohydrodynamic flow and heat transfer of two immiscible fluids, Thermal Science, 20 (Suppl. 5) (2016) S1405-S1417. [15] H. N. Zaidi, N. Ahmed, MHD convection flow of two immiscible fluids in an inclined channel with heat generation/absorption, American J. Appl. Math., 4 (2) (2016) 80-91, doi: 10.11648/j.ajam.20160402.13. [16] A. A. Mohammad, Q. W. Tao, Y. L. He, S. Bawazeer, Treatment of transport at the interface between multilayers via the lattice Boltzmann method, Numerical Heat Transfer, Part B, 67 (2015) 1-11, doi: 10.1080/10407790.2014.949563. [17] I. Ullah, S. Sharidan, I. Khan, K. L. Hsiao, Brownian diffusion and thermophoresis mechanisms in Casson fluid over a moving wedge, Results in Physics, 9 (2018), 183-194 [18] X. Guo, H. Qi, Analytical Solution of Electro-Osmotic Peristalsis of Fractional Jeffreys Fluid in a Micro-Channel, J. Micromachines, 8 (12) (2017) 341, doi: 10.3390/mi8120341. [19] M. Zhao, S. Wang, H. Wang, U. S. Mahabaleshwar, Darcy-Brinkman biothermal convection in a suspension of gyrotactic microorganisms in a porous medium, Neural Comput. & Applic., (2017) doi: 10.1007/s00521-017-3137-y. [20] J. Hristov, A transient flow of a non-Newtonian fluid modelled by a mixed time-space derivative: An improved integral-balance approach,” in Mathematical Methods in Engineering—Theory, edited by K. Taş, D. Baleanu, and J. A. T. Machado, Springer International Publishing, 2018.
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[21] J. Hristov, Multiple integral-balance method: Basic idea and an example with Mullins model of thermal grooving, Thermal Science, 21 (3), (2017) 15551560. [22] X. P. Wang, H. Qi, H. Y. Xu, Transient electro-osmotic flow of generalized second-grade fluids under slip boundary conditions, Can. J. Phys., 95 (12) (2017) 1313-1320, DOI: 10.1139/cjp-2017-0179. [23] X. Wang, H. Qi, Y. Bo, Analytical and numerical study of electroosmotic slip flows of fractional second grade fluids, Comm. Nonlinear Sci. Numer. Simul. 50 (2017) 77-87, DOI: 10.1016/j.cnsns.2017.02.019. [24] X. Yang, H. Qi, X. Jiang, Numerical analysis for electroosmotic flow of fractional Maxwell fluids, Appl. Math. Lett., 78, (2018) 1-8, DOI: 10.1016/j.aml.2017.10012. [25] D. Brian, Integral Transforms and Their Applications (third ed.) New York: Springer, 2002. [26] R. N. Bracewell, The Fourier Transform and its Applications (Third ed.) Boston: Mc Graw-Hill, 2000. [27] J. Abate, P. P. Valko, Multi-precision Laplace transform inversion, Int. J. Numer. Meth. Engng., 60 (2004) 979-993, doi:10.1002/nme.995. [28] B. Dingfelder, J. A. C. Weideman, An improved Talbot method for numerical Laplace transform inversion, Numer. Algor., 68 (2015) 167-183, doi: 10.1007/s11075-014-9895-z.
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Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates Muhammad Danial Hisham , Abdul Rauf , Dumitru Vieru , Aziz Ullah Awan PII: DOI: Reference:
S0577-9073(18)30854-2 https://doi.org/10.1016/j.cjph.2018.10.009 CJPH 666
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
20 June 2018 8 September 2018 17 October 2018
Please cite this article as: Muhammad Danial Hisham , Abdul Rauf , Dumitru Vieru , Aziz Ullah Awan , Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.10.009
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ACCEPTED MANUSCRIPT Highlights
Unsteady flows of two immiscible Maxwell fluids are studied. New analytical solutions for velocity and shear stress are determined. Semi-analytical solutions are obtained using Talbot algorithm. The developed method can be generalized for n-Maxwell fluids.
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Corresponding author: [email protected]
ACCEPTED MANUSCRIPT Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates Muhammad Danial Hisham1, Abdul Rauf2, Dumitru Vieru3, Aziz Ullah Awan1 1 University of the Punjab, Lahore, Pakistan 2 Air University Multan, Pakistan 3 Technical University of Iasi, Romania
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Abstract. Unsteady flows of two immiscible Maxwell fluids in a rectangular channel bounded by two moving parallel plates are studied. The fluid motion is generated by a time-dependent pressure gradient and by the translational motions of the channel walls in their planes. Analytical solutions for velocity and shear stress fields have been obtained by using the Laplace transform coupled with the finite sine-Fourier transform. These analytical solutions are new in the literature and the method developed in this paper can be generalized to unsteady flows of n-layers of immiscible fluids. By using the Laplace transform and classical method for ordinary differential equations, the second form of the Laplace transforms of velocity and shear stress are determined. For the numerical Laplace inversion, two accuracy numerical algorithms, namely the Talbot algorithm and the improved Talbot algorithm are used.
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1. Introduction
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Key words. Immiscible Maxwell fluids; analytical solutions; Talbot algorithm; integral transforms.
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One of the states of matter is called phase and it can be a gas, a liquid or a solid. Flows of several phases are known as multiphase flows. Two-phase flows are the simplest multiphase flows and can be solid-liquid flow, gas-solid flow, gas-liquid flow and liquid-liquid flow. Flows of immiscible liquids have important practical applications. For example, the flows of oil and water are important in oil recovery processes or in reducing of resistance to flow in pipelines. Dispersive flows, liquid extraction processes and co-extrusion flows in polymer processing industry are other applications of the immiscible liquidliquid flows. Interesting problems of multiphase flows are reported by Bird et al. [1]. The pressure-driven motion of two immiscible fluids in an inclined channel has been investigated by Redapangu et al. [2] under consideration of the viscosity and density gradients. The authors used the multiphase lattice Boltzmann approach. They found that the increasing of viscosity ration between fluids the displacement rate decreases. Hasnain and Alba [3] studied the buoyant displacement flows of two immiscible viscous fluids in an inclined twodimensional channel. The regularized lattice Boltzmann model for immiscible two-phase flows of power-law fluids have been studied by Ba et al. [4]. The closed forms solutions for the magneto-hydrodynamic Couette steady flows with heat transfer of two-immiscible fluids through a horizontal channel with isothermal boundaries under the influence of electric and magnetic fields have
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been obtained by Nikodijevic et al. [5] and Zivojin et al. [6]. Satpathi et al. [7] have investigated the unsteady laminar flow of two immiscible fluids in a channel due to a time-dependent pressure gradient by considering a Maxwell fluid and a Newtonian fluid. Using the Laplace transform, they obtained approximate solutions for fluid velocities. Also, a numerical scheme is developed in order to obtain numerical solutions. Raju and Rao [8] have studied the unsteady two-layered fluid flows through a rectangular channel with porous walls in a rotating system. The flow is generated by a constant pressure gradient and an oscillating channel wall. Heat transfer in the peristaltic transport of an incompressible magneto-hydrodynamic second grade fluid through a porous medium in an inclined asymmetric channel has been analyzed by Ramesh and Devakar [9]. Using the regular perturbation method, authors obtained solutions for the stream function, temperature and pressure gradient. Shen et al. [10] studied the unsteady natural convection flow and heat transfer of fractional Maxwell viscoelastic nanofluid in magnetic field over a vertical plate by considering the time-fractional generalized Cattaneo heat flux. Unsteady flow of two immiscible Newtonian fluids through a porous medium between two parallel plates has been investigated by Agarwal and Kumar [11]. One of the plates is moving with constant velocity and, after the fluids reach the stationary state, the plate is suddenly stopped. Roch-Carrier et al. [12] studied the flows of three immiscible fluids in a microchannel consisting of two parallel plates under the influence of an electric field. Interesting topics of fluids flows are analyzed in the references [13-21]. Unsteady electro-osmotic flows of second grade and Maxwell fluids with fractional constitutive equations have been investigated in the references [2224]. In the present paper, unsteady flows of two immiscible Maxwell fluids between two moving parallel plates are studied. The fluid motion is influenced by a time-dependent pressure gradient applied in the flow direction. The wall motions are translational motions with time-dependent velocities in their planes. Analytical solutions for fluid velocity and shear stress have been obtained by employing the Laplace transform coupled with the finite sineFourier transform. The obtained analytical solutions are new in the literature and the method developed in this paper can be generalized to unsteady flows of n-layers of immiscible Maxwell fluids. Using the Laplace transform and classical method for ordinary differential equations, second forms of the Laplace transforms of velocity and shear stress are determined and the semianalytical solutions are obtained by using two accuracy numerical algorithms, namely the Talbot algorithm and the improved Talbot algorithm for the numerical inversion of the Laplace transforms. A numerical comparison between analytical and semi-analytical solutions is carried out. Also, the influence of the system parameters on the fluids flow has been numerically and graphically analyzed. 2. Mathematical formulation of the problem. The flow domain is D {( x, y, z) (, ) [0, h2 ] (, )} ,with the channel walls situated in planes y 0 and y h2 0 . The channel flat wall
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y [0, h1 ], h1 h2 flows a Maxwell fluid with the density 1 ,viscosity 1 , relaxation time 1 1 / G1 , G1 the elastic modulus, velocity u1( y, t ) and the shear stress 1 ( y, t ) . In the region y [h1 , h2 ], flows a Maxwell fluid with the density 2 ,viscosity 2 , relaxation time 2 2 / G2 , velocity u2 ( y, t ) and the
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shear stress 2 ( y, t ) .
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Fig. 1. Flow geometry
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The fluids are assumed to be incompressible and immiscible and the flow is unsteady, one-dimensional and fully developed. Motions of the fluids are generated by the movements of the two walls of the channel and by the timedependent pressure gradient in the flow direction. Based on the assumptions about velocity fields, the continuity equation is identically satisfied and the governing equations of motion are [5, 7, 11]
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For the Maxwell fluid in region y [0, h1 ], h1 h2
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u1 p 1 , t x y
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(1)
1 u 1 1 , t y For the Maxwell fluid in region y [h1 , h2 ],
u2 p 2 , t x y
(2)
(3)
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2 2
2 u 2 2 . t y
(4)
Along with the above equations, we consider the following initial, boundary and the interface fluid-fluid conditions: (5)
u1(0, t ) U 0 f1 (t ), u2 (h2 , t ) U 0 g1 (t ),
(6)
u1(h1 , t ) u2 (h1 , t ), 1 (h1 , t ) 2 (h1 , t ) .
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uk ( y,0) 0, k ( y,0) 0, k 1, 2 ,
(7)
Introducing the non-dimensional variables
u h t h p x y , y , t 12 , uk k , k 2 k , k k , k 1, 2, p 2 , h2 h2 h2 U0 1U 0 k 1U 0 2 h t h 2t h 1 1 2 1 , 2 2 2 1 , a0 2 , b0 2 , f(t)= f1 2 , g(t)= g1 2 , h 1 , h2 h2 1 1 h2 1 1
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2 u b0 2 , t y
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u1 p 1, t x y
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with the following non-dimensional conditions:
uk ( y,0) 0, k ( y,0) 0, k 1, 2 ,
(13)
u1 (0, t ) f (t ), u2 (1, t ) g (t ),
(14)
u1 (h, t ) u2 (h, t ), 1 (h, t ) 2 (h, t ) .
(15)
3. Analytical solution In order to determine analytical solutions of the Eqs. (9)-(12) with conditions (13)-(15) we use the finite sine-Fourier transform with respect to
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p (t ) 1.5 0.5cos(5t 3 / 4) cos(5t / 3) x
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The diagram of the function (t ) is given in figure 2. This function is a periodic function with the period t0 6 / 5 .
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Fig. 2. Diagram of the pressure gradient
p (t ) x
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Applying the Laplace transform to Eqs. (9)-(12) and using the initial conditions (13), we obtain the problem in transformed domain
1 ( y, s) , y
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1 u1 ( y, s) , 1 1s y
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b0 u2 ( y, s) , 1 2 s y with the boundary and interface transformed conditions 2 ( y, s)
(20)
u1 (0,s) f (s), u2 (1,s) g (s),
(21)
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(22)
In the above relation ( y, s) ( y, t ) exp( st ) dt denotes the Laplace 0
transform of function ( y, t ) [25]. The finite sine-Fourier transform of the function ( y, s), y [a,b] is defined as [26]
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n( y a) , b a, n 1, 2,... , ba
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Eliminating 1 between Eqs. (17) and (18), respectively 2 between (19) and (20), we obtain equations for transformed velocities
2u1 ( y, s) , y [0, h] , y 2
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s(1 1s)u1 ( y, s) (1 1s)( s)
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2 u2 ( y , s ) (26) a0 s(1 2 s)u2 ( y, s) (1 2 s) ( s) b0 , y [h,1] . y 2 Applying the finite sine-Fourier transform to Eqs. (25), (26) and using conditions (21) and (22), we obtain the following expressions of the Fourier transform of fluid transformed velocities: (1) n n n f ( s) 1 1s 1 (1) n ( s ) u (h, s) , 2 2 1 n s(1 1 s) n s(1 1 s) n s(1 1 s) 2n n n , n 1, 2,..., h
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b0nu2 (h, s) (1) n b0n g ( s) 1 (1) n (1 2 s) ( s) , n a0 s(1 2 s) b0n2 a0 s (1 2 s ) b0n2 a0 s (1 2 s ) b0n2 n n , n 1, 2,.... 1 h
(28)
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Function (27) can be written in the suitable form s(1 1s) 1 (1) n 1 u1n ( s ) f (s) f ( s ) u1 (h, s ) n n [ s(1 1 s) n2 ] n (1) n s(1 1 s) 1 1s 1 (1) n u ( h , s ) ( s ), 1 2 n [ s(1 1 s) n ] n s(1 1 s) n2
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(29)
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and, by using the pairs of functions
h y 1 , y [0, h], 1n , h n y (1) n 1 2 ( y ) , y [0, h], 2 n , h n and applying the inverse Fourier transform to Eq. (29) we obtain 1 ( y )
s(1 1s) sin( n y ) h y y 2 f ( s) u1 (h, s) f ( s ) 2 h h h n 1 n [ s (1 1 s) n ] (1) n s(1 1 s)sin( n y) 2 2 1 (1) n (1 1s) sin( n y) u1 (h, s) ( s ) . h n [ s (1 1 s) n2 ] h n s(1 1 s) n2 n 1 n 1
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Now, from (18) and (31), we obtain
(31)
1 u1 ( y, s) T11 (y, s )u1 (h, s ) T12 (y, s ) f ( s ) T13 ( y, s) ( s), 1 1s y
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(1) n s(1 1 s) cos( n y ) 1 1 2 , h(1 1 s) s(1 1 s) 2n n 1
T12 ( y, s )
s(1 1 s) cos( n y ) 1 1 2 , h(1 1 s) s(1 1 s) 2n n 1
(32)
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A similar manner leads to
(1) n a0 s(1 2 s) (1) n 1 1 g (s) g ( s ) u2 (h, s) 2 n n [a 0 s(1 2 s) b0n ] n n a0 s(1 2 s) 1 2 s 1 (1) u2 (h, s) ( s), 2 n [a 0 s(1 2 s) b0n ] n a 0 s(1 2 s) b0n2
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u2 n ( s )
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By using the pairs function-sine Fourier transform 1 y 1 3 ( y ) , y [h,1], 3n , 1 h n yh (1) n 1 4 ( y ) , y [h,1], 4 n , 1 h n and applying the inverse Fourier transform to Eq. (33), we obtain
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(33)
(34)
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u2 (y, s)
2u (h, s) a0 s (1 2 s) sin(n ( y - h)) 1 y yh u2 (h, s) g ( s) 2 1 h 1 h 1 h n 1 n [a0 s(1 2 s) b0n2 ]
(35) 2 g(s) (1) n a0 s(1 2 s) sin(n ( y - h)) 2( s) [1 ( 1) n ](1 2 s) sin(n ( y - h)) [a s(1 s) b 2 ] 1 h 1 h n 1 n [a0 s(1 2 s) b02n ] n 1 n 0 2 0 n
From (20) and (28), we have
T21 ( y, s)
u2 ( y, s ) T21 ( y, s)u2 (h, s) T22 ( y, s) g(s)+T23 ( y, s) ( s) y
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2 (y, s ) b0
b0 a0 s(1 2 s) cos(n ( y - h)) 1 2 , (1 h)(1 2 s) a0 s(1 2 s) b02n n 1
T23 ( y, s)
(36)
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b0 (1) n a0 s(1 2 s) cos(n ( y - h)) T22 ( y, s) 1 2 , (1 h)(1 2 s) a0 s(1 2 s ) b02n n 1
2b0 [1 (1) n ]cos(n ( y - h)) . 1 h n 1 a0 s (1 2 s ) b02n
u1 (h, s ) u2 (h, s )
A1 ( s ) , A0 ( s )
(37)
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A0 ( s ) T11 (h, s) T21 (h, s),
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Using the interface liquid-liquid conditions (22), we obtain the following expression of the velocity on the interface y h
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A1 ( s ) T12 (h, s ) f(s)+T22 (h, s ) g(s)+ T13 (h, s ) T23 (h, s ) ( s ).
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1 L1 21 (t ; m, n), s (1 ms) n ms L1 2 (t ; m, n) 1 (t ; m, n), s (1 ms) n 1 ms L1 1 (t ; m, n) 2 (t ; m, n), s (1 ms) n s (1 ms ) L1 (t ) 2n1 (t ; m, n), s (1 ms) n
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(38)
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t exp 2m sinh t 1 4mn , 1 (t ; m, n) 2m 1 4mn
1 lim 1 (t; m, n) lim 2 (t; m, n) exp( nt) . m 0 m 0 2
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t 1 4mn t 2 (t ; m, n) exp . cosh 2m 2m Functions given by Eq. (39) satisfy the property
(39)
(40)
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Using Eqs. (31), (35) and (38), we obtain for velocities u1 ( y, t ), u2 ( y, t ) the following expressions: (h y ) f (t ) y u1 ( y, t ) u1 (h, t ) h h
2 (1) n sin( n y) u1 (h, t ) 2 2nu1 (h, t) 1 (t ; 1 , 2n ) h n 1 n
(41)
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2 sin( n y) f (t ) 2 2n f (t ) 1 (t; 1 , 2n ) h n 1 n
( y h) g (t ) 1 y u2 (h, t ) 1 h 1 h
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u2 ( y , t )
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2 [1 (1) n ]sin( n y) (t ) 1 (t ; 1 , 2n ) 2 (t; 1 , 2n ) , h n 1 n
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2b0 2 b 2 (1) n sin(n ( y - h)) nu2 (h, t) 1 (t ; 2 , 0 2n ) u2 (h, t ) 1 h n 1 n a0 a0
(42)
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2b0 2 b 2 (1) n sin(n ( y - h)) n g (t ) 1 (t ; 2 , 0 2n ) g (t ) 1 h n 1 n a0 a0 [1 (1) n ]sin(n ( y - h)) b b 2 (t ) 1 (t ; 2 , 0 2n ) 2 (t; 2 , 0 2n ) , a0 (1 h) n 1 n a0 a0
t
where f1 (t ) f 2 (t ) f1 (t ) f 2 () d is the convolution product of the functions 0
f1 (t ) , f 2 (t ) and u1 (h, t ) u2 (h, t ) is given by
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u1 (h, t ) u2 (h, t )
A1 (t ) A0 (t )
T12 (h, t ) * f (t ) T22 (h, t ) * g (t ) [ T13 (h, t ) T23 (h, t )]* (t ) , T11 (h, t ) T21 (h, t )
(43)
where
t 2 1 2 (t; 1 , 2n ) 1 (t; 1 , 2n ) , exp h1 h 1 1 n 1
T12 (h, t )
t 2 1 exp (1) n 2 (t; 1 , 2n ) 1 (t; 1 , 2n ) , h1 h 1 1 n 1
T13 (h, t )
4 [(1) n 1]1 (t; 1 , 2n ), h n 1
T21 (h, t )
t b0 2b0 exp (1 h) 2 2 (1 h) 2
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T11 (h, t )
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t b0 2b0 T22 (h, t ) exp (1 h) 2 2 (1 h) 2
n 1
2
(t; 2 ,
(1) n 1
n
b0 2 b n ) 1 (t; 2 , 0 2n ) , a0 a0
b0 2 b0 2 2 (t; 2 , n ) 1 (t; 2 , n ) ,(45) a0 a0
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4b0 b [1 (1) n ]1 (t; 2 , 0 2n ). a0 (1 h) n 1 a0
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T23 (h, t )
(44)
4. Semi-analytical solution.
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In this section, we determine a new form of solutions to the problem given by Eqs. (9)-(15) using the Laplace transform coupled with the classical method for the ordinary differential equations. The general solutions of equations (25) and (26) are
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1 u1 ( y, s) C1 ( s) exp y w1 ( s) C2 ( s) exp y w1 ( s) ( s), s
1 u2 ( y, s) D1 ( s) exp y w2 ( s) D2 ( s) exp y w2 ( s) ( s), a0 s where, a w1 ( s) s(1 1s), w2 ( s) 0 s(1 2 s) . b0
(46)
(47)
Now, Eqs. (18), (20) and (46) lead to the following form of the Laplace transforms of the shear stresses i ( y, t ), i 1, 2 :
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1 ( y, s)
1 C1 ( s) w1 ( s) exp( y w1 ( s)) C2 ( s) w1 ( s) exp(y w1 ( s)) , 1 1s
(48)
b0 2 ( y, s) D1 ( s) w2 ( s) exp( y w2 ( s)) D2 ( s) w2 ( s) exp(y w2 ( s)) . 1 2 s Using the boundary conditions (21) and the interface conditions (22), we have
Q4 ( s) R1 ( s ) Q2 ( s ) R2 ( s ) 1 , C2 ( s ) f ( s ) ( s ) C1 ( s ), Q1 ( s)Q4 ( s ) Q2 ( s)Q3 ( s ) s
D1 ( s )
Q1 ( s ) R2 ( s ) Q3 ( s ) R1 ( s ) 1 , D2 ( s) g ( s) ( s) exp w2 ( s) Q1 ( s )Q4 ( s ) Q2 ( s )Q3 ( s ) a0 s
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C1 ( s )
D1 ( s ) exp 2 w2 ( s) , where,
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(49)
Q1 ( s ) 2sinh h w1 ( s) , Q2 ( s) exp (h 2) w2 ( s) exp h w2 ( s) , Q3 ( s )
2 cosh h w1 ( s ) 1 1s
, Q ( s)
a0b0
1 2s
4
exp (h 2)
w2 ( s) exp h w2 ( s) ,
( s) (1 a0 ) ( s) ( s ) R1 ( s ) f ( s ) exp h w1 ( s ) g ( s ) , (50) exp (h 1) w2 ( s) a0 s a0 s s
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a0b0 1 1 exp h w1 ( s ) R2 ( s ) ( s) f ( s) ( s) exp ( h 1) w2 ( s) . g ( s) a0 s 1 1s 1 2s s
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The Laplace transforms given by Eqs. (46), (48)-(50) are complicated, therefore it is difficult to obtain the inverse Laplace transforms of these functions by the analytical methods. In order to obtain the inverse Laplace transforms of the above functions, we use two accuracy numerical algorithms, namely the fixed Talbot algorithm [27] and the improved Talbot algorithm [28] for numerical Laplace transform inversion. Let F ( y, s) be the Laplace transform of function f ( y, t ) . Using the Talbot idea for the Laplace transform inversion, the function f ( y, t ) is approximated by [27] M 1 r exp(rt ) f ( y, t ) F ( y , r ) Re exp(tz (k )) F ( y, z (k )) 1 i(k ) , (51) M 2 k 1 where 2M r , z () r(cot i ), (, ), 5t (52) k () ( cot 1) cot , k . M
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Using the improved Talbot algorithm, the inverse Laplace transform is given by [28] f ( y, t )
1M exp(tz1 (k )) F ( y, z1 (k )) i 1 (k ) , t k 1
(53)
where
M i cot() , [, ], t
(54)
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z1 ()
(2k 1) 1 () ( cot() 1) cot(), k . M
In the above relations, M , , , , are constants to be specified by the user.
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5. Numerical results and discussions
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In this paper we have studied the unsteady laminar one-dimensional flow of two immiscible Maxwell fluids in a rectangular channel bounded by two parallel plates. The bottom plate and the upper plate have translational motions with time-dependent velocities in their planes. The fluid motion is influenced by a time-oscillating pressure gradient applied in the flow direction. Analytical solutions for the velocity and shear stress fields have been obtained by using the Laplace transform coupled with the finite sine-Fourier transform. These analytical solutions are new in the literature and the method developed in this paper can be generalized to the flow of n-layers of immiscible Maxwell fluids. Analytical solutions obtained in Section 3 can be easily customized for the Newtonian fluids. If the fluid situated in the domain y [0, h] is a Newtonian fluid, then the relaxation time 1 becomes zero. By using Eq. (30), Eq. (32) is written for Newtonian fluids as 2 (1) n 1 n2 T11N ( y, s ) cos( n y ), h n 1 s n2
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T12N ( y, s )
2 n2 cos( n y ), h n 1 s n2
(55)
2 1 (1) n n2 cos( n y ) h n 1 s n2 The inverse Laplace transforms of the above relations, for y h are given by T13N ( y, s )
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(56)
Using Eq. (43), the velocity on the interface is given by
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2 n f (t ) exp( 2nt ) sin( n y) h n 1
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T12N (h, t )* f (t ) T22 (h, t )* g (t ) [ T13N (h, t ) T23 (h, t )]* (t ) u (h, t ) u2 (h, t ) (57) T11N (h, t ) T21 (h, t ) Now, using properties (40), the velocity for the Newtonian fluid has the following analytical expression: 2 u1N ( y, t ) (1) n 1 n u1N (h, t) exp( 2nt ) sin( n y) h n 1 N 1
(58)
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2 [1 (1) n ]sin( n y) (t ) exp( 2nt ) , h n 1 n In a similar manner, velocity of the second Maxwell fluid can be customized for the Newtonian fluid by making the relaxation time 2 equal to zero into Eqs. (36) and (42). Solutions for the following problems are determined: flows of two-layered Maxwell fluids, one fluid is Newtonian fluid and other one is Maxwell fluid, respectively both fluids are Newtonian fluids. In order to have the validation of the analytical solutions, we have determined semi-analytical solutions by using the Laplace transform coupled with the classical method for the ordinary differential equations. The Laplace transform u ( y, s) of the velocity u ( y, t) and the Laplace transform ( y, s) of the shear stress ( y, t ) have complicated analytical expressions, therefore, the inverse Laplace transform can not be easily obtained through analytical methods. In order to obtain the inverse Laplace transforms we have used two accuracy numerical algorithms, namely the fixed Talbot algorithm and the improved Talbot algorithm for numerical inversion of the Laplace transforms. In the non-dimensional forms of the governing equations appear four characteristic parameters, namely, the non-dimensional relaxation times 1 and 2 , respectively a0 2 the ratio of fluids density and b0 2 the ratio of 1 1 fluids viscosity. In order to study the fluids behavior, the numerical computations are made by taking [7] Kg Ns N 1 , 2 [0.05,1] 2 , G1 , G2 1 2 , 1 1000 3 , a0 0.45, b0 0.35 . m m m To numerical simulations, we used for the bottom wall velocity f (t ) sin(t ) , respectively for the upper plate velocity g (t ) sin(2t ) .
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In Fig. 3 are plotted curves corresponding to the velocities and shear stresses of both fluids, versus variable y for different values of the time t for 1 0.15 and 2 0.25 . The first Maxwell fluid with velocity u1 ( y, t ) flows in the layer y [0,0.4] and the second Maxwell fluid with velocity u2 ( y, t ) flows in the layer y [0.4,1] . It is observed from Fig. 3 that in neighbor of the position y 0.62 the shear stress changes its sign, therefore it becomes braking force. As consequence, the fluid velocity increases till to the maximum value attains in the position where the shears stress changes its sign. a The influence of the ratio of fluids kinematic viscosities 0 1 is b0 2 analyzed by graphs given in Fig. 4. It is clear that, for the increasing values of 1 the ratio , the viscous force in the first fluid increases, therefore the 2 maximum value of the fluid velocity is decreasing. The influence of the ratio h of the widths of two regions of flow has been analyzed by graphs presented in Fig. 5 at different values of the time t. It is observed that, if the first layer is thinner, the fluid located in the second layer has higher velocity. This is due to the interface velocity which is bigger than the velocity from the plate y 0 . For small values of the spatial coordinate y, the velocity of the first fluid has ery small variations wihle, the velocity of the second fluid has large variations and attains a maximum close to the wall y 1. In order to make a comparison between numerical results obtained with the analytical solution (41) and with the semi-analytical solutions (46) coupled with the numerical algorithms (51) and (53), we use the following notations: Let u1a (y,0.75) be the numerical values of the velocity u1 (y,0.75) given by Eq. (46) using the formula (51), u1b (y, 0.75) -the numerical values of the velocity u1 (y,0.75) given by Eq. (46) using the formula (53), respectively u1c (y,0.75) the numerical values of the velocity given by analytical solution (41). The absolute errors
u1a (y,0.75) u1b (y,0.75) , u1a (y,0.75) u1c (y,0.75) ,
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u1b (y,0.75) u1c (y,0.75) are given in Table 1 for yk 0.2k , k 0,1,...,15. For numerical calculations we used for Eqs. (51)-(54) following values of the parameters: M 25, 0.9, 0.4, 0.55, 0.01.
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u1a (y,0.75) u1b (y,0.75)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30
1.357.10 -4 1.341.10 -4 2.678.10 -4 3.723.10 -4 4.348.10 -4 4.480.10 -4 4.112.10 -4 3.301.10 -4 2.149.10 -5 7.970.10 -5 6.001.10 -4 1.888.10 -4 2.932.10 -4 3.637.10 -4 3.958.10 -4 3.904.10
u1a (y,0.75) u1c (y,0.75)
-5
u1b (y,0.75) u1c (y,0.75)
-12
-5
1.357.10 -4 7.737.10 -3 1.475.10 -3 2.044.10 -3 2.436.10 -3 2.620.10 -3 2.575.10 -3 2.302.10 -3 1.814.10 -3 1.140.10 -4 3.216.10 -4 5.920.10 -3 1.549.10 -3 2.501.10 -3 3.407.10 -3 4.237.10
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1.455.10 -4 6.396.10 -3 1.208.10 -3 1.672.10 -3 2.002.10 -3 2.172.10 -3 2.164.10 -3 1.972.10 -3 1.599.10 -3 1.060.10 -4 3.816.10 -4 4.032.10 -3 1.256.10 -3 2.138.10 -3 3.011.10 -3 3.847.10
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y
Fig. 3. Velocity and shear stress profiles for a0 0.35, b0 0.45 and different values of the time t. 16
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Fig. 4 Velocity profiles for different values of ratio a0 / b0 , 1 0.15, 2 0.25
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Fig. 5 Velocity profiles for different values of the ratio h of widths of two regions for a0 0.525, b0 0.35 , 1 0.15, 2 0.25 and different values of the time t .
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6. Conclusions Unsteady laminar one-dimensional flows of two immiscible Maxwell fluids in a rectangular channel bounded by two moving parallel plates have been studied. The fluid motion is generated by the both plates and by a timedependent pressure gradient which acts in the flow direction. Analytical solutions for the velocity and shear stress fields have been obtained by using the Laplace transform coupled with the finite sine-Fourier transform. These analytical solutions are new in the literature and the method developed in this paper can be generalized to the flow of n-layers of immiscible Maxwell fluids. Solutions for the flow of Maxwell fluids have been particularized for Newtonian fluids be making the relaxation times to be equal to zero. Thus, are determined solutions for the flow of two immiscible liquids, both fluids being Maxwel fluids, one fluid is Newtonian fluid and another one is Maxwell fluid or, both fluids are Newtonian fluids. It was found that the fluid velocity attains a maximum value near the wall y 1. The maximum value of the velocity is decreasing if the ratio of the kinematic viscosities increases. For small values of the ratio of the widths of two regions, the second fluid flows faster than the fluid situated near the wall y 0 .
References
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