Thin film flow of magnetohydrodynamic (MHD) pseudo-plastic fluid on vertical wall

Applied Mathematics and Computation 245 (2014) 544–556

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Applied Mathematics and Computation

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Thin film flow of magnetohydrodynamic (MHD) pseudo-plasticfluid on vertical wall q

http://dx.doi.org/10.1016/j.amc.2014.07.0470096-3003/� 2014 Elsevier Inc. All rights reserved.

q The research was partially supported by Higher Education Commission of Pakistan.⇑ Corresponding author.

E-mail address: [email protected] (M.K. Alam).

M.K. Alam a,⇑, A.M. Siddiqui b, M.T. Rahim a, S. Islam c, E.J. Avital d, J.J.R. Williams d

a National University of Computer & Emerging Sciences, 25000 Peshawar, Pakistanb Pennsylvania State University, York Campus, 1031 Edgecombe Avenue, York, PA 17403, USAc Department of Mathematics, Abdul Wali Khan University, 3953 Mardan, Pakistand School of Engineering & Material Sciences, Queen Mary University of London, UK

a r t i c l e i n f o a b s t r a c t

Keywords:Lifting and drainage problemsPseudo-plastic fluid modelMHD fluidNonlinear Marching techniqueAdomian decomposition methodAdomian polynomial

The effect of the transverse magnetic field on the thin film flow of pseudo-plastic fluid forlifting and drainage on a vertical (wall) surface is investigated. Approximate solutions areobtained analytically and numerically for the resulting non-linear and inhomogeneousordinary differential equations arising from the constitutive equation of the pseudo-plasticfluid. Physical quantities such as the velocity profile, volume flux, average velocity, shearstress, vorticity and the force exerted by fluid on the belt surface respectively for lift anddrainage are calculated. The condition for the net upward flow for lift case has also beeninvestigated. The effect of different parameters such as the magnetic effect M2, the non-Newtonian parameter b, the Stokes number St on velocity profiles and shear stress are pre-sented via graphs. The analysis for the analytical solution is carried out for small value of b.The analogy of the velocity profiles of MHD pseudo-plastic and MHD Newtonian fluidsreveals that the MHD pseudo-plastic fluid drains down the upward moving belt faster thanthe MHD Newtonian fluid whereas in drainage, the behavior of the velocity profiles is viceversa. In other words, the applied magnetic field generally reduces the flow velocity in thelifting case, while, in the drainage case, the magnetic field and the velocity field are shownto have a direct relation.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

In view of the importance of drainage phenomena, the necessary understanding of its physics is described: when a fluid isdisposed to a vertical rigid object, it adheres to it and drains down the object under the conditions such that the gravitationaland viscous forces dominate the inertial forces due to which fluid film formed in contact with free surface drain down underthe action of gravity only. This type of drainage is termed ‘‘free drainage’’ [1] and it consists of an extent of fluid bounded byan object and with a free surface (usually air). The thickness d is much shorter than the length of the object so that the flowtakes place in the longer dimensions under the action of gravity. The flow velocity perpendicular to the plate is much smallerthan the main flow velocity. Hence, we can consider it as one dimensional flow. Drainage is abundant in nature i.e. themotion of rain drop down a window pane, in eye the tear films, flow of water on stalactites hanging from the roof of alimestone cave, the acclivity of buoyant magma below the solid rocks, the spreading of lava on volcanoes and found many

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M.K. Alam et al. / Applied Mathematics and Computation 245 (2014) 544–556 545

applications in industry i.e. tertiary oil recovery, fabrication of microchip and in many coating processes [1,2,4]. Jeffreys [3]was the first who investigated the drainage of Newtonian fluid by assuming the whole process as a simple balance betweenviscous and gravitational forces i.e. by neglecting the accelerating term. His reported thickness was valid only for large valuesof time. In recent years, there has been a great deal of interest in understanding the behavior of non-Newtonian fluids as theyare used in many engineering processes. Also, non-Newtonian fluids are intensively studied by mathematicians, essentiallyfrom the point of view of differential equations theory. Common examples of such fluids are slurries, pastes, gels, moltenplastics and lubricants containing polymer additives. Various food stuffs such as honey and tomato sauce, the biological flu-ids like blood and synovial fluid naturally found in cavities of synovial joints also belong to the general class of the non-Newtonian fluids [5]. The simplest model of non-Newtonian fluids is the power law model and a special case of the powerlaw equation, with n < 1, is the pseudo plastic model. Commonly encountered non-Newtonian fluids like polymer solutions,paper pulps, detergents, oils and greases may be classified by the pseudo plastic fluid model [6].

Nowadays a lot of interest has been shown by researcher towards the study of the thin film flows of non-Newtonian fluidsdue to its wide spread applications in nonlinear sciences and engineering industries. The work on the thin film flows of non-Newtonian fluids under various configurations is relatively of recent origin. Siddiqui et al. [7] discussed the thin film flows ofa third grade fluid down an inclined plane. In [8] they examined the thin film flows of Phan-Thein–Tanner (PTT) fluids on amoving belt. Siddiqui et al. [9] also considered the same flow of non-Newtonian fluids on a moving belt. The thin film flow ofa fourth grade fluid down a vertical cylinder is also analyzed by Siddiqui et al. [10]. Adomian decomposition method [11–19]plays an important role and many researchers have applied Adomian decomposition method to a wide class of linear andnon-linear ordinary differential equations, partial differential equations and integral equations. Further the ADM resultswere compared with numerical solution in which both were agree very well. J.S. Duan introduced new techniques for theAdomian polynomials [20–23].

2. Basic equations

The basic equations governing the motion of an isothermal, homogeneous and incompressible fluid are:

div V ¼ 0; ð1Þ

qDVDt¼ qf � gradP þ div S; ð2Þ

where V is the velocity vector, q is the constant density and f is the body force per unit mass, P denotes the dynamic pres-sure, D

Dt denotes the material derivative which is defined as

Dð�ÞDt¼ @

@tð�Þ þ ðV � rÞð�Þ:

S is the extra stress tensor which for pseudo-plastic fluid model [25] is defined as

Sþ k1 Srþ1

2ðk1 � l1ÞðA1Sþ SA1Þ ¼ g0A1; ð3Þ

where g0 is the zero shear viscosity, k1 is the relaxation time and l1 is the material constant.The first Rivlin–Ericksen tensor A1 is defined as

A1 ¼ ðgradVÞ þ ðgradVÞT : ð4Þ

The contravariant convected derivative denoted by super imposed r over S is defined as

Sr¼ DS

Dt� ðgradVÞT Sþ SðgradVÞn o

: ð5Þ

3. Formulation of the electrically conducted paint film flow down a vertical wall by gravity

Consider steady, incompressible, parallel, laminar flow of electrically conducting paint flowing down on an infinite ver-tical wall. Also assume that the flow of the paint film on a infinite vertical wall is laminar and uniform. The thickness of thepaint film is assumed to be d and gravity acts in the negative z-direction (downward) which means that gz ¼ �g. The bound-ary conditions for the problem are.

(i) There is no-slip at the wall

at x ¼ 0; w ¼ 0: ð6Þ

(ii) At the free surface, the shear is negligible, thus

at x ¼ d; Sxz ¼ 0; ð7Þ

where Sxz is the shear stress component of the pseudo-plastic fluid.

546 M.K. Alam et al. / Applied Mathematics and Computation 245 (2014) 544–556

The velocity field with magnetic effect for the stated problem defines as

V ¼ ð0;0;�rB20wðxÞÞ; ð8Þ

and extra stress tensor as

S ¼ SðxÞ: ð9Þ

By inserting (8) in (1) and (9), the continuity Eq. (1) is identically satisfied and Eq. (2) takes the following form

0 ¼ dSxx

dxþ qf1; ð10Þ

0 ¼ dSxz

dxþ qf3 � rB2

0wðxÞ; ð11Þ

where f1 and f3 are components of body force in x and z directions, respectively.Since z-axis is in upward direction and gravity acts in the negative z-direction (downward) which means that gz ¼ �g, so

the above equations become

0 ¼ dSxx

dx; ð12Þ

0 ¼ dSxz

dx� qg � rB2

0wðxÞ: ð13Þ

The non-zero components of S are obtained by inserting Eq. (4), (5) and (8) in Eq. (3)

Sxx ¼�g0 k1 � l1

� �dwdx

� �2

1þ k21 � l2

1

� �dwdx

� �2 ; ð14Þ

Szz ¼g0 k1 þ l1

� �dwdx

� �2

1þ k21 � l2

1

� �dwdx

� �2 ; ð15Þ

Szx ¼g0

dwdx

� �1þ k2

1 � l21

� �dwdx

� �2 : ð16Þ

Substituting the value of Szx in Eq. (13), we get

ddx

g0dwdx

� �1þ ðk2

1 � l21Þ dw

dx

� �2

" #¼ qg þ rB2

0wðxÞ: ð17Þ

Applying the non-dimensional parameters w� ¼ wU0

and x� ¼ xd in Eq. (17), we get

ddx

dwdx

1þ b dwdx

� �2

" #¼ St þM2wðxÞ; ð18Þ

with

dwdx¼ 0 at x ¼ 1;

w ¼ 0 at x ¼ 0;

where b ¼ k21�l2

1ð ÞU20

d2 denotes the non-Newtonian parameter, St ¼ qgd2

leff U0represents the Stokes number and M2 ¼ rB2

0d

leffis the MHD

parameter. Eq. (18) can be written as a second order differential equation as follows

d2w

dx2 � bdwdx

� �2 d2w

dx2 � b2Stdwdx

� �4

� 2bStdwdx

� �2

�M2b2wðxÞ dwdx

� �4

� 2bM2wðxÞ dwdx

� �2

�M2wðxÞ ¼ St; ð19Þ

with the wall and free surface boundary conditions, respectively;

w ¼ 0 at x ¼ 0;dwdx¼ 0 at x ¼ 1:

We observe that Eq. (19) subject to the above boundary conditions is a non-linear second order differential equation. It is awell-posed problem, but difficult to calculate its exact solution because of its high degree of non-linearity. Therefore, weapply the Adomian decomposition method (ADM) and numerical method for the solution of this problem.


3.1. Solution derivation of the paint film flow problem by the Adomian decomposition method

Consider the inhomogeneous non-linear differential Eq. (19) in Adomian’s operator form

Lwþ Rwþ Nw ¼ g; 0 6 x � 1;

subject to the mixed set of inhomogeneous Dirichlet–Neumann boundary conditions

wð0Þ ¼ 0;dwdxjðx¼1Þ ¼ 0;

where

L ¼ d2

dx2 ; R ¼ �M2; g ¼ St; Nw ¼X5

i¼1

Niw;

with

N1w ¼ b1dwdx

� �2 d2w

dx2

!; N2w ¼ b2

dwdx

� �4

; N3w ¼ b3dwdx

� �2

; ð20Þ

N4w ¼ b4dwdx

� �4 d2w

dx2

!; N5w ¼ bwðxÞ dw

dx

� �2

; ð21Þ

where b1 ¼ �b; b2 ¼ �b2St ; b3 ¼ �2bSt ; b4 ¼ �M2b2; b5 ¼ �2bM2. Observe that we have quadratic, quartic and productnon-linearities involving the solution and the first, second order derivatives.

We shall proceed with the solution using the Duan–Rach modification of the Adomian decomposition method [24].

Lw ¼ g � Rw� Nw;

L�1Lw ¼ L�1g � L�1Rw� L�1Nw;

we have chosen L�1 ¼R x

0

R x1 ð�Þdxdx, where the limits of integration are chosen so that to incorporate as many of the boundary

values as possible.

L�1Lw ¼ w�U;

where LU ¼ 0;Uð0Þ ¼ 0; dUdx ð1Þ ¼ 0, Upon substitution of L�1Lw, we obtain

w ¼ Uþ L�1g � L�1Rw� L�1Nw:

Next we evaluate U and L�1 prior to a suitable recursion scheme.

L�1Lw ¼Z x

0

Z x

1

d2w

dx2

!dxdx ¼ w�wð0Þ � x

dwdxð1Þ ¼ w; thus U ¼ 0; ð22Þ

L�1g ¼Z x

0

Z x

1Stdxdx ¼ St

1� xð Þ2 � 12

!: ð23Þ

Upon substitution of the obtained formulas for U; L�1g and using R ¼ �M2, we obtain

wðxÞ ¼ St1� xð Þ2 � 1

2

!þM2L�1w� L�1Nw; ð24Þ

w ¼X1n¼0

wn; Nw ¼X1n¼0

A�;

N1w ¼X1n¼0

An; N2w ¼X1n¼0

Bn; N3w ¼X1n¼0

Cn; N4w ¼X1n¼0

Dn; N5w ¼X1n¼0

En;

and hence

A� ¼X1n¼0

An þX1n¼0

Bn þX1n¼0

Cn þX1n¼0

Dn þX1n¼0

En:

Upon substitution of the Adomian decomposition series and the series of the Adomian polynomials in Eq. (24), we obtain

w ¼X1n¼0

wn ¼ St1� xð Þ2 � 1

2

!þM2L�1

X1n¼0

wn � L�1X1n¼0

A�n: ð25Þ


Next we consider the classic Adomian recursion scheme

w0 ¼ St1� xð Þ2 � 1

2

!; ð26Þ

wnþ1 ¼ M2L�1X1n¼0

wn � L�1X1n¼0

A�n; ð27Þ

wnþ1 ¼ M2Z x

0

Z x

1wndxdx�

Z x

0

Z x

1A�ndxdx; n P 0: ð28Þ

The general form of mixed set of inhomogeneous Dirichlet–Neumann boundary conditions can be expressed as

dwnþ1

dxð1Þ ¼ 0; wnþ1ð0Þ ¼ 0; n P 0: ð29Þ

We note that the Adomian polynomial A� for the non-linear terms (20) and (21) are as follows

A0 ¼ b1dw0

dx

� �2 d2w0

dx2

!; A1 ¼ 2b1

dw0

dx

� �dw1

dx

� �d2w0

dx2

!; ð30Þ

B0 ¼ b2dw0

dx

� �4

; B1 ¼ 4b2dw0

dx

� �3 dw1

dx

� �; ð31Þ

C0 ¼ b3dw0

dx

� �2

; C1 ¼ 2b3dw0

dx

� �dw1

dx

� �; ð32Þ

D0 ¼ b4w0ðxÞdw0

dx

� �4

; D1 ¼ b4w1ðxÞdw0

dx

� �4

þ 4b4w0ðxÞdw0

dx

� �3 dw1

dx

� �; ð33Þ

E0 ¼ b5w0ðxÞdw0

dx

� �2

; E1 ¼ b5w1ðxÞdw0

dx

� �2

þ 2b5w0ðxÞdw0

dx

� �dw1

dx

� �: ð34Þ

On setting n ¼ 0 in Eq. (28), we get the first component as follows

w1 ¼ M2Z x

0

Z x

1w0dxdx�

Z x

0

Z x

1A�0dxdx; ð35Þ

or

w1 ¼ M2Z x

0

Z x

1w0dxdx�

Z x

0

Z x

1A0 þ B0 þ C0 þ D0 þ E0½ �dxdx: ð36Þ

On inserting the values of A0; B0; C0; D0; E0 in Eq. (36), we get the following form

w1 ¼ M2Z x

0

Z x

1St

1� xð Þ2 � 12

!" #dxdx�

Z x

0

Z x

1b1

dw0

dx

� �2 d2w0

dx2

!þ b2

dw0

dx

� �4

þ b3dw0

dx

� �2"

þ b4w0ðxÞdw0

dx

� �4

þ b5w0ðxÞdw0

dx

� �2#

dxdx:

On solving the above integration, we arrive at

w1ðxÞ ¼M2St

6x� 1ð Þ4

4�M2St

2x2

2þ b1

S3t

3x� 1ð Þ4

4þ b2

S4t

4x� 1ð Þ6

6þ b3

S3t

3x� 1ð Þ4

4

þ b4S5t

114

x� 1ð Þ8

8� 1

10x� 1ð Þ6

6

" #þ b5S3

t1

10x� 1ð Þ6

6� 1

6x� 1ð Þ4

4

" #þ f1ðxÞ þ f2; ð37Þ

where f1 and f2 are constants of integration which can be determined through boundary conditions (29) by setting n ¼ 0 thatis

dw1

dxð1Þ ¼ 0; w1ð0Þ ¼ 0:

Thus the solution w1ðxÞ has the following form

w1ðxÞ ¼ W1ð1� xÞ2 � 1

2

!þW2

ð1� xÞ4 � 14

!þW3

ð1� xÞ6 � 16

!þW4

ð1� xÞ8 � 18

!: ð38Þ

Similarly, on inserting n ¼ 1 in Eq. (28), we get the second component problem as follows

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Fig. 1.profile


w2 ¼ M2Z x

0

Z x

1w1ðxÞdxdx�

Z x

0

Z x

1A�1dxdx; ð39Þ

or

w2 ¼ M2Z x

0

Z x

1w1dxdx�

Z x

0

Z x

1A1 þ B1 þ C1 þ D1 þ E1½ �dxdx: ð40Þ

On inserting the values of A1;B1;C1;D1; E1 in Eq. (40), we get the following form

w2ðxÞ¼W5ðx�1Þ4�1

12

!þW6

ðx�1Þ6�130

!þW7

ðx�1Þ8�156

!þW8

ðx�1Þ10�190

!þW9

ðx�1Þ14�1182

!: ð41Þ

Substituting the values of (26), (38) and (41) in Eq. (25), the solution of the differential Eq. (19) takes the form

wðxÞ ¼ St þW1ð Þ ðx� 1Þ2 � 12

!þ W2 þW5ð Þ ðx� 1Þ4 � 1

4

!þ W3 þW6ð Þ ðx� 1Þ6 � 1

6

!

þ W4 þW7ð Þ ðx� 1Þ8 � 18

!þW8

ðx� 1Þ10 � 190

!þW9

ðx� 1Þ14 � 1182

!þ � � � ð42Þ

which is the velocity profile for the drainage of the pseudo-plastic fluid in the presence of transverse magnetic field down thevertical wall. Here, Wi; 1 6 i � 9 are constants and are given in Appendix A. According to Eq. (42) the ADM solution’s accu-racy increases by increasing the number of solution components.

In this chapter the flow behavior during the paint film flow problem of an electrically conducting pseudo-plastic fluid inthe presence of transverse magnetic field on an infinite vertical wall has been modeled and analyzed using ADM and Numer-ical approaches. Excellent agreement between the two solutions was achieved as can be observed in Fig. 1 and Table 1.Results obtained are shown graphically in Figs. 2–7. Figs. 2–4 are plotted in order to see the effects of M2; St and

b ¼ k21�l2

1ð ÞU20

d2 on the velocity field wðxÞ. Figs. 2 and 3 give the effect of the magnetic field M2 and the non-Newtonian param-eter b on the velocity field wðxÞ, respectively. In Fig. 2, it is observed that the velocity profile wðxÞ increases with the increasein M2. Also in Fig. 3 the non-Newtonian parameter b has the same effect as the M2 has on the velocity field wðxÞ. Fig. 4 showsthe influence of the Stokes number St on the dimensionless velocity profile wðxÞ. It is noticed that the velocity profileincreases by decreasing the Stokes number St . Table 1 shows the ADM and numerical comparison for the paint film flowof pseudo-plastic fluid keeping St ¼ 0:1; b ¼ 0:5 and M2 ¼ 0:5. The Numerical values are given in Tables 2–6 for different val-ues of non-Newtonian parameters b, Stokes number St and MHD effect M2, which illustrate that for small values of non-New-tonian parameter there is significant improvement in the solution of the velocity profile. The numerical solution is alsopresented for the paint film flow problem which fully support the ADM solution and the whole detail is given below. Thenumerical results are valid for all values of fluid parameters. The graph are plotted for 0 6 b < 1.

4. Numerical solution

A numerical solution of Eq. (19) is obtained in order to provide verification for the ADM solution and a further insight intothe solution behavior. The scheme is coupled with the shooting method due to the nature of the boundary conditions being

0 0.2 0.4 0.6 0.8 1

NumericalADM

0

0.01

0.02

0.03

0.04

0.05

0 0.2 0.4 0.6 0.8 1

NumericalADM

Numerical and ADM solutions (a) by choosing We ¼ 0:1; a ¼ 0:5;/ ¼ 0:3; St ¼ 0:8 and (b) by choosing We ¼ 1:5; a ¼ 0:5;/ ¼ 1; St ¼ 0:1 on velocityfor the drainage problem.

Table 1ADM and numerical comparison for the paint film flow keeping St ¼ 0:1; b ¼ 0:5 and M2 ¼ 0:5using the Section 4.1.

x ADM solution Numerical solution Error

0.1 �8.12732255E�03 �8.11754062E�03 9.78193374E�060.2 �1.52915700E�02 �1.52752691E�02 1.63008899E�050.3 �2.15294718E�02 �2.15089891E�02 2.04827834E�050.4 �2.68729153E�02 �2.68498820E�02 2.30332424E�050.5 �3.13491631E�02 �3.13246636E�02 2.44994928E�050.6 �3.49810172E�02 �3.49557170E�02 2.53001799E�050.7 �3.77869537E�02 �3.77612052E�02 2.57484930E�050.8 �3.97812310E�02 �3.97551613E�02 2.60697114E�050.9 �4.09739733E�02 �4.09475593E�02 2.64140246E�051.0 �4.13712263E�02 �4.13443637E�02 2.68626600E�05

0.0 0.2 0.4 0.6 0.8 1.00.000

0.002

0.004

0.006

0.008

0.010

0.012

x

Vx

M 0.5

M 0.4

M 0.3

M 0.2

M 0.1

Fig. 2. The effect of M2 on the dimensionless velocity profile wðxÞ keeping St ¼ 0:2 and b ¼ 1.

0.0 0.2 0.4 0.6 0.8 1.00.000

0.002

0.004

0.006

0.008

0.010

0.012

x

Vx

0.5

0.4

0.3

0.2

0.1

Fig. 3. The effect of b on the dimensionless velocity profile wðxÞ keeping St ¼ 0:2 and M2 ¼ 0:1.


imposed on both sides of the domain of interest, i.e. x ¼ 0 and 1. However, marching Eq. (19) numerically in the x directioncan cause difficulties when the second derivative terms disappear from the equation as it happens at dw

dx

� �¼ 1

b. This occurs athighly non-linear situations for b P 1 that are not part of the current investigation. However, the shooting method whichbasically replaces the two boundary conditions imposed on each side of the domain by two boundary conditions imposedon the same side of the domain may reach a high magnitude of dw

dx , while seeking the correct boundary conditions. Therefore,in order to avoid such situations, Eq. (18) has been integrated analytically to yield a first order differential integral equationas follows;

dwd~x¼ St~xþM2

Z ~x

0wðnÞdn

!1þ b

dwd~x

� �2" #

; ð43Þ

where ~x ¼ 1� x. ~x is used in Eq. (43) instead of x in order to simplify the integral term while already fulfills the free surfacecondition of dw

d~x ¼ 0 at ~x ¼ 1 when deriving Eq. (43). The boundary condition at the wall wð~x ¼ 1Þ ¼ 0 is still to be fulfilled.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

x

Vx

St 1

St 0.8

St 0.6

St 0.4

St 0.2

Fig. 4. The effect of St on the dimensionless velocity profile wðxÞ keeping b ¼ 0:1 and M2 ¼ 0:5.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

x

Vx

0.70.60.50.40.30.20.1

Fig. 5. The effect of b on the shear stress Sxz keeping St ¼ 0:1 and M2 ¼ 1.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

x

Vx

M 0.7M 0.6M 0.5M 0.4M 0.3M 0.2M 0.1

Fig. 6. The effect of M2 on the shear stress Sxz keeping St ¼ 0:5 and b ¼ 0:1.


The solution was marched in the ~x direction from 0 to 1 using a second order Runge–Kutta method where the marchingwas achieved using the dominant derivative term. Such strategy has been used successfully in non-linear acoustics [26]. Thenumerical marching requires the value of wð~x ¼ 0Þ in order to initiate the marching. Thus a Runge–Kutta method was usedby guessing that value and iterating it until achieving wð~x ¼ 1Þ ¼ 0 by the end of the numerical marching. The bisection iter-ative procedure has been employed and for this purpose it was found sufficient to assume wð~x ¼ 0Þ lies between 0 and 1 inorder to promote convergence in the bisection iterative procedure. Further details on the numerical schemes are givenbelow.

4.1. Numerical solution procedure

The numerical solution procedure is based on the numerical scheme by selecting the dominant derivative term in the Eq.

(43). We identified the dominant derivative terms dwdx or bðdw

dxÞ2

and used that to marched forward while evaluated the other

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

Vx

St 0.7St 0.6St 0.5St 0.4St 0.3St 0.2St 0.1

Fig. 7. The effect of St on the shear stress Sxz keeping b ¼ 0:5 and M2 ¼ 0:1.

Table 2Component solutions of the ADM for the paint film flow problem for the fluid parameters St ¼ 0:2; b ¼ 0:1;M2 ¼ 1.

x Zeroth component 1st Component 2nd Component 3rd Component

0.0 0.00 0.00 0.00 0.000.1 �0.019 �0.0123074 �0.0123295 �0.03139840.2 �0.036 �0.0228215 �0.0228574 �0.05897020.3 �0.051 �0.0317076 �0.0317516 �0.08289150.4 �0.064 �0.0391118 �0.0391601 �0.01033160.5 �0.075 �0.0451612 �0.0452116 �0.01203770.6 �0.084 �0.0499634 �0.0500147 �0.01341860.7 �0.091 �0.0536069 �0.0536584 �0.01448320.8 �0.096 �0.0561601 �0.0562118 �0.01523860.9 �0.099 �0.0576724 �0.0577241 �0.01568991.0 �0.100 �0.0581733 �0.0582249 �0.0158399

Table 3Component solutions of the ADM for the paint film flow problem for the fluid parameters St ¼ 0:2; b ¼ 0:4, M2 ¼ 0:5.


0.0 0.00 0.00 0.00 0.000.1 �0.019 �0.017076 �0.0171127 �0.03618560.2 �0.036 �0.0322762 �0.0323391 �0.06843080.3 �0.051 �0.0456259 �0.0457067 �0.0968050.4 �0.064 �0.0571481 �0.0572405 �0.01213410.5 �0.075 �0.0668628 �0.0669622 �0.01420640.6 �0.084 �0.0747873 �0.0748905 �0.01589930.7 �0.091 �0.0809358 �0.0810408 �0.01721440.8 �0.096 �0.0853196 �0.0854252 �0.01815280.9 �0.099 �0.0879466 �0.0880524 �0.01871551.0 �0.100 �0.0888217 �0.0889275 �0.018903



0.0 0.00 0.00 0.00 0.000.1 �0.019 �0.0172089 �0.0172262 �0.03626230.2 �0.036 �0.0325034 �0.032533 �0.06857970.3 �0.051 �0.0459174 �0.0459554 �0.09700630.4 �0.064 �0.0574811 �0.0575246 �0.01215770.5 �0.075 �0.0672209 �0.0672676 �0.01423210.6 �0.084 �0.075159 �0.0752075 �0.01592620.7 �0.091 �0.081314 �0.0813634 �0.01724180.8 �0.096 �0.0857002 �0.0857498 �0.01818040.9 �0.099 �0.0883278 �0.0883775 �0.01874321.0 �0.100 �0.0892029 �0.0892526 �0.0189307




0.0 0.00 0.00 0.00 0.000.1 �0.019 �0.0168090 �0.0168895 �0.03604510.2 �0.036 �0.0318199 �0.0319570 �0.06815400.3 �0.051 �0.0450408 �0.0452161 �0.09642780.4 �0.064 �0.0564798 �0.0566795 �0.012089800.5 �0.075 �0.0661443 �0.0663587 �0.014158000.6 �0.084 �0.0740414 �0.0742637 �0.015848600.7 �0.091 �0.0801770 �0.0804030 �0.01716260.8 �0.096 �0.0845559 �0.0847833 �0.01810070.9 �0.099 �0.0871818 �0.0874096 �0.01866331.0 �0.100 �0.0880569 �0.0882847 �0.0188508



0.0 0.00 0.00 0.00 0.000.1 �0.019 �0.0169427 �0.0170005 �0.03611310.2 �0.036 �0.0320484 �0.0321471 �0.06828890.3 �0.051 �0.0453337 �0.0454603 �0.09661220.4 �0.064 �0.0568143 �0.0569588 �0.01211150.5 �0.075 �0.0665039 �0.0666592 �0.01418170.6 �0.084 �0.0744147 �0.0745758 �0.01587350.7 �0.091 �0.0805568 �0.0807206 �0.01718800.8 �0.096 �0.0849381 �0.0851030 �0.01812620.9 �0.099 �0.0875646 �0.0877297 �0.01868891.0 �0.100 �0.0884397 �0.0886048 �0.0188764


term using the backward finite difference scheme. This method has been used successfully to solve the non-linear soundpropagation problems using the Westervelt equation [26].

1. First order approachAt x ¼ xi where xi ¼ ði� 1ÞDx;Dx ¼ 1

N�1, where Dx is the spatial marching step and N is the number of grid points. Weevaluated dw

dx using first order backward scheme;

dwðxiÞdx

¼ wðxiÞ �wðxi � DxÞDx

þ OðDxÞ: ð44Þ

Option 1. If bj dwdx j

26 1, then the Eq. (43) can be seen as weakly non-linear and is marched forward as follows

wðxi þ DxÞ ¼ wðxiÞ þ Dx Stxi þM2Z xi

0wðxiÞdx

� �1þ b

wðxiÞ �wðxi � DxÞDx

� �2" #

;

where the integral termR xi

0 wðxiÞdx is computed using the Trapezoidal formula.Option 2. If bj dw

dx j2 P 1, then the Eq. (43) can be seen as strongly non-linear and is marched forward as follows

wðxi þ DxÞ ¼ wðxiÞ þ Dx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1b

wðxi � DxÞ �wðxiÞDx Stxi þM2 R xi

0 wðxiÞdxh i� 1

24

35

vuuut : ð45Þ

However, if the right hand side of Eq. (45) is an imaginary number, then we used Option 1 to marched Eq. (43).2. Extension to second order approachHere, the marching method is the second order Runge–Kutta for the Eq. (43).2.1. We calculate dw

dx using the second order backward finite difference scheme

dwðxiÞdx

¼ wðxiÞ � 4wðxi � DxÞ þ ðwi � 2DxÞ2Dx

þ OðDxÞ2: ð46Þ

For the velocity profile wðxÞ we computed w xi þ Dx2

� �at x ¼ xiþ1

2and proceeded as in the first order forward method.


26 1, then the Eq. (43) can be seen as weakly non-linear and is marched forward as follows

w xi þDx2

� �¼ wi þ

Dx2

Stxi þM2Z xi

0wðxiÞdx

� �1þ b

wðxiÞ � 4wðxi � DxÞ þ ðwi � 2DxÞ2Dx

� �2" #

:



2 P 1, then the Eq. (43) can be seen as strongly non-linear and is marched forward as follows

w xi þDx2

� �¼ wðxiÞ þ �X1;

where

�X1 ¼Dx2b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðxiÞ � 4wðxi � DxÞ þ ðwi � 2DxÞ

2DxStxi þM2

Z xi

0wðxiÞdx

� �� 1

s:

However, if the right hand side of the above Equation is an imaginary number, then the above equation is marched usingOption 1.

2.2. Further, we calculate dwdx at x ¼ xi þ Dx

2 using the backward finite difference scheme and computed wðxi þ DxÞ atx ¼ xi þ Dx

dwb

dxjx¼xiþDx

2¼

8w xi þ Dx2

� �� 9wðxiÞ þ ðwi � DxÞ

3Dxþ OðDxÞ2: ð47Þ

Proceed with the same approach as before as discussed above.Option 1. If bjðdwb

dx Þx¼xiþDx2j2 6 1, then the Eq. (43) can be seen as weakly non-linear and is marched forward as follows

wðxi þ DxÞ ¼ wi þ Dx � X1;

where

X1 ¼ Stxi þDx2þM2

Z xiþDx2

0wðxiÞdx

" #1þ b

dwb

dx

� �2

jx¼xiþ1

2

" #: ð48Þ

Option 2. bjðdwbdx Þx¼xiþDx

2j2 > 1, then

wðxi þ DxÞ ¼ wðxiÞ þ Dx � X2; ð49Þ

where

X2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1b

dwbdx

iþ1

2

Stxiþ12þM2 R xiþDx

20 wðxiÞdx

h i� 1

264

375

vuuuut :

Again, if the right hand side of the above Equation is an imaginary number, then the Eq. (49) is marched using Option 1.

4.1.1. Flow rate and average velocity of the thin film flowThe flow rate per unit width is given by the formula

Q ¼Z 1

0vðxÞdx; ð50Þ

equivalently

Q ¼ � St

3þW1

3þW2

5þW3

7þW4

9þW5

5þW6

7þW7

9þW8

99þ W9

195

� �þ � � �

The average velocity V for drainage problem is given by

V ¼ Q :

4.1.2. Shear stress on the beltThe expression for the shear stress on the belt is obtained using Eq. (16)

Sxzjx¼0 ¼St þW1 þW2 þW3 þW4 þW5 þW6 þW7 þ W8

9 þW913

b St þW1 þW2 þW3 þW4 þW5 þW6 þW7 þ W89 þ

W913

� �2 � 1:

Using the above expression we obtain the shear stress exerted by the wall on the fluid for the paint film flow problem. Theeffect of the fluid parameters such as the magnetic effect M2, the non-Newtonian parameter b, the Stokes number St on theshear stress are discussed through graphs. Figs. 5–7 deal with the shear stress effect on the wall under the influence of fluidparameters. In all these three figures the shear stress Sxz has a direct relation with all the fluid parameters.


4.1.3. Vorticity for the problemThe vorticity function x (z-component of vorticity vector) of the flow is given by

x¼ StþW1ð Þ x�1ð Þþ W2þW5ð Þðx�1Þ3þ W3þW6ð Þ x�1ð Þ5 W4þW7ð Þ x�1ð Þ7þW8ðx�1Þ9

9

!þW9

ðx�1Þ13

13

!þ��

" #k;

where k is the unit vector in the z-direction.

Appendix A. Constants defined for the solution of the drainage problem

b1¼�b;

b2¼�b2St ;

b3¼�2bSt;

b4¼�M2b2;

b5¼�2bM2;

W1¼�M2 St

2;

W2¼ M2 St

6þb1

S3t

3þb3

S2t

3�b5

S3t

6

!;

W3¼ b2S4

t

5þb5

S3t

10�b4

S5t

10

!;

W4¼ b4S5

t

14;

W5¼ �14

M2S4t �M2S2

t b3þ1724

M2S3t b5�

112

S5t b1b5�

130

S6t b2b5�

112

S4t b3b5þ

13S7t b4b5

1680þ 1

40S5

t b25

!;

W6¼7

12M2S4

t þ16

S6t b1�2M2S4

t b2þ16

S5t b3þ

2924

M2S5t b4�

112

S7t b1b4�

130

S8t b2b4�

112

S6t b3b4

�

þ13S9t b

24

1680�3

4M2S3

t b5�1

12S6

t b5þ1

40S7

t b4b5þ2Stb3M2St

6þ1

3S3

t b1þ13

S2t b3�

16

S3t b5

!

�S2t b5

M2St

6þ1

3S3

t b1þ13

S2t b3�

16

S3t b5

!!;

W7¼ � 512

M2S4t �

13

S6t b1þ

110

S7t b2�

13

S5t b3�

54

M2S5t b4�

120

S8t b4þ

124

M2S3t b5þ

1360

S6t b5þ

112

S5t b1b5þ

112

S4t b3b5

�

� 124

S5t b

25þ4S3

t b2M2St

6þ1

3S3

t b1þ13

S2t b3�

16

S3t b5

!�2S4

t b4M2St

6þ1

3S3

t b1þ13

S2t b3�

16

S3t b5

!

þS2t b5

M2St

6þ1

3S3

t b1þ13

S2t b3�

16

S3t b5

!þ2Stb3

15

S4t b2�

110

S5t b4þ

110

S3t b5

� ��S2

t b515

S4t b2�

110

S5t b4þ

110

S3t b5

� �!;

W8¼1

12M2S4

t þ16

S6t b1�

15

S7t b2þ

16

S5t b3þ

124

M2S5t b4þ

19140

S8t b4þ

112

S7t b1b4þ

1984

S6t b3b4�

1160

S6t b5þ

130

S6t b2b5

�

�109840

S7t b4b5þ

160

S5t b

25þ2S4

t b4M2St

6þ1

3S3

t b1þ13

S2t b3�

16

S3t b5

!þ4S3

t b215

S4t b2�

110

S5t b4þ

110

S3t b5

� �

�2S4t b4

15

S4t b2�

110

S5t b4þ

110

S3t b5

� �þS2

t b515

S4t b2�

110

S5t b4þ

110

S3t b5

� ��;

W9¼1

10S7

t b2�17

140S8

t b4þ67

210S8

t b2b4�67

420S9

t b24þ

120

S6t b5þ

163S7t b4b5

1680þ2S4

t b415

S4t b2�

110

S5t b4þ

110

S3t b5

� � !

� ðx�1Þ12�1122

!þ 1

28S8

t b4þ17

112S9

t b24

� �:


References

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Thin film flow of magnetohydrodynamic (MHD) pseudo-plastic fluid on vertical wall

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