# Unsteady Stagnation-Point Flow of a Viscoelastic Fluid in the Presence of a Magnetic Field

Authors: F. Labropulu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

The unsteady two-dimensional stagnation point flow of the Walters B' fluid
impinging on an infinite plate in the presence of a transverse magnetic field is examined and
solutions are obtained. It is assumed that the infinite plate aty=0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number and the Weissenberg number.


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## Body

## 1. Introduction

During the past few decades, non-Newtonian
viscoelastic fluids have become more and more important industrially. Among
these fluids are the fluids of diffferential type such as the Walters B' fluid
[1]. Behaviour of viscoelastic fluids cannot be accurately described by the
Newtonian fluid model. The equations of motion of non-Newtonian fluids are
highly non-linear and one order higher than the Navier-Stokes equations. For
this reason, boundary conditions in addition to the non-slip condition are
required to have a well-posed problem. Only in some special cases where the
higher order nonlinear terms in these equations can be neglected thereby
reducing their order, are the “no-slip” condition sufficient to yield
unique solutions. In general, Rajagopal [2], Rajagopal and Gupta [3], Rajagopal
[4] and Rajagopal and Kaloni [5] have shown that the absence of this additional
boundary condition leads to non-unique solutions for problems involving the
flow of fluids of differential type such as Walters' B' fluid in a bounded
domain.The flow of an incompressible viscous fluid over a
moving plate has its importance in many industrial applications. The extrusion
of plastic sheets, fabrication of adhesive tapes and application of coating
layers onto rigid substrates are some of the examples. If a magnetic field is
present, viscous flows due to a moving plate in an electro-magnetic field, ie
magnetohydrodymanic (MHD) flows, are relevant to many practical applications in
the metallurgy industry, such as the cooling of continuous strips and filaments
drawn through a quiescent fluid and the purification of molten metals from
non-metallic inclusions.In the history of fluid dynamics, considerable
attention has been given to the study of 2-D stagnation point flow. Hiemenz [6]
derived an exact solution of the steady flow of a Newtonian fluid impinging
orthogonally on an infinite flat plate. Stuart [7], Tamada [8] and Dorrepaal
[9] independently investigated the solutions of a stagnation point flow when
the fluid impinges obliquely on the plate. Beard and Walters [10] used
boundary-layer equations to study two-dimensional flow near a stagnation point
of a viscoelastic fluid. Dorrepaal et al. [11] investigated the behaviour of a
viscoelastic fluid impinging on a flat rigid wall at an arbitrary angle of
incidence. Labropulu et al. [12] studied the oblique flow of a viscoelastic
fluid impinging on a porous wall with suction or blowing. The Hiemenz flow of a
Newtonian fluid in the presence of a magnetic field was first considered by Na
[13] and later by Ariel [14]. The flow of the non-Newtonian Walters' B' fluid
in the presence of a transverse magnetic field was studied by Ariel [15]. Attia
[16] investigated the steady flow of a non-Newtonian second grade fluid at a
stagnation point with heat transfer in an an external uniform magnetic field.Furthermore, the unsteady or time dependent viscous
flow near a stagnation-point has also been widely investigated. Glauert [17]
and Rott [18] studied the stagnation-point flow of a Newtonian fluid when the
plate performs harmonic oscillations in its own plane. Srivastava [19]
investigated the same problem for a non-Newtonian second grade fluid using the
Karman-Pohlhausen method. Labropulu et al. [20] used series methods to solve the
unsteady stagnation point flow of a viscoelastic fluid impinging on an
oscillating flat plate.In this work, the two-dimensional unsteady stagnation
point flow of the Walters B' fluid impinging on an infinite plate in the
presence of a magnetic field is examined and solutions are obtained. It is
assumed that the infinite plate aty=0 is making harmonic oscillations in its own
plane and the magnetic field is transverse or perpendicular everywhere in the
flow field. Solutions for small and large frequencies of the oscillations are
obtained using a finite difference technique.


## 2. Flow Equations and Boundary Conditions

We consider the two-dimensional flow of an
incompressible non-Newtonian Walters' B' fluid against an infinite plate normal
to the flow in the presence of a magnetic field. Thex-axis is along the plate and the y-axis is normal to the plate. We assume that
the plate makes harmonic oscillations on its own plane with velocity in the x-direction equal to aeiωt where a and ω are constants. The unsteady two-dimensional
flow of a viscous incompressible non-Newtonian Walters B' fluid in the presence
of a magnetic field is governed by (see Beard and Walters [10] and Na
[13])(2.1)∂u∂x+∂v∂y=0,(2.2)∂u∂t+u∂u∂x+v∂u∂y+1ρ∂p∂x=ν∇2u−α1ρ{∂∂t(∇2u)+(u∂∂x+v∂∂y)∇2u−∂u∂x∇2u−∂u∂y∇2v−2[∂u∂x∂2u∂x2+∂v∂y∂2u∂y2+(∂u∂y+∂v∂x)∂2u∂x∂y]}−σB0ρu,(2.3)∂v∂t+u∂v∂x+v∂v∂y++1ρ∂p∂y=ν∇2v−α1ρ{∂∂t(∇2v)+(u∂∂x+v∂∂y)∇2v−∂v∂x∇2u−∂v∂y∇2v−2[∂u∂x∂2v∂x2+∂v∂y∂2v∂y2+(∂u∂y+∂v∂x)∂2v∂x∂y]},where u=u(x,y,t), v=v(x,y,t) are the velocity components, p=p(x,y,t) is the pressure, ν=μ/ρ is the kinematic viscosity, ρ is the fluid density, σ is the electrical conductivity, B0 is the magnetic field and α1 is a measure of the viscoelasticity of the
fluid. It is assumed that σB0≪1,
so that it is possible to neglect the effect of the induced magnetic field.For this problem, the boundary conditions are given
by(2.4)u=aeiωt,v=0aty=0,u=cx,v=−cyasy→∞,where c>0 is a constant which has the units of inverse
time. The quantity U=cx is the velocity of the fluid outside the
boundary layer.Continuity (2.1) implies the existence of a
streamfunction ψ(x,y,t) such that(2.5)u=∂ψ∂y,v=−∂ψ∂x.Substitution of (2.5) in
(2.2) and (2.3) and elimination of pressure from the resulting equations using pxy=pyx yields(2.6)∂∂t(∇2ψ)+α1ρ∂∂t(∇4ψ)−∂(ψ,∇2ψ)∂(x,y)−α1ρ∂(ψ,∇4ψ)∂(x,y)−ν∇4ψ+σB0ρ∂2ψ∂y2=0.Having obtained a solution of (2.6), the velocity components are given by (2.5).The boundary conditions in terms of the streamfunctionψ(x,y,t) take the form(2.7)∂ψ∂y=aeiωt,∂ψ∂x=0aty=0,∂ψ∂y=cx,∂ψ∂x=cyasy→∞.The shear stress componentτ12 is given by(2.8)τ12=μ{∂2ψ∂y2−∂2ψ∂x2}−α1[∂ψ∂y(∂3ψ∂x∂y2−∂3ψ∂x3)−∂ψ∂x(∂3ψ∂y3−∂3ψ∂x2∂y)+2∂2ψ∂x∂y∂2ψ∂y2+2∂2ψ∂x2∂2ψ∂x∂y].


## 3. Solutions

Following Glauert [17], we seek a solution in the
form(3.1)ψ=cxf(y)+aeiωtg(y)The boundary conditions take the
form(3.2)f(0)=f′(0)=0,g′(0)=1,f′(∞)=1,f(∞)=y,g′(∞)=0.Using (3.1) in (2.6), we
obtain(3.3)νf(iv)+c(ff′′′−f′f′′)+α1cρ(ff(v)−f′f(iv))−σB0ρf′′=0,νg(iv)−iωg′′+α1ρiωg(iv)+c(fg′′′−f′′g′)+αcρ(fg(v)−f(iv)g′)−σB0ρg′′=0.Non-dimensionalizing
using(3.4)η=cνy,f(y)=νcF(η),g(y)=νcG(η)we get(3.5)F(iv)+FF′′′−F′F′′+We(FF(v)−F′F(iv))−MF′′=0,G(iv)+FG′′′−F′′G′+We(FG(v)−F(iv)G′)−iωcG′′−iωWecG(iv)−MG′′=0,where We=a1c/μ,
the Weissenberg number, is the ratio of the elastic effects over the viscous
effects and M=σB0/ρc, is the Hartmann's number.The asymptotic behaviour ofF(y) far away from the plate (outside the boundary
layer) is given by(3.6)F(y)~y−Aasy→∞,where A is a constant that accounts for the boundary
layer displacement.Using this asymptotic behaviour, we
obtain(3.7)F′(∞)=1,F′′(∞)=F′′′(∞)=F(iv)(∞)=0Integrating (3.5) once with respect
to η and using the conditions at infinity, we have
(see [10])(3.8)F′′′+FF′′−F′2+We(FF(iv)−2F′F′′′+F′′2)−MF′=−1−M,F(0)=0,F′(0)=0,F′(∞)=1,(3.9)G′′′+FG′′−F′G′+We(FG(iv)−F′G′′′+F′′G′′−F′′′G′)−iωc(G′+WeG′′′)−MG′=0,G′(0)=1,G′(∞)=0.System (3.8) with M=0 has been solved numerically by many authors
(Beard and Walter [10], Ariel [21]). Using the shooting method with the finite
difference technique described by Ariel [21], we find that F′′(0)=1.23259 when We=0 and M=0 which is in good agreement  with the values
obtained by Hiemenz [6] and Glauert [17]. Numerical values of F′′(0) for different  values of We and M are shown in Table 1. These values are in good
agreement with the values obtained by Ariel [21] when M=0,
Ariel [15] for all M and Labropulu et al. [20] when M=0.
These values are also in good agreement with the values obtained by Attia [16]
when K=We=0.0.
Figure 1 shows the profiles of F′ for various We when M=0.
Figure 2 shows the profiles of F′ for various M when We=0.
Figure 3 depicts the profiles of F′ for various M when We=0.2.
We observed that as the elasticity of the fluid and the Hartman's number
increase, the velocity near the wall increases.Table 1
Numerical
values ofF′′(0),ϕ0′(0), ϕ1′(0) and ϕ2′(0) for different values of M and We=0.
MF′′(0)ϕ0′(0)ϕ1′(0)ϕ2′(0)0.01.23259−0.811318−0.493070.09452760.51.41976−1.05648−0.409990.09454881.01.58533−1.2615−0.3571490.05821492.01.87353−1.60113−0.292280.04005422.251.93895−1.67603−0.2808050.020476Figure 1
Variation ofF′(η) for various We and M=0.Figure 2
Variation ofF′(η) for various M and We=0.Figure 3
Variation ofF′(η) for various M and We=0.2.Lettingϕ(η)=G′(η),
then system (3.9) becomes(3.10)ϕ′′+Fϕ′−F′ϕ+We(Fϕ′′′−F′ϕ′′+F′′ϕ′−F′′′ϕ)−iωc(ϕ+Weϕ′′)−Mϕ=0,ϕ(0)=1,ϕ(∞)=0.The only parameter in system (3.10)
is the frequency ratio ω/c.
Series solutions will be developed, valid for small and large values of ω/c, respectively.


### 3.1. Small Values ofω/c

Consider the case whereω=0,
which implies that the plate velocity has the constant value a.
Letting ϕ=ϕ0,
then system (3.10) gives(3.11)ϕ0′′+Fϕ0′−F′ϕ0+We(Fϕ0′′′−F′ϕ0′′+F′′ϕ0′−F′′′ϕ0)−Mϕ0=0,ϕ0(0)=1,ϕ0(∞)=0.This system is solved numerically using a shooting
method and it is found that forWe=0 and M=0, ϕ0′(0)=−0.811318 which is in good agreement with the value
obtained by Glauert [17]. Numerical values of ϕ0′(0) for different values of We and M are shown in Table 1. These values are in good
agreement with the values obtained by Labropulu et al. [20] for M=0.
Figure 4 depicts the profiles of ϕ0 for various values of We and M=0.
Figure 5 shows the profiles of ϕ0 for various M when We=0.1.Figure 4
Variation ofϕ0(η) for various We and M=0.Figure 5
Variation ofϕ0(η) for various M and We=0.1.For small but non-zero values ofω/c, we let(3.12)ϕ(η)=∑n=0∞(iωc)nϕn(η)=ϕ0(η)+iωcϕ1(η)+(iωc)2ϕ2(η)+⋯.Substituting (3.12) into (3.10), we get for n≥1(3.13)ϕn′′+Fϕn′−F′ϕn+We(Fϕn′′′−F′ϕn′′+F′′ϕn′−F′′′ϕn)−Mϕn=ϕn−1+Weϕn−1′′,ϕn(0)=0,ϕn(∞)=0.This system can be solved numerically either by using
the perturbation technique or by a finite difference scheme. Numerical
integration of system (3.13) for n=1 using a finite difference technique gives for We=0 and M=0, ϕ1′(0)=−0.49307 which is in good agreement with Glauert's value
[17]. Numerical values of ϕ1′(0) for different values of We and M are shown in Table 1. These values are in good
agreement with Labropulu et al. [20] for M=0.
Figure 6 shows the profiles of ϕ1 for various values of We and M=1 and Figure 7 depicts the profiles of ϕ1 for various values of M and We=0.2.Figure 6
Variation ofϕ1(η) for various We and M=1.Figure 7
Variation ofϕ1(η) for various M and We=0.2.Numerical integration of system (3.13) for n=2 using a finite difference technique gives for We=0 and M=0, ϕ2′(0)=0.0945488 which is in good agreement with Glauert's
value [17]. Numerical values of ϕ2′(0) for different values of We and M are shown in Table 1. These values are in good
agreement with Labropulu et al. [20] when M=0.
Figure 8 depicts the profiles of ϕ2 for various values of We and M=1 and Figure 9 shows the profiles of ϕ2 for various values of M and We=0.2.Figure 8
Variation ofϕ2(η) for various We and M=1.Figure 9
Variation ofϕ2(η) for various M and We=0.2.The oscillating component of the shear stress on the
wall is given by(3.14)τ12ρa2=cνa2eiωt[ϕ0′(0)+iωcϕ1′(0)−WeF′′(0)],where F′′(0),ϕ0′(0),ϕ1′(0) and ϕ2′(0) are given in Tables 1, 2, 3, and 4 for different values
of We and M.
When We=0 and M=0,
the value of the shear stress on the wall is in good agreement with the value
obtained by Glauert [17].Table 2
Numerical
values ofF′′(0),ϕ0′(0), ϕ1′(0), and ϕ2′(0) for different values of M and We=0.1.
MF′′(0)ϕ0′(0)ϕ1′(0)ϕ2′(0)0.01.36946−0.86708−0.5473010.0658510.51.56814−1.11269−0.478790.03342951.01.74431−1.31954−0.437190.01703922.02.05118−1.66405−0.3903210.00143892.252.12082−1.74024−0.38277−0.0008256Table 3
Numerical
values ofF′′(0),ϕ0′(0), ϕ1′(0), and ϕ2′(0) for different values of M and We=0.2.
MF′′(0)ϕ0′(0)ϕ1′(0)ϕ2′(0)0.01.58676−0.947459−0.6338920.02219970.51.80411−1.19217−0.582324−0.00923791.01.99811−1.40046−0.553854−0.02650882.02.33916−1.75019−0.528391−0.0449822.252.41702−1.82787−0.525596−0.0480166Table 4
Numerical
values ofF′′(0),ϕ0′(0), ϕ1′(0), and ϕ2′(0) for different values of M and We=0.3.
MF′′(0)ϕ0′(0)ϕ1′(0)ϕ2′(0)0.02.11092−1.10873−0.842848−0.0761150.52.34531−1.34347−0.806961−0.114921.02.56139−1.5492−0.791711−0.139112.02.95028−1.90173−0.79062−0.1700322.253.0402−1.98085−0.79368−0.175938


### 3.2. Large Values ofω/c

Whenω/c is large, we let(3.15)Y=iωcη=iωνy.Lettingc/iω=α,
then d/dη=d/αdY and  (3.10) takes the
form(3.16)1α2d2ϕdY2+1α[FdϕdY−dFdYϕ]+1α3We[Fd3ϕdY3−dFdYd2ϕdY2+d2FdY2dϕdY−d3FdY3ϕ]−1α2ϕ−Weα2d2ϕdY2−Mϕ=0.SinceWe is small for most fluids which behave as
second order fluids (see Markovitz and Coleman [22]), we follow Srivastava [19]
and take We to be of the order of α2.
Thus, We=mα2 and (3.16) becomes(3.17)(1−m)d2ϕdY2+α[FdϕdY−dFdYϕ]+mα[Fd3ϕdY3−dFdYd2ϕdY2+d2FdY2dϕdY−d3FdY3ϕ]−ϕ−Mα2ϕ=0.The expansion forF(Y) near the wall Y=0 is(3.18)F(Y)=12Aα2Y2−16(1+M)α3Y3+124MAα4Y4−16mA2α5Y3+1120(A2−M2−M)α5Y5+⋯,where A=F′′(0).Since for large values ofω/c the parameter α is small, we let(3.19)ϕ=∑n=0∞αnϕn(Y)=ϕ0(Y)+αϕ1(Y)+α2ϕ2(Y)+⋯.The boundary conditions are(3.20)ϕ0(0)=1,ϕn(0)=0ifn≥1,ϕn(∞)=0∀n.Substituting (3.20) in (3.17) and equating the
coefficients of different powers of α to zero, we find that the boundary-value
problem for ϕ0(Y) is(3.21)(1−m)d2ϕ0dY2−ϕ0=0,ϕ0(0)=1,ϕ0(∞)=0with solution ϕ0(Y)=exp[−(Y/1−m)] provided 0≤m<1.The second equation gives thatϕ1 is zero. The next four equations for ϕ2(Y),ϕ3(Y),ϕ4(Y) and ϕ5(Y) are(3.22)(1−m)d2ϕ2dY2−ϕ2=Mϕ0,(1−m)d2ϕ3dY2−ϕ3=Mϕ1−12mAY2d3ϕ0dY3+mAYd2ϕ0dY2+(−12AY2−mA)dϕ0dY+AYϕ0,(1−m)d2ϕ4dY2−ϕ4=Mϕ2+16m(1+M)Y3d3ϕ0dY3−12m(1+M)Y2d2ϕ0dY2+[16(1+M)Y3+m(1+M)Y]dϕ0dY−[12(1+M)Y2+m(1+M)]ϕ0−12mAY2d3ϕ1dY3+mAYd2ϕ1dY2−(12AY2−Am)dϕ1dY+AYϕ1,(1−m)d2ϕ5dY2−ϕ5=Mϕ3−12mAY2d3ϕ3dY3=mAd2ϕ2dY2−(12AY2+mA)dϕ2dY+AYϕ2+16m(1+M)Y3d3ϕ1dY3−12m(1+M)Y2d2ϕ1dY2+[16(1+M)Y3+m(1+M)Y]dϕ1dY−[12(1+M)Y2+m(1+M)]ϕ1−124mAMY4d3ϕ0dY3+16mAMY3d2ϕ0dY2−(124MAY4+12mAMY2)dϕ0dY+(16MAY3+mAMY)ϕ0.Solving these equations and using the boundary
conditions, we obtain(3.23)ϕ2(Y)=−M21−mYe−(1/1−m)Y,ϕ3(Y)=−A1−me−(Y/1−m)[3+4m8Y+381−mY2+112(1−m)Y3],ϕ4(Y)=181−me−(Y/1−m){[12(3+4m)(1+M)+M2]Y+[12(3+4m)(1+M)+M2]Y2+1+M1−mY3+16(1−m)3/2Y4},ϕ5(Y)=AM81−me−(Y/1−m){−12m3+25m2−30m+941−mY+12m2−13m+98Y2+4m2−5m+66(1−m)3/2Y3+3m+112(1−m)2Y4−130(1−m)3/2Y5}provided 0≤m<1.
If m=0 and M=0,
we recover the solutions for the Newtonian fluid obtained by Glauert [13] and
if M=0,
we recover the solutions obtained by Labropulu et al. [16].The oscillating component of the shear stress on the
wall is given by(3.24)τ12ρa2=−cνa2eiωt[1α1−m+M21−mα+(3+12m−8m2)A8(1−m)α2−(3+4m)(1+M))+2M2161−mα3+⋯].If m=0 and M=0,
the shear stress is in good agreement with the result obtained by Glauert [13].


## 3.1. Small Values ofω/c

Consider the case whereω=0,
which implies that the plate velocity has the constant value a.
Letting ϕ=ϕ0,
then system (3.10) gives(3.11)ϕ0′′+Fϕ0′−F′ϕ0+We(Fϕ0′′′−F′ϕ0′′+F′′ϕ0′−F′′′ϕ0)−Mϕ0=0,ϕ0(0)=1,ϕ0(∞)=0.This system is solved numerically using a shooting
method and it is found that forWe=0 and M=0, ϕ0′(0)=−0.811318 which is in good agreement with the value
obtained by Glauert [17]. Numerical values of ϕ0′(0) for different values of We and M are shown in Table 1. These values are in good
agreement with the values obtained by Labropulu et al. [20] for M=0.
Figure 4 depicts the profiles of ϕ0 for various values of We and M=0.
Figure 5 shows the profiles of ϕ0 for various M when We=0.1.Figure 4
Variation ofϕ0(η) for various We and M=0.Figure 5
Variation ofϕ0(η) for various M and We=0.1.For small but non-zero values ofω/c, we let(3.12)ϕ(η)=∑n=0∞(iωc)nϕn(η)=ϕ0(η)+iωcϕ1(η)+(iωc)2ϕ2(η)+⋯.Substituting (3.12) into (3.10), we get for n≥1(3.13)ϕn′′+Fϕn′−F′ϕn+We(Fϕn′′′−F′ϕn′′+F′′ϕn′−F′′′ϕn)−Mϕn=ϕn−1+Weϕn−1′′,ϕn(0)=0,ϕn(∞)=0.This system can be solved numerically either by using
the perturbation technique or by a finite difference scheme. Numerical
integration of system (3.13) for n=1 using a finite difference technique gives for We=0 and M=0, ϕ1′(0)=−0.49307 which is in good agreement with Glauert's value
[17]. Numerical values of ϕ1′(0) for different values of We and M are shown in Table 1. These values are in good
agreement with Labropulu et al. [20] for M=0.
Figure 6 shows the profiles of ϕ1 for various values of We and M=1 and Figure 7 depicts the profiles of ϕ1 for various values of M and We=0.2.Figure 6
Variation ofϕ1(η) for various We and M=1.Figure 7
Variation ofϕ1(η) for various M and We=0.2.Numerical integration of system (3.13) for n=2 using a finite difference technique gives for We=0 and M=0, ϕ2′(0)=0.0945488 which is in good agreement with Glauert's
value [17]. Numerical values of ϕ2′(0) for different values of We and M are shown in Table 1. These values are in good
agreement with Labropulu et al. [20] when M=0.
Figure 8 depicts the profiles of ϕ2 for various values of We and M=1 and Figure 9 shows the profiles of ϕ2 for various values of M and We=0.2.Figure 8
Variation ofϕ2(η) for various We and M=1.Figure 9
Variation ofϕ2(η) for various M and We=0.2.The oscillating component of the shear stress on the
wall is given by(3.14)τ12ρa2=cνa2eiωt[ϕ0′(0)+iωcϕ1′(0)−WeF′′(0)],where F′′(0),ϕ0′(0),ϕ1′(0) and ϕ2′(0) are given in Tables 1, 2, 3, and 4 for different values
of We and M.
When We=0 and M=0,
the value of the shear stress on the wall is in good agreement with the value
obtained by Glauert [17].Table 2
Numerical
values ofF′′(0),ϕ0′(0), ϕ1′(0), and ϕ2′(0) for different values of M and We=0.1.
MF′′(0)ϕ0′(0)ϕ1′(0)ϕ2′(0)0.01.36946−0.86708−0.5473010.0658510.51.56814−1.11269−0.478790.03342951.01.74431−1.31954−0.437190.01703922.02.05118−1.66405−0.3903210.00143892.252.12082−1.74024−0.38277−0.0008256Table 3
Numerical
values ofF′′(0),ϕ0′(0), ϕ1′(0), and ϕ2′(0) for different values of M and We=0.2.
MF′′(0)ϕ0′(0)ϕ1′(0)ϕ2′(0)0.01.58676−0.947459−0.6338920.02219970.51.80411−1.19217−0.582324−0.00923791.01.99811−1.40046−0.553854−0.02650882.02.33916−1.75019−0.528391−0.0449822.252.41702−1.82787−0.525596−0.0480166Table 4
Numerical
values ofF′′(0),ϕ0′(0), ϕ1′(0), and ϕ2′(0) for different values of M and We=0.3.
MF′′(0)ϕ0′(0)ϕ1′(0)ϕ2′(0)0.02.11092−1.10873−0.842848−0.0761150.52.34531−1.34347−0.806961−0.114921.02.56139−1.5492−0.791711−0.139112.02.95028−1.90173−0.79062−0.1700322.253.0402−1.98085−0.79368−0.175938


## 3.2. Large Values ofω/c

Whenω/c is large, we let(3.15)Y=iωcη=iωνy.Lettingc/iω=α,
then d/dη=d/αdY and  (3.10) takes the
form(3.16)1α2d2ϕdY2+1α[FdϕdY−dFdYϕ]+1α3We[Fd3ϕdY3−dFdYd2ϕdY2+d2FdY2dϕdY−d3FdY3ϕ]−1α2ϕ−Weα2d2ϕdY2−Mϕ=0.SinceWe is small for most fluids which behave as
second order fluids (see Markovitz and Coleman [22]), we follow Srivastava [19]
and take We to be of the order of α2.
Thus, We=mα2 and (3.16) becomes(3.17)(1−m)d2ϕdY2+α[FdϕdY−dFdYϕ]+mα[Fd3ϕdY3−dFdYd2ϕdY2+d2FdY2dϕdY−d3FdY3ϕ]−ϕ−Mα2ϕ=0.The expansion forF(Y) near the wall Y=0 is(3.18)F(Y)=12Aα2Y2−16(1+M)α3Y3+124MAα4Y4−16mA2α5Y3+1120(A2−M2−M)α5Y5+⋯,where A=F′′(0).Since for large values ofω/c the parameter α is small, we let(3.19)ϕ=∑n=0∞αnϕn(Y)=ϕ0(Y)+αϕ1(Y)+α2ϕ2(Y)+⋯.The boundary conditions are(3.20)ϕ0(0)=1,ϕn(0)=0ifn≥1,ϕn(∞)=0∀n.Substituting (3.20) in (3.17) and equating the
coefficients of different powers of α to zero, we find that the boundary-value
problem for ϕ0(Y) is(3.21)(1−m)d2ϕ0dY2−ϕ0=0,ϕ0(0)=1,ϕ0(∞)=0with solution ϕ0(Y)=exp[−(Y/1−m)] provided 0≤m<1.The second equation gives thatϕ1 is zero. The next four equations for ϕ2(Y),ϕ3(Y),ϕ4(Y) and ϕ5(Y) are(3.22)(1−m)d2ϕ2dY2−ϕ2=Mϕ0,(1−m)d2ϕ3dY2−ϕ3=Mϕ1−12mAY2d3ϕ0dY3+mAYd2ϕ0dY2+(−12AY2−mA)dϕ0dY+AYϕ0,(1−m)d2ϕ4dY2−ϕ4=Mϕ2+16m(1+M)Y3d3ϕ0dY3−12m(1+M)Y2d2ϕ0dY2+[16(1+M)Y3+m(1+M)Y]dϕ0dY−[12(1+M)Y2+m(1+M)]ϕ0−12mAY2d3ϕ1dY3+mAYd2ϕ1dY2−(12AY2−Am)dϕ1dY+AYϕ1,(1−m)d2ϕ5dY2−ϕ5=Mϕ3−12mAY2d3ϕ3dY3=mAd2ϕ2dY2−(12AY2+mA)dϕ2dY+AYϕ2+16m(1+M)Y3d3ϕ1dY3−12m(1+M)Y2d2ϕ1dY2+[16(1+M)Y3+m(1+M)Y]dϕ1dY−[12(1+M)Y2+m(1+M)]ϕ1−124mAMY4d3ϕ0dY3+16mAMY3d2ϕ0dY2−(124MAY4+12mAMY2)dϕ0dY+(16MAY3+mAMY)ϕ0.Solving these equations and using the boundary
conditions, we obtain(3.23)ϕ2(Y)=−M21−mYe−(1/1−m)Y,ϕ3(Y)=−A1−me−(Y/1−m)[3+4m8Y+381−mY2+112(1−m)Y3],ϕ4(Y)=181−me−(Y/1−m){[12(3+4m)(1+M)+M2]Y+[12(3+4m)(1+M)+M2]Y2+1+M1−mY3+16(1−m)3/2Y4},ϕ5(Y)=AM81−me−(Y/1−m){−12m3+25m2−30m+941−mY+12m2−13m+98Y2+4m2−5m+66(1−m)3/2Y3+3m+112(1−m)2Y4−130(1−m)3/2Y5}provided 0≤m<1.
If m=0 and M=0,
we recover the solutions for the Newtonian fluid obtained by Glauert [13] and
if M=0,
we recover the solutions obtained by Labropulu et al. [16].The oscillating component of the shear stress on the
wall is given by(3.24)τ12ρa2=−cνa2eiωt[1α1−m+M21−mα+(3+12m−8m2)A8(1−m)α2−(3+4m)(1+M))+2M2161−mα3+⋯].If m=0 and M=0,
the shear stress is in good agreement with the result obtained by Glauert [13].


## 4. Conclusions

The unsteady stagnation-point flow of a viscoelastic
Walters' B' fluid in the presence of a magnetic field is examined. Results for
this flow are obtained for various values of the Hartmann's numberM and the Weissenberg number We.
At the higher frequencies, the perturbation is a shear layer, exactly as on a
plate oscillating in a fluid at rest. Figure 1 shows the variation of F′(η) for various Weissenberg numbers when the
Hartmann's number M=0.
The effect of the Weissenberg number is to increase the velocity F′(η) near the wall as it increases. Figures 2 and 3
show the variation of F′(η) for various Hartmann's number M when We=0 and We=0.2.
The effect of the Hartmann's number is to increase the velocity F′(η) near the wall as it increases. Figures 4 and 5
show that ϕ0(η) decreases near the wall as M and We are increasing. Also, from Tables 1 to 4, F′′(0) increases with Hartmann's number M and Weissenberg number We.
The reason for this behaviour is that the magnetic field B0 induces a force along the surface which
supports the motion. As a result, the velocity along the surface increases
everywhere and hence the shear stress on the wall increases with increasing
Hartmann's number and Weissenberg number.These conclusions are presented for the Walters' B'
fluid. Future works will examine the unsteady stagnation-point flow in the
presence of a magnetic field of other viscoelastic fluids.



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*Source: 573425-2009-03-05.xml*