INTRODUCTION
The studies of nonNewtonian fluids have received considerable attention because
of numerous applications in industry, geophysics and engineering. Some studies
are notably important in industries related to paper, food stuff, personal care
product, textile coating and suspension solutions. A large class of real fluids
do not exhibit the linear relationship between stress and rate of strain. Due
to the nonlinear dependence, the analysis of the behaviour of fluid motion
of non Newtonian fluids tends to be much more complicated and subtle in comparison
with that of Newtonian fluids. When the motion of a fluid is set up, the velocity
field contains transients obtained by the initial conditions. These transients
gradually disappear in time and the starting solution tends to the steadystate
solution, which is independent of the initial conditions. Several researchers
have discussed the flows of generalized Burgers’ fluid in different configurations
(Fetecau et al., 2009; Vieru
et al., 2008; Shah, 2010; Hayat
et al., 2006; Shah and Qi, 2010; Khan
et al., 2010; Xue et al., 2008; Khan
and Hayat, 2008). There are available few attempts in which the flows of
nonNewtonian fluids have been investigated in different separate cases. Such
attempts are made by Fetecau et al. (2006), Khan
et al. (2008, 2009), Hayat
(2006) and Hayat et al. (2008a,b).
The aim of the current study is to establish exact steady state solutions for
the velocity field corresponding to flow induced by a constantly accelerating
plate in generalized Burger fluid. The fluid is magnetohydrodynamic (MHD) in
the presence of an applied magnetic field and occupying a half porous space,
which is bounded by a rigid and nonconducting plate. Constitutive equations
of a generalized Burgers fluid are used. Modified Darcy’s law has been
utilized. The steadystate solution to the resulting problem is attained by
Fourier sine Transform, which contains as limiting cases the similar solutions
for Burgers’ fluid, OldroydB, Maxwell, Second grade and NavierStokes
fluids. The graphs are plotted in order to illustrate the variations of embedded
flow parameters.
FORMULATION OF THE PROBLEM
We choose a Cartesian coordinate system by considering an infinite plate at
z = 0. An incompressible fluid which occupies the porous space is conducting
electrically by exerting an applied magnetic field
parallel to the zaxis. The electric field is not taken into consideration,
the magnetic Reynolds number is small and the induced magnetic field is not
accounted. Both plate and fluid possess solid body rotation with a uniform angular
Ω about the zaxis.
The governing flow equation is given by Hayat et al. (2008a,b).
In which F = u+i,
u and are
the velocity components in x and y directions, respectively, ρ is the fluid
density, μ is the dynamic viscosity, σ is the finite electrical conductivity
of the fluid and φ, k are the porosity and permeability of the porous medium,
respectively,λ_{1} and λ_{3} are correspondingly the
relaxation and retardation times and λ_{2}, λ_{4}
are the material constants having the dimensions as the square of time.
The initial and boundary conditions for a constant accelerated plate are:
where A has dimension of .
SOLUTION OF THE PROBLEM
Introducing the following dimensionless quantities:
where v is kinematic viscocity.
The problem statement (1) reduces to:


(6) 
Upon using Fourier sine transform, Eq. 68
yield:
Solving the ordinary differential Eq. 9 and inverting the
result by means of the Fourier sine transform, we can write the velocity field
G (ξ, τ) as a sum of the steady state and transient solutions, i.e.
The steadystate solution, which is valid for large values of time, has the
form:
where
The above expressions for a MHD Burgers’ fluid (λ_{4}) in
a porous space take the form:
where
The result (Eq. 10) for a MHD OldroydB fluid (λ_{2}
= λ_{4} = 0) in a porous space takes the form:
The Eq. 22 for a MHD Maxwell fluid (λ_{2} =
λ_{3} = λ_{4} = 0) in a porous space is now of the
form:
The result (Eq. 10) for a MHD second grade fluid (λ_{1}
= λ_{2} = λ_{4} = 0) in a porous space takes the form:
with
The above expressions Eq. 2425 for MHD
viscous fluid (λ_{1} = λ_{2} = λ_{3}
= λ_{4} = 0) in a porous space now become:
where
RESULTS AND DISCUSSION
Here, we present the graphical illustrations of the velocity profiles which
have been determined for the flow due to the constant accelerated of an infinite
flat plate. The emerging parameters here are the rotating parameter w, magnetic
field parameter M and parameter of the porous medium B, the material constants
parameters are E and R. In order to illustrate the role of these parameters
on the real and imaginary parts of the velocity G, the Fig. 1
6 ha ve been displayed. In these Fig. 16
panels (a) depict the variations of [Re[G]]
for generalized Burgers’ fluid and panels (b) indicate the variations of
[Im [G]].
Figure 1a shows that the real part of the velocity profile
decreases for various values of rotation w, with respect to the increase in
.
As w increases, the velocity profile decreases. Figure 1b
indicates that the magnitude of imaginary part of the velocity profile increases
initially and later decreases for various values of rotation w, with respect
to the increase in .
As w increases, the velocity profile also increases. Similar result is obtained
(Hayat et al., 2008a,b).
Figure 2a is prepared to see the effects of magnetic on the
real part of velocity profile. Keeping R, E, B, Q, P, w, τ fixed and varying
M, it is noted that the real part of velocity profile decreases by increasing
the magnetic parameter M. Figure 2b also is prepared to see
the effects of magnetic on the imaginary part of the velocity profile. Keeping
R, E, B, Q, P, w, τ fixed and varying M, it is noted that the imaginary
part decrease initially and later increases. Similar result is obtained (Hayat
et al., 2008a).
Figure 3a indicates that the variation of porosity parameter
Keeping R, E, M, Q, P, w, τ fixed. It is found that by increase in the
porosity parameter is lead to increase the real part of the velocity profile.
Figure 3b Keeping R, E, M, Q, P, w, τ fixed and varying
M, it is noted that the imaginary part increases initially and later decrease.
Figure 4a show the effects of material parameter E of G.
Burgers’ fluid on the real part of velocity profile when R, B, M, Q, P,
w, τ are fixed. It interesting to note that by increase in the material
constant parameter E is lead to increase the real part of velocity profile.

Fig. 1: 
(a, b) The variation of velocity profile G (x, t) for various
values of rotation w when (R = 1.3, E = 1.5, B = 1, Q = 1, P = 2, M = 2,
τ = pi/2) 

Fig. 2: 
(a, b) The variation of velocity profile G (x, t) for various
values of (MHD) M when (R = 1.3, E = 1.5, B = 1, Q = 1, P = 2, w = 1, τ
= pi/2) 

Fig. 3: 
(a, b) The variation of velocity profile G (x, t) for various
values of porosity parameter B when (R = 1.3, E = 1.5, M = 2, Q = 1, P =
2, w = 1, τ = pi/2) 

Fig. 4: 
(a, b) The variation of velocity profile G (x, t) for various
values parameter E when (R = 1.3, B = 1, M = 2, Q = 1, P = 2, w = 1, τ
= pi/2) 

Fig. 5: 
(a, b) The variation of velocity profile G (x, t) for various
values parameter R when (E = 1.5, B = 1, M = 2, Q = 1, P = 2, w = 1, τ
= pi/2) 

Fig. 6: 
The variations of velocity profile G(x, t) for various fluids
when (B = 1, M = 2, w = 1, τ = pi/2) 
Figure 4b it is shown that when are fixed and by increasing
the material constant parameter R, B, M, Q, P, w, τ is lead to imaginary
part increases initially and later decrease.
Figure 5a show the effects of material parameter R of G.
Burgers’ fluid on the real part of velocity profile keeping R, E, M, Q,
P, w, τ fixed. It is found that by increase in the parameter Ris lead to
decrease the real part of the velocity profile.
Figure 5b is prepared to see the influence of material parameter
R of G. Burgers’ fluid on the imaginary part of velocity profile keeping
R, E, M, Q, P, w, τ fixed. It found that by increase in the material constant
parameter R is lead decrease the imaginary part of the velocity.
Figure 6 is prepared to show the variation of velocity profile
for various fluids in comparison of G. Burgers’ fluid. It is observe that
real part of OldroydB is quite same of G. Burgers’ fluid.
CONCLUSIONS
The steadystate solution corresponding to the motion of generalized Burgers’
fluid due to the constant acceleration of an infinite flat plate is established
by means of the Fourier sine transforms. The solution for generalized Burgers’
fluid and similar solutions (i.e., the limiting cases) for Burgers’, Oldroyd
 B, Maxwell, Second grade and NavierStokes fluids. Fetecau
(2006) presented here in a simple form in terms of the elementary exponential
and trigonometric functions. These satisfy all the above governing equations
and all the above imposed boundary conditions.
ACKNOWLEDGMENTS
Authors are thankful to the Sudanese government for financial support and MOSTI,
Malaysia for the NSF scholarship. This research is partially funded by the MOHE
research grant FRGS Vot. Nos. 78485 and 78675.