0910 0118-0904-0501-0101062,ijaridea

S. Abdul Gaffar et al., International Journal of Advanced Research in Innovative Discoveries in Engineering and Applications[IJARIDEA]
Vol.1, Issue 1,27 October 2016, pg. 7-11
7
© 2016, IJARIDEA All Rights Reserved
Mixed Convection Magnetohydrodynamic
Boundary Flow And Thermal Convectionof Non-
Newtonian Tangenthyperbolic Fluid From Non-
Iosthermal Wedgewith Biot Number Effects
S. Abdul Gaffar1
, O. Anwar Beg2
, E. Keshava Reddy3
, V. Ramachandra Prasad4
1
Department of Mathematics, Salalah College of Technology, Salalah, Oman
2
Fluid Mechanics, Spray Research Group, Petroleum and Gas Engineering Division, School of Computing,
Science and Engineering University, University of Salford, Room G77, Newton Building,
Manchester, M5, 4wt, United Kingdom
3
Department of Mathematics, Jawaharlal Nehru Technological University Anantapur, Anantapuramu, India
4
Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle, India
keshava.maths@jntua.ac.in
Abstract— A scientific investigation is created to examine the nonlinear unfaltering blended convection limit
layer stream and warmth exchange of an incompressible digression hyperbolicnon-Newtonian liquid from a
non-isothermal wedge in the nearness of attractive field. The changed preservation conditions are understood
numerically subject to physically fitting limit conditions utilizing a second-arrange precise verifiable limited
distinction Keller Box method. The numerical code is accepted with past studies. The impact of various rising
non-dimensional parameters, to be specific Weissenberg number (We), power law record (n), blended
convection parameter, weight angle parameter (m), Prandtl number (Pr), Biot number, attractive parameter
(M)and dimensionless extraneous direction on speed and temperature development in the limit layer
administration are inspected in subtle element. Moreover, the impacts of these parameters on surface warmth
exchange rate and nearby skin erosion are additionally examined. Approval with prior Newtonian studies is
introduced and amazing relationship accomplished. It is found that speed is lessened with expanding We,
while, temperature is increased. Expanding n improves speed yet diminishes temperature, a comparable
pattern was seen. An expanding M is found to decline speed however temperature increments.
Keywords— Magnetic parameter, Mixed Convection parameter, Non-Newtonian digression hyperbolic liquid,
power law index, Weissenberg number, Weight inclination parameter.
I. INTRODUCTION
This Non-Newtonian liquids have been a subject of awesome enthusiasm to analysts as of
late in light of their different applications in industry and building. Case of such liquids
incorporate coal-oil slurries, cleanser, paints, earth covering and suspensions, oil, restorative
items, custard, physiological fluids (blood, bile, synovial liquid) and so forth. Dissimilar to
the gooey liquids, the non-Newtonian liquids can't be portrayed by the single constitutive
relationship between the anxiety and the strain rate. This is because of various attributes of
such liquids in nature. When all is said in done, the numerical issues in non-Newtonian
liquids are more convoluted on account of its non-direct and higher-request than those in
gooey liquids. In spite of their complexities, researchers and specialists are occupied with
non-Newtonian liquid flow. Fetecau et al.[1] determined a precise answer for the viscoelastic
liquid stream actuated by a roundabout barrel subjected to the time subordinate shear stress.
The issues portraying the shaky helical streams of Oldroyd-B and second grade liquids were
processed by Jamil et al.[2, 3]. Tan and Masuoka[4, 5] talked about the soundness of the
Maxwell liquid in a permeable medium and inferred an accurate answer for the Stokes first
issue for an Oldroyd-B fluid.Recent examinations incorporate the Casson model [6], second-

8
arrange Reiner-Rivlin differential liquid models [7], power-law nanoscale modes [8], Eringen
smaller scale morphic models[9] and Jeffery's viscoelastic model [10].
The goal of the present study is to research the laminar limit layer stream and warmth
exchange of a Tangent Hyperbolic non-Newtonian liquid from a non-isothermal wedge with
attractive field. The non-dimensional conditions with related dimensionless limit conditions
constitute an exceedingly nonlinear, coupled two-point limit esteem issue. The impacts of the
developing thermophysical parameters, to be specific the Weissenberg number (We), power
law list (n),pressure inclination parameter (m), Mixed convection parameter, attractive
parameter (M), Biot number (γ) andPrandtl number (Pr)on speed, temperature, skin contact
number and warmth exchange rate (neighborhood Nusselt number) attributes are
concentrated on. The present issue has to the creators' learning not showed up to this point in
the experimental writing and is significant to polymeric assembling forms in substance
building.
.
II. NON-NEWTONIAN CONSTITUTIVE TANGENT HYPERBOLIC FLUID MODEL
An In the present study a subclass of non-Newtonian fluids known as Tangent Hyperbolic
fluid is employed owing to its simplicity. The Cauchy stress tensor, in Tangent Hyperbolic
non-Newtonian fluid [20] takes the form:
( )
. .
0 tanh
n
τ µ µ µ γ γ∞ ∞
  
 = + + Γ 
    (1)
where is extra stress tensor,
µ∞ is the infinite shear rate viscosity, 0µ is the zero shear
rate viscosity, Γ is the time dependent material constant, n is the power law index i.e. flow
behaviour index and is defined as
. . .1 1
,
2 2
ij ji
i j
γ γ γ= = Π∑ ∑
(2)
Where
( )( )
21
.
2
T
tr gradV gradVΠ = +
We consider Eqn. (1), for the case when
µ∞= 0
because it is not possible to discuss the problem for the infinite shear rate viscosity and since
we considering tangent hyperbolic fluid that describing shear thinning effects so Γ < 1.
Then Eqn. (1) takes the form
. . . .
0 0
. .
0
1 1
1 1
n n
n
τ µ γ γ µ γ γ
µ γ γ
      
   = Γ = +Γ −   
         
  
= + Γ −  
    (3)
III. MATHEMATICAL FLOW MODEL

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Relentless, laminar, two-dimensional, electrically-leading, incompressible stream of a
Tangent Hyperbolic liquid from a non-isothermal wedge, is considered, as outlined in Fig. 1.
An impelled attractive field, is thought to be uniform and acts radially i.e. ordinary to the
wedge surface. The x-coordinate (unrelated) is measured along the periphery of the wedge
from the most reduced point and the y-coordinate (spiral) is guided typical to the surface. The
gravitational increasing speed g, acts downwards. Attractive Reynolds number is thought to
be sufficiently little to disregard attractive incitement impacts. Lobby current and ionslip
impacts are additionally dismissed since the attractive field is frail. We additionally accept
that the Boussineq guess holds i.e. that thickness variety is just experienced in the lightness
term in the force equation.Additionally, the electron weight (for feebly directing liquids) and
the thermoelectric weight are insignificant. The outspread attractive field is produced by
passing a consistent electric current along the longitudinal (z-pivot) parallel to the wedge,
where the wedge edges end at impeccable cathodes which are associated by means of a heap.
Both wedge and Tangent Hyperbolic liquid are kept up at first at the same temperature.
Momentarily it is raised to a temperature Tw> T, the surrounding temperature of the liquid
which stays unaltered. In accordance with the methodology of Yih[42] and presenting the
limit layer approximations, the conditions for mass, force, and vitality, can be composed as
takes after:
0
u v
x y
∂ ∂
+ =
∂ ∂ (4)
( ) ( )
22 2
0
2 2
1 2 sin
2
BdUu u u u u
u v U n n g T T u
x y dx y y y
σ
ν ν β
ρ
∞
∞ ∞
 ∂ ∂ ∂ ∂ ∂ Ω 
+ = + − + Γ ± − −   
∂ ∂ ∂ ∂ ∂    (5)
2
2
T T T
u v
x y y
α
∂ ∂ ∂
+ =
∂ ∂ ∂ (6)
Where u and v are the speed segments in thex andy headings individually. The digression
hyperbolic liquid model along these lines presents a blended subsidiary (second request, first
degree) into the energy limit layer condition (5). The non-Newtonian impacts highlight in the
shear terms just of eqn. (5) and not the convective (increasing speed) terms. The fourth term
on the right hand side of eqn. (5) speaks to the warm lightness compel and couples the speed
field with the temperature field condition (6). The fifth term on the right hand side of eqn. (5)
speaks to the hydromagnetic drag.
( )At 0, 0, 0,
As , ,
w w
m
T
y u v k h T T
y
y u U Cx T T∞ ∞
∂
= = = − = −
∂
→ ∞ → = → (7)
Here T∞- free stream temperature,k- thermal conductivity, hw- convective heat transfer
coefficient,Tw- convective fluid temperature,
m
U Cx∞ =
- free stream velocity,
( )1 12m β β= −
is the Hartree pressure gradient parameter which corresponds to 1β π=Ω
for a total angle Ω of the wedge, C is a positive number. The stream function ψ is defined
by
u
y
∂ψ
=
∂ and
v
x
∂ψ
= −
∂ , and therefore, the continuity equation is automatically satisfied. In

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order to render the governing equations and the boundary conditions in dimensionless form,
the following non-dimensional quantities are introduced.
( )
( )1/2
1/2
, Re , , ,x
w
T Tx y
f
U x T TU x
σ ψ
ξ η θ ξ η
ρ ν
∞
∞ ∞∞
−
= = = =
−
( ) 3 3/2 2 2
1 0
2 2
2 Re
Pr , , ,
Re
w x
e
x
g T T x B x
Gr W M
x
β ν σν
α ν ρν
∞− Γ
= = = =
(8)
All terms are defined in the nomenclature. In view of the transformation defined in eqn.
(8), the boundary layer eqns. (5)-(7) are reduced to the following coupled, nonlinear,
dimensionless partial differential equations for momentum and energy for the regime:
( ) ( )( )
( )
21
1 ''' '' 1 ' '' ''' sin '
2 2
'
1 ' ''
e
m
n f ff m f nW f f Mf
f f
m f f
λθ
ξ
ξ ξ
+ Ω   
− + + − + + −   
   
 ∂ ∂
= − − 
∂ ∂  (9)
( )
'' 1
' 1 ' '
Pr 2
m f
f m f
θ θ
θ ξ θ
ξ ξ
 + ∂ ∂ 
+ = − −  
∂ ∂    (10)
The transformed dimensionless boundary conditions are:
'
0, 0, ' 0, 1
, ' 0, 0
At f f
As f
θ
η θ
γ
η θ
= = = = +
→∞ → → (11)
Here primes denote the differentiation with respect toη and
1/2
Rew xxh
k
γ =
- Biot number.
IV.NUMERICAL SOLUTION WITH KELLER BOX IMPLICT METHOD
The Keller-Box understood contrast strategy is actualized to take care of the nonlinear limit
esteem issue characterized by eqns. (9)–(10) with limit conditions (11). This procedure,
regardless of late advancements in other numerical strategies, remains a capable and
exceptionally exact methodology for explanatory limit layer streams. It is unequivocally
steady and accomplishes extraordinary exactness. As of late, this strategy has been sent in
determining numerous testing, multi-physical liquid progression issues. The Keller-Box
discretization is completely coupled at every progression which mirrors the material science
of allegorical frameworks – which are additionally completely coupled. The Keller Box
Scheme involves four phases.
1) Decomposition of the Nth request halfway differential condition framework to N first
request conditions.
2) Finite Difference Discretization.
3) Quasilinearization of Non-Linear Keller Algebraic Equations lastly.
4) Block-tridiagonal Elimination arrangement of the Linearized Keller Algebraic Equations
V. CONCLUSION
The Numerical arrangements have been exhibited for the lightness driven stream and
warmth exchange of Tangent Hyperbolic stream outer to an even barrel. The Keller-box
verifiable second request exact limited contrast numerical plan has been used to productively

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settle the changed, dimensionless speed and warm limit layer conditions, subject to
reasonable limit conditions. Great relationship with past studies has been exhibited vouching
for the legitimacy of the present code. The calculations have demonstrated that:
1) Increasing Weissenberg number, We,and the attractive parameter, M, lessens the speed,
though it lifts temperatures in the limit layer.
2) Increasing force law file, n, expands the speed all through the limit layer administration
while it discourages the temperature.
3) Increasing Biot number, γ, builds speed, temperature.
4) Increasing Prandtl number, Pr, diminishes speed and temperature.
5) Increasing weight inclination parameter, m expands the velocitybut lessens the
temperature.
REFERENCES
[1] Fetecau, C., Mahmood, A., and Jamil, M., Exact solutions for the flow of a viscoelastic fluid induced by a
circular cylinder subject to a time dependent shear stress, Communications in Nonlinear Science and
Numerical Simulation, 15(12), 3931–3938 (2010)
[2] Jamil, M., Fetecau, C., and Imran, M., Unsteady helical flows of Oldroyd-B fluids, Communications in
Nonlinear Science and Numerical Simulation, 16(3), 1378–1386 (2011)
[3] Jamil, M., Rauf, A., Fetecau, C., and Khan, N. A. Helical flows of second grade fluid to constantly
accelerated shear stresses. Communications in Nonlinear Science and Numerical Simulation, 16(4), 1959–
1969 (2011)
[4] Tan, W. C. and Masuoka, T. Stability analysis of a Maxwell fluid in a porous medium heated from below,
Physics Letters A, 360(3), 454–460 (2007)
[5] Tan, W. C. and Masuoka, T. Stokes first problem for an Oldroyd-B fluid in a porous half space, Physics of
Fluids, 17(2), 023101–023107 (2005)

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