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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

Mixed Convection Magnetohydrodynamic Boundary Flow And Thermal Convectionof NonNewtonian Tangenthyperbolic Fluid From NonIosthermal Wedgewith Biot Number Effects S. Abdul Gaffar1, O. Anwar Beg2, E. Keshava Reddy3, V. Ramachandra Prasad4 1 2

Department of Mathematics, Salalah College of Technology, Salalah, Oman

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], second7 © 2016, IJARIDEA All Rights Reserved

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

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: n   .  . τ =  µ∞ + ( µ0 + µ∞ ) tanh  Γγ   γ     (1)

µ

µ

where is extra stress tensor, ∞ is the infinite shear rate viscosity, 0 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 .

γ =

. .

1 γ ij γ ∑ ∑ 2 i j

Π =

ji

=

1 Π, 2

1 tr gradV + ( gradV 2

(

(2) T

)

)

2

.

µ

Where We consider Eqn. (1), for the case when ∞ = 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 n  . n  .  .  . τ = µ0  Γ γ   γ = µ0 1 + Γ γ − 1  γ      

  .  . = µ0 1 + n  Γ γ −1 γ   

(3)

III. MATHEMATICAL FLOW MODEL

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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

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:

∂u ∂v + =0 ∂x ∂y

(4)

 ∂u  ∂ u σB ∂u ∂u dU ∞ ∂u Ω + v = U∞ +ν (1 − n ) 2 + 2ν nΓ   2 ± g β sin   (T − T∞ ) − u ∂x ∂y dx ∂y ρ 2  ∂y  ∂y 2

u

2 0

2

(5)

2

u

∂T ∂T ∂T +v =α 2 ∂x ∂y ∂y

(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. ∂T At y = 0, u = 0, v = 0, − k = hw (Tw − T ) ∂y

As y → ∞,

u → U ∞ = Cx m ,

T → T∞

(7) Here T∞- free stream temperature,k- thermal conductivity, hw- convective heat transfer coefficient,Tw-

convective

fluid

m = β1 ( 2 − β1 )

temperature,

U∞ = C xm

-

free

stream

velocity,

β =Ω π

is the Hartree pressure gradient parameter which corresponds to 1 for a total angle Ω of the wedge, C is a positive number. The stream function ψ is defined

u= by

∂ψ ∂ψ v=− ∂y and ∂x , and therefore, the continuity equation is automatically satisfied. In 9

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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

order to render the governing equations and the boundary conditions in dimensionless form, the following non-dimensional quantities are introduced. σx y ψ T − T∞ ξ= , η = Re x1/ 2 , f = , θ ( ξ ,η ) = 1/2 x Tw − T∞ ρU ∞ (U ∞ xν ) Pr =

ν , α

Gr =

g β1 (Tw − T∞ ) x3

ν2

We =

,

2νΓ Re3/x 2 , x2

M=

σ B02 x 2 ρν Re x

(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: 1+ m  2 Ω (1 − n ) f ''' +   ff ''+ m 1 − ( f ') + nWe f '' f ''' + λθ sin   − Mf '  2  2

(

)

 ∂f ' ∂f  = ξ (1 − m )  f ' − f ''  ∂ξ   ∂ξ

 ∂θ ∂f  + −θ '   f θ ' = ξ (1 − m )  f ' Pr  2  ∂ξ   ∂ξ The transformed dimensionless boundary conditions are:

(9)

θ ''  1 + m 

At η = 0, As η →∞,

f = 0, f ' → 0,

f ' = 0,

θ = 1+

(10)

θ' γ

θ →0

(11)

Here primes denote the differentiation with respect to η and

γ=

1/2 x

xhw Re k

- 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 10 © 2016, IJARIDEA All Rights Reserved

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

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.

[1]

[2] [3]

[4] [5]

REFERENCES 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) 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) 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) 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) 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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