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Dual solutions for stagnation-point flow and convective heat transfer of a Williamson nanofluid past a stretching/shrinking sheet

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Abstract

In this paper, the problem of boundary layer stagnation-point flow and heat transfer of a Williamson nanofluid on a linear stretching/shrinking sheet with convective boundary condition is studied. The effects of Brownian motion and thermophoresis are considered in the energy equation. The governing partial differential equations are first transformed into set of ordinary differential equations, which are then solved numerically using Runge–Kutta–Felhberg fourth–fifth order method with Shooting technique. The characteristics of the flow and heat transfer as well as skin friction and Nusselt number for various prevailing parameters are presented graphically and discussed in detail. A comparison with the earlier reported results has been done and an excellent agreement is shown. It is found that dual solutions exist for the shrinking sheet case. Further, it is observed that the thermal boundary layer thickness increases with increase in Williamson parameter for both solutions.

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Abbreviations

A 1 :

First Rivlin–Erickson tensor

a, b, c :

Constants

Bi :

Biot number

C :

Nanoparticle volume fraction (kg/m3)

C f :

Skin friction co-efficient

(C p ) f :

Specific heat coefficient of fluid (J/kg K)

(C p ) p :

Specific heat coefficient of nanoparticles (J/kg K)

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

f :

Dimensionless velocity component

h f :

Heat transfer coefficient

I :

Identity vector

j w :

Nanoparticles mass flux

k :

Thermal conductivity (W/mK)

Le :

Lewis number

Nb :

Brownian motion parameter

Nt :

Thermophoresis parameter

Nu x :

Local Nusselt number

p :

Pressure

Pr :

Prandtl number

q w :

Heat flux

O :

Origin

Re x :

Local Reynolds number

S :

Cauchy stress tensor

Sh :

Sherwood number

T :

Fluid temperature (K)

t :

Time (s)

T f :

Surface temperature (K)

\(\vec{V}\) :

Velocity of the fluid (m/s)

u, v :

Velocity components along x and y directions (m/s)

u e :

External flow velocity

x :

Coordinate along the plate (m)

y :

Coordinate normal to the plate (m)

ν :

Kinematic viscosity (m2/s)

μ :

Dynamic viscosity (kg/m/s)

μ 0 :

Limiting viscosity at zero shear rate

μ ∞ :

Limiting viscosity at the infinite shear rate

ϕ :

Dimensionless nanoparticle volume fraction

θ :

Dimensionless temperature

ψ :

Stream function

η :

Similarity variable

τ :

Extra stress tensor

τ w :

Surface shear stress

ρ f :

Density of the base fluid (kg/m3)

ρ p :

Density of the particles (kg/m3)

\(\rho_{{f_{\infty } }}\) :

Ambient density of the fluid

Γ:

Time constant

π :

Second invariant strain tensor

γ :

Williamson fluid parameter

α m :

Thermal diffusivity

ɛ :

Stretching ratio parameter

\(\tau_{xx} , \tau_{xy} , \tau_{yy} , \tau_{yx} , \tau_{yz} , \tau_{zy} , \tau_{zz} , \tau_{xz} , \tau_{zx}\) :

are extra stress tensor components

′:

Derivative with respect to η

w :

Wall

∞:

Ambient condition

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Acknowledgments

One of the authors (B.J.Gireesha) is thankful to the University Grants Commission, India, for the financial support under the scheme of Raman Fellowship for Post-Doctoral Research for Indian Scholars in USA.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Cleveland State University, Cleveland, OH, 44115, USA

    Rama Subba Reddy Gorla & B. J. Gireesha

  2. Department of Studies and Research in Mathematics, Kuvempu University, Shankarghatta, Shimoga, Karnataka, 577451, India

    B. J. Gireesha

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  1. Rama Subba Reddy Gorla
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Correspondence to Rama Subba Reddy Gorla.

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Gorla, R.S.R., Gireesha, B.J. Dual solutions for stagnation-point flow and convective heat transfer of a Williamson nanofluid past a stretching/shrinking sheet. Heat Mass Transfer 52, 1153–1162 (2016). https://doi.org/10.1007/s00231-015-1627-y

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