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Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder

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Abstract

The unsteady natural convective couple stress fluid flow over a semi-infinite vertical cylinder is analyzed for the homogeneous first-order chemical reaction effect. The couple stress fluid flow model introduces the length dependent effect based on the material constant and dynamic viscosity. Also, it introduces the biharmonic operator in the Navier-Stokes equations, which is absent in the case of Newtonian fluids. The solution to the time-dependent non-linear and coupled governing equations is carried out with an unconditionally stable Crank-Nicolson type of numerical schemes. Numerical results for the transient flow variables, the average wall shear stress, the Nusselt number, and the Sherwood number are shown graphically for both generative and destructive reactions. The time to reach the temporal maximum increases as the reaction constant K increases. The average values of the wall shear stress and the heat transfer rate decrease as K increases, while increase with the increase in the Sherwood number.

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Abbreviations

Bu :

combined buoyancy ratio parameter

C′:

species concentration

C :

dimensionless species concentration

\(\bar C_f\) :

dimensionless average skin-friction coefficient

C f :

dimensionless local skin-friction coefficient

D :

binary diffusion coefficient

Gr C :

mass Grashof number

Gr T :

thermal Grashof number

g :

acceleration due to gravity

K :

dimensionless chemical reaction parameter

k :

thermal conductivity

k 1 :

chemical reaction parameter

\(\overline {Nu}\) :

dimensionless average Nusselt number

NuX :

dimensionless local Nusselt number

Pr :

Prandtl number

R :

dimensionless radial coordinate

r :

radial coordinate

r 0 :

radius of cylinder

Sc :

Schmidt number

\(\overline {Sh}\) :

dimensionless average Sherwood number

ShX :

dimensionless local Sherwood number

T′:

temperature

T :

dimensionless temperature

t′:

time

t :

dimensionless time

U, V :

dimensionless velocity components along the X- and R-directions

u, v :

velocity components along the x- and rdirections

X :

dimensionless axial coordinate

x :

axial coordinate

α :

thermal diffusivity

β C :

volumetric coefficient of expansion with concentration

β T :

volumetric coefficient of thermal expansion

η :

material constant

μ :

viscosity of the fluid

ν :

kinematic viscosity

ρ :

density

w:

condition on the wall

i :

designate grid point along the Xdirection

∞ :

free stream condition

j :

designate grid point along the Rdirection

n :

time step level

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Warangal, 506004, India

    H. P. Rani & G. J. Reddy

  2. Department of Mechanical Engineering, College of Advanced Technology (Industrial Liaison Research Institute), Kyung Hee University, Gyeonggi-do, 446701, Korea

    C. N. Kim

Authors
  1. H. P. Rani
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  2. G. J. Reddy
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  3. C. N. Kim
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Correspondence to C. N. Kim.

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Rani, H.P., Reddy, G.J. & Kim, C.N. Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder. Appl. Math. Mech.-Engl. Ed. 34, 985–1000 (2013). https://doi.org/10.1007/s10483-013-1722-6

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  • DOI: https://doi.org/10.1007/s10483-013-1722-6

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