Applied Mathematics and Mechanics

, Volume 34, Issue 8, pp 985–1000

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

Article

DOI: 10.1007/s10483-013-1722-6

Cite this article as:
Rani, H.P., Reddy, G.J. & Kim, C.N. Appl. Math. Mech.-Engl. Ed. (2013) 34: 985. doi:10.1007/s10483-013-1722-6

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.

Key words

couple stress fluidchemical reactionnatural convectionvertical cylinderfinite difference method

Nomenclature

Bu

combined buoyancy ratio parameter

C

species concentration

C

dimensionless species concentration

\(\bar C_f\)

dimensionless average skin-friction coefficient

Cf

dimensionless local skin-friction coefficient

D

binary diffusion coefficient

GrC

mass Grashof number

GrT

thermal Grashof number

g

acceleration due to gravity

K

dimensionless chemical reaction parameter

k

thermal conductivity

k1

chemical reaction parameter

\(\overline {Nu}\)

dimensionless average Nusselt number

NuX

dimensionless local Nusselt number

Pr

Prandtl number

R

dimensionless radial coordinate

r

radial coordinate

r0

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

Greek symbols

α

thermal diffusivity

βC

volumetric coefficient of expansion with concentration

βT

volumetric coefficient of thermal expansion

η

material constant

μ

viscosity of the fluid

ν

kinematic viscosity

ρ

density

Subscripts

w

condition on the wall

i

designate grid point along the Xdirection

free stream condition

j

designate grid point along the Rdirection

Superscript

n

time step level

Chinese Library Classification

O175.7O414.14

2010 Mathematics Subject Classification

78M2080A2080A32

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyWarangalIndia
  2. 2.Department of Mechanical Engineering, College of Advanced Technology (Industrial Liaison Research Institute)Kyung Hee UniversityGyeonggi-doKorea