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

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- Received:
- Revised:

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

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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.

### 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

*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*r*directions*X*dimensionless axial coordinate

*x*axial coordinate

### Greek symbols

*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

### Subscripts

- w
condition on the wall

*i*designate grid point along the

*X*direction*∞*free stream condition

*j*designate grid point along the

*R*direction

### Superscript

*n*time step level