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TECHNICAL PAPERS

# Axisymmetric Stagnation—Point Flow and Heat Transfer of a Viscous Fluid on a Rotating Cylinder With Time-Dependent Angular Velocity and Uniform Transpiration

[+] Author and Article Information
A. B. Rahimi

P. O. Box No. 91775-1111, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iranrahimiab@yahoo.com

R. Saleh

J. Fluids Eng 129(1), 106-115 (Jun 09, 2006) (10 pages) doi:10.1115/1.2375132 History: Received April 07, 2005; Revised June 09, 2006

## Abstract

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite rotating circular cylinder with transpiration $U0$ are investigated when the angular velocity and wall temperature or wall heat flux all vary arbitrarily with time. The free stream is steady and with a strain rate of $Γ$. An exact solution of the Navier-Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by the use of appropriate transformations for the most general case when the transpiration rate is also time-dependent but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the angular velocity of the cylinder and its wall temperature or its wall heat flux vary as specified time-dependent functions. In particular, the cylinder may rotate with constant speed, with exponentially increasing/decreasing angular velocity, with harmonically varying rotation speed, or with accelerating/decelerating oscillatory angular speed. For self-similar flow, the surface temperature or its surface heat flux must have the same types of behavior as the cylinder motion. For completeness, sample semi-similar solutions of the unsteady Navier-Stokes equations have been obtained numerically using a finite-difference scheme. Some of these solutions are presented for special cases when the time-dependent rotation velocity of the cylinder is, for example, a step-function. All the solutions above are presented for Reynolds numbers, $Re=Γa2∕2υ$, ranging from 0.1 to 1000 for different values of Prandtl number and for selected values of dimensionless transpiration rate, $S=U0∕Γa$, where $a$ is cylinder radius and $υ$ is kinematic viscosity of the fluid. Dimensionless shear stresses corresponding to all the cases increase with the increase of Reynolds number and suction rate. The maximum value of the shear stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder rotating with certain exponential angular velocity function and at particular value of Reynolds number is azimuthally stress-free. Heat transfer is independent of cylinder rotation and its coefficient increases with the increasing suction rate, Reynolds number, and Prandtl number. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration.

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## Figures

Figure 1

Schematic diagram of a rotating cylinder under radial stagnation flow in the fixed cylindrical coordinate system (r,φ,z)

Figure 2

Sample profiles of (a)f(η) function, (b)f′ and f″ functions for selected values of suction rate and Reynolds number

Figure 3

Sample profiles of g(η) for cylinder with (a) exponential angular velocity for Re=1 and selected values of suction and λ, (b) accelerating and decelerating oscillatory motion for Re=1000, S=0 and selected values of λ and β

Figure 4

(a) Real part of azimuthal velocity in terms of time for cylinder with harmonic rotation, for Re=1000, s=0. (b) Sample profiles of G(η,τ) for step-function angular velocity for selected values of time at Re=0.1, s=0.

Figure 5

(a) Azimuthal shear stress component for cylinder with exponential angular velocity. (b) Real part of azimuthal shear stress component for cylinder with accelerating and decelerating oscillatory motion, for Re=1000 and s=0.

Figure 6

Sample profiles of θ(η) for, (a) wall temperature, (b) wall heat flux, varying exponentially with time for selected values of Re, Pr, and suction rate

Figure 7

Sample profiles of θ1(η) for, (a) wall temperature, (b) wall heat flux, varying with accelerating and decelerating oscillatory function of time for s=0 and selected values of Re, Pr, γ, and δ

Figure 8

Sample profiles of θ1(η) for (a) wall temperature, (b) wall heat flux, varying with accelerating oscillatory function of time for selected values of Pr and Re, and s=0

Figure 9

Sample profiles of real part of Nusselt number in terms of Pr for (a) wall temperature, (b) wall heat flux for accelerating oscillatory function of time, for selected values of γ, δ and Re=1000, s=0

Figure 10

(a) Sample profiles of real part of Nusselt number for wall temperature with accelerating oscillatory function of time, for Pr=0.7, and s=0. (b) Sample profiles of θ(η) for constant wall heat flux and selected values of suction and blowing rate, for Re=1 and Pr=0.7.

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