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Unaxisymmetric stagnation-point flow and heat transfer of a viscous fluid on a moving cylinder with time-dependent axial velocity

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Abstract

The unsteady viscous flow and heat transfer in the vicinity of an unaxisymmetric stagnation-point flow of an infinite moving cylinder with time-dependent axial velocity and non-uniform normal transpiration \( U_{0} \left( \varphi \right) \) are investigated. The impinging free stream is steady with a strain rate \( \bar{k} \). A reduction of the Navier–Stokes and energy equations is obtained by use of appropriate similarity transformations. The semi-similar solution of the Navier–Stokes equations and energy equation has been obtained numerically using an implicit finite-difference scheme when the axial velocity of the cylinder and its wall temperature or its wall heat flux varies as specified time-dependent functions. In particular, the cylinder may move with different velocity patterns. These solutions are presented for special cases when the time-dependent axial velocity of the cylinder is a step function, a ramp, and a non-linear function. All the solutions above are presented for Reynolds numbers, \( \text{Re} = \bar{k}a^{2} /2\upsilon \), ranging from 0.1 to 100 for different values of Prandtl number and for selected values of transpiration rate function, \( S(\varphi ) = U_{0} (\varphi )/\bar{k}a \) 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 transpiration rate function. An interesting result is obtained in which a cylinder moving with certain axial velocity function and at particular value of Reynolds number is axially stress free. The heat transfer coefficient increases with the increasing transpiration rate function, Reynolds number and Prandtl number. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration rate function. It is shown that a cylinder with certain type of exponential wall temperature exposed to a temperature difference has not heat transfer.

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Abbreviations

\( a \) :

Cylinder radius

\( r \) :

Radial coordinate

\( z \) :

Axial coordinate

\( u,w \) :

Velocity components along (r, z)-axis

\( t \) :

Time

\( T \) :

Temperature

\( T_{\text{W}} \) :

Wall temperature

\( T_{\infty } \) :

Free-stream temperature

\( k \) :

Thermal conductivity

\( \bar{k} \) :

Free-stream strain rate

\( f(\eta ,\;\varphi ,\;\tau ) \) :

Function related to u-component of velocity

\( H(\eta ,\;\varphi ,\;\tau ) \) :

Function related to w-component of velocity

\( V(t) \) :

Axial velocity of cylinder

\( U_{0} (\varphi ) \) :

Transpiration

\( \text{Re} \) :

Reynolds number

\( \Pr \) :

Prandtl number

\( Nu \) :

Nusselt number

\( S(\varphi ) \) :

Transpiration rate function

\( P \) :

Non-dimensional fluid pressure

\( p \) :

Fluid pressure

\( h \) :

Heat transfer coefficient

\( q_{\text{w}} \) :

Heat flux at wall

\( \eta \) :

Similarity variable

\( \varphi \) :

Angular coordinate

\( \alpha \) :

Thermal diffusivity

\( \rho \) :

Fluid density

\( \upsilon \) :

Kinematic viscosity

\( \mu \) :

Dynamic viscosity

\( \theta \left( {\eta ,\varphi ,\tau } \right) \) :

Non-dimensional temperature

\( \sigma \) :

Shear stress

\( \tau \) :

Dimensionless time variable

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran

    Rasool Alizadeh & Mohammad Najafi

  2. Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box No. 91775-1111, Mashhad, Iran

    Asghar B. Rahimi

Authors
  1. Rasool Alizadeh
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  2. Asghar B. Rahimi
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  3. Mohammad Najafi
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Correspondence to Asghar B. Rahimi.

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Technical Editor: Francisco Ricardo Cunha.

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Alizadeh, R., Rahimi, A.B. & Najafi, M. Unaxisymmetric stagnation-point flow and heat transfer of a viscous fluid on a moving cylinder with time-dependent axial velocity. J Braz. Soc. Mech. Sci. Eng. 38, 85–98 (2016). https://doi.org/10.1007/s40430-015-0389-2

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