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Nonaxisymmetric Three-Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate

[+] Author and Article Information
Ali Shokrgozar Abbassi

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

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

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Corresponding author.

J. Fluids Eng 131(7), 074501 (Jun 24, 2009) (5 pages) doi:10.1115/1.3153366 History: Received December 25, 2007; Revised May 04, 2009; Published June 24, 2009

Abstract

The existing solutions of Navier–Stokes and energy equations in the literature regarding the three-dimensional problem of stagnation-point flow either on a flat plate or on a cylinder are only for the case of axisymmetric formulation. The only exception is the study of three-dimensional stagnation-point flow on a flat plate by Howarth (1951, “The Boundary Layer in Three-Dimensional Flow—Part II: The Flow Near Stagnation Point,” Philos. Mag., 42, pp. 1433–1440), which is based on boundary layer theory approximation and zero pressure assumption in direction of normal to the surface. In our study the nonaxisymmetric three-dimensional steady viscous stagnation-point flow and heat transfer in the vicinity of a flat plate are investigated based on potential flow theory, which is the most general solution. An external fluid, along $z$-direction, with strain rate $a$ impinges on this flat plate and produces a two-dimensional flow with different components of velocity on the plate. This situation may happen if the flow pattern on the plate is bounded from both sides in one of the directions, for example $x$-axis, because of any physical limitation. A similarity solution of the Navier–Stokes equations and energy equation is presented in this problem. A reduction in these equations is obtained by the use of appropriate similarity transformations. Velocity profiles and surface stress-tensors and temperature profiles along with pressure profile are presented for different values of velocity ratios, and Prandtl number.

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Figures

Figure 1

Three-dimensional stream surface and velocity profiles

Figure 2

Boundary layer thickness versus variation of velocity ratio

Figure 3

Typical u and v velocity components for λ=0.1

Figure 4

Typical u and v velocity components for λ=0.5

Figure 5

Typical w-component of velocity for λ=0.1

Figure 6

Typical w-component of velocity for λ=0.50

Figure 7

Temperature profile for the case of λ=0.1 and different Pr values

Figure 8

Temperature profile for the case of λ=0.50 and different Pr values

Figure 9

Surface shear-stress components on the flat plate

Figure 10

Pressure profiles for selected values of λ

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