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Research Papers: Forced Convection

# Analytic Approximations for a Strongly Nonlinear Problem of Combined Convective and Radiative Cooling of a Spherical Body

[+] Author and Article Information
A. El-Nahhas

Department of Mathematics, Helwan Faculty of Science, Helwan University, Helbawy Street, Cairo 11795, Egyptaasayed35@yahoo.com

J. Heat Transfer 131(11), 111703 (Aug 26, 2009) (6 pages) doi:10.1115/1.3154625 History: Received December 23, 2008; Revised May 04, 2009; Published August 26, 2009

## Abstract

In this paper, a nonlinear problem for combined convective and radiative cooling of a spherical body is considered. This problem represents a strong nonlinearity in both the governing equation and the boundary condition. Analytic approximations for the solution of this problem are obtained using the homotopy analysis method and via a polynomial exponential basis. Also, the effect of the radiation-conduction parameter $Nrc$ and the Biot number Bi for the temperature on the surface of the spherical body is investigated and discussed.

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## Figures

Figure 1

The h-curve of θ″=∂2θ(η,ξ)/∂η2∣η=0 for ξ=0.5, β=0, Bi=1, Nrc=0, and θa=0

Figure 2

The h-curve of θ″=∂2θ(η,ξ)/∂η2∣η=0 for ξ=0.5, β=1, Bi=0.5, Nrc=0.5, and θa=0.5

Figure 3

The 30th order analytic homotopy solution θ (as a function of τ) at η=1, and for β=0, Bi=1, Nrc=0, θa=0, and h=−0.3

Figure 4

A comparison between the 30th order analytic homotopy solution (thin line) and the exact solution (solid line) at η=1, and for β=0, Bi=1, Nrc=0, θa=0, and h=−0.3

Figure 5

The variation in the temperature θ on the surface of the spherical body (η=1) for different values of the Biot number Bi(Bi=0.5,1,2) and β=1, Nrc=0, θa=0, and h=−0.3

Figure 6

The variation in the temperature θ on the surface of the spherical body (η=1) for different values of the radiation-conduction parameter Nrc(Nrc=0.25,0.5) and β=1, Bi=0.5, θa=0.5, and h=−0.2

Figure 7

The variation in the temperature θ on the surface of the spherical body through different values of τ(τ=0.05,0.10,0.20,0.35,0.50,1) and for (Bi=1), β=1, Nrc=0, θa=0, and h=−0.3

Figure 8

The variation in the temperature θ on the surface of the spherical body through different values of τ(τ=0.05,0.10,0.20,0.35,0.50,1) and for (Bi=2), β=1, Nrc=0, θa=0, and h=−0.3

Figure 9

The variation in the temperature θ on the surface of the spherical body through different values of τ(τ=0.05,0.10,0.20,0.35,0.50,1,1.5) for the radiation-conduction parameter (Nrc=0.25) and β=1, Bi=0.5, θa=0.5, and h=−0.2

Figure 10

The variation in the temperature θ on the surface of the spherical body through different values of τ(τ=0.05,0.10,0.20,0.35,0.50,1,1.5) for the radiation-conduction parameter (Nrc=0.5) and β=1, Bi=0.5, θa=0.5, and h=−0.2

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