TY - JOUR

T1 - Laminar Convection in a Uniformly Heated Horizontal Pipe Using Symbolic Computation

AU - Guiasu, I.

AU - Raszillier, H.

AU - Nandakumar, K.

PY - 1994/12/1

Y1 - 1994/12/1

N2 - Analysis of the mixed convection heat transfer in a uniformly heated horizontal pipe, first considered by Morton [Q. J. Mech. & App!. Math. 12 (1959) 410], is extended with the aid of algebraic or symbolic computation packages. A set of four coupled, nonlinear partial differential equations that describe the effect of buoyancy-driven secondary flow on the pressure-driven axial flow are solved using a regular perturbation series expansion. A scale analysis suggests the product of Rayleigh and Reynolds numbers as the natural perturbation parameter and the series is constructed around the basic Poiseuille flow. In addition, the solutions also depend on the Prandtl number. The nonlinear problem is thus converted into an infinite series of linear partial differential equations to be solved sequentially. Using a further coordinate transformation into complex coordinates, solution of the infinite series of linear problems is reduced to one of evaluating quadratures, a task which is ideally suited for symbolic computation packages such as REDUCE MAPLE, or MATHEMATIC A. As an example the listing of a MAPLE program to solve the present problem to any desired order is appended. Although the symbolic computation program developed here is capable of solving the problem up to any desired order, the speed and memory limitations of a SUN workstation allowed us to generate analytical expressions for stream function, axial velocity and temperature only up to 10th order. The results provided by Morton for the first 3 terms are in agreement (except for some minor typographical errors) with our computer generated results. Primarily, we illustrate how the tedium of algebraic calculations can be relegated to algebraic computing machines. With the availability of faster CPU and larger memory machines, such problems can be routinely solved analytically to any desired accuracy.

AB - Analysis of the mixed convection heat transfer in a uniformly heated horizontal pipe, first considered by Morton [Q. J. Mech. & App!. Math. 12 (1959) 410], is extended with the aid of algebraic or symbolic computation packages. A set of four coupled, nonlinear partial differential equations that describe the effect of buoyancy-driven secondary flow on the pressure-driven axial flow are solved using a regular perturbation series expansion. A scale analysis suggests the product of Rayleigh and Reynolds numbers as the natural perturbation parameter and the series is constructed around the basic Poiseuille flow. In addition, the solutions also depend on the Prandtl number. The nonlinear problem is thus converted into an infinite series of linear partial differential equations to be solved sequentially. Using a further coordinate transformation into complex coordinates, solution of the infinite series of linear problems is reduced to one of evaluating quadratures, a task which is ideally suited for symbolic computation packages such as REDUCE MAPLE, or MATHEMATIC A. As an example the listing of a MAPLE program to solve the present problem to any desired order is appended. Although the symbolic computation program developed here is capable of solving the problem up to any desired order, the speed and memory limitations of a SUN workstation allowed us to generate analytical expressions for stream function, axial velocity and temperature only up to 10th order. The results provided by Morton for the first 3 terms are in agreement (except for some minor typographical errors) with our computer generated results. Primarily, we illustrate how the tedium of algebraic calculations can be relegated to algebraic computing machines. With the availability of faster CPU and larger memory machines, such problems can be routinely solved analytically to any desired accuracy.

KW - Morton problem

KW - mixed convection

KW - symbolic computation

UR - http://www.scopus.com/inward/record.url?scp=3142528111&partnerID=8YFLogxK

U2 - 10.1080/10618569408904507

DO - 10.1080/10618569408904507

M3 - 文章

AN - SCOPUS:3142528111

SN - 1061-8562

VL - 3

SP - 193

EP - 216

JO - International Journal of Computational Fluid Dynamics

JF - International Journal of Computational Fluid Dynamics

IS - 3-4

ER -