Normal mode analysis of the fully developed free convection flow in a vertical slot with open to capped ends

Abstract

The fully developed free convection flow in a differentially heated vertical slot with open to capped ends investigated recently by Bühler (Heat Mass Transf 39:631–638, 2003) and Weidman (Heat Mass Transf Online First, February 2006) is revisited in this paper. A new method of solution of the corresponding fourth order boundary value problem, based on its reduction to “normal modes” by a complex matrix similarity transformation is presented. As a byproduct of the method, some invariant relationships involving the heat flux and the shear stress in the flow could be found.

Appendices

Appendix A

Table 1 Overview of the values of α_1,2, A _1,2 and Z _1,2 for the limiting cases of open (θ₀ = 0) and capped (θ₀ = 1/2) slot, respectively

Appendix B

Table 2 Overview of the expressions of u(y), θ(y), Q, s _±  and q _±  for the limiting cases of open (θ₀ = 0) and capped (θ₀ = 1/2) slot, respectively

Appendix C

The conduction limit (m = 0) is obtained as the leading order terms of the following equations obtained by Taylor expansions of Eqs. (21)–(24) to powers of m.

$$u{\left(y \right)} = \frac{G}{6}{\left({\frac{1}{4} - y^{2}} \right)}{\left[ {y + 3{\left({\frac{1}{2} - \theta _{0}} \right)}} \right]},$$

(38)

$$\theta {\left(y \right)} = \frac{1}{2} - \theta _{0} + y,$$

(39)

$$Q = \frac{G}{{12}}{\left({\frac{1}{2} - \theta _{0}} \right)},$$

(40)

$$s_{+} \equiv {u'} {\left({+ 1/2} \right)} = \frac{G}{2}{\left({\theta _{0} - \frac{2}{3}} \right)},$$

(41)

$$s_{-} \equiv {u'} {\left({- 1/2} \right)} = - \frac{G}{2}{\left({\theta _{0} - \frac{1}{3}} \right)},$$

(42)

$$q_{+} \equiv - {\theta'} {\left({+ 1/2} \right)} = - 1,$$

(43)

$$q_{-} \equiv - {\theta'} {\left({- 1/2} \right)} = - 1.$$

(44)

Equations (38)–(40) coincide with Eqs. (16)–(18) of Bühler [1], as well as with Eqs. (21)–(23) of Weidman [2].