Laminar mixed convection on a horizontal plate of finite length in a channel of finite width

Abstract

The steady two-dimensional laminar mixed-convection flow past a horizontal plate of finite length is analysed for large Péclet numbers, small Prandtl numbers and weak buoyancy effects. The plate is placed in a channel of finite width, with the plane walls of the channel being parallel to the plate. The temperature of the plate is assumed to be constant. The hydrostatic pressure difference across the wake behind the plate is compensated by a perturbation of the inviscid channel flow. This outer flow perturbation affects the temperature distribution in the thermal boundary layer at the plate and the heat transfer rate, respectively. Solutions in closed form are given. The forces acting on the plate due to the potential flow perturbation are also determined.

Appendix

1.1 Appendix A: Potential-flow solution

To find the solution of Laplace’s equation that satisfies Eqs. 23–25 and the condition u = 1 as x → −∞ as boundary conditions, functional analysis is applied. Using the mirror method to satisfy the boundary condition at the channel walls, i.e. Eq. 25, a successive sequence of virtual channels is considered, cf. Fig. 11. Then, following an idea that is well-known from aerodynamics, cf. [39, 44], Cauchy’s integral theorem

Fig. 11

Path of integration for the application of Cauchy’s integral theorem

$$ f\left( z \right) = {\frac{1}{2\pi i}}\oint {{\frac{f\left( \varsigma \right)}{\varsigma - z}}} {\text{ d}}\varsigma ,\quad \left( {z = x + iy,\;\varsigma = \xi + i\eta } \right) $$

(64)

is applied to the analytic function

$$ f\left( \varsigma \right) = w\left( \varsigma \right)h\left( \varsigma \right), $$

(65)

where w(ς) = (u − 1) − iv is the complex velocity perturbation, while h(ς) is an analytic auxiliary function that will be suitably defined in the course of the analysis. As indicated in Fig. 11, the closed path of integration leads to infinity and around cuts representing the plates and the wakes, respectively, where the complex velocity is discontinuous. With regard to extending the integration also downstream to infinity, although the region of validity of the integral relationships for the wake is limited, cf. the justification given in Appendix B.

Assuming h(ς) to be bounded as ς → ∞, one obtains from Eqs. 64 and 65 the following expression for w(z):

$$ w\left( z \right) = {\frac{1}{2\pi ih\left( z \right)}}\int\limits_{0}^{\infty } { \, \sum\limits_{m = - \infty }^{ + \infty } {{\frac{{f\left( {\xi ,mb + } \right) - f\left( {\xi ,mb - } \right)}}{{\left( {\xi + mib} \right) - z}}}} } \, {\text{d}}\xi . $$

(66)

Apart from being bounded at infinity, h(ς) has to satisfy further conditions. Since, for symmetry reasons, the relations

$$ u\left( {\xi ,mb \pm } \right) - 1 = \left( { - 1} \right)^{m} \left[ {u\left( {\xi ,0 \pm } \right) - 1} \right]; $$

(67)

$$ v\left( {\xi ,mb} \right) = \left( { - 1} \right)^{m} v\left( {\xi ,0} \right) $$

(68)

hold, the auxiliary function is required to satisfy the conditions

$$ h\left( {\xi ,mb - } \right) = - h\left( {\xi ,mb + } \right)\quad {\rm for}\;0 \le \xi \le 1 $$

(69)

in order to achieve that the jump discontinuity f(ξ, mb+) − f(ξ, mb−) at the plates no longer depends on the a priori unknown tangential velocity. Similarly, the conditions

$$ h\left( {\xi ,mb - } \right) = + h\left( {\xi ,mb + } \right)\quad {\rm for}\;\xi > 1, $$

(70)

guarantee that f(ξ, mb+) − f(ξ, mb−) in the wake regions no longer depends on the vertical velocity perturbation. With Eqs. 67–70 together with the boundary conditions at the plate and at the wake, Eq. 24 and 23, respectively, the integrand in Eq. 66 vanishes for 0 ≤ ξ ≤ 1, and Eq. 66 reduces to

$$ w\left( z \right) = {\frac{{i{\text{ Ri}}}}{2\pi \, h\left( z \right)}}\int\limits_{1}^{\infty } { \, \sum\limits_{m = - \infty }^{ + \infty } {{\frac{{\left( { - 1} \right)^{m} h\left( {\xi ,mb + } \right)}}{{\left( {\xi + mib} \right) - z}}}} } \, {\text{d}}\xi . $$

(71)

Finally, according to the mirror method, the flow field in the virtual channels with even numbers of m is the same as in the physical channel (m = 0), while it is turned upside down in the other virtual channels. Thus it is required that

$$ h\left( {\xi ,mb + } \right) = h\left( {\xi ,0 + } \right), $$

(72)

and, after expressing the infinite sum in Eq. 71 as

$$ \sum\limits_{m = - \infty }^{ + \infty } {{\frac{{\left( { - 1} \right)^{m} }}{{\left( {\xi + mib} \right) - z}}}} = {\frac{1}{\xi - z}} + 2\left( {\xi - z} \right)\sum\limits_{m = 1}^{\infty } {{\frac{{\left( { - 1} \right)^{m} }}{{\left( {mb} \right)^{2} + \left( {\xi - z} \right)^{2} }}}} = {\frac{\pi }{b}}\,{\frac{1}{{\sinh \left[ {\pi \left( {\xi - z} \right)/b} \right]}}}, $$

(73)

cf. [45], Eq. 71 becomes

$$ w\left( z \right) = {\frac{{i{\text{ Ri}}}}{2bh\left( z \right)}}\int\limits_{1}^{\infty } {{\frac{{h\left( {\xi ,0 + } \right){\text{ d}}\xi }}{{\sinh \left[ {\pi (\xi - z)/b} \right]}}}} . $$

(74)

An auxiliary function that satisfies all required conditions is

$$ h\left( z \right) = \sqrt {{\frac{z}{1 - z}} \, {\frac{{\prod\limits_{m = 1}^{\infty } {\left[ {1 + \left( {z/mb} \right)^{2} } \right]} }}{{\prod\limits_{m = 1}^{\infty } {\left\{ {1 + \left[ {\left( {1 - z} \right)/mb} \right]^{2} } \right\}} }}}} = \sqrt {{\frac{{\sinh \left( {\pi \, z/b} \right)}}{{\sinh \left[ {\pi \left( {1 - z} \right)/b} \right]}}}} , $$

(75)

where a well-known representation of hyperbolic functions in terms of infinite products, cf. [24], p. 85, has been used. Inserting Eq. 75 into Eq. 74 gives the following complex velocity perturbation:

$$ [u\left( z \right) - 1] - iv\left( z \right) = - {\frac{\text{Ri}}{2b}}\sqrt {{\frac{{\sinh \left[ {\pi (1 - z)/b} \right]}}{{\sinh \left( {\pi z/b} \right)}}}} \, \int\limits_{1}^{\infty } {\sqrt {{\frac{{\sinh \left( {\pi \, \xi /b} \right)}}{{\sinh \left[ {\pi \left( {\xi - 1} \right)/b} \right]}}}} \, } {\frac{{{\text{d}}\xi }}{{\sinh \left[ {\pi \left( {\xi - z} \right)/b} \right]}}}. $$

(76)

With z = x + iy and y → 0, one easily obtains the velocity perturbation in the plane of the plate. Firstly, for 0 < x < 1, the real part of Eq. 76 gives Eq. 26 with Eq. 27 for the tangential velocity perturbation at the plate. Secondly, for x > 1, the imaginary part of Eq. 76 leads to Eq. 62 for the vertical velocity component behind the plate.

1.2 Appendix B: Effects of the break-down of the wake analysis far downstream

According to a well-known result of boundary-layer theory (cf. [2], pp. 187–190) the non-dimensional thickness of the laminar thermal wake is of the order of $\sqrt {x/{\text{Pe}}}$. Thus, the thermal wake thickness becomes comparable to the channel width, and the boundary-layer analysis ceases to be valid, for values of x that are of the order of, or larger than, x _w , with x _w  = b ²Pe. The integration in the potential-flow solution Eq. 27, however, extends to infinity. Thus it is required to show that the wake region that is beyond the limits of applicability of the boundary-layer theory contributes very little to the integral in Eq. 27. Expanding the hyperbolic functions for ξ/b ≫ 1, while x/b remains of order 1, it is easily shown that the integrand is of the order of exp(− πξ/b). Integrating from ξ = x _w to ∞ gives a term of the order of b exp(−πbPe), which is exponentially small for Pe ≫ 1 and b = O(1) or larger.