Abstract
The steady twodimensional laminar mixedconvection 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.
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Notes
Since logarithmic terms are split off in the asymptotic expansions performed in [6], the subscripts differ from the present ones. In particular, the terms U _{1h }, P _{1h } and V _{1h } in the present expansions are the same as U _{2h }, P _{2h } and V _{2h }, respectively, in [6]. When comparing the present expansions with those performed in [6], it should also be noted that all subscripts in Eq. 33 of [6] ought to read 2c.
For the reason given in footnote 2, the present term \( \tilde{\Uptheta }_{1h} \left( \eta \right) \) is to be compared with \( \tilde{\Uptheta }_{2h} \left( \eta \right) \) of [6]. Note that a coefficient \( 4/\sqrt \pi \) is erroneously missing in Eq. 48 of [6]. However, the correct equation has been used in the analysis, in particular in Eq. 49 of [6], as well as for the numerical evaluations as given in Fig. 5 of [6]. There is, however, a minus sign missing in Fig. 5 of [6], where \(  \tilde{\Uptheta }_{2h} , \) rather than \( \tilde{\Uptheta }_{2h} \), is plotted.
Abbreviations
 A(b) :
 b :

Channel width (referred to the plate length L)
 C :
 c _{ p } :

Isobaric specific heat capacity of the fluid
 D :

Linear differential operator, cf. Eq. 36
 F(η):

Auxiliary function, cf. Eq. 39
 f(ς):

Analytic function, cf. Eq. 65
 g:

Gravity acceleration
 \( \tilde{g}\left( X \right) \) :
 h(ς):

Analytic auxiliary function, cf. Eq. 65
 k :

Thermal conductivity of the fluid
 L :

Plate length
 Nu:

Nusselt number
 P :

Nondimensional pressure difference in the boundary layer with respect to the ambient pressure
 p :

Nondimensional pressure difference in the outer (potential) flow with respect to the ambient pressure
 Pe:

Péclet number
 Pr:

Prandtl number
 \( \dot{Q} \) :

Overall heat flow rate, cf. Eq. 20
 \( \dot{q} \) :

Local heat flux at the upper surface of the plate
 Re:

Reynolds number
 Ri:
 T :

Absolute temperature
 U :

Nondimensional horizontal velocity component in the boundary layer, cf. Eqs. 3, 4, 6
 u :

Nondimensional horizontal velocity component in the outer (potential) flow
 V :

Nondimensional vertical velocity component in the boundary layer, cf. Eq. 1
 v :

Nondimensional vertical velocity component in the outer (potential) flow
 w(ς):

Complex velocity perturbation, cf. Eq. 65
 X :

Horizontal coordinate for the boundary layer
 x :

Horizontal coordinate (referred to the plate length L)
 x _{w} :

Reference value for the wake analysis, cf. Appendix B
 Y :

Vertical coordinate for the boundary layer, cf. Eq. 1
 y :

Vertical coordinate (referred to the plate length L)
 z :

Complex coordinate, cf. Eq. 64
 α :

Thermal diffusivity of the fluid
 β :

Thermal expansivity of the fluid
 \( \tilde{\gamma } \) :

Vortex distribution, cf. Eq. 27
 ε :

Auxiliary parameter
 ς :

Complex coordinate, cf. Eq. 64
 η :

Similarity variable for the boundary layer, cf. Eq. 15
 Θ:

Nondimensional temperature difference, cf. Eq. 2
 \( \tilde{\Uptheta } \) :

Reduced nondimensional temperature difference, cf. Eqs. 42, 43
 ρ :

Density of the fluid
 ξ :

Variable of integration
 B :

Buoyancy, cf. Eq. 56
 c :

Due to circulation, cf. Eq. 32
 h :

Due to perturbation of the hydrostatic pressure, cf. Eq. 32
 L :

Lift, cf. Eq. 58
 p :

At the plate, cf. Eq. 2; also overall Nusselt number, cf. Eq. 51
 S :

Leadingedge suction, cf. Eq. 59
 X :

Local value of the Nusselt number, cf. Eq. 47
 0:

Leading term of the asymptotic expansion, cf. Eq. 13
 1:
 ∞:

Freestream value
 (1):

Firstorder perturbation, cf. Eq. 47
 (1c):

Firstorder perturbation due to circulation, cf. Eq. 48
 (1h):

Firstorder perturbation due to hydrostatic pressure, cf. Eq. 48
 *:

Critical value, cf. Eq. 55
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Acknowledgments
The authors should like to thank Professor H. Steinrück for many fruitful discussions on the problem. Furthermore, the authors are grateful to an anonymous reviewer for his comments that led to various improvements of the paper. Financial support of the project by Androsch International Management Consulting GmbH is also gratefully acknowledged.
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Dedicated to Professor K. Stephan on the Occasion of his 80th Birthday.
Appendix
Appendix
1.1 Appendix A: Potentialflow 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 wellknown from aerodynamics, cf. [39, 44], Cauchy’s integral theorem
is applied to the analytic function
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):
Apart from being bounded at infinity, h(ς) has to satisfy further conditions. Since, for symmetry reasons, the relations
hold, the auxiliary function is required to satisfy the conditions
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
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
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
and, after expressing the infinite sum in Eq. 71 as
An auxiliary function that satisfies all required conditions is
where a wellknown 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:
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 breakdown of the wake analysis far downstream
According to a wellknown result of boundarylayer theory (cf. [2], pp. 187–190) the nondimensional 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 boundarylayer analysis ceases to be valid, for values of x that are of the order of, or larger than, x _{ w }, with x _{ w } = b ^{2}Pe. The integration in the potentialflow 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 boundarylayer 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.
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Müllner, M., Schneider, W. Laminar mixed convection on a horizontal plate of finite length in a channel of finite width. Heat Mass Transfer 46, 1097–1110 (2010). https://doi.org/10.1007/s0023101006792
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DOI: https://doi.org/10.1007/s0023101006792
Keywords
 Nusselt Number
 Lift Force
 Mixed Convection
 Richardson Number
 Potential Flow