Editing Boundary layer (section)

== Heat and mass transfer ==
In 1928, the French engineer [[André Lévêque]] observed that convective heat transfer in a flowing fluid is affected only by the velocity values very close to the surface.<ref>{{cite journal
|author= Lévêque, A.
|year=1928
|title=Les lois de la transmission de chaleur par convection
|journal=Annales des Mines ou Recueil de Mémoires sur l'Exploitation des Mines et sur les Sciences et les Arts qui s'y Rattachent, Mémoires
|volume=XIII
|issue=13
|pages=201–239
|language=fr}}</ref><ref name="McMahon">{{cite web
|title=André Lévêque p285, a review of his velocity profile approximation
|author=Niall McMahon
|url=http://www.computing.dcu.ie/~nmcmahon/biography/leveque/leveque_approximation.html
|url-status=dead
|archive-url=https://web.archive.org/web/20120604004223/http://www.computing.dcu.ie/~nmcmahon/biography/leveque/leveque_approximation.html
|archive-date=2012-06-04
}}</ref> For flows of large Prandtl number, the temperature/mass transition from surface to freestream temperature takes place across a very thin region close to the surface. Therefore, the most important fluid velocities are those inside this very thin region in which the change in velocity can be considered linear with normal distance from the surface. In this way, for
:<math>u(y) = U \left[ 1 - \frac{(y - h)^2}{h^2} \right] = U \frac{y}{h} \left[ 2 - \frac{y}{h} \right] \;, </math>
when <math> y \rightarrow 0</math>, then
:<math>u(y) \approx 2 U \frac{y}{h} = \theta y, </math>
where ''θ'' is the tangent of the Poiseuille parabola intersecting the wall.
Although Lévêque's solution was specific to heat transfer into a Poiseuille flow, his insight helped lead other scientists to an exact solution of the thermal boundary-layer problem.<ref name="Martin2002">{{cite journal
|author=Martin, H.
|year=2002
|title=The generalized Lévêque equation and its practical use for the prediction of heat and mass transfer rates from pressure drop
|journal=Chemical Engineering Science
|volume=57
|issue=16
|pages=3217–3223
|doi=10.1016/S0009-2509(02)00194-X|bibcode=2002ChEnS..57.3217M
}}</ref> Schuh observed that in a boundary-layer, ''u'' is again a linear function of ''y'', but that in this case, the wall tangent is a function of ''x''.<ref>{{cite journal
|author=Schuh, H.
|year=1953
|title=On Asymptotic Solutions for the Heat Transfer at Varying Wall Temperatures in a Laminar Boundary Layer with Hartree's Velocity Profiles
|journal=Journal of the Aeronautical Sciences
|volume=20
|issue=2
|pages=146–147
|doi=10.2514/8.2566}}</ref> He expressed this with a modified version of Lévêque's profile,

:<math>u(y) = \theta(x) y. </math>

This results in a very good approximation, even for low <math>Pr</math> numbers, so that only liquid metals with <math>Pr</math> much less than 1 cannot be treated this way.<ref name="Martin2002"/>
In 1962, Kestin and Persen published a paper describing solutions for heat transfer when the thermal boundary layer is contained entirely within the momentum layer and for various wall temperature distributions.<ref>{{cite journal
|author1=Kestin, J.  |author2=Persen, L.N.
 |name-list-style=amp|year=1962
|title=The transfer of heat across a turbulent boundary layer at very high prandtl numbers
|journal= International Journal of Heat and Mass Transfer
|volume=5
|issue=5
 |pages=355–371
|doi=10.1016/0017-9310(62)90026-1|bibcode=1962IJHMT...5..355K
 }}</ref> For the problem of a flat plate with a temperature jump at <math>x = x_0</math>, they propose a substitution that reduces the parabolic thermal boundary-layer equation to an ordinary differential equation. The solution to this equation, the temperature at any point in the fluid, can be expressed as an incomplete [[gamma function]].<ref name="McMahon"/> [[Hermann Schlichting|Schlichting]] proposed an equivalent substitution that reduces the thermal boundary-layer equation to an ordinary differential equation whose solution is the same incomplete gamma function.<ref>{{cite book
|author=Schlichting, H.
|year=1979
|title=Boundary-Layer Theory
|publisher=McGraw-Hill
|place=New York (USA)
|edition=7}}</ref> Analytic solutions can be derived with the time-dependent [[Self-similarity|self-similar]] Ansatz for the incompressible boundary layer equations including heat conduction.<ref>{{Cite journal |last1=Barna |first1=Imre Ferenc |last2=Bognár |first2=Gabriella |last3=Mátyás |first3=László |last4=Hriczó |first4=Krisztián |date=2022 |title=Self‑similar analysis of the time‑dependent compressible and incompressible boundary layers including heat conduction |url=https://link.springer.com/article/10.1007/s10973-022-11574-3 |journal=Journal of Thermal Analysis and Calorimetry |volume=147 |pages=13625–13632 |arxiv=2101.08990 |doi=10.1007/s10973-022-11574-3}}</ref>

As is well known from several textbooks, heat transfer tends to decrease with the increase in the boundary layer. Recently, it was observed on a practical and large scale that wind flowing through a photovoltaic generator tends to "trap" heat in the PV panels under a turbulent regime due to the decrease in heat transfer.<ref>{{cite journal | url=https://doi.org/10.1038/s44172-023-00119-7 | title=Energy losses in photovoltaic generators due to wind patterns | year=2023 | last1=Rossa | first1=Carlos | journal=Nature Communications Engineering | volume=2 | issue=66 | page=66 | doi=10.1038/s44172-023-00119-7 | bibcode=2023CmEng...2...66R | pmc=10956078 }}</ref> Despite being frequently assumed to be inherently turbulent, this accidental observation demonstrates that natural wind behaves in practice very close to an ideal fluid, at least in an observation resembling the expected behaviour in a flat plate, potentially reducing the difficulty in analysing this kind of phenomenon on a larger scale.