Marangoni equations is derived; conditions in which

Marangoni boundary layers are as mathematical modeling dealt with for ‡uids with a
Prandtl number of order one. A …rst order boundary layer approximation of the volume and
surface phase …eld equations is derived; conditions in which a Marangoni boundary layer
is established are determined in terms of the problems data 15. When the appropriately
de…ned Reynolds number is large enough the Marangoni boundary layers are the edge dissi-
pative ‡ows or boundary layer type and thin dissipative layers may form near unrestricted
surfaces by Pop et al. 16. Marangoni ‡ows are persuaded by surface tension gradients at
the edge of immiscible ‡uids; due to the increased importance of surface forces and greater
extensions and interfaces such ‡ows become relevant in a microgravity environment. The nu-
merical results and mathematical structure show that the existence of Marangoni boundary
layers under nonisobaric conditions is still an open problem 17. The steady boundary layers
can be formed along the interface of two immiscible ‡uids in surface driven ‡ows that may be
generated not only with Marangoni e¤ects, but also with the existence of the buoyancy e¤ects
due to gravity and external pressure gradient by Chamkha et al. 18.The exponential tem-
perature of radiation e¤ects and particle shape by utilizing Marangoni boundary layer ‡ow
and heat transfer of copper-water nano‡uid driven have been examined in 19. The suction
and injection in a nano‡uid via Marangoni-driven boundary layer ‡ow has been investegated
by Remeli et al. 20. There are various numerical studies on Marangoni boundary layers
in several geometries such as those by AlMudhaf and Chamkha 21, Magyari and Chamkha
22, 23