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

2

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