Flow and heat transfer in a Maxwell liquid film over an unsteady stretching sheet in a porous medium with radiation
Flow and heat transfer in a Maxwell liquid film over an unsteady stretching sheet in a porous medium with radiation
Shimaa E. Waheed 0 1
0 Present Address: Department of Mathematics and Statistic, Faculty of Science, Taif University , Taif, Kingdom of Saudi Arabia
1 Department of Mathematics, Faculty of Science, Benha University , Banha 13518 , Egypt
A problem of flow and heat transfer in a non-Newtonian Maxwell liquid film over an unsteady stretching sheet embedded in a porous medium in the presence of a thermal radiation is investigated. The unsteady boundary layer equations describing the problem are transformed to a system of non-linear ordinary differential equations which is solved numerically using the shooting method. The effects of various parameters like the Darcy parameter, the radiation parameter, the Deborah number and the Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. It is observed that increasing values of the Darcy parameter and the Deborah number cause an increase of the local skin-friction coefficient values and decrease in the values of the local Nusselt number. Also, it is noticed that the local Nusselt number increases as the Prandtl number increases and it decreases with increasing the radiation parameter. However, it is found that the free surface temperature increases by increasing the Darcy parameter, the radiation parameter and the Deborah number whereas it decreases by increasing the Prandtl number.
Maxwell fluid; Liquid film; Thermal radiation; Unsteady stretching sheet; Porous medium
-
Background
The flow and heat transfer within a thin liquid film due to the stretching surface in
otherwise quiescent fluid are important because of their wide applications in a number
of industrial engineering processes. Examples may be found in the cooling of a large
metallic plate in a cooling path, design of various heat exchangers, wire and fiber
coating, manufacturing plastic films, continuous casting, crystal growing,artificial fiber,
reactor fluidization, chemical processing equipment, a polymer sheet, polymer
extrusion, annealing and tinning of copper wires, etc. The flow problem within a liquid film
of Newtonian fluid on an unsteady stretching surface where the similarity
transformation was used to transform the governing partial differential equations describing the
problem to a non-linear ordinary differential equation with an unsteadiness parameter
first are studied by
Wang (1990)
. Many authors
(Usha and Sridharan 1995; Andersson
Andersson 2008; Noor and Hashim 2010; Mahmoud 2010; Ray and Mazumder 2001;
Abel et al. 2009)
investigated thin liquid film under different situations.
Many important fluids, however, such as molten plastics, polymers, etc., are
nonNewtonian in their flow characteristics. The flow of non-Newtonian fluids are finding
increasing applications in several manufacturing processes. Flow of a thin liquid film
of a power-law fluid caused by the unsteady stretching of a surface studied numerically
by
Andersson et al. (1996
) and analytically by
Wang and Pop (2006)
. The thin film flow
problem with a third grade fluid on an inclined plane hase beed investigated by
Siddiqui et al. (2008).
Chen (2007)
examined the effect of Marangoni convection of the flow
and heat transfer within a power-law liquid film on unsteady stretching sheet.
Siddiqui
et al. (2007
) presented the thin film flow of two non-Newtonian fluids namely, Sisko and
an Oldroyd 6-constant fluid on a vertical moving belt.
Hayat et al. (2008
) presented an
exact solution for the thin film flow problem of a third grade on an inclined plane. The
problem of the flow and heat transfer in a thin film of power-law fluid on an unsteady
stretching surface has been investigated by Chen (2003, 2006) where he also studied the
effect of viscous dissipation on heat transfer in a non-Newtonian thin liquid film over
an unsteady stretching sheet. The flow and heat transfer problem of a second grade fluid
film over an unsteady stretching sheet has been presented by
Hayat et al. (2008
). Abel
et al. (2009) investigated the effect of non-uniform heat source on MHD heat transfer in
a liquid film over an unsteady stretching sheet. Sajid et al. (2009) presented exact
solutions for thin film flows of a micropolar fluid down an inclined plane on moving belt and
down a vertical cylinder.
Mahmoud and Megahed (2009)
investigated the effects of
variable viscosity and thermal conductivity on the flow and heat transfer of an electrically
conducting non-Newtonian power-law fluid within a thin liquid film over an unsteady
stretching sheet in the presence of a transverse magnetic field.
Non of the above authors deals with the problem involving the thermal radiation on
the flow and heat transfer in a liquid film on unsteady stretching surface. Thermal
radiation effects may play an important role in control (...truncated)