Summary
Exact boundary layer similarity solutions are developed for flow, friction and heat transfer on a continuously accelerated sheet extruded in an ambient fluid of a lower temperature.
Melt-spinning, polymer and glass industries and the cooling of extruded metallic plates are practical applications of this problem.
Results for skin-friction and heat-transfer coefficients are given. Larger acceleration is accompanied by larger skin-friction and heat-transfer coefficients. Rapid cooling of the sheet is accompanied by a larger Nusselt number.
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Abbreviations
- \(\bar b\) :
-
sheet width
- c :
-
dimensionless constant
- c f :
-
local skin friction coefficient
- F :
-
dimensionless transformed stream function
- G :
-
dimensionless transformed temperature
- \(\bar h_x \) :
-
local heat transfer coefficient
- \(\bar k\) :
-
fluid thermal conductivity
- \(\bar L\) :
-
length of deformation zone
- m :
-
exponent of surface speed variation
- q :
-
exponent of surface temperature variation
- T :
-
dimensionless temperature
- \(\bar T_s \) :
-
sheet surface temperature
- \(\bar T_0 \) :
-
solidification temperature
- \(\bar T_\infty \) :
-
ambient temperature
- \(\bar t\) :
-
sheet thickness
- u :
-
velocity component along the sheet
- u s :
-
sheet surface velocity
- \(\bar u_0 \) :
-
wind up velocity
- v :
-
velocity component normal to the sheet
- x :
-
dimensionless coordinate along the sheet
- y :
-
dimensionless coordinate normal to the sheet
- Nu:
-
Nusselt number,\(Nu = \bar h_x \bar L/\bar k\)
- Pr:
-
Prandtl number,\(Pr = \bar \mu \bar c_p /\bar k\)
- Re:
-
Reynolds number,\(Re = \bar \mu _0 \bar L/\bar v\)
- ɛ:
-
ɛ=Re−0.5
- η:
-
dimensionless similarity coordinate
- \(\bar \mu \) :
-
dynamic viscosity
- \(\bar v\) :
-
kinematic viscosity
- \(\bar \varrho \) :
-
fluid mass density
- \(\bar \varrho _s \) :
-
sheet mass density
- \(\bar \tau _w \) :
-
wall shear stress
- ψ:
-
dimensionless stream function
References
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Abdelhafez, T.A. Laminar thermal boundary layer on a continuous accelerated sheet extruded in an ambient fluid. Acta Mechanica 64, 207–213 (1986). https://doi.org/10.1007/BF01450395
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DOI: https://doi.org/10.1007/BF01450395