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Abstract:

 Flow in the boundary
layer on moving solid surfaces was investigated by Sakiadis 1.The boundary
layer is different from that in Blasius flow past a flat plate. Erickson, Fan
and Fox 2 this problem extended to the case when the transverse velocity at
the moving surface is non-zero and is such that similar solutions exist. They existing
numerical solutions of the boundary layer equations for various values of the
parameters. These investigations have a bearing on the problem of a polymer
sheet extruded continuously from a die and are based on the implicit assumption
that the moving sheet is inextensible. However, situations may arise in the
polymer industry when one deals with a stretching plastic sheet as pointed out
by McCormack and Crane 3.

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We Present analyze where the momentum, heat and mass
transfer in the boundary layer over a stretching sheet subject to suction or
blowing. We are not however aware of any experimental data with which our
theory may be compared.

Analysis

Consider the case of a flat sheet issuing from a thin slit
at x = 0, y = 0 and subsequently being stretched, as in a polymer processing application
(Fig.1). Assuming boundary layer approximations the equations of continuity,
momentum, energy and diffusion in the usual notations are;

ux + vy = 0  ———–(1)

u.ux + v.uy = v.uyy
———-(2)

u.Tx + v.Ty = ?.Tyy ———-(3)

 ————(4)

Respectively. Viscous dissipation is neglected in equation
(3) and the fluid properties are assumed constant over the range of temperature
and compositions considered, CA being the concentration of the
species A in the flow. In deriving equation (4), it is assumed that the species
A is present in a small concentration, the molecular diffusivity D being taken
constant. The boundary conditions are

u = cx, v = vw , T = Tw ,CA
= CAw at y = 0

then u?0 ,T?T? ,CA?CA? as y??    ———-(5)

where vw ,Tw ,T? , CAw
and CA? are constants.

It should be noted that this problem is a straight forward
extension of the work of Erickson et. al 2.

The only difference lies in the surface speed. Erickson et. al
used a constant speed for the surface while we use the linear speed given in
equation (5), c being a positive constant. Here the velocity of the sheet at
the slit exit is zero which may be a good approximation to the case where the
stretching in a real situation is considerable. In actual practice a stretching
plastic sheet may not always conform to the linear speed assumed here.

The analysis now follows closely that in ref2. Equation (1)
implies a stream function ?(x,y) given by

 and

Where ? = (cv)1/2xf(?) and ? = (c/v)1/2y
————–(6)

Here f is not a function of x and the similarity variable ?
depends on y only. We find from equation (6)

u = cxf1(?) ,v = -(cv)1/2f(?) ——-(7)

Where a prime denotes differentiation. Substitution of equation
(7) in equation (2) gives

————(8)

Subject to the following boundary conditions derived from
equations (5), (6) and (7) as

f1(0) = 1, f(0) = -vw(cv)1/2
, f1(?) = 0 ————(9)

The solution of equation (8) satisfying equation (9) is

f1 = e-?? , f =

  ————–(10)

Provided that ? > 0 and vw = -(cv)1/2   ,f(0) =

 ——-(11)

Clearly ? 0 ,
while ? > 1 corresponding to suction vw

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