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.

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 ? < 1 corresponding to blowing vw > 0 ,

while ? > 1 corresponding to suction vw < 0. It may be seen
from equations (6) and (10) that the boundary layer has a uniform thickness of
0 as in a stagnation point flow. Thus increase in suction causes progressive
thinning of the boundary layer while the reverse is true for blowing. If suction
or blowing were absent (? = 1), the thickness of the boundary layer would be of
0. Thus the physical interpretation of ? defined in (6) is that it represents
the distance normal to the sheet in units of the boundary layer thickness.
To solve equations (3) and (4), we next assume
-----------(12)
Substitute equations 7 , 10 and 12 in equation 3 it gives
------------(13)
Equation for
being same with P? replaced by Sc.
The boundary conditions are derived from equations 5 and 12 as
(0) = 1,
(?) =0 ,
(0) =1 ,
(?) =0 -------------(14)
The solution of equation 13 satisfying equation 14 is
or
--------(15)
With the same expression for
in which P? is replaced by Sc.
When P?=1
the above integral can be expressed as
--------------------(16)
When P? ?1, the integrals in equation
15 can be expressed as incomplete Gamma functions. We have computed
using an algorithm due to Bhat-tacharjee4 and the
results are shown graphically in Fig.2 For fixed values of ? and
P?
temperature decreases with increases in blowing. Table 1 shows the values of
versus ? for several values of P?
with ?
= 0.4
Following exactly the analysis of ref.22, we find that the
total quantity of mass leaving the sheet of width C per unit time over a length
x measured from the origin is
-----------(17)
Using equation 15, we find following expression for the
dimensionless heat transfer coefficient
;
or
----------------(18)
We have computed
or
for several
values of ?
and P?
or Sc and the results are shown in Table.2. The relationship among CAw
, CA? and ? can be found in the same manner as
that in ref.2 as follows;
With a similar relation for the temperature field.
Figures:
Fig.1 – Boundary layer on a stretching flat porous surface.
References:
1. Sakiadis, B.C.,AIChE
J.
2. Erickson. L.E. Fan, L.T., and Fox, V.G., Ind. Eng.Chem.
Fundam.
3. MeCormack, P.D., and Crane, L., Physical fluid
dynamics.
4. Bhattacharjee, G.P., Applied statistics.