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.
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
u.Tx + v.Ty = ?.Tyy ———-(3)
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
Where ? = (cv)1/2xf(?) and ? = (c/v)1/2y
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
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 =
Provided that ? > 0 and vw = -(cv)1/2 ,f(0) =
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.