# Boundary Layer Flow past a Stretched Plate with Heat Transfer.

Introduction

Due to application of boundary layer flow past a stretched surface or moving surface in many processing process such as metal extrusion, glass fibre production, continuous casting, hot rolling, manufacturing of plastic or rubber sheets and crystal growing, a number of scholars as Altan et al. (1991), Fisher (1976), Tadmor and Klein (1970) and Ali (1995) paid their attention to study the boundary layer flow with heat transfer. Boundary layer flow over a stretched surface moving with a constant speed was initially studied by Sakiadis (1961a) and obtained numerical solution using similarity transformation. Later, Erickson et al (1996) extended the work of Sakiadis by introducing suction or blowing at the surface of the stretched plate and he investigated its effect on the heat transfer. Tsou et al. (1967) confirmed the results of Sakiadis experimentally also.

None of the above referred investigators has obtained series solution of boundary layer flow of incompressible viscous fluid past a stretched plate. We made an attempt to obtain a series solution. The convergence of the series solution has been discussed under the limitation of coefficients and other parameters.

Formulation of the problem

Let x-axis be along the moving plate and y-axis to be perpendicular to the direction of motion of the plate. The equations governing the boundary layer flow are:

u [partial derivative]u/[partial derivative]x + v [partial derivative]u/[partial derivative]y = v [[partial derivative].sup.2]u/[[partial derivative][y.sup.2] (momentum equation) (1)

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (continuity equation) (2)

The relevant boundary conditions are

y=0, u=[u.sub.0], v=0 (3)

y [right arrow] [infinity], u=0, [u.sub.y]=0 (4)

where n is kinematic viscosity and [micro]v are horizontal and vertical components of velocity field, respectively.

To solve this problem, we define similarity variable by [eta] = y/[square root of 2kvx]

Setting u = [u.sub.0] f([eta]) and using continuity equation together with boundary condition at y = 0, we have

v = [u.sub.0] [square root of (kv/2x)] {[eta]f ([eta]) - f ([eta])} (5)

Substituting u and v in the momentum equation (1), we get

f ([eta])+ [u.sub.o] k f([eta]) f ([eta]) = 0 (6)

Boundary conditions (3) and (4) reduce to

[eta] = o, f = 1,f = = 0

[eta] [right arrow] [infinity], f [right arrow] 0 (7)

Equation (6) is non-linear ordinary differential equation of third order. K. Marwah (2001) solved this differential equation assuming [u.sub.o] small by perturbation method. We try the series solution of this problem in the following form due to Sachedev, P.L. (1991)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where c = [u.sub.o] k

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On substituting the series for f([eta]), [f.sup.[??]]([eta]) and [f.sup.[??]]([eta]) in (6), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

For i = 1, [b.sub.1]a - [b.sub.1]a = 0 which is an identity. Since a is to be determined by the boundary conditions, we may choose |[b.sub.1]|=1 without loss of generality.

For i = 2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The recurrence formula comes out to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

If |a| <1 then, we have to find c for which |bi| [pound] l, i [sup.3] 2 and the series (8) converges uniformly for any g>0. So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

only if |c| [less than or equal to] 3 i.e., -3 [less than or equal to] c [less than or equal to] 3. As c = [u.sub.o] k where [u.sub.o] > 0 and by definition of has well as k can neither be zero nor be negative. Therefore only possible value of c is positive i.e., c[member of] [0,3]. Hence for 0 < c [less than or equal to] 3 and [|b.sub.i|] [less than or equal to]1, i=1, 2, ...., there is two parameter family of solutions of (6), satisfying the condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for g>0. Other conditions from the (7) are

F(0) = [gamma]/C + [gamma] [[infinity.summation over (i-1)] [b.sub.i][a.sup.i] (12)

F(0) = [[infinity].summation over i=1] [ib.sub.1] [a.sup.i] = 1 (13)

These are two uncoupled transcendental equations for g and a. If equation (12) is solved for 'a' with the assumption |a| <1, then equation (13) provides the value of [gamma]. We determine the sign of [b.sub.i] 's. Recalling equation (11), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The coefficient of [b.sub.i-k] [b.sub.k] in (14) retains the same sign as c. Since c [??] ]0,3[, so we are left to consider the case where,

c > 0 and [b.sub.1] =1, [b.sub.i] >0 (i=2,3,...)

Now, we have to guess the value of 'a'. So, using the equation (12) after canceling out [gamma], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

For |a| < 1 and c [not equal to] 0 the equation (15) again puts one more condition on c as 0 < c < 1. Using two inequalities for 'a', we get the values of 'a' lying in

[-1,1] [intersection] (-1/1-c, 1/1-c) = (-1,1)

Thus a [member of](-1, 1). Hence, [gamma] can be found by (13) under the following constraints.

0 < c < 1, a [member of] (-1,1), [b.sub.1] = 1,[b.sub.i][member of] (0,1) i=2,3,4, ....

From the relation (13), we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where bracketed series is arithmetico- geometric. So,

|[gamma][|.sup.2] (|a|/1-|a[|.sup.2]) > 1 |[gamma][|.sup.2] > 1-|a[|.sup.2]/|a| (16)

which implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

but [gamma] > 0. Hence, the value of [gamma] > 1-|a|/[square root of |a|]

Now, only problem is to find negative root a =[a.sub.o] under the conditions on c, [b.sub.i] and [gamma]. The requirement of negative value of 'a' has been supported by the following two physical facts.

f'(o)=1 and f' ([infinity])=0 that is; f'([eta]) is decreasing function of h, so f" (0) <0, and The infinite series for f(h) converges only for negative value of 'a'.

Therefore, we restrict the value of 'a' in ]-1, 0[. Hence, the series (8) provides a unique solution of boundary layer flow past a stretched plate under the following conditions on parameters.

c [member of] (0,1) [b.sub.i][member of](0,) i=1,2,... (18)

[gamma] > [square root of 1-|a[|.sub.2]/[square root of |a| for a [??]]-1 0[

Heat problem

The boundary layer equation governing the flow of heat between stretched plate and fluid is

u [partial derivative]T/ [partial derivative]x + v [partial derivative]T/[partial derivative]y = [K.sub.0]/[mu][C.sub.p] [[partial derivative].sup.2]T/[partial derivative][y.sup.2] (19)

with boundary conditions

[gamma] = 0, T = [T.sub.p] [gamma] [right arrow] [infinity], T = [T.sub.[infinity]] (20)

where [K.sub.o] thermal conductivity, [mu] is viscosity, [C.sub.p] is specific heat at constant pressure, [T.sub.p] is temperature of plate and [T.sub.[infinity]] is temperature surrounding fluid.

Since temperature field is varying with y only as suggested by boundary conditions. So the equation (19) reduces to

v [partial derivate]T/[partial derivate]y = [K.sub.0]/[rho][C.sub.p]/ [[partial derivate].sub.2]T/[partial derivative][y.sup.2] (21)

Now, defining dimensionless temperature as

[theta] = T-[T.sub.[infinity]]/[T.sup.p]-[T.sub.[infinity]] (22)

and changing the independent variable y to h together with substitution for v from the equation (5), we have

k Pr [U.sub.0] {[eta] f'([eta])-f([eta])} d[theta]/d[eta]=[d.sup.2][theta]/d[[eta].sup.2] (23)

where Pr is Prandtl number, and now boundary conditions as

[eta] = 0, [theta] = 1 [eta] [right arrow] [infinity], [theta] [right arrow] 0 (24)

Integrating the equation (23), we have

[theta]'([eta]) = Aexp{kPr[U.sub.0]([eta] f([eta])-2 [integral] f([eta]d[eta])} (25)

Putting the expression for f from the equation (8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Again integrating from [eta] to [infinity], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using boundary conditions (24), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

In (27) both the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are convergent for any [b.sub.[??]] [member of] (0,1],

i=1,2,... and [for all][a.sub.[??]][member of](-1,0). Let the sum of these series be [S.sub.0] and [S'.sub.0] at some [eta] = [[eta].sub.0] respectively, then equation (27) becomes

[theta]([eta]) = exp{-5 Pr [U.sub.0][gamma](1/c - [S.sub.0])[eta]} (28)

for c [member of] (0,1) and 1/c - [S.sub.0] > 0.

Nusselt number

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Discussion and conclusions

The analysis done so far to seek the solution of boundary layer flow past a stretched plate in terms of infinite series form enables us to verify the convergence of series appeared for f(h). We apply ratio test as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as n [right arrow] [infinity], only when [b.sub.n+1] <[b.sub.n] i.e. [{b.sub.n}] is monotonically decreasing sequence. Therefore,

f([eta]) = [gamma]/c + [gamma] [[infinity].summation over (i-1)][b.sub.i][a.sup.i][e.sup.-iy[eta]]

represents the solution if the parameters satisfy the following conditions.

c [member of] (0,1),

[b.sub.1]=1, [b.sub.i]>0 but [b.sub.i] [member of] (0,1),i=2,3,4,... and [{b.sub.i}] is monotonic decreasing sequence, and

[gamma]>[square root of 1-|a[|.sup.2]/|a|] for a [member of](-1,0).

In the light of conclusion 1, one can find the value of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

so that this boundary value problem be converted into the following initial value problem as:

[f.sup.[??]][eta] + [u.sub.o] k f [f.sup.[??]] = 0 [eta]=0, f=0, [f.sup.[??]]=1, [f.sup.[??]]=1

where [gamma] can be calculated under the prescribed conditions on g, bi and a, as the sum of infinite series given in the equation (29).

Under the following conditions

(i) c[member of](0,1),

[b.sub.1]=1, [b.sub.i]>0 but [b.sub.i][member of](0,1), i=2,3,4,.... and [{b.sub.i}] is monotonically decreasing sequence, and [member of] > [square root of 1-|a[|.sub.2]/|a|] for a [member of](-1,0),

we can prove the convergence of f([eta]) = [gamma]/c + [gamma][[infinity].summation over (i-1)][b.sub.i][a.sub.i][e.-i[gamma][eta]]. by proving the convergence of alternating series

[[infinity].summation over (i-1)][b.sub.i][a.sup.i] [e.sup.-I[gamma][eta]] For

i) [u.sub.n+1]=[b.sub.n+1][a.sup.n+1][e.sup.-(n+1)[gamma][eta] <[b.sub.n][a.sup.n][e.sup.-n][gamma][eta]=[u.sub.n] " n

and,

ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, by Leibnitz test, series [[infinity].summation over (i-1)] [b.sub.i][a.sup.i] [e.sup.-[gamma][eta]] is convergent. Hence, f([eta]) [gamma]/c + [gamma] [[infinity].summation over (i-1)] [b.sub.i][a.sup.i] [e.sup.-[gamma][eta]] is convergent.

Temperature field is a function of Prandtl number Pr. As Prandtl number increases temperature field decreases , i.e. Pr suppresses heat transfer.

Nusselt number gets accelerated by Prandtl number Pr. This rssult is in agreement with result of Naseem Ahmad and A. Mubeen (1995) and N. Ahmad, G.S. Patel and B. Siddappa (1990).

Acknowledgement

We pay our sincere thanks to Professor P.L.Sachedev of Department of Mathematics, I.I.Sc. Banglore and Professor Z.U.Ahmad of Department of Mathematics, AMU Aligarh for their encouragement and valuable suggestions to carry out the present work.

References

Altan T, S On and H. Gegel, 1991. American Society of Metals, Metals Park, OH.

Fisher, E.G. 1976. Extrusion of plastics, Wiley, New York.

Tadmor, Z and I. Klein, 1970. Engineering Principles of Plasticating Extrusion, Polymer Science and Engineering Series, Van Nostrard Reinhold, New York.

Ali, N.E. 1995. Int. J. Heat and Fluid Flow, 16: 280.

Sakiadis, B.C. 1961a. AICHE J., 7: 26.

Erickson, L.E., L.T. Fan and V.G. Fox, 1996. Indust. Engg. Chem., 5: 19.

Tsou, F.K. E.M. Sparrow and R.J. Goldstein, 1967. Heat Mass Transfer, 10: 219.

Kavita Marwah, 2001. Ph.D. Thesis, Chapter 4, JMI.

Sachedev, P.L., 1991. Non-linear Ordinary Diff. Eqs. And Their Applications, pp: 119-123, Marcel Dekker.

N.Ahmad and A Mubeen, 1995. Int. Comm. Heat and Mass Transfer, 22(6): 895-906.

N.Ahmad, G.S. Patel and B. Siddappa, 1990. ZAMP, Vol. 41.

Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India.

Naseem Ahmad,: Boundary Layer Flow past a Stretched Plate with Heat Transfer, Adv. in Nat. Appl. Sci., 1(1): 7-14, 2007

Corresponding Authot: Naseem Ahmad, Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India. E-mail: naseem_mt@yahoo.com
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