ACTA MECHANICA 9 by Springcr-Verlag 1986

Aeta Mcchanica 64, 179--185 (1986)

Hydromagnetic

Flow and Heat

Transfer over a

Continuous, Moving, Porous, Flat Surface By K. Vajravelu, Orlando, Florida With 3 Figures

(Received December 30, 1985)

Summary Exact solutions for hydromagnetic boundary-layer flow and heat transfer over a, continuous, moving, fla~ surface with uniform suction and internal heat generation/ absorption are obtained. Flow of this type represents a new class of boundary-layer problems, with solutions substantially different from those for boundary-layer flow on a flat surface of finite length. These solutions are even exact solutions of the complete NavierStokes equations and the energy equation.

1.

Introduction

A study of boundary-layer behavior on continuous solid surfaces has attracted the attention of relatively few researchers, although the analysis of such flows finds application in different areas such as the aerodynamic extrusion of plastic sheets, the boundary-laye r along material-handling conveyers, the cooling of an infinite metallic plate in a cool bath, and the boundary-layer along a liquid film in condensation processes. In view of these applications, Sakiadis [1] has studied theoretically the boundary-layer on a continuous semi-infinite sheet moving steadily through an otherwise quiescent fluid environment. The boundary-layer solution of Sakiadis resulted in a skin friction of about 30 percent higher than that of Blasius [2] for the flow past a stationary flat plate. Later, for different values of the Prandtl number, experimental and theoretical studies were made by Tsou et al. [3]. In these studies, the authors have not included the effects of magnetic field and suction on the flow and heat-transfer characteristics; such effects have definite bearing on the field of aeronautical engineering. Moreover, there is an appreciable temperature difference between the plate and the ambient fluid and, thus, one needs to consider the temperature-dependent heat sources or sinks, which m a y exert strong influence on the heat-transfer characteristics.

180

K. Vajravelu:

The analysis of temperature field, as modified by generation or absorption of heat in moving fluids, is important in view of several physical problems such as: (a) Problems dealing with chemical reactions [4], (b) Problems concerned with dissociating fluids [5]. In fact, literature is replete with examples dealing with heat transfer in laminar flow of viscous fluids. Heat generation has been assumed to be constant, or a function of space variable by some authors. Others have considered directly the frictional heating and the expansion effects. Sparrow and Cess [6] have obtained solutions of the steady flow and heat transfer of the stagnation point flow, taking into account the temperature-dependent heat generation. Foraboschi and Federico [7] have assumed a volumetric rate of heat generation as Q=:Q0(T--T0)

when

T~To ]

Q=0

when

T
(0)

in their study of the steady-state temperature profiles for linear parabolic-and piston-flow in circular tubes. The relation (0), as described by Foraboschi and Federico, is valid as an approximation of the state of some exothermic process having T0 as the onset temperature. To the best knowledge of the author, very little attention has been given to the study of the effects of magnetic field, suction, and internal heat generation/ absorption on the flow and heat-transfer characteristics. This analysis is, therefore, an attempt to investigate the effects of magnetic field, suction, and internal-heat generation/absorption on the continuous fiat surface moving in a quiescent fluid environment problem. The volumetric rate of heat generation (or absorption) is taken

as

Q ~ (T--Too), where Too is the ambient temperature. The flow and heat-transfer characteristics are found to depend on dimensionless numbers M [ = ~,(~Bo2/~V02] and a [----- Q~/ Vo~QCp],which arise, respectively, from the magnetic field and the internal-heat generation. For the governing equations of the problem, closed form solutions are obtained and are evaluated numerically for several sets of values of the parameters. The contributions of the new dimensionless numbers, M and cr to the flow and heat-transfer characteristics are found to be quite significant.

2. Formulation and Solution of the Problem Consider a long, continuous sheet (for example, a polymer sheet or filament extruded continuously from a die, or a long thread traveling between a feed roll and a wind-up roll) which issues from a slot, as shown in Fig. 1. Let us make the assumptions that a certain time has elapsed after initiation of motion, so that steady-state conditions prevail and flow disturbances created by the roll are neglected. An observer fixed in space will note that the boundary-layer on the

Hydromagnetio Flow and Hea~ Transfer

181

U==O

I

Y

/

f

t --,.N,..[.,..I..,,.I..,..,_,..,..,..L.., ." /

/,e

slot

=

Ow

x

~

~

Vo : c o n s t

Fig. 1. Schematic r e p r e s e n t a t i o n of t h e b o u n d a r y layer on a continuous moving surface

sheet originates at the slot and grows in the direction of the sheet. The boundarylayer behavior here appears to be different from what would be expected if the sheet were considered as a moving, flat plate of finite length on which the boundary-layer would grow in the direction opposite to the direction of motion of the plate. Let us investigate the boundary-layer flow of an electrically conducting incompressible fluid (with electric conductivity a) over a continuous moving, flat surface, with Bo an imposed, uniform magnetic field perpendicular to the surface. The boundary-layer equations for flow and heat-transfer (with internal-heat generation/absorption) are, in the usual notation, ~U __aB~ e U + v ~-~'

u - ~U ~ + v ~U y ~U

~V

-~ + ~

3T

~T)

QOp U - ~ -t- V-ff-~

(1)

= 0,

(2) ~2T

(3)

= Q(T -- Too) + k ~y~,

where the induced magnetic field is neglected (which is justified for small magnetic Reynolds number flows, see Shercliff [8]). The boundary conditions for the velocity and temperature fields are U=

U~,

U --~ O ,

V= V0=const<0,

T=Tw

at

Y=0

/

T-+Too

as

Y-~.

!

(4)

Making use of the assumption that the velocity and temperature fields are independent of the distance parallel to the surface (which is used by Meredith a n d Griffith [9] and Schlichting [10]), Eqs. (1), (2), (3) and the boundary eondi-

182

K. Vajravelu:

tions (4) can be written as dU --Vo ~y -

aBo 2 - - u + ~

d2U v d y ~,

(5>

dT d~T --oCpVo ~ = Q ( T -- Too) + g -~-~-~,

U=

U~,

T=

U->O,

T~

at

Y=0,

T - + T~o

as

Y --> ~ .

(6)

}

(7),

Defining nondimensional variables u = U/U,.,

y = YVo/%

0 -----( T - - T~)/(T,~ - - Too),

Eqs. (5) and (6) and conditions (7) can be written as d2u du dy---~ + ~y

M u = 0,

(8}

d20 I dO dy 2 -(- Pr ~yy + a Pr 0 = O,

u=

1,

u->0,

O = 1 at

y=O,

0-+0

y->c~,

as

(9)~ /

(10}

where M = avBo2/O.Vo 2, the magnetic parameter, Pr = #Up/k, the Prandtl number, = Qv/Vo2~Cp, the heat source/sink parameter.

Solving Eqs. (8) and (9) with conditions (10) exactly, we get u(y)

(11)

--= e - I]2[I + ( I + 4 M } W : ] y ,

O(y) = e -1/2EPr+(Pr2-4~Pr)*Iqy.

(12)

The shear stress (or the skin friction) and the heat transfer coefficient (or theNusselt number) at the surface are defined, respectively, in nondimensional forms, as ~' =

_

du

1 [1 + (1 + 4M)1/2],

~VwV---~ = U u . = o

dO N u = kVo(Tw - - T ~ ) =

_

y=0

(13}

2

1

~ [Pr + (Pr 2 -- 4~

pr)i/2].

(14)

The velocity field, temperature field, skin friction and heat transfer coefficient are calculated for air (Pr ~- 0.714) and water (Pr = 6.750) with M = 1, 2, 3, 4, 5 and = 0.15, 0, --0.15, --0.30, --0.45, --0.60. The numerical results thus obtained are presented in Figs. 2 and 3. I t should be mentioned here that the problem of Sakiadis [1] is a special case of this analysis for M and c~ equal to zero.

Kydromagnetie Flow and Keat Transfer

183

3. Discussion of the Besults The values of the velocity field, u, for M = 0, 1 and 3 are summarized in Fig. 2(a). From Fig. 2(a), it is observed that u decreases considerably as M increases. In the magnetic field case, the velocity field satisfies the boundary condition (at infinity) for smaller values of y than that in its counterpart. Fig. 2(b) describes the behavior of the skin friction (absolute) coefficient at the surface. :From Fig. 2(b) it is evident that the skin friction coefficient is an increasing function of M.

1,0

~

O,B

M:O M:I

0,6

(a)

M=3

0.4

0.2

O

!

2

0

8

4

5

Y 3.0

2.6

2.2

ITI

(b) 1,8

1.4

1.0

0

I

I

1

2

I

t

3

4

'

-

5

M

Fig. 2. (a) Velocity profiles. (b) 8kin friction coefficient

184

K. Vajravelu: 1.0

0.8 0.6 8

(a)

-0.45 0.4

0.2

0

|

0

1

2

3

4

5

Y 7.5

6.0 4.5

~ 5 0

INul

(b) Pr=0.714

8.0

1.5 0 -0.60

I -0.45

i -0.30

I

-0.15

I

I

0

0.15

Fig. 3. (a) Temperature profiles for Pr = 0.714. (b) Heat transfer coefficient

The values of temperature, O, for c~ = 0.15, --0.15, --0.45 are summarized in Fig. 3(a) for air (Pr = 0.714). From Fig. 3(a) it can be observed t h a t the values of 0 have increased considerably in the presence of heat sources. The opposite phenomenon is observed in the presence of heat sinks. Also, in the case of heat sinks, the t e m p e r a t u r e field satisfies the boundary condition for smaller values of y t h a n t h a t in the case of heat sources. Fig. 3(b) describes the behavior of the rate of heat transfer (absolute) at the surface. F r o m Fig. 3(b) it can be observed t h a t the rate of heat-transfer p a r a m e t e r is an increasing function of Pr. However, opposite is the case with the heat source p a r a m e t e r ~.

ttydromagnefio Flow and Heat Transfer

18~

4. Conclusions 1. T h e fluid v e l o c i t y u h a s l a r g e r v a l u e s in t h e a b s e n c e o f m a g n e t i c field thar~ in i t s presence. 2. T h e fluid t e m p e r a t u r e 0 h a s l a r g e r v a l u e s i n t h e p r e s e n c e of h e a t s o u r c e s t h a n in t h e i r a b s e n c e . 3. S k i n f r i c t i o n (absolute) i n c r e a s e s w i t h i n c r e a s i n g M . 4. T h e r a t e of h e a t t r a n s f e r (absolute) increases w i t h a n i n c r e a s i n g P r b u t decreases w i t h a n i n c r e a s i n g ~.

References [1] Sakiadis, B. C. : Boundary-layer behavior on continuous solid surfaces. A. I., C h . E . , J1. 7, 26 (1961); 7, 221 (1961). [2] Blasius, H. : Grenzsehichten in Flfissigkeiten mit kleiner Reibung. Z. Math. Phys. 56, 1 (1908). [3] Tsou, F., Sparrow, E. M., Goldstein, R.: Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transfer 10, 219 (1967). [4] Veron, M.: Bull. Tech. Soc. France, Const. Babcock and Wilcos, No. 21, Paris 1948. [5] Lighthill, M. J. : F. M. 1958, Aero. Res. Council, July 1956. [6] Sparrow, E.M., Cess, R . D . : Temperature-dependent heat sources or sinks in a stagnation point flow. Appl. Sci. Res. A 10, 185 (1961). [7] Foraboschi, F. P., Federico, I. D. : Heat transfer in laminar flow of non-Newtonian heat-generating fluids. Int. J. Heat Mass Transfer 7, 315 (1964). [8] Shercliff, J. A.: A text book of magnetohydrodynamics. Pergamon Press 1965. [9] Meredith, F. M., Griffith, A . A . : Modern developments in fluid dynamics. Oxford University Press 1938. [10] Schlichting, H.: Die Grenzsehicht mit Absangung und Ausblasen. Luftfahrtforschung 19, 179 (1942).

K. Va]ravelu Department o/Mathematics University o/Central Florida, Orlando, FL 32816 U.S.A.