Acta Mechanica 93, 169-177 (1992)

ACTA MECHANICA 9 Springer-Verlag 1992

On heat transfer to pulsatile flow of a viscoelastic fluid A. K. Ghosh, Calcutta, India, and L. Debnath, Orlando, Florida (Received November 2, 1990; revised June 4, 1991)

Summary. A study is made of a problem of heat transfer to pulsatile flow of a viscoelastic fluid between two parallel plates of which the upper one is at a temperature higher than the lower one. The solutions for the steady and the fluctuating temperature distributions are obtained. The rate of heat transfer at the plates is also calculated. Numerical solutions are discussed with graphical representations. It is shown that the elasticity of the fluid significantly increases the temperature in the boundary layers near the plates. The magnitude of heat transfer at the plates is also greatly affected by the elasticity of the fluid and the Eckert number.

1 Introduction The generation of heat due to friction and the variation of pressure usually exert a large effect on the process of cooling, and these factors often make a warmer wall heated instead of being cooled. In order to understand some physical phenomena like transpiration cooling and gaseous diffusion, problems of fluid flow in a channel or pipe have received considerable attention in recent years. The exact solutions of several problems associated with heat transfer in viscous fluids have been obtained, and they are reported in Schlichting's book [1]. In recent years, considerable attention has been given to problems of heat transfer to pulsatile fluid flows. The solutions of these problems play an important role in the study of blood flow in arteries. Radhakrishnamacharya and Maiti [2] have made an investigation of heat transfer to pulsatile viscous fluid flow in a porous channel. The main purpose of this paper is to study a problem of heat transfer to pulsatile flow in a viscoelastic fluid bounded by impervious rigid parallel plates separated by a distance h. With the assumption that the upper plate is at a temperature higher than the lower one, the solutions for the steady and fluctuating temperature distributions are obtained. The rate of heat transfer at the plates is also calculated. Numerical solutions are discussed with graphical representations. It is shown that elastic properties of the fluid significantly increase the temperature in the boundary layers near the plates. The magnitude of heat transfer at the plates is also greatly affected by the elasticity of the fluid and the Eckert number.

2 Mathematical formulation We consider the pulsatile flow of fluid between two infinitely long parallel plates, a distance h apart, which is driven by the unsteady pressure gradient in the form

1@ Q 0x

- A{1 + e exp (/cot)}

(2.1)

170

A.K. Ghosh and L. Debnath

where A is a known constant, e is a suitably chosen positive quantity and co is the frequency. We suppose that the motion is slow and hence all second order quantities may be neglected. This study is based upon the Oldroyd model of a viscoelastic fluid [3], and the properties of such a fluid are specified by three constants, t/o, of the dimension of viscosity, and 21, 22 of dimensions of time. The equations of state relating to stress tensor Pik and the rate of strain tensor

1

elk = ~ (Ui,k + Uk,i) of such fluids are of the form

(2.2)

Pik = P}k -- PSik

i+21

P~k=2t/o

1+22

(2.3)

eik

where ul denotes the velocity vector, 51k is the Kronecker delta, Pik is the part of the stress tensor related to the change of the shape of a material element, and p is an isotropic pressure. The liquid (eu = 0) designated by the above model behaves as a viscous liquid if t/0 > 0 and 21 = 22. In a rectangular Cartesian coordinate system, the x-axis is taken along the lower plate at y = 0 and the y-axis is normal to this plate. The lower plate at y = 0 and the upper plate at y = h are maintained at constant temperatures To and T1 ( > To) respectively. The equations of motion combined with constitutive equations of the viscoelastic fluid are given by

lq-21~ a7= 0=--

1+21

0 5

5 ~x q-v lq-22~7 @2

--

(2.4)

(2.5)

ay

where u is the fluid velocity in the x-direction. The energy equation is

(aT) = ~ - -a2T+ # (a.?

0cp ~ -

ay 2

(2.6)

kay/

where ~o,cp, z, #, v are respectively the density, specific heat, thermal conductivity, coefficient of dynamic viscosity and coefficient of kinematic viscosity, and 21 and 22 are the relaxation and retardation times respectively. The boundary conditions are u = 0,

T = To

at

y -= 0

(2.7.1)

u = 0,

T = T1

at

y -- h.

(2.7.2)

The solution of (2.4) subject to the conditions (2.7) is written as u* -

U Ah2/v

-

Uo + e u l exp (it),

z = cot

(2.8)

where

(1 - . ) , i [ul = - ~ 2

L1 -

(2.9) sinh (1 + i) mtl + sinh (1 + i) m(1 - q)-]

sinh (1 + i) m

J'

(2.1o)

On heat transfer to pulsatile flow

171

with y r/= ~ ,

fiR m-- ]~,

R2--

coh 2 V '

(2.11) f2

_

1 + iF1 1 + iF1F2'

_

F1 = 21e)

22

and

F2 = . ~ (< 1).

It is noted that the results for viscous fluid correspond to the case 22 = 21, i.e. F2 = 1, independent of the values of Ft. Introducing (2.11) and the non-dimensional temperature T-To

0-

T1 - To'

(2.12)

Eq. (2.6) becomes

R2 ~0 1 020 (Ou*~2 & - p~ ~r/2 + Ec \ - ~ j

(2.13) A2h 4

where Pr = #cp is the Prandtl number, and Ec -

v:c,,(T~

-

To)

is the Eckert number. In terms

of 0, the boundary conditions are 0= 0

at

t/= 0

(2.14.1)

0= 1

at

t / = 1.

(2.14.2)

In view of (2.8) the temperature 0 can be assumed in the form O(tl, t) = Oo(r/) + eF(r/) exp (iv) + eZG(q) exp (2iz).

(2.15)

Substituting (2.15) and u* in (2.13), equating harmonic terms, retaining coefficients of e2 and solving the corresponding equations for 0o, F(r/) and G(r/) with the help of (2.14), it follows that 0o(t/)=q

1 -3r/+4t/2 1 + ~EcPr -(

-

-

2r/3)],

(2.16)

F(r/) -

L(0) s i n h ~ [sinh N r / + sinh N(1 - r/)] + L(r/),

L(r/) =

R 2 ( ~ 2 N ~ sinh M

1/--

{cosh Mr/-- cosh M(1 - r/)}

2M

1

(2.18t

M2 ~-N2 {sinh M r / + sinh M(1 - t/)} , PrEc(1 - i) m

L(0) = L(1) = R 2 ( ~ 5- 7 ~ i sinh M

[1 - cosh M + M2 _ N~ sinh M ,

where

_ Rfl M = (l + i) m ,

N = (l + i) n,

m

~,

( 1 +iF1 ~1/2, fl =

n = R ( ~ ) 1/2,

(2.17)

-1-_ i~1F 2/]

F1=~o21,

F2 = 22/21,

(2.19)

172

A.K. Ghosh and L. Debnath

and G(t/) = Q(0) sinh 1/2 N(1 - r/) + Q(1) sinh 1/2 Nt/ _ Q(q),

(2.20)

sinh ]/2 N

M2prEc

[-.1 - cosh M

Q(t/) - 2R 4 sinh 2 M k +

1 + cosh 2M - 2 cosh M 2(2M 2 _ N2 ) cosh 2Mr/

N2

sinh 2M - 2 sinh M sinh 2Mr/l, 2(2M 2 - N 2)

(2.21)

M2P~E~ [-1 - cosh M Q(0) = Q(1) - 2R~, sinh e M k N2

3 Rate

1 + cosh 2M _- _2cosh M7 2(2M 2 -- n 2) ]'

(2.22)

of heat transfer

The rate of heat transfer per unit area at the plate, t / = 0 is given by

qoh Co)

0o'-

~(T1-To)-

~,=o

dG

dF

=1+

P~Ec _ eei~,, ~ NL(O)

m

(si~

24

+ R2(M~ - ~ s i n h N

(1 -- cosh N)

~---N

2(1-

coshM) + ~ - s i n h M

q- 82e2iO)tIQ(O) ]/2N (1 - cosh ]/2 N) [sinh ]/2 N

M3PrEc 2R4(sinh 2 M) (2M 2 - -

N 2)

(sinh 2M -- 2 sinh M ) I

(3.1)

= (0o').=o -t- 8 IDol cos (cot -I- ~o) -I- -.where

Do = Dot + iDol and tan ~o = Dol/Dor.

(3.2)

Similarly, the rate of heat transfer per unit area at the plate t / = 1 is given by

Qa'-

q,h ~(T1- To) -

~ .=1

P,.Ec 24 + eei~ I NL(O)(1 - cosh N) - R~(~-~YV5 ~

{M2+N2

M ~-

N ~ (1 - cosh M)

sinh } 1

5-

On heat transfer to pulsatile flow

_

173

NQ(O)N (i-cosh ~ N) +

M2PrEc

e%+.+,Lsinh[l~

[1 - cosh M

2R r sinh 2 M

)

N2

2M 2 _ N2 (1 - 2 cosh M sinh 2M + 2 sinh M cosh 2M)

(3.3)

= (0o')n= 1 + e IDll cos (cot + cq) + .--, where D1 =

Dlr + iDli.

tan cq = Dll/Dlr

and

(3.4)

4 Numerical results and discussions For the problem under investigation 0o represents the steady temperature distribution in the fluid containing one linear term corresponding to the fluid at rest and added to it a biquadratic term which arises due to viscous friction. The expression for 0o given by (2.16) remains the same for both a viscous and a visco-elastic fluid of Oldroyd type under similar conditions. The temperature profiles corresponding to 0o are shown in Fig. 1 for various values of P,Ec. Regarding the rate of heat transfer in the steady-state condition the reversal of heat flux from the fluid to the hotter plate takes place when PrEc > 24 which, in turn, makes the hotter plate more hot. In fact, the value of PrEcprovides a measure of the amount of heat generated due to friction which, in the present case, increases with the increase of the pressure gradient. As a result, if the temperature difference between the plates is fixed, heat flows from the hotter plate to the fluid as long as the pressure gradient does not exceed a certain value, i.e., for PrE~> 24. This phenomenon is important for cooling at high pressure gradients. However, if instead of pressure gradients, the motion in the fluid is produced otherwise, such as in the case of steady Couette flow of viscous or visco-elastic fluids under constant pressure, the critical value of PrE~ for the reversal of heat flux at the hotter plate is found as 2. Such a reversal of heat flux occurs only when the motion of the upper (hotter) plate exceeds a certain velocity provided the temperature difference between the plates remains constant. This phenomenon is also important for cooling at high

'8

PrEc:1.0 PrEc:10j~/-~S S

+

.2

~

[

.4

l

w

.6

Fig. 1. Steady temperature profiles

l

,8

O/Z/

i

I

1-0

i

I

1-2 eo

#

i

I

1.4

i

I

1-6

i

I

1.8

i

I

2.0

l

I

2-2

i

!

2-4

174

A.K. Ghosh and L. Debnath

<4

9

9

,&---

2

.6 9

.4

~

] ~p

3. <~t =Trk

~--...._.-----..._~ ~

P~o~oo.R:,.E~=,

.2

I

--10

I

-.08

I

I

r

-.06

I

-.04

,

--02

0.0

-02 8t

.04

-06

t

08

I

I

.10

I

I

.12

Fig. 2. Unsteady temperature profiles in a viscous fluid

velocity and is reported in Schlichting [1]. We, therefore, conclude that the critical value of PrEc for the reversal of heat flux at the hotter plate is much higher in the case of cooling at high pressure gradients compared to its value for cooling at high velocity. The effect of Eckert number Ec on the steady heat transfer coefficients is shown in Table 1. Fixing Pr to 100, R to 1.0 and Ec to 1.0 or 3.0, the instantaneous temperature profiles are plotted in Figs. 2 - 5 to find the effect of changing values of the elastic parameters F1 and Fz. Figure 2 represents the instantaneous temperature profiles for a viscous fluid where Fz is taken as unity and Ec = 1.0. It has already been pointed out that the results for F2 = 1 always represent the case of a viscous fluid irrespective of the values of F1. The non-steady temperature profiles for Ec = 1, F1 = 0.1 F2 = 0.01 are shown in Fig. 3. Comparison of Fig. 2 and Fig. 3 shows that the presence of the elasticity of the fluid increases the temperature in a region near the plate and diminishes the same at the central part of the channel. Further, from Fig. 4, we see that the temperature in a viscoelastic fluid increases rapidly with the increase of Ec which is similar to that in a viscous fluid. Figure 5 shows that the increase of temperature near the plate occurs mainly due to the increase of the relaxation time of the fluid while the increase in retardation time of the fluid produces a slight decrease of temperature at the central part of the channel. The effect of changing elastic parameters and changing Ec on the amplitude and phase of the rate of heat transfer is shown in Tables 2, 3 and 4. In Tables 2 - 4, the case F2 = 1 represents the result for viscous fluid. It is observed that the increase in F2, the retardation time of the fluid, decreases the amplitude of heat transfer at the colder plate till the viscous result is reached when F2 = 1. On the other hand, the increase of Fz increases the amplitude of the rate of heat transfer at the warmer plate till the viscous result is reached when F2 = 1.

Table 1. (Pr = 100, R = 1)

(0o')n=o = (0o')~-1 =

Ec=l

Ec=3

Ec=5

5.17 -3.17

13.51 -11.51

21.85 -19.85

O n h e a t transfer to pulsatile flow

175

1.0

4

.8

3 1

__..A---~

2

,

.3~

1. ~ t :0'0

c_-6 ~ ''''F4

3. cot =71"/2

.4

-2

I

f

-.10

--12

I

I

i

-.08

I

-'r

I

--06

,

-.04

~

--02

~

0.0

-

-

-

_2

-

-

I

i

I

I

I

i

-02

.04

_

~

I

-06

I

.08

I

~

-10

et Fig. 3. U n s t e a d y t e m p e r a t u r e profiles in a viscoelastic fluid

1~ [-

>

C~ ,

/

-.32

--24

-.16

.

~

3. 60t= ~'/2

~

~

-.08

0.0

P~=loo,R=l,E~=3

.08 Ot

.16

.24

.32

.40

.48

Fig. 4. U n s t e a d y t e m p e r a t u r e profiles in a viscoelastic fluid

T a b l e 2. (Pr = 100, R = 1, E c = 1)

F1

F2

[Do[

IDI[

tan %

t a n ~1

0.01

0.001 0.01 1.0

3.693 66 3.693 63 3.690 09

1.315 86 1.315 93 1.323 46

- 0.67015 -0.67028 - 0.684 52

- 446.246 21 - 4 8 7 . 7 5 9 80 52.893 45

0.1

0.001 0.01 1.0

3.742 86 3.742 54 3.690 09

1.259 93 1.260 63 1.323 46

- 0.549 01 - 0.55018 - 0.684 52

- 4.938 08 - 4.989 49 52.893 45

I'

-12

I

t

.14

176

A. K.

1. Fl :O-01.F2

Ghosh

and

L. Debnath

= 0.001

2. Fl = 0.1 , F2 = 0.001 3. Fl

= O.Ol.FZ

= 0.01

4. Fl ~0.1 , F2 z 0.01

Fig. 5. Unsteady

temperature

profile

in a viscoelastic

fluid

Table 3. (P, = 100, R = 1, EC = 3)

Fl

F2

0.01

0.001 0.01 1.0

0.1

0.001 0.01 1.0

PII

tan x0

tan c(~

11.08099 11.08090 11.07027

3.947 59 3.947 80 3.97040

-0.67015 -0.67028 -0.68452

-446.24621 487.759 81 52.89345

11.22858 11.22762 11.07027

3.77080 3.78190 3.97040

-0.549015 -0.550 180 -0.684520

-4.93808 -4.98949 52.89345

Table 4. (P, = 100, R = 1, EC = 5)

F,

F2

ID,1

IP,I

tan a,

tan 0+

0.01

0.001 0.01 1.0

18.468 32 18.468 16 18.45045

6.51932 6.57966 6.61734

-0.67015 -0.67028 -0.68452

-446.24621 -487.75981 52.89345

0.1

0.001 0.01 1.0

18.71431 18.71271 18.45045

6.29967 6.30317 6.617 34

-0.54901 -0.55018 -0.68452

-4.93808 -4.98949 52.89345

There is a phase lag at both the plates when the fluid is viscoelastic. But for viscous fluid phase lead exists at the warmer plate and phase lag at the colder plate. The increase of the Eckert number E, increases the amplitude of heat transfer at the plates both for viscous and viscoelastic fluids while the phase at the plates remains unaffected by the increase of E,. Finally, for a fixed value of the retardation time parameter Fz, the increase of the relaxation time parameter F1 increases the amplitude of heat transfer at the colder plate and diminishes the same at the warmer one. The phase value at the plates however diminishes significantly in both the cases.

On heat transfer to pulsatile flow

177

Acknowledgements This work was partially supported by the University of Central Florida. Authors express their grateful thanks to the Editor for suggesting some changes.

References [1] Schlichting, H.: Boundary layer theory, 6th. ed. New York: McGraw-Hill 1966. [2] Radhakrishnamacharya, G., Maiti, M. K.: Heat transfer to pulsatile flow in a porous channel. Int. J. Heat Mass Transfer 20, 171-173 (1977). [3] Oldroyd, G.: Quart. J. Mech. Appl. Math. 4, 281-292 (1951). Authors' addresses: A. K. Ghosh, Department of Mathematics, Jadavpur University, Calcutta-700032, India, and L. Debnath, Department of Mathematics, University of Central Florida, Orlando, FL-32 816, U.S.A.