Cmjv01i01p0033 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(1), 33-40 (2009).

FREE CONVECTIVE NON-NEWTONIAN FLOW BETWEEN TWO CO-AXIALCYLINDERS U.K.Tripathy1, Rosalin Dash2 and S.M.Patel3 1

Department of Mathematics, University College of Engineering, Burla-768017(India) 2 Department of Mathematics, Gandhi Institute for Technology and Management, Gangapada, BBSR-754022(India) 3 Department of Mathematics, Sundergarh Engg. School, Sundergarh (India) ABSTRACT An approximate analysis of free convective flow of a nonNewtonian liquid between two co-axial cylinders has been carried out. The equation of motion and energy including viscous dissipative terms are a pair of simultaneous non-linear ordinary differential equations. These equations under appropriate boundary conditions have been solved by the fourth order Runge-kutta method. The truncation errors involved in this method of solution have been determined for one set of values of parameters and have been noted to be O (10-5). It has been observed that the elastic parameter has greater influence on the velocity field than that on the temperature field. The effect of other parameters Pr(Prandatl number), E (Eckert number) etc. on flow and temperature field has been depicted through graphs.

1. INTRODUCTION The free convective viscous flow between vertical heated plates was investigated by Ostrach1. In the above paper frictional heating has been taken into account. Tau2 has studied the free convective viscous flow, neglecting the viscous dissipative terms, through a vertical circular pipe when it is heated or cooled uniformly. Nanda and Sharma 3 considered the same problem including the effect of frictional heating and distributed

sources and sinks. In the recent years free convective laminar flow of a non Newtonian liquid has gained substantial importance and has attracted the attention of several researchers. Emery, et al.4 has studied the problem of free convection through vertical plane layers of nonNewtonian power law fluids. Ouldhadda, D., Il Idrisoi. A and Asbik. M5 has investigate numerically the laminar flow and heat transfer in a pseudo plastic nonNewtonian falling liquid film on a horizontal

[ 34 ] cylinder for constant heat flux and isothermal boundary condition taking the inertia terms into account. The effects of operational parameters on heat transfer are examined and discussed in detail. Free convection effects on flow past a moving infinite vertical plate was discussed by Revanker6 and Muthukumarswamy7 have examined separately the flow and heat transfer of a viscous incompressible fluid past an impulsively started vertical plate. More recently, Hayat et.al.8 discussed the flow of visco-elastic fluid past an oscillating plate, in which they discussed the influence of suction or injection on velocity distribution. In many problems particularly those involving the cooling of electrical and nuclear component, the wall heat flux is specified. In such problem over heating burn out and melt down are very important issues. The problem with prescribed heat flux is special case of the vast analytical accessible class problem. Pantokraters9 have investigated the convection flow with prescribed heat flux condition. 2. Mathematical analysis : The aim in this problem is to investigate the free convective flow of an elastico-viscous liquid between two co-axial cylinders including the effects of frictional heating. Here it is assumed that the inner cylinder is moving parallel to itself in vertical upward direction with constant velocity w while the outer one is held fixed. the rheological equation of state of the elasticoviscous model considered in this problem is due to Noll10. The stress-strain rate relation of this model is as follows:

~ Pij  1 Pij  2  eij  4 c e j ej (1) Where

p ij ~ Pij   P ij , k , v k  P ik v i , k  P kj v j , t k  P ij v k , k (2) In the above relation the Pij are the stress tensors, eij are the rate of deformation tensors, vi is the velocity component in the ith direction, 1 is the time of relaxation, µ is the coefficient of viscosity and µc is cross-viscous co-efficient. It is assumed to be a fully developed free convective flow between the annual spaces of two concentric vertical cylinders which are considered to be infinitely long. The work was carried out through cylindrical polar co-ordinates ( r ,  , z ) . The common axis of cylinders is assumed to coincide with the zaxis and let the radii of the cylinders be a and b ( a  b) . The velocity components u , v , w compatible with the equation of continuity are given by :

u  0, v  0, w  w ( r ) Since the buoyancy force is only due to gravity acting vertically downwards, the components of the extraneous forces are

Fr  0, F  0, Fz  f z If   is the temperature at any point, then using the usual Boussinesq's approximation, the equation of motion and energy including viscous dissipative terms can be written as: 3 3 d 2 w 1 dw 1  c  d  dw  1  dw    f z           2 dr r dr   dr  dr  r  dr    (3)

 d 2  1 d    dw       1 c 2 r dr k  dr  k dr

 dw     0 (4)  dr 

[ 35 ] Where  is the volumetric co-efficient of thermal expansion and k is the thermal conductivity of the fluid. The temperature of the inner and the outer cylinders are assumed to be  b  and  a  respectively where  b   a ( 0) . Assuming the inner cylinder is moving parallel to itself with a constant velocity w and the outer one is fixed, the boundary conditions for equations (3) and (4) can be given as :

w  w r  b     b

w  0 ra    0

On introducing the following non -dimensional quantities:

h r b w ,  ,w  b ab w 2 f z h  b   w2    ,G  , Rc  1 c 2  b w h h  a  b,  

E

 w2 , Pr  c  b c k

In equation (3) and (4) they become

 d  dw  3 d 2w  dw   Rc    d 2 1   d  d  d 

  dw     1    d 

   G  0 

(5)

4  dw  3 d 2  d  dw     E P    R   r  r     0 (6) d 2 1   d  d   d  

And the corresponding boundary conditions are

   0 : w  1,   1    1 : w  0,  0

(7)

Where c is the specific heat and G, Rc,Pr and E respectively denote the Grashof number, the elastic parameter, Prandtl number and the Eckert number. Equation (5) and (6) subject to the boundary conditions (7) are solved by fourth order Runge-Kutta Gill technique and preceded with step by step integration as illustrated in Ralston and Wilf11. Defining the following variables

Y1   , Y2  w, Y3   , Y4   , Y5  w (8) Where primes denote differentiation with respect to . Using the variables (8) in (5) and (6) the following five simultaneous equations are obtained

    Y1  1   Y2  Y5   Y3  Y4    2 4 Y4  Y4  E Pr Y5  Rc Y5 1  Y1    Y5  2  1  Rc Y5  GY3     1  Y1  Y5  2 (9)  1  3RcY5







 

[ 36 ] The boundary conditions (7) accordingly can be written as:

Y1 0  0, Y2 0  1, Y3 0  1  (10) Y1 1  1, Y2 1  0, Y3 1  0 The step by step integration of (9) is performed in (0,1) with step length h=0.1 and subsequently h=0.2. To start with the integration, rough values of Y4(0) and Y5(0). Were supplied and these were corrected by a self iterated corrective procedure as given by Fox12. The numerical integration is performed on IBM-1130 digital computer. The truncation- error involved in the method of solution is shown in Table (1). To estimate the truncation -error the following formula based on Richardsons extrapolation technique(cf. Carnahan, B., et al.13) is given by :

TnY  h  n1  Yn 1  Y h  n1 

Figure-1 VELOCITY DISTRIBUTION FOR DIFFERENT VELUE OF Rc. E=0.01,

Pr  1, G  5.0,   0.2

Y h  n1  Y 2 h  n1 (11) 2r 1

Where Yn1 is the true ordinate at x n1 , Y h  n1 and Y  2 h  n 1 are the values obtained with spacing h and 2h respectively calculated at two steps starting at xn 1 and r is the order of the Runge-Kutta formula. In this case r being 4 and h=0.1. It is noted that for a given set of values of parameters the truncation-error in the calculation of the velocity w and the temperature  for   0.2, 0.4, 0.6, 0.8 are of





the order O 10 5 (ref. Table 1).

Figure-2 VELOCITY DISTRIBUTION FOR DIFFERENT VELUE OF G. E=0.01, Pr=1.0,

Rc  0.01,   0.9

[ 37 ] Table 1

E  0.01, Pr  1.0, Rc  0.01, G  5.0,   0.5 Truncation error in 

Truncation error in w Order of the error

w .1

w .2 

0.2

0.74674

0.74652 1.410-5 O 10 5

0.4

0.51834

0.51783 3.410-5

0.6

0.30214

0.30139 5.110-5

0.8

0.13484

0.13375 7.210-5

Error

 O10 O10 O10

5 5 5

   

 .1

 .2 

Order of the error

Error

 O10 O10 O10

0.76354 0.76346 0.510-5

O 10 5

0.54847 0.54816 2.010-5

5

0.35229 0.35181 3.210-5 0.16803 0.16727 4.810-5

Table 2 Effect of Rc over 

E  0.01 , Pr  1, G  5.0,   0.2

 Rc

0.0

0.05

0.10

0.0 0.2 0.4 0.6 0.8 1.0

1.000 0.78407 0.56758 0.37703 0.18498 0.000

1.000 0.77454 0.55689 0.34750 0.14595 0.000

1.000 0.76501 0.53720 0.31797 0.10685 0.000

Table 3 Effect of Pr over w E  0.01 , Rc = 0.01, G = 5.0,  = 0.8

 Pr

2.0

5.0

10.0

0.0 0.2 0.4 0.6 0.8 1.0

1.000 0.74622 0.52571 0.33093 0.15667 0.000

1.000 0.74536 0.52447 0.32974 0.15591 0.000

1.000 0.74344 0.52241 0.32775 0.15463 0.000

5 5

   

[ 38 ]

Figure-3

Figure-5

TEMERATURE DISTRIBUTION FOR DIFFERENT VALUE OF G. E=0.01,

TEMERATURE DISTRIBUTION FOR DIFFERENT VALUE OF E.

Pr  1.0, Rc  0.01,   0.2

Figure-4

Pr  1.0, G  5, Rc  0.01,   0.2

Figure-6

VELOCITY DISTRIBUTION FOR DIFFERENT VALUE OF E.

VELOCITY DISTRIBUTION FOR DIFFERENT VALUE OF  . E. = 0.01,

RC  0.01, Pr  0.7, G  5.0,   0.8

Pr  1.0, Rc  0.01, G  5.0

[ 39 ] 3. CONCLUSION

Figure-7 TEMERATURE DISTRIBUTION FOR DIFFERENT VALUE OF . E=0.01, Pr = 1.0 Rc = 0.01, G = 5.0

The effects of the parameters Rc, E, Pr, G and  (the parameter characterizing the gap between two cylinders) have been shown through graphs and tables. The effect of Rc on w and  is shown through figure (1) and Table (2). When Rc increases it is noted that both velocity and temperature decreases for all . The effect of Rc both on w and  are remarkable at the outer cylinder. The percentage decrease in w and  for =0.8 is nearly 53% and 44% respectively. Figures (2) and (3) show as the Grashof number increases, w and decreases and  increase. Figures (4) and (5) show the effect of the parameter due to frictional heating of the liquid. For large E both velocity and temperature decline steadily. However, the effect of E on  is more noticeable then on w. Figures (6) and (7) represent that the larger is the gap between the two cylinders both velocity and temperature are decreased. Table (3) shows that for large Pr, w decreases. When Pr is comparatively large, the temperature in the annular space becomes gradually zero, negative and thereafter attains the value zero at the outer cylinder (Fig. 8). REFERENCES

Figure-8 TEMERATURE DISTRIBUTION FOR DIFFERENT VALUE OF Pr . E = 0.1, G = 5.0, Rc = 0.01,  = 0.2

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