CrossMark

Available online at www.sciencedirect.com

ScienceDirect

Procedía Engineering 90 (2014) 510 - 516

Procedía Engineering

www.elsevier.com/locate/procedia

10th International Conference on Mechanical Engineering, ICME 2013

Non-Newtonian Mixed Convection Flow from a Horizontal Circular Cylinder with Uniform Surface Heat Flux

Sidhartha Bhowmick a, Md. Mamun Mollab'*, Mustak Miac, Suvash C. Sahad

aDept. of Mathematics, Jagannath University, Dhaka-1100, Bangladesh b cDept. of Electrical Engineering & Computer Science, North South University, Dhaka-1229, Bangladesh dInstitute of Future Environments, School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia

Abstract

Mixed convection laminar two-dimensional boundary-layer flow of non-Newtonian pseudo-plastic fluids is investigated from a horizontal circular cylinder with uniform surface heat flux using a modified power-law viscosity model, that contains no unrealistic limits of zero or infinite viscosity; consequently, no irremovable singularities are introduced into boundary-layer formulations for such fluids.. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear systems of partial differential equations are solved numerically applying marching order implicit finite difference method with double sweep technique. Numerical results are presented for the case of shear-thinning fluids in terms of the fluid temperature distributions, rate of heat transfer in terms of the local Nusselt number.

Keywords: Non-Newtonian fluid; mixed convection; boundary-layer; horizontal circular cylinder; heat flux; implicit finite difference method

1. Introduction

In numerous engineering areas related with pseudo-plastic fluids, a vital role is played by mixed convective fluids from a horizontal circular cylinder with uniform surface heat flux. In 1977, an excellent experimental research on non-Newtonian fluids was given by Boger [1]. He demonstrated a set of data for the pseudo-plastic fluids. A

* Corresponding author. Tel.:+88-02-8852000, Ext. 1519; fax: +88-02-8823030. E-mail address:mmamun@northsouth.edu, mmamun@gmail.com

1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering

and Technology (BUET)

doi: 10.1016/j .proeng .2014.11.765

theoretical analysis for boundary-layer non-Newtonian power-law fluids is considered firstly by Acrivos [2]. After that, a large number of papers have been published, due to their wide relevance in pseudo-plastic fluids, those were impractical. The reason is that they used the traditional power-law viscosity correlation that viscosity becomes infinite for small shear rates or vanishes for the limits of large shear rates, which are giving the unrealistic physical results; however, a few references, which can be used as starting points for a more extensive search [3-7], are listed here.

Merkin [8] considered the mixed convective flow from a horizontal circular cylinder with constant temperature considering that the stream is flowing in the upward vertical direction; solutions were based on the non-dimensional parameter X = gfiqwa 2Re"1/2/(kUK2) = Gr/Re512. Solutions were obtained for both small X, where forced convection effects dominate and for large X, where natural convection effects dominate.

The proposed modified power-law correlation is outlined for the shear-thinning fluids (n = 0.6) along with the Newtonian fluids (n = 1.0) in Fig. 2 and this model is prepared based on the available experimental data for the non-Newtonian fluids (see Boger [1]). The constants in the proposed model, that fits measured viscosity data well, can be fixed with available measurements and are described in detail in Yao and Molla [9]. The boundary-layer formulation on a flat plate and the associated heat transfer for two different heating conditions are described and numerically solved for non-Newtonian fluid in Yao and Molla [9-11]. But still now, there is no such kind of realistic research for mixed convection of non-Newtonian fluids from a horizontal circular cylinder with uniform surface heat flux.

In the present research, the behavior of shear-thinning and Newtonian fluids on the mixed convection laminar flow from a horizontal circular cylinder with constant surface heat flux are studied by choosing the power-law index as n (= 0.6, 1.0) to fully demonstrate the performance of a pseudo-plastic fluid comparing with Newtonian fluid. The numerical solutions are based on the buoyancy parameter X, Prandtl number Pr and power-law index n. The results refer to the position of the boundary-layer separation, Nusselt number as well as to the temperature profiles near the lower stagnation point (X = 1). Results are given in the form of tables and graphs, as well.

Nomenclature

a Radius of the circular cylinder Uw Free stream velocity

C Constant u, v Velocity components along the x, y axes,

respectively

Cp Specific heat at constant pressure U, V Dimensionless fluid velocities in the X, Y

directions, respectively

D Non-dimensional viscosity of the fluid x, y Cartesian coordinate measured along the surface

of the cylinder and normal to it respectively

g Acceleration due to gravity X Axial direction along the circular cylinder

Gr Grashof number Y Pseudo-similarity variable

k Thermal conductivity of the fluid

K Dimensional constant Greek symbols

n power-law index a Thermal diffusivity

Nu Local Nusselt number P Thermal expansion coefficient

Pr Prandtl number y Shear rate

qw Heat flux X Ratio between Gr and Re5'2

Q Heat transfer M Dynamic viscosity

Re Reynolds Number 0 Dimensionless temperature of the fluid

T Dimensional temperature of the fluid p Fluid density

T Ambient temperature V ( jU / p ) kinematic viscosity

ue (x ) Velocity of the potential flow V1 Reference viscosity of the fluid

2. Formulation of the problem

A steady two-dimensional laminar mixed convection boundary-layer flow of a non-Newtonian fluid over a horizontal circular cylinder of radius 'a' with uniform surface heat flux and a distributed heat source of the form — TCD)sin(x / a) (x measuring distance from the lower stagnation point) has been considered. The viscosity depends on shear rate and is correlated by a modified power-law. We consider shear-thinning position of non-Newtonian fluids. It is assumed that the surface heat flux qw is applied to the cylinder; is the ambient

temperature of the fluid and T is the temperature of the fluid. The design considered is as shown in Fig. 1. The inviscid flow along the cylinder develops instantly and its velocity outside the boundary-layer is given by ue (x) = Uw sin(x / a) , where Um is the free-stream velocity.

/|\/|\/|\/|Vj\

Fig.1: Physical model and coordinate system

n = 1.0

n = 0.6

10-3 10-2 10-1 100 101 102 103 104 105 10'

Fig. 2: Modified power-law correlation

Under the above assumptions, the boundary-layer equations governing the flow and heat transfer are du ^civ _ o

I —du _du P\ + v

^ dx dy

_dT _dT u + v^^ =

■■y (,f - T.)s,n( ^

pCp dy2

Where u and V are velocity components along the x and y axes, p is the fluid density, ^ is the dynamic viscosity of the fluid in the boundary-layer region, g is the acceleration due to gravity, ft is the coefficient of thermal expansion, k is the thermal conductivity and Cp is the specific heat at constant pressure. The kinematic viscosity v = ¡x/p is correlated by a modified power-law, which is

K du n-1 du

v = — for yx <

P dy dy

The constants y1 andy2are threshold shear rates, which are given according to the model of Boger [1], K is the dimensional constant, for which dimension depends on the power-law index n. The values of these constants can be determined by matching with measurements. Outside of the preceding range, viscosity is assumed to be constant; its value can be fixed with data given in Fig. 2 [12].

The boundary conditions for the present problems are _ _ dT _

u = 0, v = 0, - k— = qw at y = 0 (5a)

u ^ ue (x), T ^ Tx as y (5b)

We establish non-dimensional dependent and independent variables according to,

y = y Re1/2,

Gr=gPw4

Re = -

: - v =-

U / U a

u„ =-

= sinx,0 = T T'\ Re1/2

(iqwa / k)

ViPCp k

Where, V1 is the reference viscosity at^1, 6 is the non-dimensional temperature of the fluid, Re is the Reynolds number, Gr is the Grashof number and Pr is the Prandtl number. The physical meaning of

gfiqwa Re kUl

■2t> „-1/2

represents the ratio of the length scale that the non-Newtonian effect becomes apparent

and the length scale that the natural convection effect grows dominant. For larger G5r , it takes shorter distance for

the effect of natural convection becomes dominant. Using equation (6) in equations (1)-(4) yield the non-dimensional equations and applying the parabolic transformations (7) lead to the following equations (8)-(9):

X = x, Y = y, U = -, V = v, 0 = 0 x

dU dV X — + U + — = 0 dX dY

,rTT8U dU rr2 sinXcosX ^82U dU 8D A(9smX XU-+ V-+ U =-+ D—- +--+ -

xu^+V ™ = -L ^

8X 8Y Pr 8Y2

-Re1/2 ^ dy

8Y2 8Y 8Y

The length scale associated with the non-Newtonian power-law is

,P) Vj-

The corresponding boundary conditions are

at Y = 0

U = 0, V = 0, sin X

0 as Y

Y1 ^ M ^ ^2

where, y — X

(13a) (13b)

The correlation (14) is a modified power-law correlation first presented by Yao and Molla [11]. This correlation describes that if the shear rate lies between the threshold shear rates yl and y2, then the non-Newtonian viscosity,

D, varies with the power-law of y. On the other hand, if the shear rate does not lie within this range, then the non-Newtonian viscosities are different constants, as shown in Fig. 2. This is a property of many measured viscosities.

Table 1. Numerical values of heat transfer, Q = Nu Re 1/2for shear thinning fluid (n = 0.6) for Pr = 50 and different values of 1.

xi -5.0 -2.0 0.0 2.0 5.0 15.0

0.0000 2.11370 2.27444 2.35250 2.41819 2.50157 2.70771

0.4014 2.17729 2.50464 2.64251 2.75386 2.89125 3.22175

0.8028 2.44335 2.62322 2.75936 2.92106 3.29327

1.2042 2.10638 2.40727 2.58995 2.78813 3.20966

1.6057 1.96119 2.27419 2.53826 3.02531

2.0071 1.78851 2.19311 2.76447

2.4085 1.75314 2.43209

2.7925 2.03948

3.1416 1.35948

Equations (8-10) are discretized by a central-difference scheme for the diffusion term and a backward-difference scheme for the convection terms. Finally, we get a tri-diagonal algebraic system of equations, which can be solved by implicit finite difference with double sweep technique. The normal velocity is directly solved from the continuity equation. The computation is started at X=0 and marches to downstream to X=3.1416. After several test runs, converged results are obtained using AX = 0.0025 and AY = 0.005.

Table 2. Numerical values of heat transfer, Q = Nu Re 1/2for Newtonian fluid (n = 1.0) for Pr = 50 and different values of 1.

xi -5.0 -2.0 0.0 2.0 5.0 20.0

0.0000 2.11370 2.27444 2.35250 2.41819 2.50157 2.78726

0.4014 2.05510 2.23155 2.31418 2.38283 2.46917 2.76101

0.8028 1.82490 2.09617 2.19688 2.27630 2.37276 2.68399

1.2042 1.82728 1.98847 2.09486 2.21317 2.55983

1.6057 1.63391 1.82775 1.99249 2.39457

2.0071 1.43432 1.71538 2.19550

2.4085 1.96603

2.7925

3.1416

In practical applications, the physical quantities of principle interest is the local Nusselt number Nu, which is

Nu Re~1/2 =-1--(15)

0(X ,0) ( )

3. Result and Discussion

In this paper, it has been investigated the problem of laminar mixed convection two-dimensional boundary-layer flow and heat transfer from a horizontal circular cylinder with constant surface heat flux for the case of non-Newtonian power-law fluid of shear thinning (n = 0.6) case along with Newtonian fluid (n = 1.0) for the value of the Prandtl number Pr = 50 and 100 and negative and positive values of the mixed convection parameter I (from -5 to 20). The cases when I > 0 (qw > 0) and I < 0 (qw < 0) are considered. The non-dimensional viscosity, D, is given by the modified power-law correlation, which is plotted in Fig. 2 as a function of the non-dimensional shear rate y. Numerical results are expressed the rate of heat transfer as a form of the local Nusselt number, Nu Re~1/2 along with the temperature distribution of the fluids.

It can be seen from the Tables 1and 2 that the boundary-layer separates from the cylinder for all I < 0 (opposing flow) and for some I > 0 (assisting flow). Opposing flow (I < 0) brings the separation point close to the lower stagnation point. For sufficiently large negative values of I (< -5) there will not be a boundary-layer on the cylinder.

1.0 (Newtonian fluid).

fluid) and (b) n = 1.0 (Newtonian fluid), respectively.

Increasing X holdup the separation and that separation cannot be available entirely in 0 < X < n for sufficiently large positive values of X, those are clearly designated in the Tables. For the case of shear-thinning fluid (n = 0.6), it is found the point of separation at X = n for X = 15 but for the Newtonian fluid case the point of separation occurs at X = 30. The Tables also show the values of Nu Re~1/2at different positions of X and different values of X at Pr = 50 for the shear-thinning (n = 0.6) fluid and the Newtonian fluid (n = 1.0), respectively. At Table 1, for non-Newtonian fluid the heat transfer initially increased at a certain X and then decreased up to the boundary-layer separation points. But at Table 2, for the Newtonian fluid, it is apparently observed that for all values of X the rate of heat transfer, Nu Re~1/2are highest at the lower stagnation points and reduce continuously up to the boundary-layer separation points.

—1/2

The variation of Nu Re are demonstrated in Figure 3 for five different values of X (= -5, 0, 2, 5, 10) at Pr = 100 for the non-Newtonian power-law fluid of shear-thinning case (n = 0.6) and the Newtonian fluid (n = 1.0),

—1/2

respectively. At Figure 3(a), for shear-thinning fluid, the heat transfer Nu Re for all values of X is constantly large for a certain X and then decreases to the boundary-layer separation points. Other than at Figure 3(b), the values for the Newtonian fluid and in heat transfer case are the highest at the lower stagnation points and nonstop decrease

up to the boundary-layer separation points. It is clearly show that the boundary-layer separates from cylinder for all X > 0 (qw > 0) and for some X < 0 (qw < 0). Opposing flow (X < 0) brings the separation point close to the lower stagnation point and for sufficiently large negative values of X (< -5) there will not be a boundary-layer on the cylinder. Increasing X delays the separation and that separation can be suppressed completely in 0 < X < n for sufficiently large positive values of X, those are clearly indicated in the Figure 3.

In Figure 4 the temperature distribution as a function of Y at the selected location (X = 1) for the case of non-Newtonian power-law fluid of shear-thinning (n = 0.6) and Newtonian fluid (n = 1.0) are offered for four different values of X (= -5, 0, 2, 10) at Pr = 100. It shows that, for shear-thinning fluid (n = 0.6), the variation of temperature in the boundary-layer decreases rapidly at the leading edge when ever at the downstream region it decreases slowly; accordingly, the boundary-layer is thickened. On the other hand, for Newtonian fluid (n = 1.0), the variation of temperature in the boundary-layer decreases less rapidly at the whole stream region and the boundary-layer is relatively thinned than shear-thinning fluid, as the fluid becomes more sticky. We may bring to a close that in both case (n = 0.6 and 1.0), the fluid temperature for large X is smaller than that for small X.

4. Conclusions

• Opposing flow (X < 0) brings the separation point close to the lower stagnation point and increasing X holdup the separation.

— 1/2

• For shear-thinning fluid, the heat transfer Nu Re for all values of X are constantly ahead for a very little certain X then raise it at a definite X and then decrease to the boundary-layer separation points, but for the Newtonian fluid, the heat transfer are the highest at the lower stagnation points and nonstop decrease up to the boundary-layer separation points.

• In the boundary-layer the variation of temperature, for shear-thinning fluid (n = 0.6), decreases rapidly at the leading edge when ever at the downstream region it decreases slowly; but the other hand, for Newtonian fluid (n = 1.0), the variation of temperature in the boundary-layer decreases less rapidly at the whole stream region.

References

[1] D. V. Boger, Demonstration of upper and lower Newtonian fluid behavior in a pseudo plastic fluid, Nature, vol. 265 (1977), pp. 126-128.

[2] A. Acrivos, A Theoretical Analysis of Laminar Natural Convection Heat Transfer to Non-Newtonian Fluids, AIChE Journal, vol. 6 (1960), No. 4, pp. 584-590.

[3] F. H. Emery, S. Chi, and J. D. Dale, Free Convection Through Vertical Plane Layers of Non-Newtonian Power Law Fluids, Journal of Heat Transfer, vol. 93 (1970), pp. 164-171.

[4] T. V. W. Chen, and D. E. Wollersheim, Free Convection at a Vertical Plate with Uniform Flux Conditions in Non-Newtonian Power Law Fluids, Journal of Heat Transfer, vol. 95 (1973), pp. 123-124.

[5] Z. P. Shulman, V. I. Baikov, and E. A. Zaltsgendler, An Approach to Prediction of Free Convection in Non-Newtonian Fluids, International Journal of Heat and Mass Transfer, vol. 19 (1976), No. 9, pp. 1003-1007.

[6] M. J. Huang, J. S. Huang, Y. L. Chou, and C. K. Cheng, Effects of Prandtl Number on Free Convection Heat Transfer from a Vertical Plate to a Non-Newtonian Fluid, Journal of Heat Transfer, vol. 111 (1989), pp. 189-191.

[7] W. A. Khan, J. R. Culham, and M. M. Yovanovich, Fluid Flow and Heat Transfer in power Law Fluids across Circular Cylinders: Analytical Study, Journal of Heat Transfer, vol. 128 (2006), pp. 870-878.

[8] J. H. Merkin, Mixed Convection from a horizontal Circular cylinder, Int. J. Heat Mass Transfer, vol. 20 (1977), pp. 73-77.

[9] L. S. Yao, and M. M. Molla, Non-Newtonian Fluid Flow on a Flat Plate, 1: Boundary Layer, Journal of Thermophysics and Heat Transfer, vol. 22 (2008), No. 4, pp. 758-761.

[10] M. M. Molla, and L. S. Yao, Non-Newtonian Fluid Flow on a Flat Plate, 2: Heat Transfer, Journal of Thermophysics and Heat Transfer, vol. 22 (2008), No. 4, pp. 762-765.

[11] M. M. Molla, and L. S. Yao, The Flow of Non-Newtonian Fluids on a Flat Plate With a Uniform Heat Flux, ASME J. Heat Transfer, vol. 131 / 011702 (2009), pp. 1-6.

[12] S. Bhowmik, M. M. Molla, and L. S. Yao, Non-Newtonian Mixed Convection Flow from an Isothermal Horizontal Circular Cylinder, Numerical Heat Transfer: Part A, (2013) (Accepted).