VAJRAVELU K.

Department of Mathematics, Department of Mechanical, Material and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, USA, E-mail: kuppalapalle.vajravelu@ucf.edu

PRASAD K. V., RAJU B. T.

Department of Mathematics, Central College Campus, Bangalore University, Bangalore 560 001, India

Effects of variable fluid properties on the thin film flow of Ostwald-de Waele fluid over a stretching surface*

VAJRAVELU K.

Department of Mathematics, Department of Mechanical, Material and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, USA, E-mail: kuppalapalle.vajravelu@ucf.edu

PRASAD K. V., RAJU B. T.

Department of Mathematics, Central College Campus, Bangalore University, Bangalore 560 001, India

(Received July 3, 2012, Revised August 23, 2012)

We investigate, in this paper, the effects of thermo-physical properties on the flow and heat transfer in a thin film of a power-law liquid over a horizontal stretching surface in the presence of a viscous dissipation. The fluid properties, namely the fluid viscosity and the fluid thermal conductivity, are assumed to vary with temperature. Using a similarity transformation, the governing partial differential equations with a time dependent boundary are converted into coupled non-linear Ordinary Differential Equations (ODEs) with variable coefficients. Numerical solutions of the coupled ODEs are obtained by a finite difference scheme known as the Keller-box method. Results for the velocity and temperature distributions are presented graphically for different values of the pertinent parameters. The effects of unsteady parameter on the skin friction, the wall temperature gradient and the film thickness are presented and analyzed for zero and non-zero values of the temperature-dependent thermo-physical properties. The results obtained reveal many interesting features that warrant further study on the non-Newtonian thin film fluid flow phenomena, especially the shear-thinning phenomena.

Thin film flow, variable fluid properties, viscous dissipation, finite difference method

Introduction

All the above investigators restricted their analyses to Newtonian fluids. However, the fluids employed in material processing or protective coatings are, in general, non-Newtonian, and thus the surface flow of non-Newtonian liquid thin film is widely occurring in various industrial applications, for example, in polymer and plastic fabrication, food processing and in coating equipment. The simplest and the most commonly applicable model is the Ostwald-de Waele model. For this power-law fluid model, the rheological equation of the state between the stress components τijand strain components eijis defined by

Motivated by these applications, in this article, the authors examine the effects of the variable fluid properties, namely variable thermal conductivity and the fluid viscosity on the flow and heat transfer of a power-law liquid thin film induced by an accelerating unsteady stretching surface in the presence of viscous dissipation. In contrast to the work of Abel et al.[4]the present work considers the combined effects of variable fluid properties and the viscous dissipation on the non-Newtonian (power law liquid) fluid flow and heat transfer. Highly nonlinear coupled partial differential equations governing the flow and heat transfer are reduced to a system of coupled non-linear ordinary differential equations by a similarity transformation, and are solved numerically by the Keller box method for different values of the governing parameters. Typical results for the temperature and velocity profiles, freesurface temperature, the dimensionless film thickness and the heat transfer rate are presented through figures and tables for selected values of the unsteady parameter and the power law index. The numerical results are compared for the special case, namely Newtonian fluid and found to have good agreement with earlier studies. The numerical solutions presented in this study are helpful to understand the flow and heat transfer mechanism of the liquid film and find applications to technological and manufacturing industries (such as polymer extrusion).

1. Flow analysis

Consider an unsteady, laminar, flow and heat transfer in a thin liquid film on a horizontal sheet which issues from a slot as shown in Fig.1.

Fig.1 Physical model and geometry

The fluid is assumed to obey the power law model. The fluid motion within the thin film arises due to the stretching of an elastic porous sheet. The x-axis is taken along the surface in the direction of the motion and they -axis perpendicular to it. The thermo-physical fluid properties of the ambient fluidare assumed to be isotropic and constant, except for the fluid viscosity and the fluid thermal conductivity which are assumed to vary as a function of temperature, that is,

Bothaand Trare constants and their values depend on the reference state and the thermo-physical property of the fluid, i.e.,δis a small parameter reflecting a thermal property of the fluid (a constant): In general,a >0for liquids and a <0for gases. Also,θris a constant which is defined as

It is worth mentioning here that as δ→0, i.e., if μ=μ0(constant), then θr→∞. It is also important to note that θris negative for liquids but positive for gases. This is due to the fact that viscosity of a liquid usually decreases with increasing temperature while it increases for gases. The continuous sheet is assumed to have a surface velocity Usand a prescribed surface temperature Tswhich vary with the horizontal coordinatex and timet in the following forms:

whereuandv are the velocity components along the x and y directions respectively,ρis the density, and τxyis the shear stress. In the present problem, we have ?u/?y≤0throughout the entire boundary layer since the streamwise velocity component u decreases monotonically with the distance y from the moving surface. The shear stress is given by

whereK is the consistency coefficient and also called as absolute viscosity(K=μ),γ=(K(=μ)/ ρ)is the kinematic viscosity,σis the electrical conductivity, and n is the flow behavior indexnamely power-law index. Asn deviates from unity, the fluid becomes non-Newtonian, for example n<1 and n >1corresponding to shear thinning (pseudo plastic) and shear thickening (dilatant) fluids respectively, and for n =1, the fluid is simply the Newtonian fluid. In Eq.(10),cpis the specific heat at constant pressure and k( T )is the temperature dependent variable thermal conductivity. Substituting Eqs.(2)-(7) and (11) into Eqs.(9)-(10) we obtain

In the derivation of the above governing equations, the conventional boundary layer approximation has been invoked. This is justified by the assumption that the film thicknessh is much smaller than the characteristic lengthL (in the direction along the sheet). The mass conservation Eq.(4) then implies that the ratio (v/ u)between the two velocity components is of order (h/ L2). Also, the stream-wise diffusion of momentum and the thermal energy are of order (h/ L2), smaller than the corresponding diffusion perpendicular to the sheet. For this reason, the stream-wise diffusion terms are neglected in Eqs.(12) and (13). Assume that the interface of the planar liquid film is smooth and free of surface waves, and the viscous shear stress and the heat flux vanish at the adiabatic free surface, then the boundary conditions becomes

where Usand Tsare the surface velocity and temperature of the stretching sheet, respectively. The special forms of Usand Tsdefined in Eqs.(6) and (7) respectively helps us to develop a new similarity transformation which transforms the governing partial differential equations in to a set of non-linear ordinary differential equations. Further, it should be noted that the end effects and the gravity are negligible, and the surface tension is sufficiently large such that the film surface remains smooth and stable throughout the motion. We introduce the following dimensionless variable f(ξ)and θ( ξ)as well as the similarity variableξas

In Eq.(16) the stream function ψ(x, y, t)is defined by u=?ψ/?yand v=-?ψ/?x , such that the continuity Eq.(8) is satisfied automatically, andβis an unknown constant denoting the dimensionless film thickness. For ξ=1at the free surface we have from Eq.(18)

which gives

In terms of these new variables, the momentum and the energy equations together with the boundary conditions becomeand

where a prime denotes differentiation with respect to ξanddenotes a determinant. The parameters S,Pr,θrandEc are respectively, the dimensionless measure of the unsteadiness, the generalized Prandtl number, the Eckert number and the variable fluid viscosity parameter, which are defined as

The parameter βdenotes the dimensionless film thickness andβis not known at this moment, which must be determined as a part of the problem (present). Also, it should be noted that the transformed kinematic constraint is different from the corresponding equation in Ref.[3] and this difference disappears for n= 1. The dimensionless film thickness is a constant for fixed values of S and n, and the actual film thickness depends on timet and the stream-wise locationx . From Eq.(18), we find that the film thickness h( x, t) can be expressed as

For practical purposes, the physical quantities of interest include the velocity componentsuandv , the local skin friction coefficient Cf xand the local Nusselt number Nux. These quantities can be written as

x number.

2. Numerical solution method

Table 1 Variation of dimensionless film thickness βand skin friction f′′(0)with unsteady parameter S for n =1.0 when Pr =1.0,ε=0.01,Ec =0.01 and θr→∞

Table 2 Variation of dimensionless film thickness βand skin friction f′′(0)and wall-temperature gradient θ′(0)for different values of the physical parameters

3. Results and discussion

In this study our focus is to study the effects of different governing parameters, namely the unsteady parameterS , the power-law index parametern, the variable thermal conductivity parameterε, and the variable viscosity parameter θr, on the hydrodynamic flow and heat transfer of the power-law liquid above the stretching surface in the presence of viscous dissipation. In order to illustrate how the thermo-physical properties modify the structure of the non-Newtonian thin film flow, the velocity and the temperature fields are presented graphically in Figs.2-6. Changes in the skin friction, the dimensionless film thickness and the wall temperature gradient for several sets of pertinent parameters are recorded in Table 2. For this hydrodynamic problem, the solution exists only for0≤S≤2. Wang[12]noticed the critical value of S =2 for Newtonian fluid. It may be noted here that as S→0 we obtain the analytical solution for an infinitely thick fluid layer (i.e.,β→∞), whereas the limiting case of S→2represents a liquid film of infinitesimal thickness (i.e.,β→0). In the case of non-Newtonian fluids, the present calculations show that the critical value of S =1.0 for shear thinning fluids and the critical value of S =2.5 for shear thickening fluid whenβ→0. However, it is difficult to perform these calculations for the limiting case ofβ→∞.

Fig.2(a) Transverse velocity profiles for different values ofn and s with ε=0.01,Ec =0.01,Pr =1.0 and θr= –1.0

Fig.2(b) Horizontal velocity profiles for different values of θrandnwith ε=0.01,Ec =0.01,Pr =1.0 and S= 0.4

3.1 Velocity field

The transverse velocity profile f and the horizontal velocity profiles f′are shown graphically in Figs.2(a)-2(d) for different values ofS,nand θr. The general trend is that,f increases monotonically whereasf′decreases monotonically as the distance increases from the slit. The effect of increasing values ofS is to increase f and f′. The effect of increasing power-law index parameter n is to reduce the horizontal velocity and thereby reduces the boundary layer thickness, i.e., the thickness is much large for shear thinning (pseudo plastic) fluids (0

Fig.2(c) Horizontal velocity profiles for different values of S and n with ε=0.01,Ec =0.01,Pr =1.0 and θr=

Fig.2(d) Horizontal velocity profiles for different values of S andnwith ε=0.01,Ec =0.01,Pr =1.0 and θr= –1.0

These results are in good agreement with the physical situations. Further, the effect of increasing values of the fluid viscosity parameter θris to decrease the momentum boundary layer thickness. Also, as θr→0, the boundary layer thickness decreases and the velocity distribution asymptotically tends to zero (see Fig.2(d)). This is due to the fact that, for agiven fluid, whenδis fixed, smaller θrimplies higher temperature difference between the wall and the ambient fluid. The results presented in this paper demonstrate clearly that θr, the indicator of the variation of fluid viscosity with temperature, has a substantial effect on the horizontal velocity f′and hence on the skin friction.

3.2 Temperature field

In Figs.3-6, the numerical results for the temperature θ( ξ)for several sets of values of the governing parameters are presented. The general trend is that the temperature distribution is unity at the wall. With changes in the physical parameters the temperature decrease with the distance from the elastic sheet, for all values of the governing parameters. Further, the effect of increasing values of S and the power-law index n is to reduce the temperature and hence reduce the thermal boundary layer thickness.

Fig.3(a) Temperature profiles for different values ofS and n with ε=0.01,Ec =0.01,Pr =1.0 and θr=–1.0

Fig.3(b) Temperature profiles for different values of θrand n with ε=0.01,Ec =0.01,Pr =1.0 and S= 0.4

This behavior is very much noticeable in shear thinning and shear thickening fluids as shown in Fig.3(a). From Fig.3(b), we notice that the effect of increasing values of the fluid viscosity parameter θris to enhance the temperature.

Fig.4(a) Temperature profiles for different values ofεand n with S =0.8,Ec =0.01,Pr =1.0 and θr=0

Fig.4(b) Temperature profiles for different values of εand n with S =0.8,Ec =0.01,Pr =1.0 and θr= –1.0

This is due to the fact that an increase in the fluid viscosity parameter θrresults in an increase in the thermal boundary layer thickness. The effects ofε on the temperature profile in the boundary layer for both zero and nonzero values of fluid viscosity parameter (→0and≠0) are depicted in Figs.4(a) and 4(b), respectively. From these figures, we observe that the temperature distribution is lower throughout the boundary layer for zero values ofε as compared with non-zero values ofε.

This is due to the fact that the presence of temperature-dependent thermal conductivity results in reducing the magnitude of the transverse velocity by a quantity ?K( T)/?yand this can be seen from the energy equation. This behavior holds for all types of fluids considered, namely pseudo plastic, Newtonian, and dilatant fluids. Comparison of Fig.4(a) with Fig.4(b) reveals that the effect of the fluid viscosity parameter is to enhance the thermal boundary layer thickness. Figures 5(a) and 5(b) present the temperature distribution θ( ξ)andξfor different values of Ec for zero and nonzero values of fluid viscosity parameter (→0and≠0) respectively.From these figures we see that the effect of increasing Ec is to increase the temperature distribution θ( ξ). This is in conformity with the fact that energy is stored in the fluid region as a consequence of dissipation due to viscosity and elastic deformation.

Fig.5(a) Temperature profiles for different values of Ec and n with S =0.8,ε=0.01,Pr =1.0 and θr=0

Fig.5(b) Temperature profiles for different values of Ec and n with S =0.8,ε=0.01,Pr =1.0 and θr=–1.0

The variations of temperature profile θ( ξ)with ξfor various values of the modified Prandtl number Pr are shown in Figs.6(a) and 6(b) for both zero and nonzero values of fluid viscosity parameter (→0 and≠0) respectively. Both figures demonstrate that an increase inPr results in a monotonic decrease in the temperature distribution and it tends to zero as the distance increases from the elastic sheet.

It is noteworthy that the temperature vanishes at the free surface for sufficiently high Prandtl numbers. At high Prandtl numbers, the thermal boundary layer is contained within the lower part of the liquid film and the temperature gradients vanish adjacent to the free surface. This is fully analogues to the situation in a steady state aerodynamic boundary layer in an infinite fluid medium at high Prandtl number. That is, the thermal boundary layer thickness decreases for higher values of the Prandtl number. This holds well for all values ofnand fw.

Fig.6(a) Temperature profiles for different values of Pr and n with S =0.8,ε=0.01,Pr =1.0 and θr=0

Fig.6(b) Temperature profiles for different values of Pr and nwith S =0.8,ε=0.01,Pr =1.0 and θr=–1.0

3.3 Flow and heat transfer characteristics

The values of f′′(0),θ′(0),S and θrare recorded in Table 2. It is interesting to note thatβ, f′′(0)and θ′(0)decreases gradually with increasingS . This is true for all values ofn . Further, the effect of increasing nand θris to enhanceβ, f′′(0)and decreases θ′(0). Form Table 2 we see that the effect ofEcandεis to decreases the magnitude of the wall temperature gradient, whereas the effect ofPr is to enhance it. This true for all values of the power-law index parameter n.

4. Concluding remarks

The effects of the thermo-physical properties on the non-Newtonian power-law fluid thin film flow and heat transfer characteristics due to an accelerating surface have been carried out numerically. The results obtained are useful for material processing industries, such as wire and fiber coating, foodstuff processing, extrusion and polymer processing. Some of the interesting results are as follows:

(1) The critical values of Soare 1.3 when n= 0.8 and 3.0 when n =1.2. This observation holdseven the variable fluid properties are considered.

(2) The temperature distribution broadens when the power law index n decreases or when the unsteady parameterS increases.

(3) The dimensionless temperature within the fluid film decreases rapidly for higher values of the Prandtl numberPr .

(4) Unlike in the Newtonian fluid model, here the dimensionless surface temperature is enhanced in the pseudo plastic fluid, while it is decreased in the dilatants fluid case.

(5) Also, approaches zero for sufficiently large Pr but it becomes unity as the Prandtl numberPr approaches zero.

Acknowledgement

The authors appreciate the constructive comments of the reviewer which led to definite improvement in the paper. One of the authors (KVP) is thankful to the University Grants Commission, New Delhi for supporting financially under Major Research Project (Grant No. 41-790/2012 (SR)).

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10.1016/S1001-6058(13)60333-9

* Biography: VAJRAVELU K. (1949-), Male, Ph. D., Professor