Natural Convection Flow of Power-Law Fluids over a Heated Horizontal Plate Surface

1 Introduction

The present paper discusses about the steady two-dimensional laminar natural convection boundary layer flow of a power-law fluid over a heated horizontal semi-infinite plate surface. The heated face of this plate is set up in the upward direction for generating the upward buoyancy forces owing to the variable density of the fluids above the plate surface. Indeed, the effect of this buoyancy force is the origin of the natural convection flow for the present flow problem. This flow is found in many heat transfer processes of nature and is applied to many technological applications. For natural convection boundary layer flow over a heated horizontal plate surface, the theoretical, experimental as well as the numerical analyses are carried out widely by the research community. However, the first theoretical work on the free convective heat transfer from a horizontal plate surface was investigated by Stewartson [1]. Later Gill et al. [2] revisited this flow problem and they also interpreted the Stewartson’s result that the boundary layer solution exists only when the heated surface faces upward. Following this pioneering work of Stewartson [1], various features (under various conditions) of this flow problem were extensively investigated by many authors such as Rotem and Claassen [3], Ackroyd [4], Wickern [5], Daniels [6], Noshadi and Schneider [7], Martorell et al. [8], Kozanoglu and Lopez [9], Samanta and Guha [10], Dholey [11] and the references therein.

All the above studies are, however, restricted to the flows of Newtonian fluids under various situations. We note that almost all the fluids used as well as produced by the industry are non-Newtonian in nature and referred to as rheological fluids. Indeed, non-Newtonian fluids are used industrially for various reasons like drag reduction and modification of effective heat flux. Most importantly, the boundary layer flow characteristics of such fluids can be adequately described by the two-parameters power-law model [also known as the Ostwald-de Waele model (Bird et al. [12])] as this model is compatible with the existing boundary layer assumptions (Dholey [13]). In this model, the first parameter k(>0) is the power-law consistency index while the second parameter n(>0) represents the flow behaviour (or power-law) exponent. An important feature of the rheological fluids is that they have the propensity towards the low Grashof and Reynolds number owing to their large apparent viscosities. Accordingly, in engineering practices laminar flow situations are encountered more often than Newtonian fluids. Because of the growing interest on the rheological fluids in different processing and manufacturing industries, considerable attention has been paid by the research community on understanding their flow characteristics which include mainly the skin-friction coefficient and surface heat flux as well as the fluid pressure on the plate surface.

A closer look into the available literature reveals that the theoretical work concerning natural convection flows for non-Newtonian fluids was first performed by Acrivos [14]. On the contrary, the experimental work on natural convection heat transfer in a non-Newtonian fluid was presented by Reilly et al. [15]. We note that Acrivos’ contribution is fundamental to the work of Reilly et al. [15]. It is applicable to any two-dimensional surface and Acrivos [14] provides us with various examples, but he does not specifically mention the horizontal plate (but he does not exclude either). In any case of Acrivos [14] the wall temperature is kept constant and no analysis is made for the imposed constant surface heat flux case. Recently, Guha and Pradhan [16] have reported the similarity solution for the laminar natural convection flows for non-Newtonian fluids over a horizontal plate maintaining a constant surface temperature. The similarity solution which were found in (40) of their analysis is the simple representation (without any discussion) of the similarity variable for natural convection flows of Newtonian fluids used by Rotem and Claassen [3], Samanta and Guha [10] and many others. The notable fact is that the (40) of Guha and Pradhan [16] does not follow the definition of the similar solution. For this reason, the results of their analysis are questionable. On the other hand, Bahmani and Kargarsharifabad [17] investigated the laminar natural convection of power-law fluids over a horizontal plate only for the constant heat flux case. We assure that the contents of the current analysis and those by Guha and Pradhan [16], and Bahmani and Kargarsharifabad [17] are completely different. Most importantly, there are no such studies in any theoretical and numerical field of work on the laminar natural convection flow for non-Newtonian fluids with different types of temperature boundary conditions on the plate surface in the literature. This provides the motivation for the present study. The main aim of this study is, therefore, to verify the existence of similarity solution of the natural convection boundary layer flow for non-Newtonian power-law fluids, and if the solution exists, then to explore the underlying physics of the flow for various values of power-law index n which will be discussed in the corresponding sections of this paper.

In this paper, we have investigated the two-dimensional steady natural convection boundary layer flow of power-law fluids over a semi-infinite horizontal heated plate facing upward. The plate is subjected to either prescribed surface temperature or variable surface heat flux. The concerning issue of constant surface temperature case for an incompressible viscous fluid is also considered and our numerical results validated with the corresponding results reported by Stewartson [1], Gill et al. [2] and Rotem and Claassen [3]. However, the prime motive of this study is to assess the effects of the physical parameters which include power-law index n, Grashof number Gr and the new dimensionless parameter S on the skin-friction coefficient, given through f′′(0), Nusselt number, given through – θ′(0), thermal as well as the momentum boundary layer of this flow dynamics. The main focus of our present analysis is to ascertain the wall temperature parameter r for which the similarity solution exists in both cases of prescribed surface temperature and variable surface heat flux.

2 Mathematical Formulation and Dimensionless Analysis

We consider the leading edge of the plate surface as the origin, while the axes (x and y) are along and perpendicular to the surface as sketched in Figure 1. Here the plate temperature T_w is always higher than the surrounding fluid temperature T∞ as the heated surface of the plate is facing upward. Hence the effect of gravity on the heated fluid of variable density, i.e. the buoyancy forces due to the density differences between the adjoining and surrounding fluid of the plate surface together with a pressure drop in the x-direction originates this natural convection flow.

Figure 1:

Schematic diagram of the problem shows the velocity and temperature distributions in natural convection flow from a heated horizontal plate surface.

Thus we see that the fluid motion is induced by the buoyancy forces which originates due to the small temperature gradient (density gradient) in a fluid over the plate surface. As a consequence, a body force proportional to the density difference arises into this kind of flow problem. As the fluid motion of the natural convection flow is small, the dissipation term in the thermal boundary layer equation can be neglected, and a body force due to gravity must be included in the velocity boundary layer equation in the problem.

Using Prandtl boundary layer approximations as well as Boussinesq approximation [the inclusion of the body force (gravity) into the v-momentum equation] and disregarding the dissipation term in the energy equation, the governing equations for this natural convection flow problem are given by (p. 148 of Schlichting and Gersten [18])

where (u, v) are the velocity components along x- and y-directions, respectively. And T denotes the fluid temperature within the viscous layer. Also a∞(=λ∞/(ρ∞cp)) and ρ∞ are, respectively the thermal diffusivity and density of the fluid at temperature T∞. Again c_p and λ∞ denote the specific heat and thermal conductivity of the fluid at temperature T∞. Here, G_y is the body force on the fluid of density ρ owing to the gravitational acceleration g in a medium of fluid density ρ∞, which is given by the Archimedes principle as

where Δρ=(ρ(T)−ρ∞) represents the density variation with respect to the reference density ρ∞. Using Taylor series expansion of the density function ρ(T) up to the first order derivative term (as the density gradient is very small), we can rewrite the buoyancy term of (3) as

where β∞=−[(dρ/dT)/ρ]T=T∞ is the coefficient of thermal expansion at temperature T∞. Furthermore, τx^y^ is the shear stress in its dimensional form. It was shown by Schowalter [19] that in two-dimensional flows of non-Newtonian power-law fluids the dimensionless shear stress τxy(=τx^y^/ρVIN2) can be expressed as

provided the velocity gradient is always positive, and the Ostwald-de Waele power-law model (Bird et al. [12]) is adopted. A similar expression of τxy was also found in a recent paper published by Dholey [13]. However for n = 1, (7) coincides with the constitutive equation for an incompressible viscous fluid with k = μ, μ being the dynamic coefficient of viscosity. Therefore, the variation of n from unity implies the degree of variation from the Newtonian fluid behaviour. The fluid is called a shear-thinning (or pseudoplastic) fluid when 0<n<1, whereas a shear-thickening (or dilatant) fluid when n > 1.

In order to make the governing (1)–(4) dimensionless, we introduce the following boundary layer transformations

where the dimensionless quantities V_IN and Gr are the characteristic velocity and the Grashof number, appropriate to the natural convection flow for power-law fluids. It is noticeable that the expressions of V_IN and Gr are reduced to the classical definitions of the characteristic velocity and Grashof number for the natural convection flow of Newtonian fluids (see 10.165 and 10.118 of Schlichting and Gersten [18]). Moreover, l is the length scale and B(>0) is a constant which will be (Tw−T∞) in the CST case. The constant B will be specified for other boundary conditions on the plate surface e.g. prescribed surface temperature (PST case) and variable surface heat flux (PHF case) in Section 4.

Substituting the buoyancy term (Gy/ρ∞) and the shear stress τxy as given in (6) and (7) and the non-dimensional quantities of (8) into (1)–(4), we obtain the dimensionless governing equations for this flow problem as

where S(=a∞(kρ∞)5(n−2)n+4((gβ∞B)(5n−16)l−(5n+8))n−12(n+4)) is a dimensionless quantity which emerges from this dimensionless analysis as a new fundamental parameter. It is worthwhile to remark at this point that for Newtonian fluids (n=1) this parameter will be the reciprocal of the usual Prandtl number Pr(=ν/a∞), where ν denotes the kinematic viscosity co-efficient of the fluid. Another important thing to note here is that the governing (10) conceives the parameter Gr for non-Newtonian fluids (n≠1) in contrast with the corresponding equation of Newtonian fluids (n = 1) where Gr is used only in the boundary layer hypothesis (Dholey [20]).

We consider the dimensionless temperature θ(η) as

where r is the power-law variation of surface temperature parameter. It is worth noting that in case of CST r = 0 while in other two cases r ≠ 0.

3 Limitations of the Similarity Solution

Here, we make an attempt to find the necessary conditions for which the current flow problem will be self-similar. Accordingly, we consider the forms of the similarity variable and transformations as follows:

where (p, q, r) are real constants while (c1,c2,c3) are arbitrary but positive constants. The dimensionless stream function ψ^∗ is described as u∗=∂ψ∗/∂y∗ and v∗=−∂ψ∗/∂x∗, which normally satisfies the mass conservation (9). Hence the velocity components (u∗,v∗) are given by

which transform the governing (10)–(12) into the following self-similar equations

provided that the conditions n(p − 2q) − q = 2p − 2q − 1 = q + r − 1 and p + q − 1 = 0 are fulfilled. Indeed, these conditions are essential for the existence of similarity solutions of the present natural convection flow problem. However on resolving the first equalities, we get the values of p and q as p=(2nr−r+3)/(n+4) and q=(nr−2r+2)/(n+4). With these values of p and q, the last condition will be valid either for

1. n = 1, whatever be the value of r or

2. r=1/3, whatever be the value of n.

We have already pointed out that the main aim of this study is to investigate the convective flow behaviours of non-Newtonian power-law fluids (i.e. for n ≠ 1) over a heated horizontal semi-infinite plate facing upwards. Beside this, we have for the CST case r = 0 and for the other two cases r ≠ 0. Hence, from the above mathematical verification, we have come to this conclusion that the similarity solution does not exist in the CST case and for the other two cases it will exist only for a particular value of r=1/3, viz., Tw−T∞=Bx∗1/3. Here after, we will consider only the fixed value of r(=1/3) for the remaining part of this analysis unless stated otherwise. It is noteworthy to mention here that the natural convection flow for Newtonian fluids (i.e. for n = 1) over a heated horizontal semi-infinite plate subjected to variable as well as constant surface temperature was investigated by Dholey [20].

4 Boundary Conditions

First of all, the velocity boundary conditions are given by (u,v)=(0, 0) at y = 0 and (u,u∞,p)→(0, 0, 0) as y → ∞. Hence, we obtain the boundary conditions for f and g by using (8), (15) and (16) as follows: f(0)=0, f′(0)=0 and f′(∞)=0,g(∞)=0.

Second, the temperature boundary conditions on the plate surface (y = 0) are given by T=Tw for the CST case; T=Tw(=T∞+Ax∗r) for the PST case; qw=−λ∞(∂T/∂y∗)w=Dx∗(r−q) for the PHF case and T→T∞ as y → ∞ for all the cases. Here, q_w is the surface heat flux and the positive constant B in (8) that was unknown so far, is now A for PST case and (Dc2/λ∞) for PHF case. Hence the boundary conditions of g are obtained from (13), (14) and (18) as follows: g′(0)=(c2/c3) and g′(∞)=0 for both CST and PST cases; g′′(0)=−(c2/c3) and g′(∞)=0 for the other case. Finally, the boundary conditions of θ can be obtain from (18) with the help of the boundary conditions of g as given above.

The well-suited boundary conditions of f, g and θ for three different cases of surface temperature variation considered in this flow problem are given by

1. For constant surface temperature (CST case)

   (20)f(0)=0,f′(0)=0,g′(0)=c2c3 andθ(0)=1 and f′(∞)=0,g(∞)=0,g′(∞)=0 and θ(∞)=0.

2. For prescribed surface temperature (PST case)

   (21)f(0)=0,f′(0)=0,g′(0)=c2c3 andθ(0)=1 and f′(∞)=0,g(∞)=0,g′(∞)=0 and θ(∞)=0.

3. For prescribed power law heat flux (PHF case)

   (22)f(0)=0,f′(0)=0,g′′(0)=−c2c3 andθ′(0)=−1 and f′(∞)=0,g(∞)=0,g′(∞)=0 and θ(∞)=0.

It is found that the boundary conditions (20) and (21) for the CST and PST cases are identical. Although, the governing boundary layer (17)–(19) is not identical in these cases as r = 0 for the CST case while r ≠ 0 in the PST case. Obviously, the case of uniform heat flux gives us the relation q=r. Using this along with the value of q in terms of n and r as given in Section 3, we obtain q=r=1/3 which is independent of n. We note that this case was considered by Wickern [5] only for the Newtonian fluids (i.e. for n=1).

When n = 1, r = 0 and c1=c2=c3=1, (17)–(20) one reduced to the (11) and (12) of Stewartson [1]. For n = 1, c2=c3 and r=c, (17)–(19) corroborate with the (6) and (7) of Gill et al. [2]. Again, for n = 1, r = 0 and c1=c2=c3=1, (17)–(20) are identical with the (21)–(23) and (20) of Rotem and Claassen [3].

5 Physical Quantities of Interest

From the practical point of view, the important quantities of physical interest for the present investigation are the assessment of the skin-friction coefficient C_f and Nusselt number Nu which can be expressed as follows:

Skin-Friction Coefficient: The skin-friction coefficient C_f is given by

This shows that the skin-friction coefficient C_f is directly proportional to the dimensionless wall velocity gradient (f′′)n which is always positive for this flow problem.

Nusselt Number: The local Nusselt number at the wall is defined as

Hence the average values of the Nusselt number are obtained as

Here θ′(0)=−1 for the PHF case only.

6 Results and discussion

Equations (17)–(19) subjected to (21) for the PST case and (22) for the PHF case were solved numerically by using the standard fourth-order Runge-Kutta method together with the conventional shooting technique for various values of the governing parameters: (a) Power-law index n, (b) Grashof number Gr and (c) the dimensionless number S. In our numerical study, Gr and S are varied from 0.01 to 100 for a range of values of n from 0.1 to 2. All the numerical results in this analysis are based on the values of c1=c2=c3=1, unless stated otherwise. In order to validate this numerical technique, we have compared our present results (A) with the corresponding results published by Stewartson [1] (B), Gill et al. [2] (C) and Rotem and Claassen [3] (D) in the CST case only. The comparison results which are presented in Tables 1 and 2 assure the reliability of our numerical technique.

Tables 1 and 2 show the comparison of f′′(0), −θ′(0), −g(0) and f(∞) obtained from the present analysis (A) with those of (B), (C) and (D).

Physically, the values f′′(0), −θ′(0) and −g(0) give the wall shear stress, surface heat flux, dynamic pressure on the plate surface, respectively, while f(∞) represents the free stream functional value at the outer boundary layer region.

From the foregoing analysis it is clear that the characteristics features of the present flow problem depend highly on the values of the physical parameters n, Gr and S. And for Newtonian fluids, i.e. for n = 1 the present flow problem is independent of Gr but highly dependent on the values of Pr. In this case, the parameter S reduces to the reciprocal of the usual Prandtl number Pr which characterises the natural convection boundary layer flow [see (17)–(19)]. On the other hand, Grashof number Gr is frequently used in both the mechanisms of heat and mass transfer to study the situations involving natural convection flow for non-Newtonian fluids. The non-Newtonian behaviours of this type of fluids increase with an increasing value of n. Most importantly, the similarity solution of this flow problem exists in both cases of PST and PHF, and it will happen only under a definite value of the wall temperature parameter r=1/3. Hence we can explore the main results of this analysis with the fixed value of r(=1/3) and discuss these results with correlated physics in the following two cases where we will show that the Grashof number differentiates the thermal responses of the pseudoplastic fluids (n<1) from the dilatant (n>1) fluids in comparison with the Newtonian (n=1) fluids.

6.1 Prescribed Surface Temperature (PST Case)

To illustrate the effects of n, Gr and S on this flow problem with prescribed surface temperature conditions, we have presented the main results in the forms of Figures 2–7 and Tables 3 and 4 which allow us to assess the correct behaviour of this flow dynamics.

Figure 2:

Velocity profiles f′(η) against η for some values of: (a) n when Gr=2 and S = 0.5, (b) Gr when S = 0.5 and n = 0.3, (c) Gr when S = 0.5 and n = 2, (d) S when Gr=10 and n = 0.3 and (e) S when Gr = 10 and n = 2 in the power-law variation of surface temperature (PST) case.

Figure 3:

Dimensionless temperature profiles θ(η) against η for various values of: (a) n when Gr = 2 and S = 0.5, (b) Gr with two definite values of n (=0.3 and 2) when S = 0.5 and (c) S with two definite values of n (=0.3 and 2) when Gr = 10 in the PST case.

Figure 4:

Variation of f′′(0) versus n for some values of: (a) Gr when S = 0.5 and (b) S when Gr = 2 in the PST case. For n = 1, the value of f′′(0) remains same (0.6953) for all values of Gr.

Figure 5:

Variation of – θ′(0) versus n for some values of: (a) Gr when S = 0.5 and (b) S when Gr = 2 in the PST case. For n = 1, the value of – θ′(0) remains same (0.5941) for all values of Gr.

Figure 6:

Streamlines for three different n values at various distances x^∗(=0.2, 0.4, 0.6, 0.8, 1.0) along the plate surface when Gr = 2 and S = 0.5: (a) n = 0.5, (b) n = 1.0 and (c) n = 1.5 are in the PST case.

Figure 7:

Pressure p^∗ for three different n values at various distances x^∗(=0.2, 0.4, 0.6, 0.8, 1.0) along the plate surface when Gr = 2 and S = 0.5: (a) n = 0.5, (b) n = 1.0 and (c) n = 1.5 are in the PST case.

Table 3:

Values of – g(0) for some values of n and Gr when S = 0.5 in the power-law variation of surface temperature (PST) case.

n	Gr→0.50	1.0	2.0	5.0	10.0	25.0	50.0	100.0
0.2	1.2653	1.3774	1.4918	1.6806	1.8486	2.1087	2.3404	2.6094
0.5	1.1854	1.2300	1.2755	1.3426	1.3994	1.4252	1.5381	1.6056
0.8	1.1423	1.1547	1.1785	1.1954	1.1982	1.2282	1.2446	1.2560
1.0	1.1245	1.1245	1.1245	1.1245	1.1245	1.1245	1.1245	1.1245
1.2	1.1113	1.1026	1.0942	1.0833	1.0753	1.0651	1.0576	1.0503
1.6	1.0930	1.0733	1.0549	1.0323	1.0165	0.9974	0.9841	0.9718
2.0	1.0811	1.0549	1.0311	1.0030	0.9841	0.9621	0.9475	0.9343

We begin our discussion with Figure 2a which exhibits the variation of the horizontal velocity profiles f′(η) against η for different values of n when Gr=2 and S = 0.5. We find that for a given value of n, first the velocity f′(η) increases and attains its highest value and then decreases and finally approaches zero for large values of η depending upon the values of n. Actually, f′(η) retrieved the outer boundary condition for large values of η [see (21)]. An interesting feature of this flow is that for an increasing value of n the velocity nearer to the plate surface (i.e. for small values of η) increases whereas it decreases after a definite value of η as reflected in this figure. The wall velocity gradient as well as the apparent viscosity of the fluid has an indirect involvement in this conspicuous behaviour of the flow which is described as follows. In fact, the wall velocity gradient which is positive, inherently increases with an increasing value of n which causes an increase in the fluid velocity very close to the plate surface [see (7)]. Simultaneously, the apparent viscosity of the fluid reduces with n, i.e. the momentum increasing effect of the fluid diminishes well into this flow dynamics which reduces the velocity boundary layer thickness in an appreciable amount.

Figure 2b displays the same variation as above but for various values of Gr in case of a pseudoplastic fluid (n=0.3) when S = 0.5. One can readily observe that the velocity decreases with an increase in Gr owing to the decrease of the wall velocity gradient. Here the maximum (highest) velocity is at a point where the velocity gradient is zero. However, this maximum velocity will continue up to a definite value of η, which is dependent on Gr, by the pulling forces of the comparatively higher viscosity of the pseudoplastic fluids and finally f′(η) decreases to retrieve the boundary conditions at infinity. As a result, a plateau region is manifested in the neighbourhood of the maximum velocity which essentially increases the momentum boundary layer thickness. Thus we see that both the span of the plateau region and the boundary layer thickness increase with an increase in Gr. The reverse analysis holds true for a dilatant fluid (n=2). This is exhibited in Figure 2c where the maximum velocity appears with a sharp peak. The sharpness as well as the peakedness of the velocity increase with the increase in Gr owing to the increase of the velocity gradient near the wall (see Fig. 4a). Moreover, the peak point where the velocity is maximum, comes closer to the wall with an increasing value of Gr which essentially makes the velocity boundary layer thinner.

Another explanation can also be made concerning the above mentioned results which arises out from Figure 2b,c as follows. It is well known that the natural convection boundary layer flow has no initial reference velocity. In fact, this type of (small velocity) flow problem is characterized by Grashof number Gr instead of Reynolds number Re. Most importantly, for Newtonian fluids (n=1) this natural convection boundary layer flow is independent of Gr, where it is used in the boundary layer hypothesis (see (7)–(10) of Dholey [11]). But for the present flow problem, where we mainly consider the values of n which are not equal to one, is significantly affected by the parameter Gr. If one can write (10) of the present study, separately, for pseudoplastic (n<1) and dilatant (n>1) fluids then it will be clear from those revised equations that the behaviour of Gr towards this flow problem are of different types: one is the assisting flow (means Gr acts like Re) for dilatant fluids and other is the opposing flow for pseudoplastic fluids.

Figure 2d depicts the x-component of velocity profiles f′(η) against η for various values of S when Gr=10 in case of a pseudoplastic fluid (n=0.3). On the contrary, Figure 2e displays the same variation for a dilatant fluid (n=2) with the same values of S and Gr as considered in Figure 2d. From both figures it is clear that the x-component of velocity increases with an increase of S. But the characteristics features of the velocity profiles vary accordingly with the types of the fluids, which means that the velocity profiles are different for different kinds of fluids owing to the variance of viscosity of the fluids with n. A closer scrutiny at Figure 2d reveals that for a small value of S (≈0.25 and below) the horizontal velocity profile decreases very slowly and finally it satisfies the free boundary condition at a large value of η. Physically, for decreasing values of S the momentum increasing effect (very high apparent viscosity) is more pronounced into this flow field which tends to increase the velocity boundary layer in a considerable amount. The reverse description holds true for dilatant fluids which is shown in Figure 2e.

The variation of the characteristic temperature profiles θ(η) with η for various values of n, Gr and S are delineated in Figure 3a–c. These figures clearly exhibit that for a given value of n, irrespective of the types of fluids, temperature distribution is a monotonic decreasing function of the distance from the surface leading edge and finally the temperature profiles tends to be zero as this distance increases. Figure 3a displays the variation of θ(η) against η for some values of n when Gr=2 and S = 0.5. As expected, the temperature at a given location decreases with the increase of n and simultaneously the thickness of the thermal boundary layer η∞ decreases. For pseudoplastic fluids (n<1), the thickness of the thermal boundary layer becomes thicker which in essence decreases the wall temperature gradient. This is an inherent thermal characteristic behaviour to the class of shear-thinning fluids and after in effect the temperature at a given location decreases with an increasing value of n as one may observe from this figure. Thus we see that a small value of n is more capable of stimulating the heat penetration effect into the fluid body as the distance η∞ from the surface leading edge increases.

The effect of Gr on the temperature profiles θ(η) in cases of both shear-thinning (n<1) and shear-thickening (n>1) fluids is shown in Figure 3b. This figure clearly demonstrates the opposite effects (dual nature) of Grashof number Gr on shear-thinning (n<1) fluids to that of shear-thickening (n>1) fluids in respect with the Newtonian fluids (n=1) for which the present flow problem is independent of Gr. In this figure, we have included this value of n(=1) only to bring out the opposite nature of Gr on the variation of temperature profiles for non-Newtonian fluids as well. It is readily seen that for a pseudoplastic fluid, temperature at a given position increases with an increasing value of Gr, whereas an opposite mode is found in the variation of temperature for a dilatant fluid. This is due to the fact that the increasing value of Gr, in case of pseudoplastic fluids, leads to make the thicker thermal boundary layers which essentially increases the wall temperature gradient resulting in an increase in the temperature within the flow. The reverse description holds true for dilatant fluids. Finally, we conclude that the duality of Gr on the temperature field appears to be consistent with that of the velocity field.

Figure 3c presents the variation of θ(η) against η for various values of S when Gr = 10 and for two fixed values of n as labelled on this figure. This figure discloses that for both values of n(=0.3 and 2) temperature at a given position increases with S. This is consistent with the fact that the wall temperature gradient increases with the increase in S resulting in an increase in the thermal boundary layer. As an effect, lower rate of heat transfer occurs at the plate surface which in turn leads to the increase in temperature within the flow. A simple observation of this figure reveals that the thermal boundary layer increasing effect of S is more prominent (pronounced) in dilatant fluids (n>1) than that of pseudoplastic fluids (n<1) owing to the variance of viscosity for different kinds of fluid rheology. Although, the temperature within the flow is always higher for pseudoplastic fluids (n<1) than for dilatant fluids (n>1).

Figure 4a exhibits the change of skin-friction coefficient f′′(0) as a function of n for some values of Gr when S = 0.5. It is an evidence to support the fact that the solutions of the governing (17)–(22) are independent of Gr for Newtonian fluids (i.e. for n=1) which was pointed out earlier. It is noticeable that for a dilatant fluid (n>1) the wall shear stress, given through f′′(0), is increased with the increase in Gr and this effect has very pronounced views for higher values of n(>1). On the contrary, an opposite trend is observed in case of pseudoplastic fluids (n<1). Hence we can infer that the Grashof number Gr plays a dual role in this flow dynamics: one in the assisting flow for dilatant fluids and other in the opposing flow for pseudoplastic fluids which validated our earlier result as obtained from Figure 2. Besides this, one can see that for moderately lower values of Gr the wall shear stress f′′(0) is a monotonic decreasing function of n, whereas for sufficiently large values of Gr it is an increasing function. This anomalous behaviour of f′′(0) with n is closely related to the variation of viscosity for various kinds of fluids in conjunction with Gr which appears in (17) as a power of (1−n)/2. In Figure 4b, we have shown the same variation for several values of S when Gr=2. This figure discloses that for a given value of n the wall shear stress continuously increases with the increase in S and this effect is more and more pronounced with the decrease in the value of n. For a given value of S (not too small) the wall shear stress reduces with n while for a small value of S (≈ 0.1 and below), in our bare eyes, it seems that the wall shear stress is almost equal for all values of n. But actually the data/resultant very minutely and distinctly shows that for a small value of S (≈ 0.1 and below) it increases very slowly with an increasing value of n.

The reduced Nusselt number – θ′(0) as plotted in Figure 5a suggests that the surface heat flux is independent of Gr for Newtonian fluids (n=1) and for pseudoplastic fluids (n<1) it decreases with Gr, whereas it increases for dilatant fluids (n>1). Here the surface heat flux – θ′(0) is positive as the heat is transferred always from the surface to the fluid. This figure also suggests that for moderately lower values of Gr (≈ 0.1 and below) the surface heat flux – θ′(0) is a monotonic decreasing function of n, whereas for sufficiently large values of Gr (≈ 10 and above) it is an increasing function. From Figure 5b it is easily perceptible that for a given value of n the surface heat flux reduces with an increasing value of S. For a small value of S (≈ 1 and below) the surface heat flux increases significantly with an increasing value of n. On the other hand, for higher values of S (≈ 10 and above) surface heat flux remains practically constant (decreases very slowly) for all values of n considered for the present study.

We have plotted the streamlines ψ^∗ as given in (15) for three distinct values of n(=0.5, 1.0 and 1.5 which are the representative values of pseudoplastic, Newtonian and dilatant fluids, respectively) in Figure 6a–c when Gr=2 and S = 0.5. Here we have considered the above set of values of n as the main objective of this study is to assess the impact of fluid rheology on the natural convection flows for non-Newtonian fluids and to compare these flows with the corresponding Newtonian fluid flows as well. It is easily and directly perceptible from this figure that the distance, normal to the plate surface, required to unite with the outer edge of the boundary layer is distinct for different fluid rheology. And this distance becomes lesser and lesser with the increase of n which essentially decreases the velocity boundary layer thickness. This is authenticated our earlier result as presented in Figure 2. Another important result which emerges from these figures is that the streamlines grow with the distance x^∗ from the leading edge of the plate surface as the stream function ψ^∗ is directly involved with (x∗)p where 0⩽x^∗⩽1 and p=2/3.

To investigate the deeper insight of this flow dynamics, we present in Figure 7a–c the graph of the pressure distribution within the flow for three distinct n values at various positions x^∗ on the plate surface when Gr=2 and S = 0.5. The originality which comes from these figures is that the pressure is lowest on the plate surface (x∗,η=0) and this lowest value of the surface pressure increases in the η-direction and finally merges with the free pressure value (p∗=0) asymptotically. This is consistent with the fact that the heat is always transferred from the surface to the fluid and therefore the fluid density closer to the plate surface is lesser than that of the ambient fluid which essentially gives in return the buoyancy forces in the η-direction. Besides this, the surface pressure decreases along the plate surface, i.e. with the increase of x^∗. From (11), (14), (15) and (18) it is obvious that the pressure gradient g′ is closely associated with the surface temperature which increases along the plate surface. This increasing surface temperature gives rise to a pressure drop region in the x^∗-direction which generates the boundary layer flow parallel to the plate surface. Hence we can draw an interesting inference that the flow separation can never arise in this type of flow problem as the pressure reduces along the plate surface in all the situations for existing the flow. We note that all the above results are true for all types of fluids. However, a moderate change in the Newtonian fluid rheology (n=1) has a great impact on the pressure field and hence on the whole flow dynamics which includes mainly the mass as well as heat transfer characteristics for this flow problem. A closer look at these figures ensures that for any given position on the plate surface (x∗, η=0) the lowest value of the surface pressure – g(0) increases or decreases accordingly as n< or >1 as compared with the corresponding surface pressure value for n = 1. The underlying physics behind such conspicuous behaviour of n on the surface pressure will be described for the results presented in Table 3, which gives the surface pressure values – g(0) for some values of n and Gr when S = 0.5.

The listed values in Table 3 manifest that the surface pressure is always negative as T_w>T∞ which means that the surface temperature is always higher than the fluid temperature owing to the heated surface of the plate facing in the upward direction. An important result which can be obtained from this table is that the surface pressure value –g(0) enhances with the increase of Gr for pseudoplastic fluids (n<1), whereas it reduces for dilatant fluids (n>1). Besides this, it gets the fixed value 1.1245 when n = 1 in which the governing (17) will be free from Gr. In fact, Grashof number Gr has a tendency to make the larger thermal boundary layer for pseudoplastic fluids in comparison with the Newtonian fluids which causes the decrease of the surface heat transfer rate and resulting in the increase of the surface pressure value (see Fig. 3b). The reverse explanation is true for dilatant fluids. Another important result is that for any given value of Gr the surface pressure decreases steadily with the increase of n due to the increase of the surface heat transfer rate.

Table 4 gives the values of –g(0) for some values of n and S with Gr=2 in the PST case. From this table it is observed that the surface pressure value –g(0) increases with S for all types of fluids and it decreases with n for any value of S considered in the present study. These results are as expected.

Table 4:

Values of – g(0) for some values of n and S when Gr=2 in the PST case.

n	S → 0.01	0.02	0.04	0.10	0.20	1.0	2.0	5.0
0.2	1.0572	1.1053	1.1606	1.2567	1.3506	1.6726	1.8614	2.3304
0.5	0.6709	0.7590	0.8213	0.9534	1.0718	1.4820	1.7580	2.3074
0.8	0.5134	0.5853	0.6715	0.8102	0.9417	1.3978	1.7074	2.2961
1.0	0.4554	0.5379	0.6224	0.7543	0.8895	1.3659	1.6882	2.2923
1.2	0.4154	0.4944	0.5719	0.7140	0.8520	1.3434	1.6747	2.2896
1.6	0.3633	0.4325	0.5171	0.6608	0.8028	1.3144	1.6569	2.2858
2.0	0.3312	0.4019	0.4832	0.6281	0.7729	1.2967	1.6457	2.2835

6.2 Prescribed Power Law Heat Flux (PHF Case)

Here, the wall temperature θ(0) will change with the physical parameters, which govern this flow dynamics, in comparison with the PST case where it secures the fixed value one [see (21)]. Although, there are variable wall temperatures, the characteristics features of the mass and heat transfer follow the similar patterns as shown in the PST case.

The change of the similarity profiles f′(η) caused mainly by the fluid rheology is depicted in Figure 8a–c. The values of all the three governing parameters (n, Gr and S) are mentioned in each figure. The results which we can observe from these figures are same as that obtained from Figure 2 in the PST case. Therefore, there is no need to repeat the physical discussion of these results. However, a comparative analysis reveals that the horizontal component of velocity f′(η) is higher in the PHF case than in the PST case. In PHF case, the surface temperature θ(0) varies with the physical parameters as well as the plate surface gets the higher temperature values in contrast with the PST case where it remains constant, which is unity [see Fig. 11 and (21)]. This higher surface temperature modifies the flow characteristics which we have already found in the PST case. In fact, the higher surface temperature generates more buoyancy forces which lead to the development of the flow parallel to the plate surface as one can observe from these figures. This higher surface temperature also increases the temperature within the flow (in comparison with the PST case) which is manifested in Figure 9a–c. Figure 9a which is depicted for a comparatively higher value of Gr=2 when S = 0.5, clearly exhibits that the surface temperature as well as the temperature for any given position within the flow decreases with the increase of n. It is found that in case of dilatant fluids the surface temperature decreases with an increasing value of Gr, whereas an opposite result is found in case of pseudoplastic fluids which is presented in Figure 9b. From Figure 9c one can readily observe that the surface temperature increases continuously with the increase of S in cases of both types of fluids and this increasing effect is significantly influenced by the increase in the value of n.

Figure 8:

Velocity profiles f′(η) against η for some values of: (a) n when Gr = 2 and S = 0.5, (b) Gr with two definite values of n (=0.3 and 2) when S = 0.5 and (c) S with two definite values of n (=0.3 and 2) when Gr = 10 in the power-law variation of surface heat flux (PHF) case.

Figure 10a,b show how the wall gradient f′′(0) changes with n for various values of Gr and S which are indicated in the figures. These figures exhibit the similar qualitative features which one can observe in Figure 4a,b for the PST case. A careful examination on Figures 4 and 10 unveils that the wall gradient is higher in the PHF case than in the PST case due to the higher wall temperature in the PHF case. This higher wall temperature creates much more buoyancy force over the plate surface which accelerate the horizontal component of velocity and thereby increases the velocity gradient near the wall.

To illustrate the influence of fluid rheology on the wall temperature, we have delineated Figure 11a,b for some values of S and Gr as mentioned in each figure. From these figures it is clear that the wall temperature is always higher in the PHF case as compared to the PST case where its value is one. Figure 11a discloses that the wall temperature increases or decreases accordingly as n< or >1 with an increasing value of Gr, while its value remains unchanged for n = 1. This result authenticates our previous result as obtained from Figure 9b. But the result which we have found from Figure 9a is valid only for the values of Gr (≈1 and above) and below this value of Gr the wall temperature increases continuously with the increase of n as one can observe from Figure 11a. On the other hand, Figure 11b shows that for a given value of n the wall temperature increases with the increase of S. In addition to this, for a given value of S (≈1 and below) the wall temperature decreases with n and above this value of S the wall temperature increases.

Figure 9:

Dimensionless temperature profiles θ(η) against η for various values of: (a) n when Gr = 2 and S = 0.5, (b) Gr with two definite values of n (=0.3 and 2) when S = 0.5 and (c) S with two definite values of n (=0.3 and 2) when Gr = 10 in the PHF case.

Figure 10:

Variation of f′′(0) versus n for some values of: (a) Gr when S = 0.5 and (b) S when Gr = 2 in the PHF case. For n = 1, the value of f′′(0) remains same (0.9021) for all values of Gr.

Figure 11:

Variation of wall temperature θ(0) versus n for some values of: (a) Gr when S = 0.5 and (b) S when Gr = 2 in the PHF case. For n = 1, the value of θ(0) remains same (1.5433) for all values of Gr.

The same characteristic features which we have observed in Figures 6 and 7 were also found for the variations of the streamlines as well as the pressure distribution profiles, and therefore these contour plots are not presented herein. Most importantly, the surface pressure which essentially depends on the surface temperature is even more pronounced for PHF case than for the PST case owing to the comparatively higher wall temperature in the PHF case. A comparative study of the values of the surface pressure as presented in Tables 3 and 4 for PST case and in Tables 5 and 6 for PHF case assures this result.

Table 5 gives the values of –g(0) for some values of n and Gr when S = 0.5 in the PHF case. Following the same physical reasons as mentioned in the discussion of the results presented in Table 3 for PST case, here also the value of the surface pressure –g(0) increases with Gr in case of pseudoplastic fluids whereas it decreases for dilatant fluids but it remains unchanged for Newtonian fluids. Besides this, for a given value of Gr the surface pressure –g(0) decreases with an increasing value of n.

Table 5:

Values of – g(0) for some values of n and Gr when S = 0.5 in the power-law variation of surface heat flux (PHF) case.

n	Gr→0.5	1.0	2.0	5.0	10.0	25.0	50.0	100.0
0.2	1.6946	1.9090	2.3999	2.6105	2.9417	3.5511	4.0971	4.7403
0.5	1.6438	1.7412	1.8472	2.0028	2.1349	2.3288	2.4850	2.6677
0.8	1.6076	1.6518	1.6693	1.7534	1.7658	1.7918	1.8277	1.8649
1.0	1.5913	1.5913	1.5913	1.5913	1.5913	1.5913	1.5913	1.5913
1.2	1.5777	1.5545	1.5319	1.5030	1.4818	1.4546	1.4347	1.4153
1.6	1.5565	1.5007	1.4486	1.3851	1.3410	1.2875	1.2504	1.2160
2.0	1.5410	1.4636	1.3937	1.3119	1.2573	1.1940	1.1514	1.1136

Table 6 gives the values of –g(0) for some values of n and S when Gr=2 in the PHF case. Table 6 clearly exhibits that for a given value of n, irrespective of the types of fluids, the surface pressure –g(0) increases with S. In addition to this, for a given value of S the surface pressure decreases with an increase in the non-Newtonian characteristic of the fluids.

Table 6:

Values of – g(0) for some values of n and S when Gr=2 in the PHF case.

n	S → 0.01	0.02	0.04	0.10	0.20	1.0	2.0	10.0
0.2	1.2697	1.3779	1.4549	1.6574	1.8299	2.5348	3.1074	6.5738
0.5	0.6824	0.7966	0.9380	1.1986	1.4064	2.3379	3.0626	6.8461
0.8	0.4440	0.5515	0.6888	0.9312	1.2219	2.2172	3.0300	7.0544
1.0	0.3877	0.4638	0.6098	0.8350	1.0919	2.1638	3.0143	7.2643
1.2	0.3092	0.4218	0.5288	0.7647	1.0223	2.1225	3.0015	7.2644
1.6	0.2415	0.3308	0.4506	0.6703	0.9270	2.0630	2.9821	7.4154
2.0	0.2020	0.2800	0.3902	0.6110	0.8660	2.0231	2.9684	7.5343

7 Conclusion

We have obtained the conditions for the existence of similarity solution of the two-dimensional steady Navier-Stokes equations which describes the laminar natural convection boundary layer flows of non-Newtonian power-law fluids over a heated horizontal semi-infinite plate facing upwards. The present flow analyses completely based on the Ostwald-de Waele power-law model. Adopting the similarity transformation technique, the governing partial differential equations are converted into the self-similar ordinary differential equations which conceive especially two physical parameters for non-Newtonian natural convective flows: one is the Grashof number Gr which does not appear in the corresponding equations for Newtonian fluids and the other is the dimensionless number S which reduces to the reciprocal of the usual Prandtl number Pr for Newtonian fluids. The combined influences of these parameters on this flow field are analyzed and the main findings are discussed with the associated physics. It is found that the velocity and temperature profiles as well as the skin-friction coefficient, surface heat flux and surface pressure are strongly influenced by the above parameters. We believe that the present analysis may enhance the understanding of the physical quantities like skin-friction coefficient and Nusselt number as well as the surface pressure which are the most important factors in controlling the natural convection boundary layer flows in numerous manufacturing and processing industries. Here, we have considered three different types of temperature boundary conditions on the plate surface and each case is analysed with great care. Briefly, the main results obtained from these cases are summarized as follows:

Case 1:CST case (Constant surface temperature):

The present study analytically stated that the similarity solution does not exist in the CST case.

Case 2:PST case (Prescribed surface temperature):

Exact mathematical calculations prescribed that the similarity solution exists, in this case, only under a particular value of the wall temperature parameter r=1/3. All the numerical results of this analysis are performed only for the above specified value of r. The most interesting finding of the present analysis is the characteristic behaviour of the Grashof number Gr which differentiate the thermal responses by the pseudoplastic fluids (n<1) from the dilatant fluids (n>1) relative to the Newtonian fluids (n=1) for which it remains unchanged.

Case 3:PHF case (Prescribed heat flux at the wall):

Similarity solution exists in this case only when the plate surface is held at constant heat flux situation. The qualitative features of the momentum as well as the thermal boundary layer characteristics, in this case, follow the similar patterns with higher magnitude as observed in the PST case as the surface temperature θ(0) is always higher for the PHF case than for the PST case. Surface temperature increases with the increase of S irrespective of the types of fluids. It also increases with the increase of Gr for pseudoplastic fluids whereas decreases for dilatant fluids, but remains constant for Newtonian fluids.