Open Access

Effect of variable viscosity on vortex instability of non-Darcy free convection boundary layer flow adjacent to a non-isothermal horizontal surface in a porous medium

Boundary Value Problems20122012:26

DOI: 10.1186/1687-2770-2012-26

Received: 1 October 2011

Accepted: 27 February 2012

Published: 27 February 2012

Abstract

In this article, we study the effect of variable viscosity on the flow and vortex instability of non-Darcian free convection boundary layer flow on a horizontal surface in a saturated porous medium. The wall temperature is a power function of the distance from the origin. The variation of viscosity is expressed as an exponential function of temperature. The transformed boundary layer equations, which are developed using a non similar solution approach, are solved by means of a finite difference method. The analysis of the disturbance flow is based on linear stability theory. The local Nusselt number, critical Rayleigh number and the associated wave number at the onset of vortex instability are presented over a wide range of wall to ambient viscosity ratio parameters μ*= μ w /μ. The variable viscosity effect is found to enhance the heat transfer rate and destabilize the flow for liquid heating, while the opposite trend is true for gas heating.

Keywords

vortex instability porous media variable viscosity non-Darcy free convection finite difference method

1 Introduction

The problems of the vortex mode of instability in free and mixed convection flows over horizontal and inclined heated surfaces in saturated porous media have received considerable attention (see [111]). The instability mechanism is due to the presence of a buoyancy force component in the direction normal to the surface. The importance of the problems is due to a large number of technical applications, such as material transfer associated with the storage of radioactive nuclear waste, separation processes in the chemical industry, filtration, transpiration cooling, transport processes in aquifers, and ground water pollution. Some researchers considered Darcy model [15], others considered non-Darcy model [611]. A comprehensive literature survey on this subject can be found in the recent book by Nield and Bejan [12].

All of the studies mentioned previously dealt with constant viscosity. The fundamental analysis of convection through porous media with temperature dependent viscosity is driven by several contemporary engineering applications from the cooling of electronic devices to porous journal bearings and is important for studying the variations in constitutive properties. The effect of variable viscosity for convective heat transfer through porous media has also been widely studied (e.g., [1319]). The effect of variations in viscosity on the instability of flow and temperature fields are discussed by Kassoy and Zebib [13] and Gray et al. [14]. Lai and Kulacki [15] considered the variable viscosity effect for mixed convection along a vertical plate embedded in a saturated porous medium. The effect of variable viscosity on non-Darcy, free or mixed convection flow on a horizontal surface in a saturated porous medium is studied by Kumari [16]. Jayanthi and Kumari [17] studied the same problem presented in Kumari [16] for vertical surface. Afify [18] studied the effects of non-Darcy, variable viscosity and Hartmann-Darcy number on free convective heat transfer from an impermeable vertical plate embedded in a thermally stratified, fluid saturated porous medium for the case of power-law variation in the wall temperature. In [1518], the viscosity of the fluid is assumed to vary as an inverse linear function of temperature, whereas in [19], the variation of viscosity with temperature is represented by an exponential function.

The effects of variable viscosity on the vortex instability of horizontal free and mixed convection boundary layer flows in a saturated porous medium for isothermal and non-isothermal surfaces are studied by Jang and Leu [19] and Elaiw et al. [20], respectively. However, the effect of variable viscosity on the flow and vortex instability of non-Darcian free convection boundary layer flow over a non-isothermal horizontal plate does not seem to have been investigated. This motivated the present investigation.

The purpose of this article is to examine in details the effect of temperature-dependent viscosity on the flow and vortex instability of a horizontal non-Darcian free convection boundary layer flow in a saturated porous medium with variable wall temperature. This is accomplished by considering the Ergun model equation of motion. The variation of viscosity with temperature is represented by an exponential function, which is more accurate than linear function especially for large temperature differences. The transformed boundary layer equations, which are given using a non similar solution approach, are solved by means of a finite difference method. The stability analysis is based on the linear stability theory. The resulting eigenvalue problem is solved using a finite difference scheme.

2 Analysis

2.1 The main flow

We consider a semi-infinite non-isothermal horizontal surface (T w ) embedded in a porous medium (T ) as shown in Figure 1, where x represents the distance along the plate from its leading edge, and y the distance normal to the surface. The wall temperature is assumed to be a power function of x, i.e., T w (x) = T + Ax λ , where A is a constant and λ is the parameter representing the variation of the wall temperature. In order to study transport through high porosity media, the original Darcy model is improved by including inertia. For the mathematical analysis of the problem we assume that, (i) local thermal equilibrium exists between the fluid and the solid phase, (ii) the physical properties are constant, except for the viscosity μ and the density ρ in the bouncy force, (iii) Ergun non-Darcy model [21] is considered, and (iv) the Boussinesq approximation is valid. With these assumptions, the governing equations are given by:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Fig1_HTML.jpg
Figure 1

The physical model and coordinate system.

u x + v y = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ1_HTML.gif
(1)
u + K * v u 2 = - K μ P x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ2_HTML.gif
(2)
v + K * v v 2 = - K μ P y + ρ g , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ3_HTML.gif
(3)
u T x + v T y = α 2 T y 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ4_HTML.gif
(4)

where ρ = ρ [1 - β(T - T )] is the fluid density. Note that the second term on the left-hand side of Equations (2) and (3) represents the inertia force, where K* is the inertia coefficient in Ergun model. As K* = 0, Equations (2) and (3) reduce to Darcy model.

The viscosity of the fluid μ is assumed to vary with temperature according to the following exponential form
μ = μ e A 1 T - T T w - T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ5_HTML.gif
(5)

where μ is the absolute viscosity at ambient temperature and A1 is a constant adopted from the least square fitting for a particular fluid. Formula (5) is a generalization of that used in [19], where the wall temperature is taken to be constant.

The pressure terms appearing in Equations (2) and (3) can be eliminated through cross-differentiation. The boundary layer assumption yields ∂/∂x ∂/∂y and v u. With ψ being a stream function such that u = ∂ψ/y and v = -∂ψ/∂x, the Equations (1)-(4) become
μ + 2 ρ K * ψ y 2 ψ y 2 + u μ y = - K ρ g β T x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ6_HTML.gif
(6)
ψ y T x - ψ x T y = α 2 T y 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ7_HTML.gif
(7)
The boundary conditions are defined as follows
v ( x , 0 ) = - ψ x = 0 , T ( x , 0 ) = T w = T + A x λ , u ( x , ) = 0 , T ( x , ) = T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ8_HTML.gif
(8)
By introducing the following similarity variables:
η ( x , y ) = y x R a x 1 / 3 , f ( ξ , η ) = ψ ( x , y ) α R a x 1 / 3 , θ ( ξ , η ) = T - T T w - T , ξ = x d ( 2 λ - 1 ) / 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ9_HTML.gif
(9)
Equations (5)-(7) become
μ μ = e A 1 T - T T w - T = ( μ * ) θ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ10_HTML.gif
(10)
( 1 + 2 E r R a d 2 / 3 ξ ( μ * ) - θ f ) f = - ( ln μ * ) f θ - ( μ * ) - θ λ θ + λ - 2 3 η θ + 2 λ - 1 3 ξ θ ξ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ11_HTML.gif
(11)
θ = λ θ + 2 λ - 1 3 ξ θ ξ f - λ + 1 3 f + 2 λ - 1 3 ξ f ξ θ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ12_HTML.gif
(12)

where μ * = μ w μ = e A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq1_HTML.gifis the wall to ambient viscosity ratio parameter, R a x = ρ g β K ( T w - T ) x α μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq2_HTML.gif is the local Rayleigh number, R a d = ρ g β K A d λ + 1 α μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq3_HTML.gif is the Rayleigh number based on the pore diameter and E r = K * α d ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq4_HTML.gif is the Ergun number.

The boundary conditions are transformed as follows:
f ( ξ , 0 ) = 0 , θ ( ξ , 0 ) = 1 , f ( ξ , ) = 0 , θ ( ξ , ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ13_HTML.gif
(13)
where the primes denote the derivatives with respect to η. The velocity components and the heat transfer coefficient in term of the local Nusselt number are given by:
u ( x , y ) = α R a x 2 / 3 x f ( ξ , η ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ14_HTML.gif
(14)
v ( x , y ) = - α R a x 1 / 3 3 x ( λ + 1 ) f ( ξ , η ) + λ - 2 η f ( ξ , η ) + 2 λ - 1 ξ f ( ξ , η ) ξ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ15_HTML.gif
(15)
N u x R a x 1 / 3 = - θ ( ξ , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ16_HTML.gif
(16)

2.2 The disturbance flow

The standard linear stability method yields the following:
u 1 x + v 1 y + w 1 z = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ17_HTML.gif
(17)
μ 0 u 1 + μ 1 u 0 + 2 K * ρ u 0 u 1 = - K P 1 x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ18_HTML.gif
(18)
μ 0 v 1 + μ 1 v 0 + 2 K * ρ v 0 v 1 = - K P 1 y - ρ g β T 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ19_HTML.gif
(19)
μ 0 w 1 = - K P 1 z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ20_HTML.gif
(20)
u 0 T 1 x + v 0 T 1 y + u 1 T 0 x + v 1 T 0 y = α 2 T 1 x 2 + 2 T 1 y 2 + 2 T 1 z 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ21_HTML.gif
(21)

where the subscripts 0 and 1 signify the mean flow and disturbance components, respectively.

Following the order of magnitude analysis method described in detail by Hsu et al. [1], the terms ∂u1/∂x and ∂2T1/∂x2 in Equations (17) and (21) can be neglected. The omission of ∂u1/∂x in Equation (17) implies the existence of a disturbance stream function Ψ1, such as
w 1 = Ψ 1 y , v 1 = - Ψ 1 z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ22_HTML.gif
(22)
Eliminating P1 from Equations (18)-(20) with the aid of (22) leads to
u 0 μ 1 z + ( μ 0 + 2 K * ρ u 0 ) u 1 z = μ 0 2 Ψ 1 x y + Ψ 1 y μ 0 x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ23_HTML.gif
(23)
- ( μ 0 + 2 K * ρ v 0 ) 2 Ψ 1 z 2 + v 0 μ 1 z = μ 0 2 Ψ 1 y 2 + μ 0 y Ψ 1 y + K ρ g β T 1 z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ24_HTML.gif
(24)
u 0 T 1 x + v 0 T 1 y + u 1 T 0 x - Ψ 1 z T 0 y = α 2 T 1 y 2 + 2 T 1 z 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ25_HTML.gif
(25)
As in Hsu et al. [1], we assume that the three-dimensionless disturbances for neutral stability are of the form
( Ψ 1 , u 1 , T 1 ) = Ψ ̄ ( x , y ) , ū ( x , y ) , T ̄ ( x , y ) e i a z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ26_HTML.gif
(26)
where a is the spanwise periodic wave number. Substituting Equation (26) into Equations (23)-(25) yields
ū = 1 1 + 2 K * ν ( μ * ) - θ u 0 1 i a 2 Ψ ̄ x y + ( ln μ * ) Ψ ̄ y θ x - ( ln μ * ) u 0 T ̄ T w - T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ27_HTML.gif
(27)
2 Ψ ̄ y 2 - a 2 1 + 2 K * ν ( μ * ) - θ v 0 Ψ ̄ - i a v 0 ( ln μ * ) T ̄ T w - T + ( ln μ * ) θ y Ψ ̄ y = - i a K ρ g β μ ( μ * ) - θ T ̄ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ28_HTML.gif
(28)
u 0 T ̄ x + v 0 T ̄ y + ū T 0 x - i a Ψ ̄ T 0 y = α 2 T ̄ y 2 - a 2 T ̄ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ29_HTML.gif
(29)

Equations (27)-(29) are solved based on the local similarity approximation [22], wherein the disturbances are assumed to have weak dependence in the streamwise direction (i.e., ∂/∂x ∂/∂η).

On applying the following transformations
k = a x R a x 1 / 3 , F ( η ) = Ψ ̄ i α R a x 1 / 3 , Θ ( η ) = T ̄ T w - T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ30_HTML.gif
(30)
into Equations (27)-(29) leads to the following:
F + ( ln μ * ) θ F - k 2 1 - 2 E r R a d 2 / 3 R a x 1 / 3 ξ ( μ * ) - θ G 1 F + k ( ln μ * ) R a x 1 / 3 G 1 Θ = - k R a x 1 / 3 ( μ * ) - θ Θ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ31_HTML.gif
(31)
Θ + G 3 Θ - k 2 + λ f - ( ln μ * ) f G 2 1 + 2 E r R a d 2 / 3 ξ ( μ * ) - θ f Θ - G 2 λ - 2 3 η F + 2 λ - 1 3 + ( ln μ * ) G 4 F k R a x 1 / 3 1 + 2 E r R a d 2 / 3 ξ ( μ * ) - θ f = k R a x 1 / 3 θ F , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ32_HTML.gif
(32)
with the boundary conditions
F ( 0 ) = F ( ) = Θ ( 0 ) = Θ ( ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equ33_HTML.gif
(33)
where G1 - G4 are given by
G 1 = λ + 1 3 f + λ - 2 3 η f + 2 λ - 1 3 ξ f ξ , G 2 = λ θ + λ - 2 3 η θ + 2 λ - 1 3 ξ θ ξ , G 3 = λ + 1 3 f + 2 λ - 1 3 ξ f ξ , G 4 = λ - 2 3 η θ + 2 λ - 1 3 ξ θ ξ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Equa_HTML.gif

For fixed values of ξ, λ, k, Er, Ra d , and μ*, the solution F and Θ is an eigenfunction for the eigenvalue Ra x . When μ* = 1, Er = Ra d = ξ = 0, Equations (31) and (32) reduce to those presented in [1] and when μ* = 1, ξ = 1, ξ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq5_HTML.gif, λ = 0.5 they reduce to those given in [8]. It may also remarked that for λ = Er = Ra d = ξ = 0, Equations (31) and (32) reduce to those presented in Jang and Leu [19].

3 Numerical scheme

In this section, we compute the approximate values of Ra x for Equations (31) and (32) with boundary conditions (33). An implicit finite difference method is used to solve first the base flow Equations (11) and (12) with boundary conditions (13). The results are stored for a fixed step size h, which is small enough to make an accurate linear interpolation between mesh point. The domain is 0 ≤ η ≤ η , where η is the edge of the boundary layer of the basic flow. For a positive integer N, let h = η /N and η i = ih, i = 0, 1, . . . , N. The problem is discretized with standard centered finite differences of order two, following Usmani [23]. The solution of the eigenvalue problem is achieved by using the subroutine GVLRG of the IMSL library, see [24].

4 Results and discussion

Numerical results for the local Nusselt number, neutral stability curves, critical Rayleigh numbers and associate wave numbers at the onset of vortex instability for wall to ambient viscosity ratio parameter μ* ranging from 0.1 to 10 and Ergun number Er ranging from 0.1 to 0.6 are presented. The effect of the non uniform temperature profile on the wall is also studied, and corresponds to variation of the parameter λ from 0.5 to 2. It is known that as the temperature is increased, the gas viscosity increases and the liquid viscosity decreases [19]. Therefore, for a heated wall, values of μ* > 1 corresponds to the case of gas heating, values of μ* < 1 corresponds to the case of liquid heating.

The local Nusselt number as a function ξ for various values of μ*, Er, and λ is shown in Figures 2 and 3. The heat transfer rate is observed to be more than the constant-viscosity case for liquids μ* < 1 and it has the opposite trend for gases μ* > 1. Further, a lower local Nusselt number is seen to occur with greater values of Er and ξ. Furthermore, a higher Nusselt number is seen to occur with greater values of λ.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Fig2_HTML.jpg
Figure 2

The local Nusselt number as a function of ξ for selected values of μ *.

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Fig3_HTML.jpg
Figure 3

The local Nusselt number as a function of ξ for selected values of Er and λ.

Figures 4 and 5, respectively, show the neutral stability curves, in terms of Rayleigh numbers Ra x and the dimensionless wave number k for selected values of μ* and ξ. Observe that as μ* and ξ increase, the neutral stability curves shift to a higher Rayleigh number Ra x . On the other hand, for liquid heating μ* < 1, the neutral stability curves shift to lower Rayleigh numbers and higher wave numbers, indicating a destabilization of the flow, whereas for gas heating μ* > 1, the opposite trend is true.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Fig4_HTML.jpg
Figure 4

Neutral stability curves for selected values of μ *.

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Fig5_HTML.jpg
Figure 5

Neutral stability curves for selected values of ξ.

The critical Rayleigh number and wave number are plotted as a function of Ra d for selected values of μ* and λ in Figures 6 and 7. Observe that, as the variable viscosity parameter μ* or λ increases, the critical Rayleigh R a x * https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq6_HTML.gif increases. Also, as Ra d is increased the critical Rayleigh number R a x * https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq6_HTML.gif is decreased. Further, the critical wave number k* is increased as Ra d and λ are increased and μ* is decreased. From these figures it can be seen that, the inertia effects reduce the heat transfer rate and destabilize the flow. Finally, we conclude that, the variable viscosity effect enhances the heat transfer rate and destabilizes the flow for liquid heating, while the opposite tend is true for gas heating.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Fig6_HTML.jpg
Figure 6

Critical Rayleigh number as a function of Ra d for selected values of μ * and λ.

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_Fig7_HTML.jpg
Figure 7

Critical wave number as a function of Ra d for selected values of μ * and λ.

5 Conclusions

The non-Darcy free convection flow on a semi-infinite non-isothermal horizontal plate embedded in a porous medium with variable viscosity is investigated. The effects of variable viscosity characterized by the parameter μ* on the flow and vortex instability are examined. The governing partial differential equations are transformed to a non similar form by introducing appropriate transformations and are solved numerically using an implicit finite difference scheme. The resulting eigenvalue problem is solved by using a finite difference scheme. The effects of all involved parameters on the local Nusselt number, critical Rayleigh and associated wave number are presented. The results show that, for liquid heating, the variable viscosity effect enhances the heat transfer rate and destabilizes flow; for gas heating, the opposite is true.

Nomenclature

a spanwise wave number

d mean particle diameter or pore diameter

f dimensionless base state stream function

F dimensionless disturbance stream function

g gravitational acceleration

i complex number

k dimensionless wave number

K permeability of porous medium

K* inertial coefficient in Ergun Equation

Nu x local Nusselt number

P pressure

Ra x local Rayleigh number

Ra d Rayleigh number based on the pore diameter

T fluid temperature

u, v, w volume averaged velocity in the x, y, z directions

x, y, z axial, normal and spanwise coordinates

Greek symbols

α thermal diffusivity

β volumetric coefficient of thermal expansion

η pseudo-similarity variable

θ dimensionless base state temperature

Θ dimensionless disturbance temperature

μ dynamic viscosity of the fluid

ξ non similarity parameter

λ exponent in the wall temperature variation

ν kinematic viscosity

μ* wall to ambient viscosity ratio, μ * = μ w μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-26/MediaObjects/13661_2011_Article_115_IEq7_HTML.gif

ψ stream function

Subscripts

w conditions at the wall

conditions at the free stream
  1. 0

    basic undisturbed quantities

     
  2. 1

    disturbed quantities

     

Superscripts

* critical value

' differentiation w.r.t η

Declarations

Acknowledgements

The authors were grateful to the anonymous reviewers for constructive suggestions and valuable comments, which had improved the quality of the article.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, Faculty of Science for Girls, King Khalid University
(3)
Department of Mathematics, University Collage in Makkah, Umm-Alqura University
(4)
Department of Mathematics, Faculty of Science, Al-Azhar University (Assiut Branch)
(5)
Department of Mathematics, Faculty of Science, Assiut University

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