Non-similar solution of the forced convection of laminar gaseous slip flow over a flat plate with viscous dissipation: linear stability analysis for local similar solution

Non-similar solution of the forced convection of laminar gaseous slip flow over a flat plate with viscous dissipation:

linear stability analysis for local similar solution

Elhoucine Essaghir . Youssef Haddout . Abdelaziz Oubarra . Jawad Lahjomri

Received: 6 October 2013 / Accepted: 12 May 2015 Ó Springer Science+Business Media Dordrecht 2015

Abstract Forced convection of laminar nearly in- compressible gaseous slip flow over an isothermal flat plate at low Mach number with viscous dissipation is considered. The non-similar solutions of hydrody- namical and thermal boundary layers equations with velocity-slip and temperature-jump at the wall are obtained numerically by using the implicit finite difference method. The effects of the modified boundary layer Knudsen number, i.e., the slip pa- rameter and the Eckert number on the heat transfer characteristics are presented graphically and dis- cussed. The numerical results show that for small Eckert number, the slip parameter does not have significant effect on the local heat transfer in the continuum and in slip flow regimes while for the large Eckert numbers, its effect depends that the plate being colder or warmer than the free stream. In addition, we develop a linear stability analysis, based on the traditional normal-mode approach, by assuming local parallel flow approximation, to study the effect of slip parameter on the stability of local similar solution.

This approach leads to the usual Orr–Sommerfeld equation which governs the perturbation stream func- tion satisfying slip boundary condition. This equation is solved numerically by using a powerful method

based on spectral Chebyshev collocation. For no slip flow, the results for the eigenvalues and the corre- sponding wave numbers are found in excellent agree- ment with previous available numerical calculations that supports the validity of our results. Furthermore, the neutral curves of stability in the Reynolds-wave number plane are obtained, for the first time, for the boundary layer in the slip flow regime. The results show that the effect of slip parameter is to increase the critical Reynolds numbers for instability and to decrease the most unstable wave numbers. It is concluded that the rarefaction has a stabilizing effect on the Blasius flow and suggests that the transition to turbulence could be delayed in the slip flow regime.

Keywords Slip flow Flat plate Boundary layers Forced convection Non-similar solution Local similar solution Viscous dissipation Linear stability analysis Orr–Sommerfeld equation

List of symbols

a Parameter appears in Eq. (21), a ¼

T

rM

2rM

2c cþ1

b Dimensionless adiabatic wall temperature b c Complex wave velocity, b c ¼ b c

þ i b c

,

Eq. (29)

c Dimensionless complex wave velocity, c ¼ c

þ i c

, Eq. (33)

Cp Specific heat at constant pressure (J Kg

K

)

E. Essaghir ( & ) Y. Haddout A. Oubarra J. Lahjomri Laboratory of Mechanics, Faculty of Sciences Ain Chock, Hassan II University, Maarif, B.P. 5366,

20100 Casablanca, Morocco

e-mail: [email protected]

DOI 10.1007/s11012-015-0204-2

Ec Eckert number, Ec ¼ U

=C

ðT

T

Þ f Dimensionless stream function, Eq. (12) h Heat transfer coefficient (W m

K

) k Thermal conductivity (W m

K

) K Local slip parameter, K ¼ k

M

ffiffiffiffiffiffi

U1

mx

q L Length of the plate (m)

Nu

Local Nusselt number, Eq. (25) Pr Prandtl number, Pr ¼ m=a

Re

Local Reynolds number, Re ¼ U

x=m Re

Local Reynolds number,

Re

¼ U

dðxÞ=m ¼ ffiffiffiffiffiffiffi Re

p

Re

Local Reynolds number, Re

¼ U

d

ðxÞ=m Tw Temperature at the surface of the plate, wall

temperature (K)

T j

Fluid temperature at the wall, Eq. (5) (K) T Temperature of the fluid (K)

T

Temperature of the ambient fluid (K) uj

The local wall slip velocity, Eq. (4) (m s

) u

The dimensionless x-component of the

velocity, Eq. (13)

u The x-component of the velocity (m s

) U

Free stream velocity (m s

)

v

The dimensionless y-component of the velocity, Eq. (14)

v The y-component of the velocity (m s

) x, y Distance along and normal to the wedge (m) Greek

a

Thermal diffusivity, a

¼ k=qC

(m

s

) b a Wave number (m

)

a Dimensionless wave number, Eq. (33) b Ratio of boundary layers thickness,

b ¼ d

=d ¼ R

0

ð1 f 0Þdg d Boundary layer thickness d

Displacement thickness

c Ratio of specific heats, c ¼ C

=C

/ Dimensionless complex amplitude of the perturbation of stream function Eq. (33) r Accommodation coefficient

k Mean free path (m) w Stream function (m

s

)

w b Complex amplitude of the perturbation of stream function, Eq. (26)

W

Base flow of stream function (m

s

) g Similarity variable

H Dimensionless fluid temperature, Eq. (12) m Kinematic viscosity (m

s

)

1 Introduction

The convective heat transfer over a flat plate in the slip flow regime may be one of the most fundamental problems in fluid mechanics and has been studied extensively over the decades because it can be adapted to typical practical industrial applications in processes involving the design of micro-electromechanical sys- tems (MEMS) [1]. The classical hydrodynamic prob- lem in continuum flow was solved for the first time by Blasius [2] and the corresponding thermal problem by Pohlhausen [3]. This problem stills continues today to attract attention as an active area of research [4–12]. In many cases, similar solutions of Prandtl’s boundary layer equations are well established and widely used.

However, similarity solutions of boundary layers could not always occur in certain situations [13].

Few factors could deteriorate the self-similarity of boundary layer equations and could lead to a non- similarity for hydrodynamic and thermal flow. Non- similarity of boundary layers may arise from some factors such as for example: the effect of the velocity slip and thermal jump at the wall, effect of suction or injection of fluid at the wall, variation in free stream velocity, variation in wall temperature, effect of an externally applied magnetic field on the magnetohy- drodynamic (MHD) boundary layers, effect of buoy- ancy force for flow over vertical plate in continuum or in slip flow regime with and without applied magnetic field.

The similarity solution of the hydrodynamic

boundary layer equations in the slip flow regime has

been obtained by Martin and Boyd [14] who used local

similarity approach. The slip flow model was extended

by Anderson [15] to a stretching surface and by Fang

and Lee [16] to a moving flat plate. Vedantam and

Parthasarathy [17] extended the work of Martin and

Boyd [14] with three different models of slip boundary

conditions. Aziz [18] studied a boundary layer slip

flow over a flat plate under constant heat flux by using

also local similarity approach. The similarity solution

of the boundary layer flow and heat transfer over an

isothermal stretching surface of nanofluids was pre-

sented by Noghrehabadi et al. [19]. Bhattacharyya

et al. [20] analyzed MHD boundary layer slip flow and

heat transfer over a flat plate with uniform temperature

boundary condition by using shooting method. Rah-

man [21] studied combined effects of slip flow and

convective surface heat flux on hydromagnetic bound- ary layer for a flat plate with variable fluid properties.

By assuming local similarity approach, the boundary layer equations are transformed into ordinary ones and are solved numerically. Das [22] extended the work of Rahman [21] to investigate the thermal radiation effect on the hydromagnetic flow and heat transfer over vertical porous plate and flat impermeable plate with variable fluid properties and non-uniform heat source/

sink by applying a similarity transformation. Yazdi et al. [23] analyzed the effect of viscous dissipation on the MHD slip flow with heat transfer past a permeable surface by using also the local similarity method. Cao and Baker [24] extended the local non-similar method developed by Sparrow et al. [25, 26] to study the slip effects on mixed convective flow and heat transfer from a vertical flat plate.

Most of the previous works considering the slip flow boundary layer frequently applied the approach of local similarity, or local non-similar method. The first approach consists to fix the slip parameter K at any x-location along the plate, i.e., K = constant and neglecting the variation of the velocity field and temperature field with K. This is equivalent to ignore the upstream history of the flow. Consequently, the original partial differential boundary layer equations become ordinary differential equations. It was shown that the hydrodynamic and thermal boundary layer in the rarefied flow regime, the conditions of slip velocity and jump temperature at the wall, leads to the loss of self-similarity of the Blasius flow. Lahjomri and Oubarra [27] have shown recently that the results given by the local similarity method are satisfactory only for small values of K (K B 0.05) and this approach presents a limitation when K becomes large than 0.05 in the interval of slip flow regime. However, the local non similar method although is more precise and powerful than the local similarity approach, it remains approximate method because the partial differential equations after some level of approxima- tion truncation are finally reduced also to ordinary differential equations. This method is described and explained in Refs. [25, 26]. Martin and Boyd [28]

further studied and gave, for the first time, non-similar solutions of the momentum and energy equations for laminar boundary layer over an isothermal flat plate and for Falkner–Skan flow over a wedge [29] under slip boundary conditions. The impact of the magnetic field on the wall friction of the flat plate was also

studied in the rarefied flow regime [30]. The non- similar solutions have been obtained by transforming the physical coordinates (x, y) to dimensionless non- similar coordinates (K, g) and the resulting equations have been solved with the finite difference method by using a marching scheme. Unfortunately there seem to be some errors in their transformed equations and related results, as shown by Lahjomri and Oubarra [27]. The latter authors reconsidered and revised the problem by obtaining a correlations which give functional relations of overall skin friction coefficient and Nusselt number with the Reynolds number and the slip parameter. They show particularly that, for sufficiently large Reynolds number and unlike the previous studies, the slip parameter has practically no effect on the overall skin friction coefficient in the interval of slip flow regime. The effect of slip on skin friction appears only when the flow becomes more rarefied and they concluded that the slip parameter K must be treated as variable rather than constant.

On the other hand, it is well known that the boundary layers are by nature unstable flows. The study of their stability under the effect of the rarefaction at low pressure is an important problem in engineering. It may find great interest for example in the subsonic flow over an airplane or a space vehicle reentering the earth’s atmosphere. In consequence, it is crucial to predict the axial location or local Reynolds number along the flat plate for which the laminar flow becomes unstable, and eventually to predict the transition to turbulence. Usually, for practical applications one wishes to delay this transi- tion to reduce, for example, the heat transfer to the space vehicle.

The analysis of stability for boundary layer of Blasius continuum flow over a flat plate has been also extensively investigated in the past. Tollmien [31] and Schlichting [32] obtained approximately the neutral stability curve and showed that the viscosity can have a destabilizing effect on the disturbance waves of unidirectional flows as the Reynolds number (Re) increases. Later Schubauer and Skramstad [33]

validated experimentally the result predicted by the

theory of Tollmien–Schlichting and showed that

depending on the amplitude of excited waves, the

energy of the disturbances would either start decaying

at a certain downstream location or continue to grow

and cause a breakdown of the boundary layer leading

to turbulence. Jordinson [34] used space amplification

theory and replaced the Orr–Sommerfeld differential equation by a set of difference equations and the resulting algebraic model are expressed in matrix form. An iterative scheme was then used to find the eigenvalues of the matrix. This allows computing several eigenvalues modes but the accuracy is limited for the larger one. Mack [35] integrated the two decreasing solutions of the Orr–Sommerfeld stability equation from some large value of finite distance from the wall to attain the wall. He used Gramm–Scmidt orthonormalization procedure to prevent that the two solutions, which increase in the direction of the integration, become linearly dependent. However, it is shown that this integration process is cumbersome.

To avoid the repeated process of orthogonalization, Van Tijn and Van Vooren [36] improved the nu- merical methodology of Mack by using the method of order reduction and obtained an accurate value of the critical Reynolds number for which the flow becomes unstable. Latter, numerous investigations have been proposed to solve efficiently the Orr–Sommerfeld equation for Poiseuille and boundary layers flows using more sophisticated and accurate methods. These methods are known as spectral methods (See for example [37, 38]). A good literature review and details of principle of these methods with their applications to fluid dynamics can be found in innovative book of Canuto et al. [39].

Recall that the linear stability problem of boundary layer flow reveals some difficulties compared to the Poiseuille flow through a parallel plate channel or a pipe. This is due to the fact that one of the boundary conditions for boundary layer is unbounded, whereas both boundaries for channel are at finite distances.

Other difficulty encountered, that, contrary to the channel flow, velocity profile of the baseline Blasius boundary layer is not an exact solution of the Navier–

Stokes equations (e.g. Prandtl equations) and the parallel flow assumption is only an approximation for this flow, while this assumption is valid and justified for the Poiseuille flow at a sufficient large distance from the inlet section of the channel. In this context, Canuto et al. [39] pointed out that this approximation has proven to capture many, but not all, of the qualitative features of boundary layer stability, and often provides acceptable quantitative results.

Despite numerous studies of the stability for boundary layer over a flat plate, to our knowledge, no investigation is known to have studied the effect of

slip velocity on a stability of Blasius flow. The only work in the slip flow regime was reported recently by He and Wang [40] in the case of Poiseuille flow. These authors have shown that the effect of slip at the boundary is to increase the critical Reynolds number for instability.

The present study was motivated firstly by the work of Ref. [27] to obtain non-similarity solution for the pure forced convection of gaseous slip flow over a flat plate due to combined effects of the velocity slip and thermal jump boundary conditions in the presence of viscous dissipation which it was neglected in [27]. The non-linear partial differential equations of momentum and energy according to these boundary conditions are solved numerically with the symbolic algebra software Maple by taking into account of the variation of the stream function and temperature field with the slip parameter. The numerical method is based on centered implicit finite difference method of Crank–Nicolson.

The results show that the heating due to the viscous dissipation changes drastically the temperature distri- bution which leads to affect the heat transfer rate. This analysis provides an estimate of change in heat transfer due to rarefied-flow effects for a range of slip parameter and Eckert number for incompressible gases.

As supplementary contribution, the paper reports a linear stability analysis for boundary layer over a flat plate under slip flow regime by assuming as first approach a local similarity solution for the mean flow.

This assumption leads to the usual Orr–Sommerfeld equation which governs the perturbation stream func- tion satisfying slip boundary condition. This equation is solved efficiently by means of numerical code due to Driscoll et al. [41], which is essentially based on the spectral Chebyshev collocation method. The aim of this part of the present contribution is to determine the effect of slip parameter on the stability of the flow and to establish the corresponding curves of neutral stability.

2 Analysis

2.1 Dimensionless governing non-similar boundary layers equations under steady state and slip flow conditions

We consider a steady, laminar and two-dimensional

flow of a viscous nearly incompressible gas of

temperature T

with uniform velocity U

past a horizontal semi infinite flat plate at low Mach numbers (M \ 0.3). The plate is assumed to occupy the positive x-axis with the leading edge at origin and the positive y is distance normal to the wall. The plate is maintained at constant wall temperature T

. The present study is restricted to a flow with constant physical properties with no pressure gradients and that the velocity and temperature gradients in the x-direction are small compared to those of the y-direction. The buoyancy forces caused by temperature differences are assumed to be small compared with the inertia and friction forces. Thus, the natural convection is assumed to be negligible. The transversal fluid velocity normal to the wall will also be small compared to the fluid velocity parallel to it (v«u). Under these assumptions, which are consistent with the usual boundary layer ap- proximations [42] (Prandtl theory), the form of the boundary layer equations of continuity, momentum and energy is:

ou ox þ ov

oy ¼ 0 ð1Þ

u ou ox þ v ou

oy ¼ m o

u

oy

ð2Þ

u oT ox þ v oT

oy ¼ a

o

T oy

þ m

C

ou oy

2

ð3Þ The boundary conditions representing the charac- teristics of slip flow are velocity-slip and temperature- jump conditions at the gas-wall interface. For an isothermal wall, they can be expressed respectively as [1]

uj

¼ k 2 r

r

ou oy

ð4Þ

T j

¼ T

þ k Pr

2 r

r

2c c þ 1

oT oy

ð5Þ

uj

and Tj

are the local slip velocity and the fluid temperature at the wall, k is the molecular mean free path. c is the ratio of specific heats of gas, and Pr ¼ m=a

is the Prandtl number. r

and r

are the tangential momentum and thermal accommodation coefficients.

The appropriate boundary conditions for the prob- lem can be given by Eqs. (4) and (5), the conditions at

infinity (6), (9) and the impermeable condition at the wall surface (8):

uðx; 1Þ ! U

ð6Þ

uðx; 0Þ ¼ k 2 r

r

ou oy

ð7Þ

vðx; 0Þ ¼ 0 ð8Þ

T ðx; 1Þ ! T

ð9Þ

T ðx; 0Þ ¼ T

þ k Pr

2 r

r

2c c þ 1

oT oy

y¼0

ð10Þ Following the Refs. [27, 28] and in order to seek non-similar solutions, we introduce the following non- similar dimensionless variables:

g ¼ y

dðxÞ ; K ¼ 2 r

r

k

dðxÞ ¼ 2 r

r

k

x Re

ð11Þ and the dimensionless stream function and temperature

f ðK; gÞ ¼ wðx; yÞ dðxÞU

; HðK; gÞ ¼ T ðx; yÞ T

T

T

ð12Þ where dðxÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mx=U

p in (11) is the viscous length scale of the Blasius boundary layer, g is the pseudo- similarity variable and K can be interpreted as the modified boundary layer Knudsen number or the slip parameter. Re

¼ U

x=v is the local Reynolds num- ber. In Eq. (12), wðx; yÞ is the stream function defined as u ¼ ow=oy and v ¼ ow=ox. By using (11) and (12) we obtain the components of the dimensionless velocity:

u

ðK; gÞ ¼ u

U

¼ of ðK; gÞ

og ð13Þ

v

ðK; gÞ ¼ v ffiffiffiffiffiffiffi

mU1

x

q

¼ 1

2 g of ðK; gÞ

og f ðK; gÞ þ K of ðK; gÞ oK

ð14Þ

Substituting (11)–(14) into Eqs. (4) and (5) with

transformation of the independent variables (x, y) to

(K, g), we obtain the following system of non-linear

partial differential equations:

o

f og

þ f

2 o

f og

þ K

2 of og

o oK

of og of

oK o

f og

¼ 0 ð15Þ

o

H og

þ Prf

2 oH

og þ PrK 2

of og

oH oK of

oK oH

og

þ Ec Pr o

f og

¼ 0 ð16Þ

where Ec ¼ U

=C

ðT

T

Þ is the Eckert number which expresses the ratio of the kinetic energy to the enthalpy. With these new variables, the boundary conditions (6)–(10), in the (K, g) plane, now become of ðK; 1Þ

og ¼ 1 ð17Þ

of ðK; 0Þ

og ¼ K o

f ðK; 0Þ

og

ð18Þ

f ðK; 0Þ ¼ K of ðK; 0Þ

oK ð19Þ

HðK; 1Þ ¼ 0 ð20Þ

HðK; 0Þ ¼ 1 þ a K Pr

oHðK; 0Þ

og ð21Þ

where the physical constant a in (21) is defined in nomenclature.

It can be seen that the boundary conditions (18), (19) and (21) are explicitly dependent on K, so that, these equations suggest that the problem of convective heat transfer of rarefied gas flow with slip, does not possess self-similar solutions. Thus, the stream function f(K, g) and the temperature field H(K, g) must be a function of both variables g and K and they constitute non-similar solutions of the problem. When K is equal to zero or considered as constant, the solutions of the boundary layer equations become self-similar and the partial differential (15) and (16) reduce to ordinary differential equations.

To solve the Eqs. (15), (16), with consideration of slip parameter as variable, we need additional bound- ary conditions on K. This condition can be given for large values of K (K ? ?) at the leading edge (x = 0), which corresponds to highly rarefied or free- molecular flow, considered as initial guess of the solution. The basic assumption for theoretical free molecular flow calculations is that the flow of

molecules incident on a body is undisturbed by the presence of the body [43]. This will result in a uniform velocity and free stream temperature distribution with the ambient conditions:

u

ð1; gÞ ¼ 1 ð22Þ

f ð1; gÞ ¼ g ð23Þ

Hð1 ; gÞ ¼ 0 ð24Þ

Thus, for semi-infinite flat plate in rarefied gas stream, we can note that the flow starts as a free molecular type flow at the leading edge and finally develops into a continuum flow as the fluid moves far downstream. Therefore, to solve in the slip flow region, as first approximation, Eqs. (15) and (16) should be solved and extended to a large region of K, including the transitional and free molecular flow regions [27].

Once the solutions of the system (15)–(24) are determined, we can compute the local Nusselt number which is defined as:

Nu

¼ q

kðT

T

Þ=x ¼ Re

oH og

ð25Þ where q

¼ k

is the local heat flux which is given by Fourier’s law.

2.2 Linear stability analysis for local similar solution under slip flow condition assuming local parallel flow approximation

Before, beginning the analysis of the stability, we note that in the forced convection with constant viscosity, the hydrodynamical boundary layer problem for incompressible fluid is decoupled from the thermal problem. Thus, the temperature field has no influence on hydrodynamic boundary layer stability. In this study we shall focus only on the stability analysis of the hydrodynamical problem.

As in the continuum, we use here the local parallel

flow approximation to describe a stability of the flow

in the boundary layer under slip flow regime. For

sufficiently large Reynolds number, the base flow is

assumed to be essentially nearly parallel to the x axis

and is given by the local similar solution of Blasius

flow where the slip boundary condition has to be taken

into account. The validity of the parallel assumption is

however questionable and its justification needs to be

verified experimentally. Indeed, for continuum flow,

the comparison between the results on the neutral curves of stability obtained from the experiment of Schubauer and Skramstad [33] and the linear theory due to Tollmien–Schlichting [31, 32], shows that the results obtained are generally in good agreement for large Reynolds number, while a discrepancy is noted at low Reynolds number and high frequency. In this context, we note that Bertolotti et al. [44] have used the parabolized stability equations and direct numerical simutation method to investigate the effect of non-parallelism in the Blasius boundary layer. They show that the effect of non-parallelism is to be weak and they confirm that this effect is not responsible for the discrepancy between measurement and theoretical result at low Re. According to this theory, we can conclude that the local parallel flow assumption is a reasonable approximation and allows us giving some credibility of this assumption.

In this study one assumes the existence of small two dimensional perturbations around the local similar solution considered as the basic flow and one examines whether these perturbations are amplified or damped. Since these perturbations are assumed to be small, thus the vorticity equation of the Navier–

Stokes equations can be linearized around this base flow solution which is assumed to be solution of these equations for sufficienty large Re at the order of Prandtl’s boundary layer approximation. Let us de- compose the stream function into its base value noted W

plus small perturbation noted W: b

W ¼ W

ðx; yÞ þ e Wðx; b y; tÞ ð26Þ where e « 1 is an amplitude parameter. By injecting this decomposition into the vorticity equation, one obtains the following linearized partial differential equations which govern the perturbations field

o ot r

W b

þ oW

oy o ox r

W b

o W b ox

o

W

oy

þ o W b

oy r

o ox W

oW

ox r

o oy W b

mr

r

W b

¼ e o W b ox

o oy r

W b

o W b oy

o ox r

W b

" #

ð27Þ Now, by using the local parallel flow approximation which allows us to neglect, for the base flow, the axial derivative with respect to axial distance x compared with its transversal derivative y, thus Eq. (27) becomes:

o ot r

W b

þ oW

oy o ox r

W b

o W b ox

o

W

oy

mr

r

W b

¼ e o W b ox

o oy r

W b

o W b oy

o ox r

W b

" #

ð28Þ In the linear normal mode approach with time amplification theory, the terms multiplied by e in (28) are neglected and one considers perturbations in the form

wðx; b y; tÞ ¼ /ðyÞ b exp ½ i b aðx b ctÞ ð29Þ These are assumed to be periodic in space with b a is the wave number which is real and can grow or decay in time, while b c ¼ b c

þ i b c

is complex and i is the imaginary number. The quantity b c

is the wave propagation speed of the wave perturbation, whereas b

a b c

is its amplification factor, so that b c

\0 corre- sponds to a decaying perturbation, whereas insta- bility corresponds to b c

[ 0 with an amplified perturbation. The objective is to compute b c

as a function of b a and possibly, others parameters such as the Reynolds number, the slip parameter K and b c

. Thus, linear instability implies that a specified infinitesimal perturbation will grow exponentially.

Introducing (29) in (28), the result of linearization leads to the well known Orr–Sommerfeld (OS) equation:

ðu b cÞðb /

b a

/Þ b o

u oy

/ b

¼ m

i^ a ð / b

2 a ^

/ b

þ b a

/Þ b ð30Þ where u ¼ ow

=oy is the velocity component of the base flow. The dimensionless version of (30) with the associated boundary conditions can be obtained with respect to characteristic length and velocity scales defined previously by Eqs. (11), (12) and (17)–(18):

ðu

ðK; gÞ cÞð/

a

/Þ o

u

og

/

¼ 1 iaRe

ð/

2a

/

þ a

/Þ ð31Þ /ð0Þ ¼ 0; /

ð0Þ ¼ K/

ð0Þ; /ð1Þ ¼ 0; /

ð1Þ ¼ 0

ð32Þ

where the dimensionless variables and parameters are

defined as:

u

¼ of =og; a ¼ b adðxÞ ; c ¼ b c=U

; /

¼ / b .

dðxÞU

; Re

¼ U

dðxÞ

m ¼ ffiffiffiffiffiffiffi Re

p ð33Þ

with c ¼ c

þ ic

. The Reynolds number Re

in (31) may be related to the Reynolds number based on displacement thickness d, which is defined by Re

¼ bðKÞRe

, where bðKÞ ¼ d =d ¼ R

0

ð1 f 0Þdg is a function of the slip parameter K. In the particular case for no slip flow we have the familiar Blasius constant bð0Þ ¼ 1:72078765 [42]. The values of bðKÞ for various values of K are calculated and given on Table 4 (see Sect. 3.2.3 below).

We note, only in the part of the stability analysis, the modified boundary layer Knudsen number K is considered as a constant parameter. Within this approximation, the non-similar terms in Eqs. (15) and (17)–(19) are therefore neglected and consequently the basic flow u

¼ of =og in (30) is then assumed as locally self-similar and the solution is called ‘‘local similar solution’’. In this case, the equation and boundary conditions resulting from (15) and (17)–(19) which govern the dimensionless stream function f for the basic flow are then given by:

o

f og

þ f

2 o

f

og

¼ 0 with

f ð0Þ ¼ 0; f

ð0Þ ¼ Kf

ð0Þ; f

ð1Þ ¼ 1

ð34Þ

Returning now to Eq. (31), to find a non-trivial solution for / satisfying the boundary conditions (32) unless the parameters c, a, Re

and K must satisfy the characteristic eigenvalue relation in the form:

Fðc; a; Re

; KÞ ¼ 0 ð35Þ

It should be remarked at this point that Eq. (29) can be readily extended to include oblique waves. However, according to Squire’s theorem which asserts that the wave number vector of the most unstable disturbance was generally parallel to the flow, i.e. a two-dimen- sional disturbance. Therefore, in this work, we use this assumption and consider only two-dimensional waves.

3 Results and discussion

3.1 Main results of non-similar solutions 3.1.1 Numerical method

The numerical method used solves the system of partial differential Eqs. (15)–(16) with boundary

conditions (17)–(24). The solution which is called

‘‘non-similar solution’’ can be obtained by using centered finite difference method of Crank–Nicolson with the help of the symbolic algebra software Maple.

The principle of the algorithm has been described extensively in [27], so it will not be repeated here. The numerical code is robust and has the potential to solve efficiently both non-similar and self-similarity equa- tions (see Table 2 below).

As in [27] the algorithm is used with starting from large values of K (near the leading edge, x = 0) and marching the code in the flow direction with decreas- ing K until it approaches zero (continuum flow). The numerical experiment shows that, by running several numerical codes for any fixed value of Eckert number, the suitable values for the truncated boundary g

(g = ?), K

(K = ?) and the optimum of the size of the space step, Dg, and the K step, DK, are found to be g

¼ 10:5, K

¼ 50, Dg ¼ 1=16, DK ¼ 1=32. These values give a good compromise between the time of computation and the variation of the solution at any point and ensure grid independence (for more detail see Ref. [27] ). Practically, no effect on the velocity and temperature gradients at the wall has been observed by changing the values of K

from 50 to 100. This result is illustrated on the Table 1, which shows the effect of the truncated initial condition K

on the temperature gradient at the wall, oH=og

, for Ec = 1 for example. According to table, the maximum relative error of the temperature gradient between the cases K

¼ 50 and K

¼ 100, for instance, is less than 0.07 % for all values of K in the range considered.

3.1.2 Accuracy and validation

The accuracy of the numerical solution can be

estimated by computing the total local truncation

error estimate (the sum of the time (slip parameter)

and space errors) due to discretization of stream

function f and temperature field H for three different

selected grids (Dg ¼ 1=4, DK ¼ 1=4), (Dg ¼ 1=8,

DK ¼ 1=8) and (Dg ¼ 1=16, DK ¼ 1=38), and for any

fixed value of K. The results show that the local

truncation errors are improved as the grid sizes

become smaller. The numerical uncertainties in the

fine grid solution for stream function f eðf Þ and

temperature field H eðHÞ profiles should be

calculated with total errors less or equal to eðf Þ ¼ 0:3 % and eðHÞ ¼ 0:6 % (see Ref. [27] ). These two values indicate the robustness of numerical code used and the accuracy of the solution obtained.

To check the validity of the non-similar solution we calculated the values of the local Nusselt number in the case of very weak slip flow, i.e. K ¼ 10

for different values of Eckert number. The results are compared on Table 2 with the continuum flow solution (K = 0) reported by Schlichting [42], for which the expression of the local Nusselt number was given by: Nu

Re

¼ 0:332 ffiffiffiffiffi

3

Pr

p ½1 ð1=2Þ Ec bðPrÞ, where bðPrÞ ¼ 2C

ðT

T

Þ

U

is the dimensionless adiabatic wall temperature. For the case of air, the value of b is bð0:7Þ ¼ 0:835 [42]. It can be seen that our results agree very well with those reported by Ref. [42]. The maximum relative error of Nu

Re

is less than 0.8 % for all values of Eckert number. Thus, as the slip parameter approaches zero, the no slip condition and the classical self-similar solution of the boundary layer are recovered by non- similar solution for all values of Eckert number. In the absence of dissipation, i.e., Ec = 0 and for highly rarefied flow, the numerical results of non-similar solution match also the asymptotic analytical solution given by Lahjomri and Oubarra [27] for which the expression of Nusselt number, valid for K C 5, was shown equal to: Nu

Re

¼

exp

ffiffiffiffi erfc

Pr p

K a

. For example in the case of air Pr = 0.7, a ¼ 1:167, r

¼ r

, we found from numerical non similar solution Nu

Re

¼ 0:1019 for K = 5.

This value is in good agreement with the value of analytical solution 0.1028. Relative error is of order of 0.9 %. Thus, all these above agreements constitute a validity of numerical code.

3.1.3 Thermal characteristics results

It is clear that the Eckert number does not have any effect on the solution of the hydrodynamic boundary layer for pure forced convection. The results relative to the characteristics of hydrodynamic boundary layer obtained from non-similar solution have been pre- sented and discussed in great detail in [27]. Therefore we give here only the results relating to the thermal characteristics of the flow.

Figures 1, 2 and 3 show the dimensionless tem- perature profiles HðK; gÞ ¼ ðTðx; yÞ T

Þ=ðT

T

Þ as a function of g, for Pr = 0.7, c = 1.4 (air) and for various values of slip parameter, K ¼ 10

; 0:1 and 1 with Eckert number Ec ¼ U

=C

ðT

T

Þ in the range from -4 to 4. For Ec\0, the flat plate is colder than the free stream, while for Ec [ 0 the plate is hotter. It is observed that for fixed value of K, the increasing values of Ec is to increase the temperature in the boundary layer. This is because to the heat energy stored in the gas due to the frictional heating. The thickness of the thermal boundary layer also increases in magnitude as Ec increases for all fixed value of slip parameter K. It is seen from Figs. 1 and 2, that for K ¼ 10

, K = 0.1 and for ðEc [ 2:395Þ (value corresponding to an adiabatic wall in the case of continuum regime [42] ), the boundary layer near the wall is warmer than the wall itself owing to the production of frictional heat and the local gas temperature exceeds the wall temperature (T ðx; yÞ [ T

[ T

). In such cases the hot wall ceases to be cooled by the stream of cooled air. Thus, near the wall, the heat flux is reversed and proceeds from the gas to the wall in spite of the fact that we heat the stream of cooler air. By increasing K to a value of 1 (Fig. 3), the flow becomes more rarefied and the Table 1 The effect of the truncated initial condition K

on the temperature gradient at the wall calculated from non similar solution for Ec = 1 in the range: K 2 ð10

5Þ

K

25 50 75 100

K = 10

0.1703272426 0.1703272680 0.1703272726 0.1703272743

K = 0.01 0.1711409695 0.1711409854 0.1711409884 0.1711409894

K = 0.1 0.1791119764 0.1791108649 0.1791106592 0.1791105871

K = 1 0.2286078958 0.2284799703 0.2284563164 0.2284480411

K = 2 0.1875929767 0.1873645183 0.1873223296 0.1873075740

K = 5 0.1010429121 0.1007570513 0.1007046665 0.1006863739

heating action of frictional heat is considerably reduced resulting to a decrease of the temperature within the boundary layer, because of the effects of slip velocity and temperature jump at the wall which increase as K increases. The association of these two effects with the stream of the cooled air dominates by

preventing the heating provided by the friction.

Consequently, the peak value observed for the temperature for Ec = 3 and 4 in Figs. 1 and 2, disappears completely as the flow becomes more rarefied for K = 1. In this case the maximum of the temperature occurs at the surface of the plate. On the other hand, a readily visible inflection point occurs for Table 2 Comparison of the local Nusselt number obtained by our non-similar solution for K = 10

with those reported by Schlichting [42] in continuum flow (K = 0), in the case of air, Pr = 0.7 with b(0.7) = 0.835

Ec Nu

Re

Relative errors %

Non-similarity solution (this work) K = 10

Nu

Re

¼ 0:332 ffiffiffiffiffi

3

Pr

p ½1 ð1=2Þ Ec bðPrÞ Schlichting [42] K = 0

-4 0.782153408 0.7870736 0.6

-3 0.659788180 0.6640012 0.6

-2 0.537422952 0.5409288 0.6

-1 0.415057724 0.4178565 0.7

-0.5 0.353875110 0.3563203 0.7

0 0.292692496 0.2947841 0.7

0.5 0.231509882 0.2332479 0.7

1 0.1703272680 0.1717117 0.8

2 0.047962039 0.0483393 0.8

3 -0.074403188 -0.0744330 0.04

4 -0.196768416 -0.1975054 0.4

0 1 2 3 4 5 6 7 8

η

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Θ(Κ,η)

Pr=0.7, γ=1.4, σ

=σ

K=0.00001

Ec= -4, -3, -2, -1, 0, 1, 2, 2.395, 3, 4

Fig. 1 Effect of Eckert number on the temperature distribution in the thermal boundary layer for very weak slip flow K = 10

(continuum flow)

0 1 2 3 4 5 6 7 8

η

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Θ(Κ,η)

Pr=0.7, γ=1.4, σ

=σ

K=0.1

Ec= -4, -3, -2, -1, 0, 1, 2, 2.395, 3, 4

Fig. 2 Effect of Eckert number on the temperature distribution

in the thermal boundary layer for K = 0.1

all values of K, for the curves Ec = -4 for which T

[ T

. In this case and at a large distance from the wall the temperature inside the thermal boundary layer becomes larger than the free stream temperature T ðx; yÞ [ T

. This is due to the effect of the frictional heating. This distance is reduced when the flow becomes more rarefied.

Figure 4 indicates the effect of slip parameter K on the gas temperature at the wall HðK; 0Þ for different values of Eckert number Ec. It can be seen that the temperature, as the velocity field [27], is dependent of the distance along the plate due to a non similarity and increases in magnitude as Ec increases. For Ec\2:395 with increasing K, the gas temperature at the wall decreases monotonically with K. This diminution is due to the effect of the temperature jump which dominates the increase caused by the frictional heating. For Ec 2:395 and for slightly rarefied flows, the gas temperature at the wall present a maximum for a certain value of K denoted K

followed by a large decrease as the flow becomes more rarefied when K is greater than K

. This maximum is due to the compe- tition of the effects of temperature jump and frictional heating. This value K

increases from 0 to 0.25 as Eckert number increases from 2.395 to 4.

Figure 5 shows the temperature gradient at the wall as a function of K for different value of Ec. The increase in Eckert number from -4 to 4 decreases the

Nusselt number in the slight flow regime. The reduction in heat transfer is due to the increase of thickness of thermal boundary layer. Thus, the local heat transfer varies significantly with Eckert number due to the effect viscous dissipation. For zero frictional heat (Ec = 0) and for small value of Ec ( j Ec j\0:8), the slip parameter has practically no effect on the local heat transfer in the region of continuum and in the slip flow regime. This is due to equilibrium between the associations of two different effects which act in opposite sense: the slip velocity and the temperature jump effects. The slip velocity acts to increase the heat transfer, while the temperature jump tends to decrease it. The heat transfer at the wall in highly rarefied flow will decrease due to the effect of temperature jump. For a hot flat plate and for large values of Eckert number ðEc [ 0:8Þ, the local heat transfer shows a slight increase in the slip flow region for any fixed value of Ec. In this regime, the local heat transfer is dominated by the effect of slip velocity which enhances with further increase of slip parameter K leading to an increase of the local Nusselt number.

When the flow becomes more rarefied, the local heat transfer shows a large increase followed by a decrease (a maximum) in the transitional flow region. This maximum is depending on the value of Eckert number and is due to the competition between the effects of slip velocity and the temperature jump. The value of

0 1 2 3 4 5 6 7 8

η

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Θ(Κ,η)

Pr=0.7, γ=1.4, σ

=σ

K=1

Ec= -4, -3, -2, -1, 0, 1, 2, 2.395, 3, 4

Fig. 3 Effect of Eckert number on the temperature distribution in the thermal boundary layer for K = 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

K

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Θ (Κ ,0)

Ec=-4, -3, -2, -1, -0.8, 0, 0.1, 0.8, 1, 2, 2.3952, 3, 4

Fig. 4 Effect of slip parameter K on the gas temperature at the

wall for different values of Eckert number Ec

Ec = 2.395 for which the temperature gradient at the wall vanishes in the continuum regime is maintained until a certain local distance along the plate corre- sponding to a value of K approximately equal to K ¼ K

¼ 0:03 in the slip flow regime. Beyond this value, the Nusselt number becomes positive. This value K

is depending on the value of Eckert number and it is seen to increase with increasing Ec. For example, for Ec = 3 and Ec = 4 the temperature gradient vanishes locally respectively at approximately K

¼ 0:3 and K

¼ 0:5. When Ec is higher than a value of 2.395, the local heat transfer is reversed and local Nusselt number becomes negative for K\K

leading to the heating of the flow. In this case, the temperature of the flow exceeds the wall temperature, see Fig. 1. For a flat plate colder than the free stream with ðEc\ 0:8Þ, the local heat transfer exhibits a slight decrease in the interval of slip flow regime and a large decrease is observed as the flow becomes more rarefied. This attenuation is due to the effect of the temperature jump at the wall which dominates the increase caused by the effect of the slip velocity (see Fig. 4). We can also note that for highly rarefied flow the heat transfer becomes independent of Eckert number. This result can be explained by the reduction of the velocity gradient at large slip parameter which reduces considerably the effect of viscous dissipation.

In this case the heat transfer is controlled only by the

effect of temperature jump at the wall which is dominant, resulting to a decrease of Nusselt number.

3.2 Main results of linear stability calculations for local similar solution

3.2.1 Numerical solution for the Orr–Sommerfeld equation via spectral Chebyshev collocation method

Recall that the Orr–Sommerfeld equation and the associated eigenvalues have been traditionally solved by means of shooting methods, finite difference methods and Chebyshev collocation methods. In this study, in order to solve efficiently this difficult eigenvalue problem, we have chosen to use an available optimized solver ‘Chebfun’ and ‘Chebop’

developed by the numerical analysis team of Oxford University, Driscoll, Bornemann, and Trefethen [41].

Chebfun is an open-source package for computing functions and ordinary differential equations to 15- digit accuracy and is essentially based on Chebyshev discretization for representing functions defined on finite, semi-infinite or doubly-infinite intervals and most commands are overloads of MATLAB com- mands. The advantage of Chebfun is its capacity to solve accurately and efficiently with an adaptive mesh on any size domain with a variety of boundary conditions. Chebop is a system which allows construct differential and integral operators that can be applied to Chebfun for solving for example the Orr–Sommer- feld equation. For more details on these solvers the reader could refer to [45]. Once the associated operators corresponding to (OS) equation given by Eq. (31) are defined, they can be implemented in the form of generalized eigenvalue problem:

A/ ¼ k B / ð36Þ

where the two operators A and B and the eigenvalues k are defined respectively as:

A ¼ B

Re i a u

B o

u

og

; B ¼ D

a

I and k ¼ i a c

ð37Þ I denotes the matrix identity. Then the eigenvalues and the corresponding eigenfunctions could be deter- mined by means of command ‘eigs’ function in

1E-005 0.0001 0.001 0.01 0.1 1

K

0 0.4 0.8 1.2

-0.2 0.2 0.6 1

Nu

(R e

)

1 0.1

0

2.3952 0.8

Pr=0.7, σ

=σ

, γ=1.4 PARAMETER: Ec

-1 -0.1 -0.8

2

4 -3 -4

-2

3

Continuum Flow Slip Flow Transitional Flow

Fig. 5 Effects of the Eckert number and slip parameter on the

local heat transfer

MATLAB (see Ref. [45] for more details). In this study, we use a simple domain truncation which consists to approximate the solution of the base flow and (OS) equation on the interval ½0; g

instead

½0; 1. However, to satisfy the boundary conditions at infinity requires that the integration domain to be large to obtain an accurate result. The numerical experiment shows that, by running several numerical codes with g

ranging from 10 to 100, we found that g

= 50 is an optimum value. The tests are performed by examining the effect of g

on the eigenvalues. It is found that these eigenvalues are not sensitive to g

when it becomes larger than 50. Another important numerical aspect to obtain an accurate result for any stability computation is that the base flow velocity u

and its second derivative o

u

=og

must be calculated with very high accuracy. Therefore, to avoid nu- merical errors of interpolation for calculating the base flow, the system of Eqs. (34) and (31)–(32) have been simultaneously solved via spectral method by using Chebfun and Chebops system.

3.2.2 Validation of the results

The validity of the numerical solution could be first checked, in the classical case of continuum flow, by comparing the predicted eigenvalues for the unstable mode at Re

¼ 998, a

¼ a d ^

¼ 0:308 with the accurate result of Van Stijn and Van de Vooren [36]

c ¼ 0:36412129 þ 0:00796250 i. The value found by Table 3 First five discrete eigenvalues of the Blasius boundary layer for no slip flow (K = 0) for different values of Reynolds number at a = 0.179 and comparison with those reported by Mack [35] and Theofilis [37]

Mode Mack [35] Theofilis [37] Present results

c

c

c

c

c

c

Re = 580

1 0.3641 0.0080 0.3641212880 0.0079625034 0.3641228731 0.0079597165

2 0.2897 -0.2769 0.2897243201 -0.2768738567 0.2896960170 -0.2768821341

3 0.4839 -0.1921 0.4839439094 -0.192082405 0.4839272245 -0.1920497228

4 0.5572 -0.3653 0.5572212338 -0.3653515279 0.5572193529 -0.3653315571

5 0.6862 -0.3307 0.6862882375 -0.3307860195 0.6862546351 -0.3307943397

Re = 1000

1 0.3383 0.0048 0.3382665514 0.004840916 0.3382705819 0.00483502813

2 0.2408 -0.2391 0.2408009022 -0.2391381114 0.2407974425 -0.2391346579

3 0.4155 -0.1425 0.4155502915 -0.142542988 0.4155450530 -0.1425333932

4 0.4551 -0.3187 0.4550822377 -0.3187692463 0.4550745941 -0.3187742200

5 0.5773 -0.2730 0.5773537158 -0.2731113135 0.5773498558 -0.2730494451

Re = 2000

1 0.3089 -0.0166 0.3089195489 -0.0165507607 0.3089806711 -0.0165195492

2 0.1918 -0.1961 0.1917774677 -0.1961128264 0.1917102725 -0.1961233195

3 0.3425 -0.0816 0.3425614237 -0.0816631466 0.3425436600 -0.0816470091

4 0.3553 -0.2648 0.3551064431 -0.2651917636 0.3552141782 -0.2649720387

5 0.4651 -0.2079 0.466328690 -0.2057860642 0.4652598712 -0.2080437520

Re = 5000

1 0.3283 -0.0294 0.3284006118 -0.0291462801 0.3282766097 -0.0293844491

2 0.1429 -0.1484 0.1428812040 -0.1483746767 0.1428569311 -0.1483558377

3 0.2174 -0.0456 0.2173307836 -0.0456895940 0.2173264272 -0.0456559058

4 0.2603 -0.2037 0.2649475038 -0.2080391496 0.2599802031 -0.2040507116

5 0.3471 -0.1367 0.320835917 -0.1292654871 0.3472303928 -0.1369454813

our calculation is c ¼ 0:364122946 þ 0:007958904 i.

It is clear that the agreement between both both calculations is excellent. The relative error is less than 10

for c

and less than 10

for c

and indicates the robustness of numerical code used. Table 3 shows also a comparison of our results with those obtained by by Mack [35] and Theofilis [37], in the continuum regime (K = 0), for the first five modes of discrete eigenval- ues, for the dimensionless wave number a = 0.179 and different values of Reynolds number Re

. These authors have used Gram–Scmidt orthonormalization procedure and a spectral collocation method respec- tively for solving (OS) equation. It can be seen again that the agreement between both calculations is excellent for all values of Reynolds. All of these agreements constitute a validation of the numerical results.

3.2.3 The neutral stability curves in the continuum and slip flow regime

The main information we want to derive from characteristic relation (35) is, given the value of the slip parameter K, what is the lowest value of Re

or Re

, critical Reynolds number noted Re

for which the flow becomes unstable, i.e. when c

¼ 0.

The main objective, what is the effect of slip parameter on the stability of the flow. Figure 6 shows the curves of neutral stability (c

¼ 0), for different values of the slip parameter K = 0, 0.1, 0.25, 0.5 and 0.7, obtained from numerical solution of the Orr–Sommerfeld equation coupled with bisection method. The symbols points shown on the different curves for various values of slip parameter represent the values of numerical calculations. For a given flow and fixed values of K the ðRe

; aÞ plane is divided into regions where c

is negative and regions where c

is positive. These regions are separated by the curves c

ða; Re

Þ ¼ 0, shown on Fig. 6, which are called the neutral stability curves and represent the bounds between amplified and damped perturbations. The critical values of Reynolds number and the corresponding wave length number may be deduced from these curves for each values of K. For no slip flow (K = 0), for example, the values found for critical Reynolds number is about of Re

¼ 302:06 (or Re

crit: ¼ 519:78). This value agree very well with both those obtained by Jordinson [34] i.e. Re

crit: ¼ 520 and of Van Stijn and Van de Vooren [36] Re

crit: ¼ 519:06. Below this critical value of Reynolds number the boundary layer solution is stable for each value of K. Above this value, some waves are unstable (Tollmien–Schlichting waves) and will therefore be amplified, which will eventually lead to the transition to turbulence. As the slip parameter increases, it can be seen that the neutral stability curves are strongly affected by the rarefaction. The

Table 4 Effect of slip parameter on the critical Reynolds number, the corresponding wave number and the wave propagation speed

K b(K) Re

Re

crit: a

¼ b a d c

0 1.72078765497625 302.06 519.781 0.1781 0.3951987301379359

0.1 1.62419210249600 348 565.219 0.16433 0. 3879079100816214

0.25 1.49187164500601 552.9 824.856 0.15118 0. 3774143702350947

0.5 1.30273851378879 1573.91 2050.393 0.1175 0.3667471726868981

0.7 1.17641291895026 4050.16 4764.661 0.0950 0.3712761527157454

100000 10000

1000 100

Re

=√

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26

α

K=0 (no slip flow) K=0.1 K=0.25 K=0.5 K=0.7

Fig. 6 Neutral stability curves in continuum and in the slip flow

regime

critical Reynolds number becomes larger than the value of continuum despite the fact that the boundary layer thickness d and displacement thickness d

decrease with the increase of slip parameter K (see Table 4 below). For example, by increasing K from the value 0 to 0.1, leads to an increase of about 15 % on Re

, however, the most unstable wave numbers a

decreases by 5 %. This means that the slip parameter has a stabilizing effect on the Blasius boundary layer due to the effect of slip velocity at the wall. It should be noted that these results are in line and agree qualitatively with those obtained recently by He and Wang [40] in the case of Poiseuille flow with slip boundary conditions.

Table 4 shows the effect of slip parameter on the ratio of displacement thickness to boundary layer thickness bðKÞ ¼ d

=d, and on the critical values of the corresponding Reynolds numbers Re

, Re

Crit , wave number a and the real part of the eigenvalues c

. It can be seen from this table that an increase of slip parameter K leads to an increase of critical Reynolds number and in a decrease of a.

Hence, the Blasius boundary layer velocity profile becomes more stable under rarefied flow regime.

4 Conclusion

In this study, a non-similar solution of the boundary layers equations of laminar incompressible gaseous slip flow over a flat plate at relatively low Mach number is obtained by taking into account of the viscous dissipation. Furthermore, a linear stability analysis is carried out to study the effect of slip parameter on the stability of local similar baseline flow solution. Our mail results and the corresponding conclusions can be summarized as follows:

For small fixed value of Eckert number such that j Ec j\0:8, the slip parameter does not have significant effect on the local heat transfer in the interval of continuum and slip flow regime. The effect of slip appears only when the flow becomes more rarefied.

For a flat plate warmer than the free stream and for large values of Eckert numbers ðEc [ 0:8Þ, the local heat transfer exhibits a slight increase in the interval of slip flow regime and a maximum in the transitional flow region due to the competition of the effects of slip velocity and the temperature jump.

For a value of Eckert number large than 2.395 and when the flow is slightly rarefied, an inverse in heat transfer will appear leading to the overheating of the flow near the wall.

For a flat plate colder than the free stream with ðEc\ 0:8Þ, the local heat transfer is dominated by the effect of temperature jump and undergoes a decrease when the flow becomes more rarefied for any fixed value of Eckert number.

For a slightly rarefied flow, the local heat transfer decreases with the increase of the Eckert number.

For highly rarefied flow, the effect of viscous dissipation is more reduced even for large values of Eckert number and the heat transfer becomes inde- pendent of this parameter.

On the basis of the numerical accurate results obtained from the neutral curves of stability in the slip flow regime which are validated with the existing works in the limiting continuum case. The results show that the effect of slip parameter lead to increase the critical Reynolds numbers for instability and to decrease the most unstable wave numbers. It is concluded that the rarefaction has a stabilizing effect on the Blasius flow and suggests that the transition to turbulence could be delayed in the slip flow regime.

It should be noted that, even though the local parallel flow assumption is non-rational, the analysis developed here has the capacity to give some insight about the physical phenomena of boundary layer stability with quantitative results that are acceptable to engineering accuracy. However, under rarefied regime, as we have seen previously in Sect. 2.2, the flow is certainly non-parallel within the boundary layer and the solution of the base flow is indeed non- similar. Thus, as perspective, it is desirable to consider as the base flow the solution of non-similarity instead of local similar solution. This could be performed by parabolzing the full system given by the partial differential Eqs. (27) and (15) instead to use the Orr–Sommerfeld differential equation. This approach is known as parabolized stability equations (PSE) developed for incompressible flow by Herbert [46] and Bertolotti [44] and discussed by Canuto et al. [39]. The advantage of this method is that permits the stability analysis of any base flow that depends upon both the streamwise and wall normal directions.

Acknowledgments The authors wish to thank the numerical

analysis team of Oxford University for providing us their

numerical solvers Chebfun and Chebop. We would also like to thank the anonymous Reviewers whose insightful comments have considerably improved the manuscript.

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