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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

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Abstract

Forced convection of laminar nearly incompressible gaseous slip flow over an isothermal flat plate at low Mach number with viscous dissipation is considered. The non-similar solutions of hydrodynamical 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 parameter and the Eckert number on the heat transfer characteristics are presented graphically and discussed. 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 function 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 corresponding wave numbers are found in excellent agreement 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.

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Abbreviations

a :

Parameter appears in Eq. (21), \( a = \frac{{2 - \sigma_{T} }}{{\sigma_{T} }}\,\frac{{\sigma_{M} }}{{2 - \sigma_{M} }}\,\frac{2\gamma }{\gamma + 1} \)

b :

Dimensionless adiabatic wall temperature

\( \widehat{c} \) :

Complex wave velocity, \( \widehat{c} = \widehat{c}_{r} + i\,\widehat{c}_{i} \), Eq. (29)

\( c \) :

Dimensionless complex wave velocity, \( c = c_{r} + i\,c_{i} \), Eq. (33)

\( Cp \) :

Specific heat at constant pressure (J Kg−1 K−1)

Ec :

Eckert number, \( Ec = U_{\infty }^{2} /C_{p} (T_{w} - T_{\infty } ) \)

\( f \) :

Dimensionless stream function, Eq. (12)

h :

Heat transfer coefficient (W m−2 K−1)

k :

Thermal conductivity (W m−1 K−1)

K :

Local slip parameter, \( K = \lambda \frac{{2 - \sigma_{M} }}{{\sigma_{M} }}\sqrt {\frac{{U_{\infty } }}{\nu x}} \)

L :

Length of the plate (m)

\( Nu_{x} \) :

Local Nusselt number, Eq. (25)

Pr :

Prandtl number, \( Pr = \nu /\alpha \)

\( Re_{x} \) :

Local Reynolds number, \( Re = U_{\infty } x/\nu \)

\( Re_{\delta } \) :

Local Reynolds number, \( Re_{\delta } = U_{\infty } \delta (x)/\nu = \sqrt {Re_{x} } \)

\( Re_{{\delta}^*} \) :

Local Reynolds number, \( Re_{{\delta}^*} = U_{\infty } \delta^{*} (x)/\nu \)

\( Tw \) :

Temperature at the surface of the plate, wall temperature (K)

\( \left. T \right|_{w} \) :

Fluid temperature at the wall, Eq. (5) (K)

T:

Temperature of the fluid (K)

\( T_{\infty } \) :

Temperature of the ambient fluid (K)

\( \left. u \right|_{w} \) :

The local wall slip velocity, Eq. (4) (m s−1)

\( u^{*} \) :

The dimensionless x-component of the velocity, Eq. (13)

\( u \) :

The x-component of the velocity (m s−1)

\( U_{\infty } \) :

Free stream velocity (m s−1)

\( v^{*} \) :

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

v :

The y-component of the velocity (m s−1)

x, y :

Distance along and normal to the wedge (m)

\( \alpha_{t} \) :

Thermal diffusivity, \( \alpha_{t} = k/\rho C_{p} \) (m2 s−1)

\( \widehat{\alpha } \) :

Wave number (m−1)

α :

Dimensionless wave number, Eq. (33)

β :

Ratio of boundary layers thickness, \( \beta = \delta^{*} /\delta = \int_{0}^{\infty } {(1 - f\prime )d\eta } \)

δ :

Boundary layer thickness

\( \delta^{*} \) :

Displacement thickness

γ :

Ratio of specific heats, \( \gamma = C_{p} /C_{v} \)

ϕ :

Dimensionless complex amplitude of the perturbation of stream function Eq. (33)

σ :

Accommodation coefficient

λ :

Mean free path (m)

ψ :

Stream function (m2 s−1)

\( \widehat{\psi } \) :

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

\( \varPsi_{B} \) :

Base flow of stream function (m2 s−1)

η :

Similarity variable

Θ :

Dimensionless fluid temperature, Eq. (12)

ν :

Kinematic viscosity (m2 s−1)

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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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  1. Laboratory of Mechanics, Faculty of Sciences Ain Chock, Hassan II University, Maarif, B.P. 5366, 20100, Casablanca, Morocco

    Elhoucine Essaghir, Youssef Haddout, Abdelaziz Oubarra & Jawad Lahjomri

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  1. Elhoucine Essaghir
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  2. Youssef Haddout
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  3. Abdelaziz Oubarra
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Correspondence to Elhoucine Essaghir.

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Essaghir, E., Haddout, Y., Oubarra, A. et al. 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. Meccanica 51, 99–115 (2016). https://doi.org/10.1007/s11012-015-0204-2

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