Effects of parietal suction and injection on the stability of the Blasius boundary-layer flow over a permeable, heated plate

The main concern of this paper is to investigate the effects on the stability behavior of wall suction or injection for external boundary-layer flow over a heated, porous plate for a fluid with temperature-dependent viscosity. The wall suction or injection are applied to the flow by a simple modification for the no-penetration condition and the current boundary conditions on the flat plate. Liquid-type viscosities are found to entrain both the velocity and temperature profiles closer to the plate with increasing both temperature sensitivity and suction intensity, whereas gas-type viscosities are found to exhibit the reverse effect with increasing flow injection and decreasing temperature dependence. We present then the linear stability analysis and find that increasing both the temperature dependence (from gas- to liquid-type behavior) and suction intensity of the fluid leads to increasing critical Reynolds number to a point of maximum stability. We note also that increasing both the Prandtl number (Pr) and flow suction in the liquid-type behavior results in an increased critical Reynolds number. The magnitudes of the perturbation eigenfunctions are considered before utilising them to obtain solutions to an energy balance integral. We find that the eigenfunction profiles are imitative of the narrowing of their mean flow counterparts when increasing either the temperature dependence or the flow suction. Our results are then confirmed by the energy analysis, where we find a significant reduction in the energy produced by the disturbance with increasing suction intensity and ultimately leads to a more stable flow. Overall, there is a strong destabilizing effect with increasing injection and the temperature-dependent viscosity over a heated plate. In summary, the findings indicate that increasing the wall suction and temperature dependence results in significantly more stable flows. It is worth noting that application and extension of this study are considered in the context of chemical vapor deposition reactors.

Figure 1

Schematic diagram of the physical model. Wall suction $(\downarrow )$ or injection $(\uparrow )$. Adapted from Laouar and Mezaache [17], (1) wall with suction or injection, (2) dynamic laminar/turbulent boundary layer, and (3) thermal boundary layer

Figure 2

Mean flow profiles, viscosity profiles $\overline{\mu }\left(\eta \right)$, in the cases of various suction and injection, and a range of temperature dependencies.

Figure 3

Mean flow profiles, viscosity profiles $\overline{\mu }\left(\eta \right)$. (a) $f_w=0$ and a range of $\varepsilon$, (b) $f_w=0.4$ and a range of $\varepsilon$, and (c) $f_w=0.8$ and a range of $\varepsilon$.

Figure 4

Mean flow profiles, viscosity profiles $\overline{\mu }\left(\eta \right)$. (a) $f_w=0$, and a range of $\varepsilon$. (b) $f_w=-0.4$, and a range of $\varepsilon$. (c) $f_w=-0.8$, and a range of $\varepsilon$.

Figure 5

Mean flow profiles, in cases of various suction and injection, and fixed $\varepsilon =-0.75$. (a) streamwise velocity, (b) velocity gradient, and (c) temperature.

Figure 6

Mean flow profiles, in cases of various suction and injection, and fixed $\varepsilon =0$. (a) Streamwise velocity, (b) velocity gradient, and (c) temperature.

Figure 7

Mean flow profiles, in cases of various suction and injection, and fixed $\varepsilon =0.75$. (a) Streamwise velocity, (b) velocity gradient, and (c) temperature.

Figure 8

Mean flow profiles of a range of Prandtl number values, and at a fixed value of $\varepsilon =0.25$, and fixed values of $f_w=-0.4,0.4$. (a) Streamwise velocity for injection, (b) temperature profile for injection, (c) axial velocity for suction, and (d) temperature for suction.

Figure 9

Neural curves, in cases of various suction and fixed $\varepsilon =-0.75,0,0.75$. (a) Neutral curves of suction at $\varepsilon =-0.75$. (b) Neutral curves of suction at $\varepsilon =0$. (c) Neutral curves of suction at $\varepsilon =0.75$.

Figure 10

Neural curves, in cases of a range of Prandtl number. (a) Neutral curves of Prandtl number at a fixed $f_w=-0.4$ (injection) and $\varepsilon =0.25$. (b) Neutral curves of Prandtl number at fixed $f_w=0.4$ (suction) and $\varepsilon =0.25$.

Figure 11

Perturbation eigenfunctions for a range of temperature-dependent viscosity with fixed injection $f_w=-0.4$. (a) $|\tilde{u}|$ streamwise eigenfunctions. (b) $|\tilde{v}|$ plate normal eigenfunctions. (c) $|\tilde{\mathrm{\Theta }}|$ temperature eigenfunctions.

Figure 12

Perturbation eigenfunctions for a range of temperature-dependent viscosity with fixed suction $f_w=0.4$. (a) $|\tilde{u}|$ streamwise eigenfunctions. (b) $|\tilde{v}|$ plate normal eigenfunctions. (c) $|\tilde{\mathrm{\Theta }}|$ temperature eigenfunctions.

Figure 13

(a) Component energy contributions and (b) total and component energy dissipation due to different suction values for a range of temperature dependencies. The vertical and horizontal black, dashed lines are provided as reference to $\varepsilon =0$ and to zero energy contribution, respectively.

Figure 14

(a) Component energy contributions and (b) total and component energy dissipation due to different blowing values for a range of temperature dependencies. The vertical and horizontal black, dashed lines are provided as reference to $\varepsilon =0$ and to zero energy contribution, respectively.

Figure 15

Results of the energy balance integral showing the energy contribution of the individual components for a range of suction intensities with fixed $\varepsilon$. $D_1$ and $D_2$ are the first and second terms of $V_d$.

Figure 16

Results of the energy balance integral showing the energy contribution of the individual components for a range of temperature dependencies with fixed injection $f_w$. $D_1$ and $D_2$ are the first and second terms of $V_d$.