Stability of the Blasius boundary layer over a heated plate in a temperature-dependent viscosity flow

We present complementary numerical and asymptotic studies of the flow over a heated, semi-infinite flat plate for a fluid with temperature-dependent viscosity. Liquid-type viscosities are found to entrain both the velocity and temperature profiles closer to the plate with increasing temperature sensitivity; gas-type viscosities are found to exhibit the reverse effect. A linear stability analysis is presented and we find that increasing the temperature dependence of the fluid (from gas- to liquid-type behavior) results in an increased critical Reynolds number to a point of maximum stability. Using an energy-balance approach, we determine that this behavior is primarily driven by the inviscid instability of the modified steady flow, rather than being a result of modified viscous instability effects. Application and extension of the results are considered in the context of chemical vapor deposition.

Figure 1

Sketch of a horizontal CVD reactor.

Figure 2

(a) Mean flow velocity profiles and (b) velocity gradient profiles for $\varepsilon =-0.75$ to 0.75, with each plot representing an increment of 0.25. The temperature-independent ($\varepsilon =0$) solution is represented by the black curve. The solutions have been truncated to $\eta =10$.

Figure 3

(a) Mean flow temperature profiles and (b) mean flow viscosity profiles for $\varepsilon =-0.75$ to 0.75, with each plot representing an increment of 0.25. The temperature-independent ($\varepsilon =0$) solution is represented by the black curve. The solutions have been truncated to $\eta =10$.

Figure 4

Schematic diagram showing the lower branch structure of the variable viscosity boundary layer flow considered here. The zones $\mathrm{I}$, $\mathrm{II}$, and $\mathrm{III}$ denote the upper, main and lower decks respectively. The shaded area indicates the boundary layer region whereas the unshaded area indicates the inviscid region where the base flow matches with that of the free stream. Adapted from Griffiths et al. [26].

Figure 5

Asymptotic results for the neutrally stable lower-branch mode of $\varepsilon =-0.75$ to 0.75, with each line separated by an increment of 0.25. The frequency of the disturbance $F$ is plotted against the Reynolds number $R$ on a log-log scale.

Figure 6

Neutral stability curves for $\varepsilon =-0.75$ to 0.75, with each curve separated by an increment of 0.25. Here, $F=\omega /R$ is the scaled disturbance frequency, and the axis is scaled logarithmically. The regions enclosed by the curves indicate Reynolds numbers and frequencies for which disturbances create instability in the flow.

Figure 7

Agreement between the asymptotic theory and the numerical results. The Newtonian solutions are presented in black and the case when $\varepsilon =-0.28$ (and hence $\text{Pr}=0.72$ in the asymptotic framework) is presented in blue. The large Reynolds number asymptotic results are given by the dashed curves.

Figure 8

The resulting critical Reynolds number, $R_c$, produced for a range of $\varepsilon$ values. The black lines are provided as reference to the temperature-independent ($\varepsilon =0$) results.

Figure 9

Perturbation stream function profiles for $\varepsilon =-0.75$ to 0.75, with each plot representing an increment of 0.25. The temperature-independent ($\varepsilon =0$) solution is represented by the black curve and all profiles are normalized on the maximum of this profile. The solutions have been truncated to $y=10$.

Figure 10

Streamwise eigenfunction profiles for $\varepsilon =-0.75$ to 0.75, with each plot representing an increment of 0.25. The temperature-independent ($\varepsilon =0$) solution is represented by the black curve and all profiles are normalized on the maximum of this profile. The solutions have been truncated to $y=10$.

Figure 11

Wall-normal eigenfunction profiles for $\varepsilon =-0.75$ to 0.75, with each plot representing an increment of 0.25. The temperature-independent ($\varepsilon =0$) solution is represented by the black curve and all profiles are normalized on the maximum of this profile. The solutions have been truncated to $y=10$.

Figure 12

Temperature eigenfunction profiles for $\varepsilon =-0.75$ to 0.75, with each plot representing an increment of 0.25. The temperature-independent ($\varepsilon =0$) solution is represented by the black curve and all profiles are normalized on the maximum of this profile. The solutions have been truncated to $y=10$.

Figure 13

Energy contributions for a range of temperature dependencies. The black line provides reference to the temperature-independent ($\varepsilon =0$) solution.

Figure 14

Energy dissipation due to viscosity, separated into Newtonian and temperature-dependent components. The black line provides reference to the temperature-independent ($\varepsilon =0$) solution.