Boundary-layer development beyond a critical value

Abstract

Two initial-value problems are considered involving a parameter $\alpha$, the corresponding steady states have a critical value at $\alpha =\alpha _c<0$ with steady state solutions possible only if $\alpha \ge \alpha _c$. The aim is to compare how the solution to these two problems evolves in time. For the first problem we find that a solution exists for all time with a two-region structure having a self-similar relatively thin front region moving away from the wall and an inner inviscid region, increasing exponentially in thickness at a rate proportional to $\hbox {e}^{|\alpha |\,t}$. A finite-time singularity develops in the second problem, the nature of this singularity being derived in terms of the time difference $\tau =t_s-t$, where the singularity occurs at time $t_s$, showing that the velocity becomes singular at a rate proportional to $\tau ^{-1}$ and the thickness of the boundary-layer increasing at a rate proportional to $\tau ^{-1/2}$.

Appendix

A very much simplified and dimensionless version derivation of Eq. (1).

In a porous medium with the flow given by Darcy’s law the flow near a forward stagnation point with an outer flow is given by (in the usual notation)

$$\begin{aligned} u=x\,(\beta \,\theta +1) \end{aligned}$$

(42)

where $\beta$ is a parameter measuring convective to outer flow effects. Define a stream function $\psi$ by $\psi =x\,f(y,t)$ and taking $\theta =\theta (y,t)$, as is standard for a stagnation point, the heat transfer (energy) equation gives

$$\begin{aligned} \frac{\displaystyle \partial \theta }{\displaystyle \partial t}-f\,\frac{\displaystyle \partial \theta }{\displaystyle \partial y}=\frac{\displaystyle \partial ^2 \theta }{\displaystyle \partial y^2} \end{aligned}$$

(43)

with, from (42),

$$\begin{aligned} \theta =\frac{1}{\beta }\,\left( \frac{\displaystyle \partial f}{\displaystyle \partial y}-1\right) \end{aligned}$$

(44)

Equation (43) then gives Eq. (1) on putting $\alpha =1+\beta$.

The solution develops initially in terms of the independent variable $\eta =y/t^{1/2}$, as is the case in many other problems of this type. This form can be seen, for example, in the small time behaviour for $\alpha =0.5$ in Fig. 3. The numerical integrations used the variable y to reflect the behaviour at larger times, though this did require a small time step at the start of the numerical solution.