The steady states of a combustion model, derived in a previous paper, were shown to have critical points (turning points in the bifurcation diagram) for certain ranges of parameter values. Here attention is fixed on the heat release parameter λ and the time evolution for the solution for values of λ just above its critical value 403-1 is discussed. It is shown that the solution develops a three-stage structure, with the solution both approaching and leaving the critical point on a relatively short time scale. However, the majority of the time is spent in moving slowly past the critical point, on an 403-2 time scale. The solution finally attains its values on the upper solution branch, except in the special case of the exponential approximation and when reactant consumption is neglected. Here the temperature develops a singularity at a finite time tB, of O(log(tB−t)), though the fluid velocity remains finite at tB.

The equations which govern the similarity solution for free convection boundary-layer flow above a permeable, horizontal surface in a fluid-saturated porous medium are considered. These are seen to depend on the dimensionless parameters γ and m measuring mass transport rate and the wall temperature variation, respectively. Numerical solutions are presented for a wide range of values of γ and m. Asymptotic solutions are obtained for ¦γ¦ large (for both fluid injection, γ > 0, and fluid withdrawal, γ < 0) and for m large. These are compared with the numerical solutions.

The equation which governs the similarity solution for free convection boundary-layer flow along a vertical permeable surface with prescribed surface heating and mass transfer rate is discussed. The solution is seen to depend on two non-dimensional parameters;m, the power-law exponent, and γ, the mass transfer parameter. It is shown that solutions exist for allm>−1 for γ>0 (fluid injection) whereas for γ<0 (fluid withdrawal), solution exist form>m0(γ), wherem0 is determined as a function of γ. Solutions for large mass transfer rates are obtained, for both γ>0 and γ<0. For γ>0 the form of the asymptotic solution for γ large is seen to depend on the value ofm. Solutions form large are derived, these are seen to be different depending on whether γ is positive or negative.

The free convection boundary-layer flow near a stagnation point driven by catalytic surface heating is considered. The case without fuel consumption is treated first, and it is shown that the steady state equations admit multiple solutions. Explicit expressions can be obtained for these solution branches and it is found that a hysteresis point occurs when the activation energy parameter ε = 1/5. The effect of fuel consumption is seen to be characterized by the dimensionless parameter α and numerical results are obtained for a range of values of α and ε, as well as Prandtl number σ and Schmidt number S c . Multiple solutions are again observed and analytic expressions for the bifurcation points can be found when σ = S c . For σ ≠ S c these have to be determined numerically.

The effects that blowing and suction have on the free convection boundary layer on a vertical surface with a given surface heat flux are considered. Similarity equations are derived first, their solution being dependent on the wall flux exponentn and a dimensionless transpiration parameter γ, (as well as on the Prandtl number). The range of existence of solutions is considered, with it being shown that solutions exist only forn > −1 for blowing,whereas they exist for alln >n0 for suction, wheren0 < −1 and depends on γ. The solutions for strong suction and blowing are derived. In the latter case the asymptotic structure is found to be different forn in the three ranges −1 <n < − % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3775!\[\frac{1}{4}\], −% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3775!\[\frac{1}{4}\] <n < % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabgkHiTmaaleaaleaacaaIXaaabaGaaGinaaaaaaa!3EB1!\[ - \tfrac{1}{4}\], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabgkHiTmaaleaaleaacaaIXaaabaGaaGinaaaaaaa!3EB1!\[ - \tfrac{1}{4}\] <n < % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaI3aaabaGaaGOmaaaaaaa!3DC8!\[\tfrac{7}{2}\],n % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaI3aaabaGaaGOmaaaaaaa!3DC8!\[\tfrac{7}{2}\]. Results are then obtained for the non-similarity problem of constant heat flux with a constant transpiration velocity. Solutions valid for large distances from the leading edge for both suction and blowing are derived.