The effects of wall transpiration on the unsteady free convection boundary-layer flow near a stagnation point in a heat generating porous medium

During the transport and storage of a porous material heat can be generated locally within the medium. This can set up a convective flow within the material which may help to dissipate the heat or alternatively can lead to enhanced heat generation and even thermal runaway with possibly disastrous consequences. Examples include the spontaneous ignition in stock piles of coal [1‐4], in bagasse (the cellulose waste left after the extraction of sugar from sugar cane) [5, 6] or in the wetting of cellulosic materials [7‐9]. Previously we have modelled this problem assuming local heat generation at a rate proportional to \((T-T_\infty )^p\), where \(T_\infty\) is the (constant) ambient temperature and the exponent \(p \ge 1\) [10, 11]. We have assessed how the effects of the input of heat from the boundary [12] and of an outer flow [13] control the local heat generation, finding that this depended critically on the exponent p and the magnitude of the initial heat input. More recently we have considered modified form of Arrhenius kinetics [14], again seeing that the evolution of the local internal heat generation depended on the initial heat input as well as the local heating rate, finding conditions under which the local heating died away or produced a finite-time blow up in the solution.

The power-law expression for localised heat generation has been used previously in several other contexts as a model for Arrhenius kinetics. These studies have shown at least qualitative agreement with those using the more general Arrhenius form, as seen for a specific case by comparing [12] and [14]. The use of a power-law model is motivated in part by its analytic simplicity as compared to the Arrhenius function and to avoid the ‘cold boundary problem’. In this, if an inert outer region is required, as is the case here, the Arrhenius form has to be modified, as in [14]. This can, perhaps, to lead to some artificiality in the kinetic scheme used. An alternative form for the local heating has been suggested, based more on fluid flow, is to model the local heat generation as a source term with a particular spatial variation [15‐18]. However, this approach has the drawback, at least for the present case of spontaneous combustion within a porous material, in that it does not allow the local heat generation to vary with the evolving temperature and velocity fields.

Here we consider how the input from the boundary influences the heat generation within the flow. We again take the above power-law term to model the heat generation within the boundary layer and allow for fluid injection or withdrawal from a thermally insulated surface. Previously, without any flow through the wall [12] and without an outer flow, we found that, when \(1 \le p<2\), the local heating died away. However, for \(p>2\) a finite-time blow up in the solution could occur for a sufficiently large initial input, otherwise the local heating also died out. Our aim here is to determine how this behaviour is influenced by the fluid input or withdrawal from the wall. We find that, when \(p=1\), a nontrivial steady state is attained giving a balance between the rate of local heat generation and the wall mass transfer. A similar situation applies when \(1<p<2\) provided now the rate of mass transfer through the wall is greater than some critical value, dependent on p. Otherwise the heat generation is inhibited. For \(p>2\) there is a either finite-time thermal runaway if the initial input is above some threshold value or otherwise the local heating dies away. We start by deriving our model.

We consider the unsteady, two-dimensional natural convection boundary-layer flow that can arise near a forward stagnation point in a fluid-saturated porous material in which there is local heat generation within the boundary layer at a rate proportional to \((T-T_\infty )^p\), where T is the fluid temperature and \(T_\infty\) is the ambient temperature. We assume that the bounding surface is thermally insulated and that there is normal wall velocity \(v_w\), where \(v_w\) can be either positive or negative. We further assume the flow is given by Darcy’s law and we make the standard Boussinesq approximation. The basic equations for our model are, see [12, 19‐21] for example, on \(0 \le y<\infty , \, t>0\), subject to the boundary conditionstogether with some initial condition and where we assume that \(p \ge 1\). Here x measures distance along the body surface and y normal to it, u and v are respectively the velocity components in the x and y directions. Also K is the permeability of the porous media, \(\nu\) the kinematic viscosity of the fluid, \(\beta\) the coefficient of thermal expansion, g the acceleration due to gravity, \(\alpha _m\) the effective thermal diffusivity and \(\sigma\) the heat capacity ratio of the fluid-filled porous medium to that of the fluid. In Eq. (2), S(x) is the sine of the angle between the outward normal and the downward vertical and for a stagnation-point flow \(\displaystyle S(x)=\frac{x}{\ell }\) for some length scale \(\ell\).

To make Eqs. (1–4) dimensionless we writewhere \(T_s, U_s\) are respectively temperature and velocity scales given byEquation (2) becomes (on dropping the overbars) \(u=x\,\theta\) and the continuity Eq. (1) is satisfied by introducing a streamfunction \(\psi\) defined so that \(u=\psi _y, \ v=-\psi _x\). We then write \(\psi =x\,f(y,t)\), and consequently \(\theta =\theta (y,t)\), to obtainsubject to the boundary conditionswhere \(\gamma =v_w\,R^{1/2}/U_s\). In (8), \(\gamma >0\) gives fluid injection from the boundary (blowing) and \(\gamma <0\) gives fluid withdrawal from the boundary (suction).

We find different behaviour depending on whether \(p=1, 1<p<2\) or \(p>2\), treating these cases, as well as the transition case when \(p=2\), see expression (6), separately. We start by considering the steady states of Eq. (7) as these can represent the possible large time behaviour of the full initial-value problem.

These represent possible large time limit of the initial-value problem (7, 8) and, for a general value of p, are given bywhere primes denote differentiation with respect to y. We note that, for \(p=2\), Eq. (9) can be integrated to giveon satisfying the boundary condition as \(y \rightarrow \infty\). Applying the boundary conditions on \(y=0\) then shows that the only possibility when \(\gamma \ne 0\) is the trivial solution \(f \equiv -\gamma\) with a nontrivial solution only for \(\gamma =0\).

In Fig. 1 we plot \(f'(0)\) against \(\gamma\) for \(p=1\) obtained from the numerical solution of Eq. (9). The numerical solution terminated at \(\gamma =-2\), where \(f'(0)\) becomes zero (we were only able to obtain the trivial solution \(f \equiv -\gamma\) for \(\gamma <-2\)), and proceeded to large positive values of \(\gamma\) with the values of \(f'(0)\) increasing as \(\gamma\) is increased.

We plot \(f'(0)\) against \(\gamma\) in Fig. 2 for \(p=1.5\), representative of values of p in the range \(1<p<2\). In this case we see that there is a critical value \(\gamma _c\) of \(\gamma\), where \(\gamma _c \simeq -0.5085\) giving two solution branches in \(\gamma _c<\gamma\). The upper branch solution continues to large positive \(\gamma\), with \(f'(0)\) increasing as \(\gamma\) is increased, and we find only the trivial solution for \(\gamma <\gamma _c\). This leads us to consider the critical values in more detail. To calculate \(\gamma _c\) numerically we follow the approach given in [22], for example, whereby we make a linear perturbation to Eq. (9) resulting in a linear homogeneous equation subject to homogeneous boundary conditions and it is the solution to this eigenvalue problem that determines \(\gamma _c\). We plot \(\gamma _c\) against p in Fig. 3 where we see that \(\gamma _c \rightarrow 0\) as \(p \rightarrow 2\), as perhaps might be expected, and approaches a finite negative value as \(p \rightarrow 1\), with our numerical results suggesting that \(\gamma _c\) is approaching \(-2\).

We now consider the nature of the solution as \(p \rightarrow 2\) from below. We put \(p=2-\epsilon\), with then \(\gamma =\mu \,\epsilon\), and look for a solution valid for \(\epsilon\) small by expandingAt leading order we obtain \(f_0=2b_0\,\tanh (b_0 y)\) for some constant \(b_0>0\) to be found. At \(O(\epsilon )\) we haveWe can integrate Eq. (12) to obtain, on applying the boundary conditions and the expression for \(f_0\),on using a result given in [12]. From (13) we haveExpression (14) gives \(\mu =0\) at \(b_0=0\), has \(b_0>0\) for \(b_0>\text{ e }^{-\beta }\) and has a turning point (local minimum) at \(b_0=\text{ e }^{-(1+\beta )}, \displaystyle \mu =\mu _c= -\frac{8}{3}\,\text{ e }^{-(1+\beta )}\simeq -0.79807\). HenceAsymptotic expression (15) is shown in Fig. 3 by a broken line showing good agreement with the numerically determined values as \(p \rightarrow 2\).

We now take a value of \(p=3.0\) as representative of the case when \(p>2\) and in Fig. 4 we plot \(f'(0)\) against \(\gamma\). In this case there is a positive value critical value \(\gamma _c \simeq 0.6082\) with nontrivial solutions now only for \(\gamma \le \gamma _c\). This is different to the previous case when \(1<p<2\), compare this figure with Fig. 2. The upper solution branch for this case continues to large negative values of \(\gamma\) and the lower solution branch terminates as \(\gamma \rightarrow 0\) with \(f'(0) \rightarrow 0\). We can again calculate the critical values \(\gamma _c\) and in Fig. 5 we plot \(\gamma _c\) against p when \(p>2\). We see that \(\gamma _c\) is positive throughout for this case, approaching zero as \(p \rightarrow 2\) from above, and that \(\gamma _c\) increases as p is increased. The above discussion for the asymptotic behaviour as \(p \rightarrow 2\) follows directly for this case on putting \(p=2+\epsilon\). The result is thatThis asymptotic result is shown in Fig. 5 by a broken line, again showing good agreement with the numerically determined values.

We next consider how the solution behaves with p for a fixed value of \(\gamma\), taking \(\gamma =1.0\) representative of blowing and \(\gamma =-1.0\) representative of suction. We plot \(f'(0)\) against p in Fig. 6 for these two cases, in Fig. 6a for p in the range \(1 \leqslant p<2\) and in Fig. 6b for \(p>2\). In Fig. 6a both curves start with their values at \(p=1\) as shown in Fig. 1 with, for \(\gamma =1.0, f'(0)\) increasing and becoming large and the boundary-layer thickness decreasing as p is increased with our numerical integrations suggesting that \(f'(0) \rightarrow \infty\) as \(p \rightarrow 2\). For \(\gamma =-1.0\), however, \(f'(0)\) decreases to the critical point at \(p=p_c \simeq 1.220\) for \(\gamma _c=-1.0\) shown in Fig. 3, leading to a lower solution branch in \(p<p_c\), not particularly clear in the figure.

For \(p>2\), Fig. 6b, there is now a critical value for \(\gamma =1.0\) at \(p=p_c \simeq 3.938\) giving rise to solutions only in \(p \geqslant p_c\) and also a lower solution branch in \(p>p_c\). For \(\gamma =-1.0\) the values of \(f'(0)\) increase rapidly as \(p \rightarrow 2\), similar to that seen for \(\gamma =1.0\) in Fig. 6a. In both cases the values of \(f'(0)\) appear to be approaching the same limit as p is increased.

To see how the solution for \(\gamma >0\) in Fig. 6a behaves as \(p \rightarrow 2\) from below we put \(p=2-\delta\) and look for a solution for \(\delta \ll 1\) by writing \(f=a(\delta )\, F, \, \zeta =a(\delta )\,y\) for some \(a(\delta )\) to be determined, expecting that \(a \gg 1\) for \(\delta\) small. Equation (9) becomesPrimes now denote differentiation with respect to \(\zeta\). Since \(a^{-2\delta }\sim 1-2\delta \,\log a+\cdots\), we look for a solution by expandingAt leading order we havefor some constant \(c_0>0\) to be found. To obtain a solution at next order we requireThis then gives at \(O(\delta \,\log a)\), on integrating the equation once and applying the boundary conditions as \(\zeta \rightarrow \infty\),If we now apply the boundary conditions on \(\zeta =0\), particularly that \(F_1(0)=-\gamma\), and the above expression for \(F_0\) we obtainHence

We can readily adapt this analysis to the case when \(\gamma <0\) and \(p \rightarrow 2\) from above, Fig. 6b. We now put \(p=2+\delta\) and make the above transformation. We then follow the above discussion, the only difference being in a sign change on the right-hand side of Eq. (21). This in turn leads to, on applying the boundary conditions on \(\zeta =0\),with the expression for \(f'(0)\) being the same as that given in (23).

Our discussion of the steady states suggests that we should treat the cases \(p=1, 1<p<2\) and \(p>2\) separately. We solved initial-value problem (7, 8) numerically using the scheme described in [10, 12] for example, taking as our initial condition \(\displaystyle \frac{\displaystyle \partial f}{\displaystyle \partial y}=a_0\,\text{ e }^{-y}\) and prescribing a value for \(a_0>0\).

We have considered the effect that fluid transfer through the wall, both injection and withdrawal, can have on the development the boundary layer near a forward stagnation point when there is local heat generated at a rate proportional to \((T-T_\infty )^p\) within the boundary layer. Previously we have seen [12, 13] that, without the fluid transfer, how the solution evolved depended on whether \(1 \leqslant p<2\) or \(p>2\), respectively approaching a nontrivial steady state, or either dying away or having a finite-time singularity. A similar situation arises here though the injection/withdrawal of fluid through the wall can have significant effects. For \(1 \leqslant p<2\), the solution approaches a nontrivial steady state provided the dimensionless parameter \(\gamma\) measuring the wall transfer is such that it is greater than some critical value \(\gamma _c\), dependent on p. Otherwise only the trivial state is attained at large times. When there is fluid injection, i.e. \(\gamma >0\), large temperatures can be achieved within the boundary layer, increasing as \(\gamma\) is increased, see Figs. 1, 2, 7 and 8. These critical values depend on the exponent p increasing from \(-2\) for \(p=1\) to zero as \(p\rightarrow 2\), see Fig. 3. Thus if fluid withdrawal is sufficiently large the effect is to inhibit boundary layer development. However fluid injection can greatly increase both the heat transfer and the fluid flow.

The situation is essentially different when the exponent \(p>2\). Here the effect of fluid injection is to limit the range of existence of any nontrivial steady state, see Figs. 4 and 5, whereas it is now fluid withdrawal that increases the temperatures seen within the boundary layer. However, these nontrivial steady states are unstable and the time dependent solution either evolves to the trivial state or reaches a finite-time singularity, see Fig. 10a. Which of these states is achieved depends strongly on the initial temperature input, see Fig. 10b.