Combined forced- and free-convection flows (mixed convection) are encountered in many technological and industrial applications including solar receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency shutdown, heat exchanges placed in a low-velocity environment and many more. Two-dimensional stagnation-point flows arise in the vicinity of a stagnation line resulting from a two-dimensional flow impinging on a curved surface at right angles to it and thereafter flowing symmetrically about the stagnation line. Hiemenz [1] was the first to study two-dimensional stagnation-point flows. Later Eckert [2] and Gorla [3] considered the corresponding forced-convection heat-transfer problem. Three-dimensional stagnation-point flows have been studied by Homann [4] and Smith [5] and the axisymmetric stagnation-point flow on a circular cylinder by Wang [6] and Gorla [7]. The three-dimensional flow resulting from an axisymmetric stagnation flow

impinging obliquely on a body surface has been treated by Weidman and Putkaradze [8]. The problem of axisymmetric stagnation-point flow acting on a porous flat plate oscillating transversely in its own plane has been investigated by Weidman and Mahalingam [9]. In this case a three-dimensional flow results from a stagnation-point flow on a flat plate oscillating in its own plane. Gorla [10] has studied the unsteady viscous flow in the vicinity of an axisymmetric stagnation point on a circular cylinder.

The steady mixed convection flow near the stagnation region of a vertical flat plate has been studied by Ramachandran etal. [11] and by Gorla [12] for the flow near an axisymmetric stagnation point on a slender impermeable vertical cylinder. Mixed convection flows arise when the buoyancy forces resulting from temperature differences within the flow become comparable to the pressure gradient forces arising from the forced flow. As a consequence, both the flow and thermal fields are significantly affected by the buoyancy forces. The study of thick axisymmetric free-convection boundary layers along slender bodies has been shown by Kuiken [13] to have an unusual structure at large distances along the cylinder. When the boundary-layer variables are scaled so as to be of order unity within the boundary layer, the boundary conditions that hold on the surface of the slender body are given at a value of the independent variable which is close to zero. As a result, when a perturbation analysis is used to obtain the solution at large distances along the cylinder, the body is reduced to a line at the first approximation.

Several papers have been published previously on axisymmetric mixed convection boundary-layer flows along slender bodies. Naraian and Uberoi [14, 15] and Chen [16] have studied the mixed convection boundary layer on a vertical needle. These are bodies of revolution whose diameter is of the same order as the thickness of the velocity or thermal boundary layers that develop on it. By appropriately varying the radius of the needle, the boundary-layer equations admit similarity solutions. Wang [17] found a similarity solution for the mixed convection boundary layer on an adiabatic vertical needle with a heat source at the tip, a situation that arises, for example, for a stick burning at its lower end.

The present paper considers the steady mixed convection flow that develops near an axisymmetric stagnation point on a vertical isothermal cylinder in the case when the boundary layer is thick compared to the radius of the cylinder. We start by describing the governing equations, following closely [9,12] for the forced-convection problem. This results in two ordinary differential equations for the flow and temperature fields that involve, as well as the Prandtl number $\sigma$, the two further parameters $R$, which is measure of the forced flow, and a mixed convection parameter $\lambda$. We consider both aiding flows (when the outer flow and the buoyancy forces are in the same direction, $\lambda>0$ ) and opposing flows (when the outer flow and the buoyancy forces act in opposite directions, $\lambda<0$ ). We start by giving numerical solutions to these equations for representative values of $\lambda$ and $R$, finding dual solutions for negative $\lambda$ with critical points $\lambda_{c}<0$, requiring $\lambda\geq\lambda_{c}$ for the existence of a solution. We determine how $\lambda_{c}$ varies with $R$, before considering the asymptotic limits of $\lambda\rightarrow\infty$ (free-convection limit) and $R\rightarrow\infty$ and $R\rightarrow 0$.

We consider the steady mixed convection flow near an axisymmetric stagnation point on an infinite cylinder. The cylinder is taken as mounted vertically and the flow is assumed to be axisymmetric about the $x$-axis, which measures distance along the cylinder in a vertical direction with gravity acting in the negative $x$-direction. The stagnation point is at $x=0,r=a$, where $r$ measures distance radially from the centre of the cylinder of radius $a$. The ambient fluid has a constant temperature $T_{\infty}$ and the cylinder is maintained at a temperature $T_{w}(x)=T_{0}(x/a)+T_{\infty}$. Having $T_{w}>T_{\infty}$ corresponds to assisting flow, with $T_{w}<T_{\infty}$ corresponding to opposing flow. The outer flow in this situation, taken directly from [12], is

where $U_{\infty}$ is the planar flow at large distances from the cylinder and where $u$ and $v$ are the velocity components in the $x$ and $r$ directions, respectively.

Again following [12], we introduce the variables

where $T$ is the temperature of the fluid. Applying (2) in the governing equations and making the standard Boussinesq approximation, we find that our flow is described by the similarity equations, again from [12],

subject to the boundary conditions, from (1), that

(primes denote differentiation with respect to $\eta$ ) where $\sigma$ is the Prandtl number and where

are, respectively, a Reynolds number and a mixed convection parameter, with $\lambda>0$ corresponding to assisting flow and $\lambda<0$ corresponding to opposing flow. In (6) $\nu$ is the kinematic viscosity of the fluid, $g$ the acceleration due to gravity and $\beta$ the coefficient of thermal expansion.

The parameters perhaps of most physical interest are the skin friction parameter $C_{f}$ and the Nusselt number Nu, defined as

and where $\mu$ and $k$ are the dynamic viscosity and thermal conductivity, respectively. From (2), we have that

The problem given by (3-5) has been considered previously by Gorla [12] who presented a range of numerical results for both aiding and opposing flows. The results given by [12] are only for a specific value of $R$, namely $R=100$, with critical points giving a finite range of existence for opposing flows not being identified in [12]. We start by first describing our numerical solutions to Eqs. (3) and (4) subject to boundary conditions (5) for a range of values of $\lambda$ and $R$, thus adding to the numerical treatment given by Gorla [12]. Throughout we assume that the Prandtl number $\sigma$ is of $O(1)$, our numerical results are all for the case when $\sigma=1$.

Equations (3) and (4) subject to boundary conditions (5) were solved numerically using a standard shooting method for solving boundary-value problems (D02AGF in the NAG library). In Fig. 1 we plot $f^{\prime\prime}(1)$ and $\theta^{\prime}(1)$ against $\lambda$ for $R=1,5,10$ (with $\sigma=1$ ). This figure shows that, for $\lambda<0$, there is a critical value $\lambda_{c}$ with solutions possible only for $0>\lambda\geq\lambda_{c}$ and for $\lambda>\lambda_{c}$ there are dual solutions. The value of $\lambda_{c}$ decreases as $R$ is increased, thus giving a greater range of negative $\lambda$ for possible solutions. For $\lambda>0$ there is only one solution with the values of $f^{\prime\prime}(1)$ increasing and $\theta^{\prime}(1)$ decreasing as $\lambda$ is increased (for a given value of $R$ ). For the larger values of $\lambda$, the values of $f^{\prime\prime}(1)$ for $R=1$ becomes greater than those for the larger values of $R(=5,10)$, indicating that, for sufficiently large values of $\lambda,f^{\prime\prime}(1)$ increases as $R$ is decreased. However, the values of $-\theta^{\prime}(1)$ increase as $R$ is increased (for a given value of $\lambda$ ).

In Fig. 2 we take values for $\lambda(\lambda=1,-2)$ representative respectively of aiding and opposing mixed convection and plot the corresponding values of $f^{\prime\prime}(1)$ and $\theta^{\prime}(1)$ against $R$. We see that in both cases $\theta^{\prime}(1)$ decreases as $R$ is increased. However, for $\lambda=1,f^{\prime\prime}(1)$ has a minimum value, of $f^{\prime\prime}(1)=1.6648$ at $R=0.272$, before increasing again for the larger values of $R$. For $\lambda=-2$ there is a critical value $R_{c}$ of $R\left(R_{c}\simeq 0.622\right)$ below which there are no solutions, as might be expected from Fig. 1.

To obtain a solution valid for $\lambda$ large we start by putting
$f=\lambda^{1/4}F,\quad\zeta=\lambda^{1/4}(\eta-1)$,
leaving $\theta$ unscaled. This results in the equations
$\left(1+\lambda^{-1/4}\zeta\right)F^{\prime\prime\prime}+\lambda^{-1/4}F^{\prime\prime}+R\left(FF^{\prime\prime}-F^{2}\right)+\lambda^{-1}R+\theta=0$,
$\left(1+\lambda^{-1/4}\zeta\right)\theta^{\prime\prime}+\lambda^{-1/4}\theta^{\prime}+\sigma R\left(F\theta^{\prime}-F^{\prime}\theta\right)=0$,
where primes now denote differentiation with respect to $\zeta$, subject to the boundary conditions
$F(0)=0,F^{\prime}(0)=0,\theta(0)=1,\quad F^{\prime}\rightarrow\lambda^{-1/2},\theta\rightarrow 0$ as $\zeta\rightarrow\infty$.

Equations $(10,11)$ suggest looking for a solution by expanding

We can scale the leading-order problem obtained by substituting expansion (13) in (10, 11) by putting
$F_{0}=R^{-3/4}\bar{F}_{0},\quad\bar{\zeta}=R^{1/4}\zeta$.

This results in
$\bar{F}_{0}^{\prime\prime\prime}+\bar{F}_{0}\bar{F}_{0}^{\prime\prime}-\bar{F}_{0}^{\prime 2}+\theta_{0}=0$,
$\theta_{0}^{\prime\prime}+\sigma\left(\bar{F}_{0}\theta_{0}^{\prime}-\bar{F}_{0}^{\prime}\theta_{0}\right)=0$,
subject to
$\bar{F}_{0}(0)=0,\bar{F}_{0}^{\prime}(0)=0,\theta_{0}(0)=1,\quad\bar{F}_{0}^{\prime}\rightarrow 0,\theta_{0}\rightarrow 0$ as $\bar{\zeta}\rightarrow\infty$,
where primes now denote differentiation with respect to $\bar{\zeta}$. A numerical solution of (15-17) gives, for $\sigma=1$, $\bar{F}_{0}^{\prime\prime}(0)=0.73950,\theta_{0}^{\prime}(0)=-0.59509$.

We can continue to the next order by first rescaling
$F_{1}=R^{-1}\bar{F}_{1},\quad\theta_{1}=R^{-1/4}\bar{\theta}_{1}$.

This gives, with (14),
$\bar{F}_{1}^{\prime\prime\prime}+\bar{F}_{0}\bar{F}_{1}^{\prime\prime}+\bar{F}_{1}\bar{F}_{0}^{\prime\prime}-2\bar{F}_{0}^{\prime}\bar{F}_{1}^{\prime}+\bar{\theta}_{1}=-\left(\bar{F}_{0}^{\prime\prime}+\bar{\zeta}\bar{F}_{0}^{\prime\prime\prime}\right)$,
$\bar{\theta}_{1}^{\prime\prime}+\sigma\left(\bar{F}_{0}\bar{\theta}_{1}^{\prime}+\bar{F}_{1}\theta_{0}^{\prime}-\bar{F}_{0}^{\prime}\bar{\theta}_{1}-\bar{F}_{1}^{\prime}\theta_{0}\right)=-\left(\theta_{0}^{\prime}+\bar{\zeta}\theta_{0}^{\prime\prime}\right)$,
subject to
$\bar{F}_{1}(0)=0,\bar{F}_{1}^{\prime}(0)=0,\bar{\theta}_{1}(0)=0,\quad\bar{F}_{1}^{\prime}\rightarrow 0,\bar{\theta}_{1}\rightarrow 0$ as $\bar{\zeta}\rightarrow\infty$.

A numerical solution gives, again for $\sigma=1$, that $\bar{F}_{1}^{\prime\prime}(0)=0.06888,\bar{\theta}_{1}^{\prime}(0)=-0.23452$.

From ( 9,14 ) and (18) we have

Expressions (22) show that, for sufficiently large values of $\lambda,-\theta^{\prime}(1)$ increases as $R$ is increased and that $f^{\prime\prime}(1)$ decreases as $R$ is increased, in line with Fig. 1. In Fig. 4 we plot the values of $\left(\lambda^{3}/R\right)^{-1/4}f^{\prime\prime}(1)$ and $-(\lambda R)^{-1/4}\theta^{\prime}(1)$ obtained from the numerical solution of $(3,4)$ against $\lambda$ for $R=1,10$. In both cases the asymptotic limit for $\lambda$ large given by (22) is approached as $\lambda$ increases, though more slowly for $R=10$ than for $R=1$ as might be expected from (22).

To obtain a solution for $R$ large, we follow the approach given in [8] for the forced-convection case and start by writing

and leaving $\theta$ unscaled. This results in the equations

subject to the boundary conditions given in ( 5 ) and where primes now denote differentiation with respect to $\xi$. Equations $(24,25)$ suggest an expansion in powers of $R^{-1/2}$, the leading-order term just being the forced-convection limit discussed in [8].

For convection to have an effect at leading order we require $\lambda$ to be large, specifically of $O(R)$. This leads us to put

The problem for the leading-order terms $\phi_{0},\theta_{0}$ is now

still subject to the boundary conditions that

Equations (27-29) have to solved numerically and graphs of $\phi_{0}^{\prime\prime}(0)$ and $\theta_{0}^{\prime}(0)$ for $\sigma=1$ plotted against $\mu$ are shown in Fig. 5. From these figures we see that there is a critical value $\mu_{c}$ of $\mu$ with dual solutions for $0>\mu>\mu_{c}$ and no solutions for $\mu<\mu_{c}$. For $\sigma=1$, we find that $\mu_{c}=-2.3618$, giving

For $\mu>0$ there is only one solution with both $\phi_{0}^{\prime\prime}(0)$ and $-\theta_{0}^{\prime}(0)$ increasing as $\mu$ is increased.
Expression (30) shows a linear increase in $\left|\lambda_{c}\right|$ with $R$, as noted previously about Fig.3, and to confirm this asymptotic behaviour we plot the values of $\lambda_{c}R^{-1}$ obtained from our numerical integrations to find $\lambda_{c}$ against $R$ in Fig. 6. The numerically determined values approach this asymptotic limit (shown by the broken line), though only slowly as $R$ is increased, suggesting that the $O\left(R^{-1/2}\right)$ correction has a significant effect even at moderately large values of $R$.

For $\mu$ large we can recover expressions (22) by putting

When (31) is substituted in equations (27-29) and then $\mu\rightarrow\infty$, we obtain equations (15-17) and, on using (26), the leading-order terms in (22).

The behaviour of the solution for $R$ small depends on whether $\lambda$ is small or of $O(1)$. We start with the latter case, assuming that $\lambda>0$.

Our previous numerical results were for the case $\sigma=1$ and we expect qualitatively similar behaviour when $\sigma$ is of $O(1)$, typical of gases. For liquids (water) $\sigma$ is somewhat larger and it is thus worth briefly considering the large $\sigma$ limit. Following the treatment in [20,21] we expect that, for $\sigma$ large, the solution to involve a relatively thin thermal inner layer and a thicker outer viscous flow region. In the inner region we have

with $\theta$ left unscaled. At leading order, for $\sigma$ large, we obtain

subject to

with the outer condition on $\Phi$ relaxed at this stage and where primes denote differentiation with respect to $\tau$.
Equations $(57,58)$ give

for some constant $A_{0}=A_{0}(R)$ to be determined and

in terms of confluent hypergeometric functions [22]. Expressions (56) and (60) give

Expression (61) shows that the heat transfer increases as $\sigma$ is increased, being of $O\left(\sigma^{1/3}\right)$ for $\sigma$ large.
To determine the constant $A_{0}$ we need to consider the outer region, in which we can neglect the temperature and write $\eta=1+\bar{\eta}$. This leads to

and, on matching with the inner region, that

Equations $(62,63)$ are essentially the forced-convection limit with the solution given by [8]. Their solution then gives $A_{0}$ in terms of $R$. For $R$ large [8] show that $A_{0}\sim 0.6163R^{1/2}+\cdots$, giving $\left(\frac{\mathrm{d}\theta}{\mathrm{d}\eta}\right)_{\eta=1}$ of $O\left(R^{1/2}\sigma^{1/3}\right)$ for $R$ and $\sigma$ large.

The leading-order problem (57,58,62,63) for $\sigma$ large does not give a critical value for $\lambda$. In fact, the solution is independent of $\lambda$ at leading order, with the buoyancy forces arising at $O\left(\sigma^{-1/3}\right)$ in the inner region. This suggests that the critical value $\lambda_{c}$ will, for a given value of $R$, decrease to large negative values as $\sigma$ is increased. We illustrate this in Fig. 9 with a plot of $\theta^{\prime}(1)$ against $\lambda$ for $R=1$ and $\sigma=10$ to compare with the results for $\sigma=1$. This figure clearly shows that the critical values $\lambda_{c}$ has decreased for $\sigma=10$ over that for $\sigma=1$, giving a greater range of $\lambda$ for solutions in the opposing case. The values of $\theta^{\prime}(1)$ have, for a given $\lambda$, also decreased for $\sigma=10$, in line with expression (61). In Fig. 10 we plot $\lambda_{c}$ against $\sigma$ again for $R=1$, showing that $\lambda_{c}$ decreases relatively slowly as $\sigma$ is increased, as can be expected from the above analysis for the large $\lambda$ case.

We have considered the mixed convection boundary-layer flow around an axisymmetric stagnation point. The equations for the flow and temperature fields reduce to similarity form ( $3-5$ ) and involve the three parameters, the Prandtl number $\sigma$, a Reynolds number $R$ and a mixed convection parameter $\lambda$, as defined in (6). The similarity equations were solved numerically for representative values of the parameters $R$ and $\lambda$; see Figs. 1 and 2 . The main conclusions from these numerical integrations were that, for $\lambda<0$ (opposing flow), there was a critical value $\lambda_{c}$ of $\lambda$ at which there was a saddle-node bifurcation with dual solutions for $\lambda_{c}<\lambda<0$ and no solutions for $\lambda<\lambda_{c}$. For $\lambda\geq 0$ (aiding flow) there was a single solution for all $\lambda$. An asymptotic solution for $\lambda$ large was derived, with the results summarized in (22).

The occurrence of dual solutions for opposing flows is not unexpected and is consistent with many previous studies of similarity solutions in mixed convection, see [18,19,23] for example. The critical value $\lambda_{c}$ was seen to depend on $R$ (and on $\sigma$ ), see Fig. 3. The possibilities of solutions having only a limited range of existence and the existence of dual solutions for opposing flows was not noticed by Gorla [12]. The reason for this is that the results in [12] were all for $R=100$ and our study suggests that the value of $\lambda_{c}$ for this value of $R$ is quite large, well beyond the values for the mixed convection parameter taken in [12]. A solution for $R$ large was obtained which showed that $\lambda_{c}$ is of $O(R)$ in this case, as given in (30) for $\sigma=1$. Thus for strong external flows (large $R$ ) boundary-layer flows are still possible even when there are strongly opposing buoyancy forces. For $R$ small the flow was seen to be driven predominantly by the buoyancy forces when $\lambda$ is of $O(1)$, as summarized in (47). However, for $\lambda$ small both buoyancy and the external flow have comparable effects, giving a critical value of $O(R(-\log R))$ for $R$ small, as given by (55) for $\sigma=1$. The effect of weak external flows is then to severely limit the range of $\lambda$ where there can be opposing flows.

The effect of having large values for the Prandtl number $\sigma$ is to confine the thermal effects to a thin layer, of thickness $O\left(\sigma^{-1/3}\right)$, next to the cylinder with the outer flow being essentially given by forced convection. A consequence of this is, for opposing flows, to decrease $\lambda_{c}$ to large negative values, with solutions then being possible for a large range of the mixed convection parameter.

Acknowledgements CR was supported by the CEEX grant No 2-CEx06-11-96/2006. JM and IP were both supported by a Royal Society (London) Joint Project Grant.

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