2.1 Governing equations and boundary conditions

The conservation of linear momentum equation takes the following dimensionless form:

For $x>0$ , the kinematic and dynamic boundary conditions at the free surface $z=\unicode[STIX]{x1D701}(x)$ are

A prime denotes total differentiation with respect to the independent variable. In addition, the following conditions must be satisfied:

2.2 The flow structure and solution strategy

The flow is assumed to have the base Poiseuille profile (2.1) to lowest order, and is modified when the fluid leaves the channel in the form of the wall jet. When the fluid detaches itself from the lower wall of the channel, the removal of the wall stress causes a boundary layer to form in a region near the free surface. In this region, the parabolic velocity profile adjusts itself so as to satisfy the condition of zero traction at the free surface. In the inviscid limit of infinite Reynolds number, this condition would not be imposed since there is no (viscous) mechanism for the stress singularity to diffuse, and all of the conditions of the problem would be satisfied by observing that the Poiseuille profile continues undisturbed in the jet region. However, there is no unique solution for this inviscid problem, and we can assume that the fully developed Poiseuille flow is everywhere the proper inviscid limit. With this assumption, the flow in the core of the jet is, to lowest order, not affected by the flow in the boundary-layer region near the free surface although the boundary layer is expected to induce perturbations to the base Poiseuille flow when higher-order terms are included, for the flow both upstream and downstream of the channel exit. This assumption is similar to the one made for channel and tube flows with mild (Smith Reference Smith1976a ,Reference Smith b ) and severe (Smith Reference Smith1979) constrictions, where the flow field in the core region, to leading order, satisfies the inviscid equations of motion. Further discussion and comparison between the current and Smith’s formulations are given in § 4. The reader is also referred to Sobey (Reference Sobey2000) for other asymptotic analyses on interactive boundary layers.

When the fluid exits the channel, the vanishing of lower-wall shear stress at the channel exit causes a boundary layer to develop adjacent to the free surface, which we shall denote as the free-surface layer. This thin layer and the jet contraction (and expansion further downstream of the channel exit) affect the core of the jet, with this core layer remaining predominantly of inviscid rotational character. This inviscid layer cannot, however, extend to the wall, where significant viscous shearing occurs, forcing the formation of an upper-wall layer in the vicinity of the upper wall. Similarly, the core flow must also adjust to the shearing viscous flow in the lower-wall layer upstream of the stress singularity at the exit. In other words, given the predominantly inviscid character of the core flow, the core solution cannot satisfy adherence at the lower and upper walls. The thickness of the wall layers upstream of the channel exit is not expected to be significant at large Reynolds number, but can be large otherwise, and is fundamentally important to examine as it reflects the extent of upstream influence. In fact, the thickness of the wall layers will turn out to be of $O(Re^{-1})$ . The upper-wall layer downstream of the channel exit is expected to grow rapidly with distance.

Figure 1 illustrates schematically the different flow regions for the wall jet. In each layer, different physical mechanisms dominate the flow with corresponding characteristic length scales. In particular, for the flow downstream of the channel exit, the free-surface and upper-wall layers are shear dominated, and the flow is of the boundary-layer type. In the region in between, or the core layer, both shear and elongation are in balance as a result of the predominance of the Poiseuille character of the flow upstream and downstream of the channel exit. At the channel exit, $x=0$ , the shear stress undergoes a step change from a non-zero value at the lower wall, $z=0$ , to zero at the free surface, $z=\unicode[STIX]{x1D701}(x)$ . The effect of this drop diffuses upstream inside the channel $(x<0)$ , over a distance $x_{0}$ where fully developed Poiseuille flow is recovered, and downstream ( $x>0$ ), towards the upper wall. Boundary-layer solutions are sought near the free surface and walls, but not all layers admit a similarity solution. For the flow outside the channel, a similarity solution is sought in the free-surface layer, which is matched onto the core solution. The core flow, in turn, is matched onto the solution in the core region inside the channel at the exit, as well as onto the flow in the upper-wall layer. It is important to observe that no matching is required for the similarity flow at $x=0$ between the free-surface and lower-wall layers; the flow singularity at the origin is entirely avoided in the solution process. This constitutes a major advantage of the current formulation.

The present analysis is reminiscent of the ideas advanced in Stewartson’s triple-deck theory (Sobey Reference Sobey2000), which lays down a rational basis for the investigation of separation and other nonlinear features in external flow situations. The fundamental difference between the external separation phenomenon and the current wall-jet flow through a channel is the absence of an external inviscid region in the latter case. A similar approach was followed by Smith (Reference Smith1976a ,Reference Smith b , Reference Smith1979), who examined the fully developed flow through an indented tube when the Reynolds number is large. As in the current problem, there is no external flow region, and the triple-deck formulation could not simply be carried over to Smith’s problem, but, as mentioned earlier, Smith realized that a model based on an inviscid rotational core flow region together with viscous boundary-layer-type regions near the walls could be applied to his problem.

3.1 The jet thickness and the flow in the free-surface layer

To examine the free-surface-layer structure, we let $y=z-\unicode[STIX]{x1D701}(x)$ . The scaling in the transverse direction is changed by writing $y=\unicode[STIX]{x1D700}\unicode[STIX]{x1D702}$ , where $\unicode[STIX]{x1D700}=Re^{-\unicode[STIX]{x1D6FC}}$ is the small parameter in the problem and $\unicode[STIX]{x1D6FC}$ is to be determined. Following Tillett (Reference Tillett1968), and anticipating that the height $\unicode[STIX]{x1D701}$ of the free surface is of the same order of magnitude as the boundary-layer thickness, one can write $\unicode[STIX]{x1D701}(x)=\unicode[STIX]{x1D700}h(x)$ , and henceforth work with $h(x)$ . The matching indicates that $h=O(1)$ . The following change of coordinates is introduced:

The aim is to find a solution of problem (2.2)–(2.4) in the form of a boundary-layer expansion in $\unicode[STIX]{x1D700}$ . In order to match this to the core flow, it is necessary to have $\unicode[STIX]{x1D713}\sim y^{2}$ as $\unicode[STIX]{x1D702}\rightarrow \infty$ in the free-surface layer, to lowest order in $\unicode[STIX]{x1D700}$ (see below). Therefore, $\unicode[STIX]{x1D713}$ must be of order $\unicode[STIX]{x1D700}^{2}$ . As in the case of a symmetric jet (Tillett Reference Tillett1968), the value of $\unicode[STIX]{x1D6FC}$ is determined by requiring that the convective and viscous terms balance, leading to $\unicode[STIX]{x1D6FC}=1/3$ . The streamwise and transverse velocity components are expressed as $u=\unicode[STIX]{x1D713}_{z}=(1/\unicode[STIX]{x1D700})\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D702}}$ and $w=-\unicode[STIX]{x1D713}_{x}=-\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D709}}+h^{\prime }\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D702}}$ . Throughout the paper, a prime is used to denote total differentiation (not necessarily only with respect to $x$ or $\unicode[STIX]{x1D709}$ ). Considering the fact that $u$ in the free-surface layer must match the velocity in the core layer, $u\sim 6(z-z^{2})$ , it is also inferred that $u$ must be of order $\unicode[STIX]{x1D700}$ . Consequently, $w$ is of order  $\unicode[STIX]{x1D700}^{2}$ . The momentum conservation equations (2.2) are rewritten as

The boundary conditions on the free surface $\unicode[STIX]{x1D702}=0$ are obtained from (2.3) to read

The expansion proceeds in powers of $\unicode[STIX]{x1D700}$ , so that

Similarly, $h$ is expanded as

From (3.3) to (3.6), it is concluded that $p$ is of order $\unicode[STIX]{x1D700}^{4}$ . Thus,

It is important to distinguish (3.5) from expression (4.1) for the flow in the core layer. Recalling that, to leading order, the streamfunction in the core layer is $\unicode[STIX]{x1D713}_{0}=3z^{2}-2z^{3}$ , it is useful to rewrite (4.1) in terms of the core variable $y=z-\unicode[STIX]{x1D701}$ and $h$ as

which will be referred to shortly. Now, consider the leading-order solution of problem (3.3)–(3.4). The equations reduce to

Equation (3.9b ) is not needed immediately because of the decoupling of the pressure and streamfunction to this order. Two corresponding boundary conditions are obtained from (3.4), namely

To complete the problem for $\unicode[STIX]{x1D6F9}_{2}$ , another boundary condition is required, which we will now obtain by matching the boundary-layer flow near the free surface with the core-layer flow downstream of the channel exit (see further below). The resulting condition reads

The boundary-layer growth above the free surface can be estimated from (3.9a ) and (3.11), which indicates that $\unicode[STIX]{x1D702}\propto \unicode[STIX]{x1D709}/\unicode[STIX]{x1D6F9}_{2}\sim \unicode[STIX]{x1D709}/\unicode[STIX]{x1D702}^{2}$ . This leads to $y\propto (x/Re)^{1/3}$ for the thickness of the free-surface layer. As we shall see, the boundary-layer growth is different along the walls.

The asymptotic form (3.11) for $\unicode[STIX]{x1D6F9}_{2}$ is determined by considering the application of (3.1) for $m=2$ and $n=0$ . Applying $E_{0}$ on (3.8) gives

In terms of the inner variable, $\unicode[STIX]{x1D702}$ , $E_{0}\unicode[STIX]{x1D713}=3\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D702}^{2}-2\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D702}^{3}$ . Therefore,

To leading order, the expansion for the streamfunction in the free-surface layer is obtained from (3.5) as $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D6F9}_{2}$ . Thus, $E_{0}H_{2}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D6F9}_{2}$ , which, when matched with (3.13), leads to condition (3.10). A similarity solution is carried out for $\unicode[STIX]{x1D6F9}_{2}$ , which is written here as $\unicode[STIX]{x1D6F9}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=\unicode[STIX]{x1D709}^{2/3}f_{2}(\unicode[STIX]{x1D703})$ , where $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1/3}$ is the similarity variable. The problem for $f_{2}(\unicode[STIX]{x1D703})$ is given by

where $c_{1}$ is a constant to be determined numerically. Problem (3.14) is solved as an initial-value problem using a fourth-order Runge–Kutta scheme (IMSL-DIVERK), coupled to a shooting technique. Equation (3.14a ) is integrated subject to conditions (3.14b ) and a guessed value $f_{2}^{\prime }(0)$ for the slope at the origin. The slope is adjusted until reasonable matching is achieved with the asymptotic form (3.14c ) at large $\unicode[STIX]{x1D703}$ , or, more precisely, until $f_{2}^{\prime \prime }\sim 6$ is reached. The value of $c_{1}$ is then determined upon matching the numerical solution and its asymptotic form, which turns out to be $c_{1}=0.490$ .

Next, we determine $h_{0}(x)$ in (3.6) and the boundary condition for $\unicode[STIX]{x1D713}_{1}(x,z)$ in (3.8) or (4.1) by considering the application of (3.1) for $m=2$ and $n=1$ . Applying $E_{1}$ on (3.8), we have

The terms surviving to $O(\unicode[STIX]{x1D700}^{2})$ are identified by expressing $E_{1}\unicode[STIX]{x1D713}$ in terms of $\unicode[STIX]{x1D702}=y/\unicode[STIX]{x1D700}$ , to yield

On the other hand, we note from (3.14c ) that $H_{2}\unicode[STIX]{x1D713}$ may be written in terms of the core variables as

This yields

Equating (3.16) and (3.18) leads to

The homogeneous boundary condition in (3.19) will be used in § 4 to determine the first-order term $\unicode[STIX]{x1D713}_{1}(x,z)$ in (4.1). It will turn out that $\unicode[STIX]{x1D713}_{1}(x,z)$ vanishes everywhere, which means that, to the order $\unicode[STIX]{x1D700}$ , there is no interaction between the boundary layer and the core flow.

The next step is to determine $\unicode[STIX]{x1D713}_{2}(x,z\rightarrow 0)$ in (3.8) or (4.1) by considering $m=n=2$ . Proceeding as before, we obtain

Comparison yields $\unicode[STIX]{x1D713}_{2}(x,z\rightarrow 0)\sim 0$ . In this case, $\unicode[STIX]{x1D713}_{2}(x,z)$ also turns out to vanish everywhere. Consequently, the perturbation to the Poiseuille flow in the core layer is eventually reflected in the non-vanishing value of $\unicode[STIX]{x1D713}_{3}(x,z)$ in (3.8) or (4.1), which will be determined in § 4. Pursuing the matching process to next order to determine $\unicode[STIX]{x1D6F9}_{3}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ , we have

Noting that $E_{0}H_{3}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D6F9}_{2}+\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D6F9}_{3}$ , and equating it to (3.22), leads to $\unicode[STIX]{x1D6F9}_{3}\sim -2\unicode[STIX]{x1D702}^{3}$ and confirms $\unicode[STIX]{x1D6F9}_{2}\sim 3\unicode[STIX]{x1D702}^{2}$ for large $\unicode[STIX]{x1D702}$ . A similarity solution is also possible for $\unicode[STIX]{x1D6F9}_{3}$ , which is written here as $\unicode[STIX]{x1D6F9}_{3}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=\unicode[STIX]{x1D709}f_{3}(\unicode[STIX]{x1D703})$ . The problem for $f_{3}(\unicode[STIX]{x1D703})$ becomes

Here, $c_{2}=-2.931$ , which is determined numerically. The velocity to third order becomes

Although linear, problem (3.23) is of the two-point boundary-value type; it is solved numerically following the same scheme as for problem (3.14). As in typical boundary-layer similarity problems, convergence is achieved over a relatively short range for $\unicode[STIX]{x1D703}$ values. The interval $[0,10]$ was adequate for both problems. Of particular interest are the values of the slopes at the origin, $f_{2}^{\prime }(0)$ and $f_{3}^{\prime }(0)$ , which are directly related to the velocity at the free surface. In this case, $f_{2}^{\prime }(0)=5.319$ and $f_{3}^{\prime }(0)=-5.174$ .

The jet thickness to next order is now determined by setting $m=n=3$ in (3.1). Ensuring $E_{3}H_{3}\unicode[STIX]{x1D713}=H_{3}E_{3}\unicode[STIX]{x1D713}$ leads to $h_{1}=(c_{2}/6)x^{2/3}$ . Additionally, we have the following non-homogeneous condition for the core flow:

which provides the third boundary condition, $w_{3}(x,0)=-\unicode[STIX]{x1D713}_{3x}(x,0)=-2$ , needed for the solution of $\unicode[STIX]{x1D713}_{3}(x,z)$ in the core layer (see § 4.1). The height of the free surface is then given by

Figure 2 illustrates the influence of inertia on the jet thickness (figure 2 a), the streamwise (figure 2 b) and the transverse (figure 2 c) velocity components when plotted against distance from the channel exit. The profiles in figure 2(a) suggest that the jet always contracts near the exit. Although the jet is expected to eventually expand as a result of increasing friction along the upper wall, the maximum contraction and ensuing expansion, typically illustrated for $\unicode[STIX]{x1D700}=0.5$ , may not be accurately reflected here as $x$ must be $O(\unicode[STIX]{x1D700}^{-3})$ for (3.25) to remain uniformly valid. Moreover, the current formulation does not account for any interaction between the free-surface and wall layers (see § 3.3.2 for further discussion). The location of the maximum turns out to be simply proportional to the Reynolds number, namely $x_{m}=-(3c_{1}/\unicode[STIX]{x1D700}c_{2})^{3}=0.126Re$ . Interestingly, the actual value of the maximum surface height, $\unicode[STIX]{x1D701}_{m}=\unicode[STIX]{x1D701}(x=x_{m})$ , is independent of the Reynolds number, and is given by $\unicode[STIX]{x1D701}_{m}=-3c_{1}^{2}/2c_{2}=0.123$ . In contrast, the ensuing expansion depends strongly on inertia. Figure 2(b) displays the $u(x,z=\unicode[STIX]{x1D701})$ profiles, indicating velocity growth and decay with $x$ for the contracting and expanding jet portions respectively. As the jet expands, it behaves like typical wall jets, with the velocity decreasing linearly with distance (see figure 4 in Bakke (Reference Bakke1957)). The apparent decrease of the surface velocity with increasing inertia depicted in figure 2(b) is due to the scaling used. More important is the rapid increase in elongation rate near the exit, $u_{x}(x=0,z=0)$ , as inertia decreases. Elongation is also reflected in the jump in transverse flow (see figure 2 c). Given that the shearing near the exit does not change significantly, the relatively dominant elongation rate elsewhere causes the pressure to adjust suddenly (above the surface), suggesting the potential formation of an adverse pressure gradient resulting from the balance in streamwise linear momentum, which, in turn, may be conducive to flow separation at the upper wall. Flow reversal and separation in the current problem are the result of jet rapid contraction and expansion. The potential for separation will be discussed further when the pressure and shear stress are examined along the upper wall.

Figure 2.

The influence of inertia on the jet thickness (a), streamwise (b) and transverse (c) velocity components, plotted against distance downstream of the channel exit.

Finally, the current treatment precludes a distance downstream very close to the channel exit (origin) where the stress singularity is located. In the free-surface layer, the situation is similar to classical boundary-layer flow, where the similarity solution is not valid near the singularity (leading edge). Incidentally, it is important to observe that, in addition to the stress singularity, the jet also experiences a geometrical singularity as the slope of the jet surface is discontinuous at $x=0$ . The free-surface height becomes itself discontinuous when higher-order terms are sought (Tillett Reference Tillett1968).

3.2 The flow in the lower-wall layer upstream of the exit ( $x<0$ )

To examine the boundary-layer structure near the lower wall upstream of the channel exit, we introduce the near-wall coordinates as $x=\unicode[STIX]{x1D709}$ and $y=z=\unicode[STIX]{x1D700}\unicode[STIX]{x1D702}$ . As in the case near the free surface, matching with the core flow inside the channel shows that $\unicode[STIX]{x1D713}\sim 3y^{2}=3\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D702}^{2}$ as $\unicode[STIX]{x1D702}\rightarrow \infty$ . Thus, we must have $\unicode[STIX]{x1D713}=O(\unicode[STIX]{x1D700}^{2})$ near the lower wall. In this case, the momentum conservation equations, along with the wall and upstream conditions, are

The velocity components become $u=(1/\unicode[STIX]{x1D700})\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D702}}$ and $w=-\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D709}}$ . Noting that $p$ is of order $\unicode[STIX]{x1D700}^{2}$ , the flow field expansion reads

Upon inserting (3.28) and (3.29) into (3.27), a hierarchy of problems is obtained to various orders. Each problem is of the boundary-layer type, requiring only one boundary condition in the streamwise direction. Consequently, the far-upstream condition in (3.27c,d ) is sufficient for well-posedness. This confirms that no matching is required with the free-surface-layer flow at the channel exit ( $x=0$ ). In this regard, a streamwise rescaling (of $x$ ) is not necessary unless we wish to capture the flow structure very close to the origin. Consequently, $x$ is supposed to remain at least of $O(1)$ ; the current treatment precludes a distance upstream very close to the origin. This is not the case for the flow in a constricted (dilated) channel, for instance, where $x$ is rescaled in terms of the inverse power of the indentation slope (and the Reynolds number), allowing the rapidly adjusting flow to be captured close to the inception of the constriction (dilation). In that case, the rescaling is necessary in order to determine the flow through and past the constriction (dilation) (Smith Reference Smith1976a ,Reference Smith b ). In contrast, the current flow is not continuous through the singularity. Of closer relevance to the flow in the lower-wall layer is the flow near the trailing edge of a flat plate, where $x$ needs to be rescaled only if the flow very close to the singularity is to be determined; that is, when the distance is of the same order as the boundary-layer thickness. Although an analysis similar to that of the flow near the trailing edge could be envisaged, such an extensive analysis is not necessary here. In the lower-wall layer very near the origin, as near the trailing edge, the boundary-layer assumption breaks down as both elongation and shearing become equally dominant, and the full Navier–Stokes equations must be solved (see § 3 of Talke & Berger (Reference Talke and Berger1970)). Since the boundary-layer thickness turned out to be of order $1/Re$ , our results must be assumed to be invalid when $O(Re^{-1})<x<0$ .

We now consider the matching between the lower-wall and the core layers in order to derive the additional boundary conditions needed to solve problem (3.27). We recall first that in the core layer, the perturbations $\unicode[STIX]{x1D713}_{1}(x,z)=\unicode[STIX]{x1D713}_{2}(x,z)=0$ , so that (4.1) is conveniently reproduced here as

Clearly, $E_{0}\unicode[STIX]{x1D713}=3y^{2}-2y^{3}$ , so that $H_{3}E_{0}\unicode[STIX]{x1D713}=3\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D702}^{2}-2\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D702}^{3}=3y^{2}-2y^{3}$ . Moreover, from (3.28), we have $H_{3}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D6F9}_{2}+\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D6F9}_{3}$ . Setting $H_{3}E_{0}\unicode[STIX]{x1D713}=E_{0}H_{3}\unicode[STIX]{x1D713}$ , we deduce that $\unicode[STIX]{x1D6F9}_{2}\sim 3\unicode[STIX]{x1D702}^{2}$ and $\unicode[STIX]{x1D6F9}_{3}\sim -2\unicode[STIX]{x1D702}^{3}$ for large $\unicode[STIX]{x1D702}$ . In this case, it is not difficult to show that the solutions to $O(\unicode[STIX]{x1D700}^{2})$ and $O(\unicode[STIX]{x1D700}^{3})$ , which satisfy the homogeneous conditions at the wall and far-upstream condition in (3.27c,d ), are

Thus, to $O(\unicode[STIX]{x1D700}^{3})$ , the flow retains its Poiseuille character near the lower wall.

Before proceeding to the next order, we need to determine the inner value $\unicode[STIX]{x1D713}_{3}(x<0,z\rightarrow 0)$ for the core flow problem. To this end, we note first from (3.30) that

and, from (3.31) and (3.32), that

This leads to $H_{3}E_{3}\unicode[STIX]{x1D713}=3y^{2}-2y^{3}+\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D713}_{3}(x,0)$ and $E_{3}H_{3}\unicode[STIX]{x1D713}=3y^{2}-2y^{3}$ , yielding

which is the condition needed to determine the core flow upstream of the channel exit (refer to problem (4.5) in the next section).

We turn next to finding the matching condition needed, in addition to the wall conditions, to determine the disturbances $\unicode[STIX]{x1D6F9}_{4}$ and $P_{4}$ , which reflect the deviation from Poiseuille flow. The equations for $\unicode[STIX]{x1D6F9}_{4}$ and $P_{4}$ are

In this case, $P_{4}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})\equiv P_{4}(\unicode[STIX]{x1D709})$ . Clearly, (3.36) confirms that the problem is of the boundary-layer type, requiring only one boundary condition in the streamwise direction. Taking $m=4$ and $n=3$ in (3.1), and comparing $H_{4}E_{3}\unicode[STIX]{x1D713}=3y^{2}-2y^{3}+\unicode[STIX]{x1D700}^{3}y\unicode[STIX]{x1D713}_{3y}(x,z=0)=3\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D702}^{2}-2\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D702}^{3}+\unicode[STIX]{x1D700}^{4}\unicode[STIX]{x1D702}\unicode[STIX]{x1D713}_{3z}(x,z=0)$ with $E_{3}H_{4}\unicode[STIX]{x1D713}=3\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D702}^{2}-2\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D702}^{3}+\unicode[STIX]{x1D700}^{4}\unicode[STIX]{x1D6F9}_{4}$ yields the desired matching condition, namely $\unicode[STIX]{x1D6F9}_{4}\sim \unicode[STIX]{x1D702}\unicode[STIX]{x1D713}_{3z}(x,z=0)$ for large $\unicode[STIX]{x1D702}$ , where $\unicode[STIX]{x1D713}_{3z}(\unicode[STIX]{x1D709},z=0)=\sum _{n=1}^{\infty }(A_{n}/\unicode[STIX]{x1D6FD}_{n})\text{e}^{\unicode[STIX]{x1D6FD}_{n}\unicode[STIX]{x1D709}}$ is deduced from (4.9) or (4.10). We note that $V_{n}^{\prime }(z=0)=1$ from the solution of the eigenvalue problem (4.8), with $\unicode[STIX]{x1D6FD}_{n}$ being the eigenvalues. We also note that the coefficients $A_{n}$ are obtained from matching the flow at $x=0$ in the core layer as per (4.18). The values for the first eight modes are reported in table 1 below.

Upon eliminating the pressure gradient in (3.36) and using (3.31), the problem for $\unicode[STIX]{x1D6F9}_{4}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ becomes $(\unicode[STIX]{x1D709}<0)$

Problem (3.37) is solved by seeking a solution of the form $\unicode[STIX]{x1D6F9}_{4}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=\sum _{n=1}^{\infty }(A_{n}/\unicode[STIX]{x1D6FD}_{n})\text{e}^{\unicode[STIX]{x1D6FD}_{n}\unicode[STIX]{x1D709}}F_{n}(\unicode[STIX]{x1D702})$ , where the functions $F_{n}(\unicode[STIX]{x1D702})$ are governed by

This is a two-point boundary-value problem, which is solved numerically for each mode separately. A finite-difference scheme with differed corrections (IMSL-DBVPFD) is used. Figure 3 depicts the profiles of $F_{n}^{\prime }(\unicode[STIX]{x1D702})$ for the first eight modes, which reflect a monotonic growth of $F_{n}(\unicode[STIX]{x1D702})$ and its derivatives. More importantly, we observe the significant departure of the function for the first mode (corresponding to $\unicode[STIX]{x1D6FD}_{1}=0.17$ ) from the rest, which signals the dominance of the first mode on the velocity, as will be confirmed later (see table 1 in § 4). Given the physical significance of $F_{n}^{\prime \prime }(\unicode[STIX]{x1D702}=0)$ and $F_{n}^{\prime \prime \prime }(\unicode[STIX]{x1D702}=0)$ , and their relation to the shear stress at the wall and pressure respectively, we include their values in the inset of figure 3 as well as table 1. In particular, we find that $F_{n}^{\prime \prime }(\unicode[STIX]{x1D702}=0)>0$ and $F_{n}^{\prime \prime \prime }(\unicode[STIX]{x1D702}=0)<0$ for all modes, which indicates that the wall shear stress increases upon approaching the channel exit, with the pressure gradient becoming increasingly negative, as we shall see shortly (refer to expressions (3.41) and (3.42)).

Figure 3.

The profiles of $F_{n}^{\prime }(\unicode[STIX]{x1D702})$ for the first eight modes. Also shown are the (discrete) values of $F_{n}^{\prime \prime }(\unicode[STIX]{x1D702}=0)$ and $F_{n}^{\prime \prime \prime }(\unicode[STIX]{x1D702}=0)$ in the inset. Successive points are joined for guidance.

Table 1.

Values for the first eight modes.

To the current order, the velocity components near the lower wall are then given by

The asymptote is then $u(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}\rightarrow \infty )\sim 6\unicode[STIX]{x1D700}\unicode[STIX]{x1D702}(1-\unicode[STIX]{x1D700}\unicode[STIX]{x1D702})+\unicode[STIX]{x1D700}^{3}\sum _{n=1}^{\infty }(A_{n}/\unicode[STIX]{x1D6FD}_{n})\text{e}^{\unicode[STIX]{x1D6FD}_{n}\unicode[STIX]{x1D709}}$ . Figure 4 depicts the flow development near the lower wall, where the profiles of the disturbances in velocity in the streamwise (figure 4 a) and transverse (figure 4 b) directions, $U_{3}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=\sum _{n=1}^{\infty }(A_{n}/\unicode[STIX]{x1D6FD}_{n})\text{e}^{\unicode[STIX]{x1D6FD}_{n}\unicode[STIX]{x1D709}}F_{n}^{\prime }(\unicode[STIX]{x1D702})$ and $W_{4}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=-\sum _{n=1}^{\infty }A_{n}\text{e}^{\unicode[STIX]{x1D6FD}_{n}\unicode[STIX]{x1D709}}F_{n}(\unicode[STIX]{x1D702})$ respectively, are shown at equal intervals for $-10<x<0$ . Both disturbances vanish exponentially in the limit $x\rightarrow -\infty$ . As expected, the asymptote, which reflects the extent of the (inviscid) character of the core flow, does not satisfy adherence at the lower (or upper) wall. Within the wall layer, the profiles exhibit an increase in shearing, confirming the increase in wall shear stress anticipated earlier. There is also a significant elongation as the fluid approaches the exit, which is particularly reflected in drastic departure in the $W_{4}(x=0,z)$ from zero.

Figure 4.

Flow development near the lower wall. Streamwise (a) and transverse (b) velocity disturbance profiles are shown at equal intervals for $-10<x<0$ . Asymptotes are also shown for $U_{3}$ .

Figure 5 depicts the influence of inertia on the thickness of the lower-wall layer, which is estimated as the height reached when the relative difference between $u(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ and its asymptote is less than 1 %. In this case, we roughly have the following behaviour: $z\propto Re^{-1}\text{e}^{\unicode[STIX]{x1D6FD}_{1}x}$ for the boundary-layer thickness. Figure 5 indicates that, for any $\unicode[STIX]{x1D700}$ , the height increases rather rapidly with upstream distance. The behaviour is almost exponential, contrasting the much slower $\sqrt{x}$ behaviour typical of boundary-layer growth. The height of the lower layer can be rather significant, reaching 30 % into the channel gap for $\unicode[STIX]{x1D700}=0.3$ ( $Re=37$ ) and essentially vanishing for $\unicode[STIX]{x1D700}<0.1$ ( $Re>1000$ ). The onset position of the wall layer $x_{0}$ coincides with the disturbance of $\unicode[STIX]{x1D700}^{3}\sum _{n=1}^{\infty }(A_{n}/\unicode[STIX]{x1D6FD}_{n})\text{e}^{\unicode[STIX]{x1D6FD}_{n}\unicode[STIX]{x1D709}}$ from zero. Thus, keeping the dominant mode, and assuming the onset of the wall layer to coincide with a disturbance larger than $10^{-2}$ , we roughly have the following logarithmic dependence on $Re$ :

where $\unicode[STIX]{x1D6FD}_{1}=0.17$ and $A_{1}=2.43$ . The inset in figure 5 gives the onset position for $0<Re<1000$ . Obviously, the flow tends to retain its Poiseuille character closer to the exit for larger Reynolds number. We finally note that for very small $Re$ (a range that is not necessarily accurately treated in the current formulation), the upstream recovery distance is approximately 20 times the channel gap. Figure 5 also illustrates the extent of upstream influence, which, comparatively, is essentially absent for the flow over a mild constriction in a channel (Smith Reference Smith1976a ,Reference Smith b ) and is manifested for severe constrictions (Smith Reference Smith1977, Reference Smith1979).

Figure 5.

The influence of inertia on the height of the lower-wall layer, upstream of the channel exit ( $x<0$ ). The inset shows the logarithmic dependence on $Re$ of the onset position of the wall layer.

Figure 6.

The influence of inertia on the pressure (a) and skin friction (b) along the lower wall.

Subject to its vanishing far upstream, the pressure correction becomes $P_{4}(x)=\sum _{n=1}^{\infty }(A_{n}/\unicode[STIX]{x1D6FD}_{n}^{2})\text{e}^{\unicode[STIX]{x1D6FD}_{n}x}F_{n}^{\prime \prime \prime }(\unicode[STIX]{x1D702}=0)$ . In this case, the pressure and wall shear stress are given by

The shear stress at the wall turns out to be always positive (refer to the inset in figure 3 and table 1). Therefore, there is no possibility of separation upstream of the channel exit along the lower wall. Figure 6 illustrates the influence of inertia on the development of the pressure (figure 6 a) and the shear stress along the lower wall (figure 6 b) upstream of the channel exit. The pressure remains close to the Poiseuille level, decreasing essentially linearly all the way to the exit, but reaches a negative level at the exit. In contrast, the level of shearing increases rather rapidly for the range of Reynolds number reported in the figure, almost doubling for $\unicode[STIX]{x1D700}=0.5$ ( $Re=8$ ), reflecting the flattening of the velocity profile near the exit, as shown in figure 4. We observe that both the shear stress and the pressure are singular at $x=0$ , as the fluid moves along the lower wall, and emerges along the jet free surface, experiencing vanishing shear stress and pressure. However, upstream of the exit, the pressure and shear stress are well behaved.

3.3 The flow near the upper wall upstream and downstream of the channel exit

We now examine the boundary-layer structure near the infinite upper wall, upstream and downstream of the exit. In this case, the scaling in the transverse direction is changed by writing

Matching with the core flow suggests that $\unicode[STIX]{x1D713}\sim 1$ as $\unicode[STIX]{x1D702}\rightarrow \infty$ (see below). In this case, the balance between the convective and viscous terms leads to the following problem:

The expansions of the streamfunction and pressure take the form

Near the upper wall, equation (4.1) may be written as

Clearly, $E_{0}\unicode[STIX]{x1D713}=1-3y^{2}+2y^{3}$ , so that $H_{3}E_{0}\unicode[STIX]{x1D713}=1-3\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D702}^{2}+2\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D702}^{3}$ . Moreover, from (3.45a ), we have $H_{3}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D6F9}_{0}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D6F9}_{1}+\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D6F9}_{2}+\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D6F9}_{3}$ . Setting $H_{3}E_{0}\unicode[STIX]{x1D713}=E_{0}H_{3}\unicode[STIX]{x1D713}$ , we deduce, as expected, that $\unicode[STIX]{x1D6F9}_{0}\sim 1,\unicode[STIX]{x1D6F9}_{1}\sim 0$ , $\unicode[STIX]{x1D6F9}_{2}\sim -3\unicode[STIX]{x1D702}^{2}$ and $\unicode[STIX]{x1D6F9}_{3}\sim 2\unicode[STIX]{x1D702}^{3}$ for large $\unicode[STIX]{x1D702}$ . In this case, it is not difficult to show that the solutions to $O(1),O(\unicode[STIX]{x1D700})$ , $O(\unicode[STIX]{x1D700}^{2})$ and $O(\unicode[STIX]{x1D700}^{3})$ , which satisfy the homogeneous conditions at the upper wall and far-upstream condition in (3.44c-e ), are

Thus, to $O(\unicode[STIX]{x1D700}^{3})$ , near the upper wall, as in the lower-wall layer, the flow retains its Poiseuille character. The flow field is then given by

At this stage, we need to determine the outer limit $\unicode[STIX]{x1D713}_{3}(x,z\rightarrow 1)$ for the core flow problem. To this end, we note first from (3.46) that $H_{3}E_{3}\unicode[STIX]{x1D713}=1-3y^{2}+2y^{3}+\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D713}_{3}(x,z=1)$ , which must match with $E_{3}H_{3}\unicode[STIX]{x1D713}=1-3y^{2}+2y^{3}$ , yielding (for any $x$ )

which is the condition needed to determine the core flow upstream and downstream of the channel exit (refer to problem (4.5) in the next section).

We next consider the problem for $\unicode[STIX]{x1D6F9}_{4}$ and $P_{4}$ , which is similar to (3.36) both upstream and downstream of the channel exit, but is considerably more difficult for $x>0$ . For any $x$ , upon eliminating the pressure, the governing equation and upper-wall conditions for $\unicode[STIX]{x1D6F9}_{4}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ become

In addition, two boundary conditions are needed. However, they depend on whether the upper-wall layer lies upstream or downstream of the channel exit. For any $x$ , we have $H_{4}E_{3}\unicode[STIX]{x1D713}=1-3y^{2}+2y^{3}-\unicode[STIX]{x1D700}^{3}y\unicode[STIX]{x1D713}_{3z}(x,z=1)$ , which must match with $E_{3}H_{4}\unicode[STIX]{x1D713}=1-3y^{2}+2y^{3}+\unicode[STIX]{x1D700}^{4}\unicode[STIX]{x1D6F9}_{4}$ , yielding the condition

We observe that $\unicode[STIX]{x1D713}_{3z}(x,z=1)$ is not the same for $x<0$ and $x>0$ .

4.1 The flow in the core layer upstream of the channel exit

Consider now the core flow in the region inside the channel $(x<0)$ . The solution of problem (4.6) is written as (Tillett Reference Tillett1968)

The shape functions $V_{n}$ are different from those for a free jet, and are governed by the following eigenvalue problem:

which is solved subject to the additional boundary condition $V_{n}^{\prime }(0)=1$ . In this case, $V_{n}^{\prime \prime }(0)=-2$ and $V_{n}^{\prime \prime }(1)=2V_{n}^{\prime }(1)$ . In addition, the slope at the boundary turns out to be $V_{n}^{\prime }(1)=(-1)^{n}$ , which is established numerically (we could not confirm it analytically). Table 1 shows the eigenvalues for the first eight modes. Also included in the table are the corresponding values of the expansion coefficients $A_{n}$ (to be determined shortly), and the values of $F_{n}^{\prime \prime }(0)$ and $F_{n}^{\prime \prime \prime }(0)$ (determined earlier in § 3).

The streamfunction is obtained by integrating expression (4.7) subject to $\unicode[STIX]{x1D713}_{3}(x\rightarrow -\infty ,z)=0$ , leading to

The coefficients $A_{n}$ are obtained by matching the flow at the channel exit, and will therefore be determined once the core flow is considered for $x>0$ . It is clear, upon evaluating (4.10) at the channel walls, that the core solution does not satisfy adherence. A similar situation is encountered for the flow in a channel with an indentation. Smith (Reference Smith1976a ,Reference Smith b ) obtained the disturbance in velocity in the form $A(x)u_{0}^{\prime }(z)$ , where $A(x)$ was directly expressed in terms of the channel indentation geometry, as indicated in Smith’s expressions (2.6) and (2.18). A couple of remarkable differences are worth mentioning here. Like the disturbance in (4.10), $A(x)$ must also vanish far upstream, but not exponentially. In contrast to the present wall-jet flow, no upstream influence was predicted in Smith’s problem. For an indented channel, $-A(x)$ represents the displacement of the streamlines in the core from their original Poiseuille position. In that case, the displacement of the core at a given position $x$ turned out to be exactly the average of the displacements of the wall (indentations) at that position. Comparably, the disturbance in (4.10) also represents a streamline displacement for the current jet flow, but, unlike the case of an indented channel, there is no direct or obvious link between the disturbance in velocity (or streamfunction) and the shape of the disturbed jet surface. Ultimately, the link is ensured through the values of the coefficients $A_{n}$ , which are determined upon matching the core flow at the channel exit (see the following section).

4.2 The flow in the core layer downstream of the channel exit

For $x>0$ , the solution of problem (4.6) must match the flow at the exit as well as the non-homogeneous condition in (4.6b ), and is therefore taken as a combination of streamwise exponentially decaying contributions in terms of the shape functions $V_{n}(z)$ that satisfy the homogeneous conditions and a function $U(x,z)$ that ensures that $w_{3}(x>0,z\rightarrow 0)\sim -2$ is satisfied. We then have

where $U(x,z)$ satisfies the following equations and boundary conditions:

The problem is converted to an integral equation by first noting that the equation in (4.12) can be rewritten as $u_{0}U_{xx}+u_{0}U_{zz}-u_{0}^{\prime \prime }U=u_{0}U_{xx}+(u_{0}U_{z})_{z}-(u_{0}^{\prime }U)_{z}=0$ . Integration over the interval $0\leqslant z\leqslant 1$ and use of the condition $U(x,z=1)=0$ yield

where $C_{0}$ and $C_{1}$ are constants to be determined. Thus, we let

subject to the constraint $\int _{0}^{1}u_{0}U_{2}(z)\,\text{d}z=-3$ . In this case, the constants are determined through $\int _{0}^{1}u_{0}U_{1}(z)\,\text{d}z=C_{1}$ and $\int _{0}^{1}u_{0}U_{0}(z)\,\text{d}z=C_{0}$ . Substituting (4.14) into the equation (4.12), we obtain the following problems for the coefficients $U_{0}(z),U_{1}(z)$ and $U_{2}(z)$ :

Clearly, the equations and homogeneous conditions in (4.15d,e ) and (4.15f–h ) possess an infinite number of solutions of the form $U_{1}\propto u_{0}$ and $U_{2}\propto u_{0}$ . Without loss of generality, we choose $U_{1}(z)=0$ as the solution of (4.15d,e ), leading to $C_{1}=0$ . It is not difficult to show that (4.15f–h ) admits $U_{2}(z)=-(5/2)u_{0}(z)$ as solution. We now show that (4.15a–c ) admits an analytical solution for $U_{0}(z)$ .

The problem is first reduced to $u_{0}U_{0}^{\prime }-u_{0}^{\prime }U_{0}+6=5\int _{0}^{z}u_{0}^{2}\,\text{d}z=6z^{3}(6z^{2}-15z+10)$ and integrated to obtain a general solution of the form

We note that (4.16) satisfies both boundary conditions. The constant $C$ is determined by first recalling that the slope must satisfy $u_{0}U_{0}^{\prime }-u_{0}^{\prime }U_{0}+6=6z^{3}(6z^{2}-15z+10)$ , yielding $U_{0}^{\prime }(z)=((1-2z)U_{0}(z)+6z^{5}-15z^{4}+10z^{3}-1)/(z-z^{2})$ . We observe from (4.16) that $U_{0}(z)$ is well behaved near $z=1$ . In this case, taking the limit as $z$ approaches 1, we can easily show that $U_{0}^{\prime }(z=1)\sim -U_{0}^{\prime }(z=1)$ , which indicates that $U_{0}^{\prime }(z=1)=0$ . Consequently, upon evaluating the derivative of (4.16) at $z=1$ , and setting it to zero, we deduce that $C=1$ . Finally, the expression for $U(x,z)$ becomes

Figure 10 depicts the $U(x,z)$ profiles at equal downstream intervals. The variation of $U$ with $x$ illustrates its growing dominance with distance, eventually increasing quadratically with $x$ , and dominating the exponential decay of the flow according to (4.11). This rapid growth is also reflected in figure 7, and the rapid decrease in pressure and shear stress in the upper-wall layer, as reported in figure 8.

Figure 10.

The $U(x,z)$ profiles at different downstream positions between $x=0$ and $x=1$ , at equal intervals of 0.1.

As mentioned earlier, the coefficients $A_{n}$ are obtained by matching the flow at the channel exit. In this case, equating (4.7) and (4.11) at $x=0$ leads to

which is a spectral representation of $U(x=0,z)$ in terms of the functions $V_{n}(z)$ . The orthogonality of $V_{n}(z)$ over the interval $[0,1]$ is shown in appendix A. It is not difficult to show that, upon multiplying the equations in (4.8) and (4.12) by $U_{0}$ and $V_{n}$ respectively, integrating and using the conditions therein, one arrives at $\langle V_{n}U_{0}\rangle =(1-5\langle u_{0}V_{n}\rangle )/\unicode[STIX]{x1D6FD}_{n}^{2}$ , where the brackets denote integration over the interval $0\leqslant z\leqslant 1$ . Consequently, upon projection, (4.18) yields

The values of the coefficients $A_{n}$ ultimately reflect the influence of the disturbance in the jet surface on the flow upstream of the channel exit. However, this influence remains somewhat obscure or indirect, but could be seen through the boundary condition $U(x,z=0)$ in (4.12), the latter being derived from matching the core flow to the flow in the free-surface layer. This property is reminiscent of the boundary-layer flow over a flat surface with an excrescence, where the streamline is also not directly relatable to the solid geometry. Instead, the displacement is related to the disturbance in pressure generated by the indentation. As a result, and as in the current jet flow, the boundary layer exhibits an upstream influence by adjusting to the presence of the excrescence (Smith Reference Smith1976a ,Reference Smith b ; Sobey Reference Sobey2000).

Finally, the streamfunction and streamwise velocity are determined upon integrating (4.11) and matching with (4.9) at $x=0$ to give

The pressure in the core layer is considered next.

4.3 The pressure in the core layer

We now consider the pressure distribution in the core layer, upstream and downstream of the channel exit. For $x<0$ , the pressure is determined by integrating (4.4b ) subject to appropriate boundary conditions, which are obtained as follows. First, we recall from § 3.1 that the pressure in the inner region outside the channel ( $x>0$ ) was shown to be $O(\unicode[STIX]{x1D700}^{4})$ . Consequently, upon matching the pressures in the core and free-surface layers for $x$ ${>}0$ , it is not difficult to deduce that the pressure in the core layer outside the channel vanishes at the interface between the two layers, or

Thus, at the channel exit, this gives $p_{3}(x=0,z\rightarrow 0)=0$ . Now, upon evaluating (2.1) and (4.7) at $z=0$ , (4.4a ) reduces to

Integrated (4.23) subject to $p_{3}(0,z\rightarrow 0)=0$ gives

Upon inserting expressions (2.1) and (4.7) into (4.4b ), and using (4.8), the expression for the transverse pressure gradient inside the channel becomes

which is integrated subject to condition (4.24), leading finally to

The pressure downstream of the exit is determined upon inserting expressions (2.1) and (4.20) into (4.4b ). In this case, an expression similar to (4.25) is obtained for $x>0$ , which is integrated and matched with (4.26) at $x=0$ to give

We note that, to this order, the pressure near the upper wall reduces predominantly to the Poiseuille level, matching the wall-layer pressure in § 3.3. Unlike the case of a symmetric jet, the pressure eventually increases linearly with $x$ as a result of increased shearing at the upper wall. However, this linear growth subsides gradually with distance below the wall, to eventually vanish along the jet surface. The pressure deviation at a given height in the core layer inside and outside the channel is thus given by expressions (4.26) and (4.27) respectively. Incidentally, these expressions remain valid in both the core and boundary layers as $\unicode[STIX]{x1D700}^{3}p_{3}(x,z)$ turns out to be equal to the composite pressure. In other words, the pressure everywhere is readily available and is given by $p(x,z)=\unicode[STIX]{x1D700}^{3}p_{3}(x,z)+O(\unicode[STIX]{x1D700}^{4})$ .

Figure 11.

The development of the pressure with distance at different height locations in the core region, upstream and downstream of the channel. The results are valid for any Reynolds number.

Figure 11 displays the pressure development at different heights in the core layer. The pressure decreases essentially linearly with $x$ upstream of the channel exit. The saturation of pressure profiles near the upper wall suggests that the pressure remains close to the fully developed level. However, as we saw in § 3.3, the pressure departs significantly from its core level in the upper-wall layer. The behaviour in figure 11 is very different from that predicted for a symmetric free jet, where it tends to vanish essentially for any height downstream of the channel (see figure 3 in Tillett (Reference Tillett1968)). The saturation in figure 11 also shows how little the flow near the upper wall is affected by the jet profile; only the lower half of the core layer is essentially affected. An interesting connection may be established with a shear-thinning fluid. We refer the reader to figure 11(b) of Khayat (Reference Khayat2014), which shows that the pressure follows $-12x$ behaviour for a strongly shear-thinning fluid, which is similar to the present pressure behaviour for $z>0.7$ . In both cases, it is the inviscid character of the core flow that is responsible for the linear behaviour of the pressure, as a result, however, of two distinct mechanisms: the remoteness from the free surface in the present case and shear thinning in the other case.