3.1 Flow in the inner layer (near the jet surface)

To examine the boundary-layer structure near the free surface, let $y=z-{\it\zeta}(x)$ . The scaling in the transverse direction is changed by writing $y={\it\varepsilon}{\it\eta}$ , where ${\it\varepsilon}$ is the small parameter in the problem, defined as ${\it\varepsilon}=Re^{-{\it\alpha}}$ (or $Re={\it\varepsilon}^{-1/{\it\alpha}}$ ), where ${\it\alpha}$ is to be determined. Anticipating that the height ${\it\zeta}$ of the free surface is of the same order of magnitude as the boundary-layer thickness, one can write ${\it\zeta}(x)={\it\varepsilon}h(x)$ , and henceforth work with $h(x)$ . Following Tillett (Reference Tillett1968) and Philippe & Dumargue (Reference Philippe and Dumargue1991), the following change of coordinates is introduced, namely,

(3.1a,b ) $$\begin{eqnarray}x={\it\xi},\quad z={\it\varepsilon}({\it\eta}+h).\end{eqnarray}$$

The aim is to find a solution of the transformed problem (2.3)–(2.5) in the form of an inner expansion in ${\it\varepsilon}$ . In order to match this to the outer Poiseuille flow, it is necessary to have ${\it\psi}\sim {\it\varepsilon}{\it\eta}=y$ as ${\it\eta}\rightarrow \infty$ in the inner region, to lowest order in ${\it\varepsilon}$ ; so ${\it\psi}$ must be of order ${\it\varepsilon}$ .

In order to determine the value of ${\it\alpha}$ , it is required that the convective and viscous terms balance in the governing equations. The balance in the boundary conditions simply follows. The transformed equations and boundary conditions are given in appendix A. Recalling that ${\it\psi}=O({\it\varepsilon})$ , the force balance in (A 1) is ensured upon taking ${\it\alpha}=1/2$ . Consequently, $Re={\it\varepsilon}^{-2}$ or ${\it\varepsilon}=Re^{-1/2}$ , which signals the major departure from previous formulations where ${\it\alpha}=1/3$ for non-slipping Newtonian (Tillett Reference Tillett1968; Philippe & Dumargue Reference Philippe and Dumargue1991) and non-Newtonian (Saffari & Khayat Reference Saffari and Khayat2009; Khayat Reference Khayat2014) jets.

Consequently, the problem in the inner layer reduces to

(3.2a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\it\psi}_{{\it\eta}}{\it\psi}_{{\it\xi}{\it\eta}}-{\it\psi}_{{\it\xi}}{\it\psi}_{{\it\eta}{\it\eta}}=-{\it\varepsilon}^{2}(p_{{\it\xi}}-h^{\prime }p_{{\it\eta}})+{\it\varepsilon}{\it\psi}_{{\it\eta}{\it\eta}{\it\eta}}+{\it\varepsilon}^{3}({\it\psi}_{{\it\xi}{\it\xi}{\it\eta}}-h^{\prime \prime }{\it\psi}_{{\it\eta}{\it\eta}}-2h^{\prime }{\it\psi}_{{\it\xi}{\it\eta}{\it\eta}}+h^{\prime 2}{\it\psi}_{{\it\eta}{\it\eta}{\it\eta}}),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.2b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle -{\it\psi}_{{\it\eta}}{\it\psi}_{{\it\xi}{\it\xi}}+{\it\psi}_{{\it\xi}}{\it\psi}_{{\it\xi}{\it\eta}}+h^{\prime \prime }{\it\psi}_{{\it\eta}}^{2}+h^{\prime }({\it\psi}_{{\it\eta}}{\it\psi}_{{\it\xi}{\it\eta}}-{\it\psi}_{{\it\xi}}{\it\psi}_{{\it\eta}{\it\eta}})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \qquad =-p_{{\it\eta}}-{\it\varepsilon}({\it\psi}_{{\it\xi}{\it\eta}{\it\eta}}-h^{\prime }{\it\psi}_{{\it\eta}{\it\eta}{\it\eta}})-{\it\varepsilon}^{3}\left(\frac{\partial }{\partial {\it\xi}}-h^{\prime }\frac{\partial }{\partial {\it\eta}}\right)^{3}{\it\psi},\end{eqnarray}$$

(3.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}({\it\xi},{\it\eta}=0)=0, & \displaystyle\end{eqnarray}$$

(3.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\varepsilon}h^{\prime }p({\it\xi},{\it\eta}=0)+[(1+{\it\varepsilon}^{2}h^{\prime 2}){\it\psi}_{{\it\eta}{\it\eta}}+{\it\varepsilon}^{2}h^{\prime \prime }{\it\psi}_{{\it\eta}}]_{{\it\xi},{\it\eta}=0}=0, & \displaystyle\end{eqnarray}$$

(3.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle p({\it\xi},{\it\eta}=0)+{\it\varepsilon}[2(1+{\it\varepsilon}^{2}h^{\prime 2}){\it\psi}_{{\it\xi}{\it\eta}}-h^{\prime }(1+{\it\varepsilon}^{2}h^{\prime 2}){\it\psi}_{{\it\eta}{\it\eta}}+{\it\varepsilon}^{2}h^{\prime }h^{\prime \prime }{\it\psi}_{{\it\eta}}]_{{\it\xi},{\it\eta}=0}=0. & \displaystyle\end{eqnarray}$$

${\it\psi}$

${\it\varepsilon}$

${\it\varepsilon}$

(3.4) $$\begin{eqnarray}{\it\psi}({\it\xi},{\it\eta})={\it\varepsilon}{\it\Psi}_{1}({\it\xi},{\it\eta})+{\it\varepsilon}^{2}{\it\Psi}_{2}({\it\xi},{\it\eta})+{\it\varepsilon}^{3}{\it\Psi}_{3}({\it\xi},{\it\eta})+\cdots \,.\end{eqnarray}$$

Similarly, $h$ is expanded as

(3.5) $$\begin{eqnarray}h({\it\xi})={\it\varepsilon}^{-1}{\it\zeta}({\it\xi})=h_{0}({\it\xi})+{\it\varepsilon}h_{1}({\it\xi})+\cdots \,.\end{eqnarray}$$

As to the pressure, (3.2b ) and condition (3.3c ) lead to $P_{0{\it\eta}}({\it\xi},{\it\eta})=0$ and $P_{0}({\it\xi},{\it\eta}=0)=0$ , respectively. Thus, $P_{0}({\it\xi},{\it\eta})=0$ everywhere. Similarly, $P_{1{\it\eta}}({\it\xi},{\it\eta})=0$ and $P_{1}({\it\xi},{\it\eta}=0)=0$ , leading to $P_{1}({\it\xi},{\it\eta})=0$ . Thus, it is concluded that the pressure in the inner layer is of order ${\it\varepsilon}^{2}$ , and

(3.6) $$\begin{eqnarray}p({\it\xi},{\it\eta})={\it\varepsilon}^{2}P_{2}({\it\xi},{\it\eta})+{\it\varepsilon}^{3}P_{3}({\it\xi},{\it\eta})+\cdots \,.\end{eqnarray}$$

The streamwise and transverse velocity components are now expressed as $u=(1/{\it\varepsilon}){\it\psi}_{{\it\eta}},w=-{\it\psi}_{{\it\xi}}+h^{\prime }{\it\psi}_{{\it\eta}}$ . It is obvious that $u=O(1)$ , which is also confirmed by considering the fact that $u$ in the inner region must match the velocity in the outer region: $u\rightarrow 4(S+z-z^{2})$ , and we must have $u\sim 4S$ as ${\it\eta}\rightarrow \infty$ . Clearly, $w$ is of order  ${\it\varepsilon}$ . In this case, the velocity components take the form:

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle u({\it\xi},{\it\eta})=U_{0}({\it\xi},{\it\eta})+{\it\varepsilon}U_{1}({\it\xi},{\it\eta})+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle w({\it\xi},{\it\eta})={\it\varepsilon}W_{1}({\it\xi},{\it\eta})+{\it\varepsilon}^{2}W_{2}({\it\xi},{\it\eta})+\cdots \,. & \displaystyle\end{eqnarray}$$

Here, $U_{0}={\it\Psi}_{1{\it\eta}},W_{1}=-{\it\Psi}_{1{\it\xi}}+h_{0}^{\prime }{\it\Psi}_{1{\it\eta}},U_{1}={\it\Psi}_{2{\it\eta}}$ , etc.

To leading order, (3.2a ) and (3.2b ) reduce, respectively, to

(3.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\Psi}_{1{\it\eta}}{\it\Psi}_{1{\it\xi}{\it\eta}}-{\it\Psi}_{1{\it\xi}}{\it\Psi}_{1{\it\eta}{\it\eta}}={\it\Psi}_{1{\it\eta}{\it\eta}{\it\eta}}, & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle -{\it\psi}_{1{\it\eta}}{\it\psi}_{1{\it\xi}{\it\xi}}+{\it\psi}_{1{\it\xi}}{\it\psi}_{1{\it\xi}{\it\eta}}+h_{0}^{\prime \prime }{\it\psi}_{1{\it\eta}}^{2}+h_{0}^{\prime }({\it\psi}_{1{\it\eta}}{\it\psi}_{1{\it\xi}{\it\eta}}-{\it\psi}_{1{\it\xi}}{\it\psi}_{1{\it\eta}{\it\eta}})=-p_{2{\it\eta}}-({\it\psi}_{1{\it\xi}{\it\eta}{\it\eta}}-h_{0}^{\prime }{\it\psi}_{1{\it\eta}{\it\eta}{\it\eta}}). & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\Psi}_{1}({\it\xi},{\it\eta}=0)={\it\Psi}_{1{\it\eta}{\it\eta}}({\it\xi},{\it\eta}=0)=0, & \displaystyle\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{2}({\it\xi},{\it\eta}=0)=-2{\it\Psi}_{1{\it\xi}{\it\eta}}({\it\xi},{\it\eta}=0). & \displaystyle\end{eqnarray}$$

${\it\Psi}_{1}$

(3.11) $$\begin{eqnarray}{\it\Psi}_{1}({\it\xi},{\it\eta})\sim 4S{\it\eta},\quad \text{as }{\it\eta}\rightarrow \infty .\end{eqnarray}$$

Clearly, ${\it\Psi}_{1}({\it\xi},{\it\eta})=4S{\it\eta}$ is the solution to this order. In this case, $P_{2}({\it\xi},{\it\eta}=0)=0$ . Consequently, the pressure gradient is given by $P_{2{\it\eta}}({\it\xi},{\it\eta})=-h_{0}^{\prime \prime }{\it\Psi}_{1{\it\eta}}^{2}=-16S^{2}h_{0}^{\prime \prime }$ , and $P_{2}({\it\xi},{\it\eta})=-16S^{2}h_{0}^{\prime \prime }{\it\eta}$ . Formally, the leading-order terms are

(3.12a,b ) $$\begin{eqnarray}{\it\Psi}_{1}({\it\xi},{\it\eta})=4S{\it\eta},\quad P_{2}({\it\xi},{\it\eta})=-16S^{2}h_{0}^{\prime \prime }{\it\eta}.\end{eqnarray}$$

To $O({\it\varepsilon}^{2})$ , using solution (3.12), (3.2a ) and boundary conditions (3.3) lead to

(3.13a,b ) $$\begin{eqnarray}4S{\it\Psi}_{2{\it\xi}{\it\eta}}={\it\Psi}_{2{\it\eta}{\it\eta}{\it\eta}},\quad {\it\Psi}_{2}({\it\xi},{\it\eta}=0)={\it\Psi}_{2{\it\eta}{\it\eta}}({\it\xi},{\it\eta}=0)=0.\end{eqnarray}$$

The solution of this problem is given in appendix B. The asymptotic form of ${\it\Psi}_{2}$ is obtained from (B 4) to read

(3.14) $$\begin{eqnarray}{\it\Psi}_{2}({\it\xi},{\it\eta})\sim 2{\it\eta}^{2}+\frac{{\it\xi}}{S},\quad \text{as }{\it\eta}\rightarrow \infty .\end{eqnarray}$$

This relation is sufficient to determine the shape of the jet surface in the matching process of § 4. However, the velocity along the jet surface becomes only available from the full solution of problem (3.13)–(3.14), which is outlined in appendix B. To this order, the velocity field in the inner layer is determined by recalling (3.4), (3.7) and (3.8), leading to

(3.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}({\it\xi},{\it\eta})=4{\it\varepsilon}S{\it\eta}+{\it\varepsilon}^{2}{\it\xi}f_{2}({\it\theta})+O({\it\varepsilon}^{3}), & \displaystyle\end{eqnarray}$$

(3.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle u({\it\xi},{\it\eta})=4S+{\it\varepsilon}{\it\xi}^{1/2}f_{2}^{\prime }({\it\theta})+O({\it\varepsilon}^{2}), & \displaystyle\end{eqnarray}$$

(3.15c ) $$\begin{eqnarray}\displaystyle & \displaystyle w({\it\xi},{\it\eta})=[4S+{\it\varepsilon}{\it\xi}^{1/2}f_{2}^{\prime }({\it\theta})]{\it\zeta}^{\prime }(x)+{\it\varepsilon}^{2}\left[\frac{{\it\theta}}{2}f_{2}^{\prime }({\it\theta})-f_{2}({\it\theta})\right]+O({\it\varepsilon}^{3}), & \displaystyle\end{eqnarray}$$

Figure 3.

Influence of slip length on $f_{2}^{\prime }({\it\theta})$ for $Re=81$ , with the $S\rightarrow \infty$ curve also corresponding to the asymptotic behaviour for all $S$ . Inset depicts dependence of $f_{2}^{\prime }(0)$ on $S$ . Markers in the inset indicate the five cases emphasized throughout this study.

where $f_{2}({\it\theta})$ and $f_{2}^{\prime }({\it\theta})$ are recalled from (B 5a ) and (B 5b ), respectively. Figure 3 displays the influence of slip on $f_{2}^{\prime }({\it\theta})$ . The solution for large $S$ $(S\rightarrow \infty )$ is also included, which constitutes the asymptotic behaviour $f_{2}^{\prime }({\it\theta})\sim 4{\it\theta}$ as ${\it\theta}\rightarrow \infty$ for any $S$ . The figure shows that $f_{2}^{\prime }({\it\theta})$ adjusts itself to the asymptotic linear behaviour at a shorter distance from the channel exit (origin) with increasing slip, reflecting a thinning of the boundary layer (see below). The curve saturation is also evident as $f_{2}^{\prime \prime }({\it\theta})$ reaches the constant level 4. Of particular interest is the dependence of $f_{2}^{\prime }({\it\theta}=0)$ on $S$ , which is related to the flow at the jet surface, and is easily determined from (3.15b ) upon recalling (B 5b ) and noting that $\text{erf}(x=0)=0$ . In this case:

(3.16) $$\begin{eqnarray}f_{2}^{\prime }({\it\theta}=0;S)=\frac{4}{\sqrt{{\rm\pi}S}}.\end{eqnarray}$$

Given its significance, this function is shown in the inset of figure 3. As discussed next, the decrease of $f_{2}^{\prime }({\it\theta}=0;S)$ with $S$ suggests a weakening of the flow perturbation along the jet surface with slip relative to the Poiseuille level.

Indeed, the flow along the jet surface, including the velocity and shear stress ${\it\tau}_{xz}={\it\psi}_{zz}-{\it\psi}_{xx}$ is given through the following expressions in terms of $Re$ and $S$ :

(3.17a ) $$\begin{eqnarray}\displaystyle & \displaystyle u(x,z={\it\zeta})=4\left(S+\sqrt{\frac{x}{{\rm\pi}S\,Re}}\right)+O(Re^{-1}), & \displaystyle\end{eqnarray}$$

(3.17b ) $$\begin{eqnarray}\displaystyle & \displaystyle w(x,z={\it\zeta})=4\left(S+\sqrt{\frac{x}{{\rm\pi}S\,Re}}\right){\it\zeta}^{\prime }(x)+O(Re^{-3/2}), & \displaystyle\end{eqnarray}$$

(3.17c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{xz}(x,z={\it\zeta})=4\sqrt{\frac{1}{{\rm\pi}Re\,Sx}}{\it\zeta}^{\prime }(x)+O(Re^{-1}). & \displaystyle\end{eqnarray}$$

Figure 4.

Streamwise velocity component as function of position along the jet surface for $Re=80$ and slip parameter range $S\in [0.6,1.4]$ . Also shown, is the level of velocity along the wall for each $S(x<0,z=0)$ . Curve legends are the same as in figure 3.

Figure 4 depicts the influence of slip on the streamwise velocity component along the jet surface for $Re=80$ . Also shown are the velocity levels at the wall, given by $u(x<0,z=0)=4S$ . The streamwise flow weakens relative to the velocity at the wall with increasing slip. Like inertia, slip helps retain the Poiseuille character of the jet, with the additional slippage along the jet surface behaving like $S^{-1/2}$ . As (3.17a ) suggests, $u({\it\xi},{\it\eta}=0)\sim 4S$ for large $S$ ; there is no change in the flow field upon crossing the channel exit.

At this stage, the next order in the pressure contribution can be obtained by considering (3.2b ) and corresponding boundary conditions:

(3.18a,b ) $$\begin{eqnarray}P_{3{\it\eta}}={\it\Psi}_{1{\it\eta}}{\it\Psi}_{2{\it\xi}{\it\xi}}-h_{1}^{\prime \prime }{\it\Psi}_{1{\it\eta}}^{2}-{\it\Psi}_{2{\it\xi}{\it\eta}{\it\eta}},\quad P_{3}({\it\xi},{\it\eta}=0)+2{\it\Psi}_{2{\it\xi}{\it\eta}}({\it\xi},{\it\eta}=0)=0.\end{eqnarray}$$

Noting from (3.13) that ${\it\Psi}_{2{\it\eta}{\it\eta}}=4S{\it\Psi}_{2{\it\xi}}$ , and recalling from (3.12) that ${\it\Psi}_{1{\it\eta}}=4S$ , the pressure gradient in (3.18) reduces to $P_{3{\it\eta}}=-16S^{2}h_{1}^{\prime \prime }$ , to give $P_{3}({\it\xi},{\it\eta})=-16S^{2}h_{1}^{\prime \prime }{\it\eta}-2{\it\Psi}_{2{\it\xi}{\it\eta}}({\it\xi},{\it\eta}=0)$ . In this case, recalling the solution of problem (3.13), the pressure contribution to this order becomes

(3.19) $$\begin{eqnarray}p({\it\xi},{\it\eta})=-16{\it\varepsilon}^{2}S^{2}h_{0}^{\prime \prime }{\it\eta}-4{\it\varepsilon}^{3}\left(4S^{2}h_{1}^{\prime \prime }{\it\eta}+\frac{1}{\sqrt{{\rm\pi}S{\it\xi}}}\right).\end{eqnarray}$$

Of particular interest is the pressure at the free surface to this order, which is given by

(3.20) $$\begin{eqnarray}p(x,z={\it\zeta})=-\frac{4}{\sqrt{{\rm\pi}S\,Re^{3}x}}+O(Re^{-2}).\end{eqnarray}$$

Although the inner solution to next order is not needed, its asymptotic behaviour is still required to determine $h_{1}(x)$ . This contribution to the jet profile is crucial as $h_{0}(x)$ turns out to be zero (see § 4). Only the problem statement for ${\it\Psi}_{3}({\it\xi},{\it\eta})$ is given here, with the details given in appendix B. Thus, to $O({\it\varepsilon}^{3})$ , (3.2a ) reduces to

(3.21) $$\begin{eqnarray}{\it\Psi}_{1{\it\eta}}{\it\Psi}_{3{\it\xi}{\it\eta}}+{\it\Psi}_{2{\it\eta}}{\it\Psi}_{2{\it\xi}{\it\eta}}-{\it\Psi}_{2{\it\xi}}{\it\Psi}_{2{\it\eta}{\it\eta}}={\it\Psi}_{3{\it\eta}{\it\eta}{\it\eta}},\end{eqnarray}$$

subject to the boundary conditions deduced from (3.3a ) and (3.3b ), as well as the matching condition from (C 6), namely

(3.22a−c ) $$\begin{eqnarray}{\it\Psi}_{3}({\it\xi},{\it\eta}=0)=0,\quad {\it\Psi}_{3{\it\eta}{\it\eta}}({\it\xi},{\it\eta}=0)=4Sh_{0}^{\prime \prime },\quad {\it\Psi}_{3}({\it\xi},{\it\eta})\sim -{\textstyle \frac{4}{3}}{\it\eta}^{3}\quad \text{as }{\it\eta}\rightarrow \infty .\end{eqnarray}$$

More precisely, the asymptotic behaviour is (appendix B)

(3.23) $$\begin{eqnarray}{\it\Psi}_{3}({\it\xi},{\it\eta})\sim -\frac{4}{3}\left({\it\eta}^{3}+3\frac{(2S-1)}{4S^{2}}{\it\xi}{\it\eta}+d\right),\quad \text{as }{\it\eta}\rightarrow \infty .\end{eqnarray}$$

Here $d$ is a constant whose value is of no consequence to the order sought in the current study.

Figure 5.

Influence of slip and inertia on boundary-layer height at $x=1$ : (a) streamwise velocity component is plotted against height in the inner layer for $Re=80$ , asymptotic (linear) behaviour is also shown for each slip level, (b) velocity at inner-layer edge for different Reynolds numbers. Inset shows the dependence of inner-layer height on $S$ .

3.2 Inner-layer growth

The inner-layer height is estimated by examining the variation of the velocity profile (3.15b ) with respect to ${\it\eta}$ (or ${\it\theta}$ ) at a position $x$ relative to its asymptotic form. Thus, the inner-layer thickness is determined once ${\it\theta}={\it\theta}_{\infty }(Re,S)$ , which is defined through $(f_{2}^{\prime }({\it\theta}_{\infty })-4{\it\theta}_{\infty })/4{\it\theta}_{\infty }=1\,\%$ . Recalling (B 5b ), we obtain the following transcendental equation and solution for ${\it\theta}_{\infty }$ :

(3.24) $$\begin{eqnarray}\text{e}^{-S{\it\theta}_{\infty }^{2}}+[\text{erf}(\sqrt{S}{\it\theta}_{\infty })-1.01]\sqrt{{\rm\pi}S}\,{\it\theta}_{\infty }=0,\quad {\it\theta}_{\infty }=\frac{1.38}{\sqrt{S}}.\end{eqnarray}$$

Figure 5 illustrates the influence of slip on velocity in the inner layer for $Re=80$ (figure 5 a) and the influence of inertia on boundary-layer height (figure 5 b) at $x=1$ . The asymptotic solution tends to underestimate the velocity level. The inner-layer height coincides with the level at which the asymptotic and inner velocity profiles begin to merge, corresponding to ${\it\eta}_{\infty }=1.38/\sqrt{S}$ . The locus of merging points is highlighted in figure 5(a). The loci at different Reynolds numbers are also plotted in figure 5(b), displaying the influence of inertia and slip on the velocity at the edge of the inner layer. Clearly, as the inset in figure 5(b) indicates, the inner-layer height decreases with slip, eventually vanishing for infinite slip length.

To leading order, (3.24) yields the inner-layer thickness as ${\it\delta}(x;Re,S)=1.38\sqrt{x/S\,Re}$ . The boundary-layer thickness predominantly grows with position, at a rate that decreases with inertia and slip. As expected, the boundary layer vanishes in the limit of large $S$ ; there is simply no discontinuity in shear stress. Incidentally, for gravity-driven jet flow Wilson (Reference Wilson1986), the boundary-layer height behaves like $(x/Re)^{1/4}$ , whereas for Newtonian (Tillett Reference Tillett1968) and non-Newtonian (Saffari & Khayat Reference Saffari and Khayat2009; Khayat Reference Khayat2014) jets, the boundary layer grows like $(x/Re)^{1/3}$ .

Generally, the stress singularity at the exit diffuses more slowly (over a longer distance) for the more slipping jet as a result of the thinner boundary layer. A similar argument holds for inertial effect. More precisely, since boundary-layer theory neglects viscous forces arising from streamwise velocity gradients, it must be expected to break down when shearing and elongation terms are of the same order of magnitude, that is when $\partial u/\partial x\simeq \partial u/\partial y$ , or when $x$ is of order ${\it\delta}$ . This leads to $x=O(1/S\,Re)$ , which means that the inner-layer solution never breaks down for an infinitely slipping jet! More realistically, the inner-layer solution remains valid, starting at a closer point to the channel exit, for a jet with larger slip length and Reynolds number.

Further correction is made by including the higher-order term. Upon balancing inertial and viscous terms in (3.2) and using the velocity at the jet surface (3.17a ), we have

(3.25) $$\begin{eqnarray}{\it\delta}(x;Re,S)=1.38\sqrt{\frac{x}{S\,Re}}\left[1-\frac{1}{2S}\sqrt{\frac{x}{{\rm\pi}S\,Re}}\right]+O(Re^{-3/2}).\end{eqnarray}$$

The rate of decrease of inner-layer height with $Re$ and $S$ is clearly tempered by the higher-order term, pointing to the possibility of non-monotonic growth, in contrast to the case of an adhering jet. Figure 6 depicts the influence of slip on the boundary-layer profile. Note the significant jump in height for an adhering jet (see below for further discussion on this phenomenon). As the boundary layer grows with distance, it eventually invades the entire jet width, that is when ${\it\delta}\sim O(1)$ . Expression (3.25) indicates that this happens at a distance $x\sim S\,Re$ .

Finally, it is worth observing that, for an adhering jet, the boundary-layer assumption in the inner layer is valid at large Reynolds number; the boundary-layer equations are obtained from the Navier–Stokes equations by neglecting higher-order terms in the inverse of the Reynolds number. For slipping flow, the boundary-layer equations are expected to remain accurate even when the Reynolds number is only moderate. This extended range of validity is attributed to the decrease in viscous effect resulting from slip, which ensures that even moderate inertia remains relatively dominant for the boundary-layer equations to hold. Hence, the Reynolds number can be lower for the more slipping fluid for the boundary-layer equations to remain valid. The extended range of validity also holds for shear-thinning fluids (Khayat Reference Khayat2014), which was also confirmed by Wu & Thompson (Reference Wu and Thompson1996) in their comparison between experiment and numerical solution for the flow over a flat plate. In the current study, it is found that the inner-layer flow is accurate to $O(Re^{-1/2})$ for a slipping jet, and to $O(Re^{-1/3})$ for an adhering jet (Tillett Reference Tillett1968; Khayat Reference Khayat2014).

Figure 6.

Influence of slip on boundary-layer thickness (height above jet surface) for $Re=80$ .

4.1 Core-layer equations and boundary conditions

In the core layer, which is far from the region near $z=0$ , equations (2.3) must be solved, and are conveniently rewritten here as

(4.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}_{z}{\it\psi}_{xz}-{\it\psi}_{x}{\it\psi}_{zz}=-p_{x}+{\it\varepsilon}^{2}({\it\psi}_{xxz}+{\it\psi}_{zzz}), & \displaystyle\end{eqnarray}$$

(4.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}_{x}{\it\psi}_{xz}-{\it\psi}_{z}{\it\psi}_{xx}=-p_{z}-{\it\varepsilon}^{2}({\it\psi}_{xxx}+{\it\psi}_{xzz}). & \displaystyle\end{eqnarray}$$

${\it\psi}$

$p$

(4.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}(x,z)={\it\psi}_{0}(x,z)+{\it\varepsilon}{\it\psi}_{1}(x,z)+\cdots \,, & \displaystyle\end{eqnarray}$$

(4.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle p(x,z)=p_{0}(x,z)+{\it\varepsilon}p_{1}(x,z)+\cdots \,. & \displaystyle\end{eqnarray}$$

${\it\psi}_{0}(z)$

${\it\psi}_{m}\,(m>0)$

Based on these assumptions, upon inserting expressions (4.2) into (4.1), a hierarchy of equations is obtained to each order. Some boundary conditions are readily available from (2.5) for ${\it\psi}_{m}$ , namely

(4.3a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}_{mz}(x\leqslant 0,z=0)=S{\it\psi}_{mzz}(x\leqslant 0,z=0),\quad {\it\psi}_{xm}(x\leqslant 0,z=0)=0, & \displaystyle\end{eqnarray}$$

(4.3c,d ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}_{m}(x\rightarrow -\infty ,z)\sim 0,\quad {\it\psi}_{m}(-\infty <x<\infty ,z={\textstyle \frac{1}{2}})=0. & \displaystyle\end{eqnarray}$$

Additional boundary conditions are needed outside the channel, which will be obtained by matching the core and the inner solutions as we shall see next.

It is worth noting that the character of the base flow, (2.1) or (2.2), is reminiscent of laminar tube flow with constriction at high Reynolds number. In that case as well, the fully developed (Poiseuille) profile is the flow to leading order. Smith examined the flow for fine (Smith Reference Smith1976a ), moderate (Smith Reference Smith1976b ) and severe (Smith Reference Smith1979) constrictions. The severity of the constriction is reflected in the characteristic slope of the obstacle, which is $O(Re^{-1/3})$ and $O(Re^{-1/6})$ for mild and moderate constrictions, respectively. Here $Re$ is Smith’s Reynolds number based on the tube diameter and the maximum velocity. In these situations, there occurs virtually no nonlinear upstream influence of the obstacle, and the core flow is just an inviscid rotational perturbation of the base Poiseuille flow. As it turns out, this is also the situation in the current slipping jet problem, where the contraction of the jet flow constitutes a fine to moderate constriction. In contrast, in a severe constriction, where the obstacle slope is $O(1)$ , there is significant upstream influence on the core flow. This level of constriction severity does not correspond to the current problem, but the flow field expansion takes the same form regardless of the constriction level of severity. Thus, Smith’s (Reference Smith1979) expansion (2.1) for the flow with severe constriction is the same as the current expansion (4.1). However, in contrast to (4.1), Smith’s leading-order terms in (2.1) do not exactly correspond to fully developed flow but still satisfy the inviscid equations of motion. Free-streamline theory was adopted by Smith, which is believed to yield the proper inviscid limiting form of the Navier–Stokes equations (Smith Reference Smith1979).

4.2 Matching between the inner and outer layers in the jet region

Both the stream function (velocity) and pressure, as well as stress, must match between the inner and core layers for $x>0$ . The matching process is detailed in appendix C.

The matching of the stream function is ensured by equating (C 8) and (C 10) in appendix C to yield the following relations:

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle 4Sh_{0}(x)+{\it\psi}_{1}(x\geqslant 0,z=0)=0, & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle 4h_{0}(x)+{\it\psi}_{1z}(x\geqslant 0,z=0)=0, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle 4Sh_{1}(x)+{\it\psi}_{2}(x\geqslant 0,z=0)=\frac{x}{S}, & \displaystyle\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle 4h_{1}(x)+{\it\psi}_{2z}(x\geqslant 0,z=0)=-\frac{(2S-1)}{S^{2}}x. & \displaystyle\end{eqnarray}$$

We next consider the matching of the pressure by equating (C 11) to (C 13) to give

(4.8) $$\begin{eqnarray}\displaystyle \hspace{-15.00002pt} & \displaystyle p_{0}(x\geqslant 0,z=0)=p_{0z}(x\geqslant 0,z=0)=p_{0zz}(x\geqslant 0,z=0)=p_{0zzz}(x\geqslant 0,z=0)=0, & \displaystyle\end{eqnarray}$$

(4.9a,b ) $$\begin{eqnarray}\displaystyle \hspace{-15.00002pt} & \displaystyle p_{1}(x\geqslant 0,z=0)=p_{1zz}(x\geqslant 0,z=0)=0,\quad p_{1z}(x\geqslant 0,z=0)=-16S^{2}h_{0}^{\prime \prime }(x), & \displaystyle\end{eqnarray}$$

(4.10a,b ) $$\begin{eqnarray}\displaystyle \hspace{-15.00002pt} & \displaystyle p_{2}(x\geqslant 0,z=0)=0,\quad p_{2z}(x\geqslant 0,z=0)=-16S^{2}h_{1}^{\prime \prime }(x). & \displaystyle\end{eqnarray}$$

Relations (4.4)–(4.7) and (4.8)–(4.10) will be used to generate the desired boundary conditions.

4.3 Leading- and first-order core flow

Clearly, the leading-order pressure term vanishes everywhere. This is established by noting from (4.1) that $p_{0x}(z,z)=p_{0z}(z,z)=0$ and recalling conditions (4.8). Formally, the leading-order flow ( $m=0$ ) is then given by

(4.11a,b ) $$\begin{eqnarray}{\it\psi}_{0}(x,z)=4Sz+2z^{2}-{\textstyle \frac{4}{3}}z^{3},\quad p_{0}(x,z)=0.\end{eqnarray}$$

This result is consistent with the infinite $Re$ limit solution of Euler’s equations.

To next order $(m=1)$ , the problem is governed by

(4.12a,b ) $$\begin{eqnarray}{\it\psi}_{0z}{\it\psi}_{1xz}-{\it\psi}_{0zz}{\it\psi}_{1x}=-p_{1x},\quad {\it\psi}_{0z}{\it\psi}_{1xx}=p_{1z},\end{eqnarray}$$

which also yield ${\rm\nabla}^{2}{\it\psi}_{1x}-(u_{0}^{\prime \prime }/u_{0}){\it\psi}_{1x}=0$ upon eliminating the pressure, where ${\rm\nabla}^{2}=(\partial ^{2}/\partial x^{2})+(\partial ^{2}/\partial z^{2})$ is the Laplacian operator. A prime denotes differentiation with respect to $z$ . In this case, conditions (4.3) suggest that ${\it\psi}_{1}(x,z)$ must vanish everywhere inside the channel. The additional condition for the flow outside the channel is obtained upon eliminating $h_{0}(x)$ between (4.7) and (4.8). Thus, the three conditions needed are

(4.13) $$\begin{eqnarray}{\it\psi}_{1x}(x\geqslant 0,z={\textstyle \frac{1}{2}})={\it\psi}_{1}(x=0,z)={\it\psi}_{1}(x\geqslant 0,z=0)-S{\it\psi}_{1z}(x\geqslant 0,z=0)=0.\end{eqnarray}$$

Given the homogeneity of equation and boundary conditions, the stream function must also vanish outside the channel. The pressure, which must satisfy the first condition in (4.9a,b ), also vanishes. Formally,

(4.14) $$\begin{eqnarray}{\it\psi}_{1}(x,z)=p_{1}(x,z)=0.\end{eqnarray}$$

The consequent vanishing of ${\it\psi}_{1}(x,z)$ means that, to the order ${\it\varepsilon}$ , there is no interaction between the inner layer and the core flows. Simultaneously, there is no deviation of the jet profile to this order since (4.7) and (4.14) suggests that

(4.15) $$\begin{eqnarray}h_{0}(x)=0.\end{eqnarray}$$

In this case, we conclude that ${\it\zeta}(x)=O({\it\varepsilon}^{3})$ , at least. The non-trivial contribution is made to next order $(m=2)$ , which is covered in the next section.

5.1 Flow in the core layer inside the channel

Consider the core flow in the region inside the channel $(x<0)$ . In this case, recalling (4.3) for $m=2$ , (5.2) must be solved subject to

(5.3a,b ) $$\begin{eqnarray}w_{2}(x\leqslant 0,z=0)=w_{2}(x\leqslant 0,z={\textstyle \frac{1}{2}})=0,\quad w_{2}(x\rightarrow -\infty )\sim 0.\end{eqnarray}$$

Despite the difference in the order and boundary conditions, the general form of the solution remains the same inside the channel as in Tillett (Reference Tillett1968), namely

(5.4) $$\begin{eqnarray}w_{2}(x\leqslant 0,z)=-{\it\psi}_{2x}(x\leqslant 0,z)=-\mathop{\sum }_{k=1}^{\infty }B_{k}\text{e}^{{\it\beta}_{k}x}V_{k}(z).\end{eqnarray}$$

The shape functions $V_{k}(z)$ are, however, different, and are governed by the following eigenvalue problem:

(5.5a−c ) $$\begin{eqnarray}V_{k}^{\prime \prime }(z)+\left({\it\beta}_{k}^{2}-\frac{u_{0}^{\prime \prime }(z)}{u_{0}(z)}\right)V_{k}(z)=0,\quad V_{k}(0)=V_{k}\left(\frac{1}{2}\right)=0,\quad V_{k}^{\prime }(0)=1.\end{eqnarray}$$

Problem (5.5) has the same form as the case of adhering Newtonian and non-Newtonian jets (Tillett Reference Tillett1968; Saffari & Khayat Reference Saffari and Khayat2009; Khayat Reference Khayat2014), with the slip effect implied through $u_{0}(z;S)$ . In this case, the shape functions are orthogonal, and their orthogonality is shown in appendix D. For given $S,{\it\beta}_{k}$ increases with $k$ , making convergence reasonably achievable after only a few modes, as suggested by table 1, which gives the values of ${\it\beta}_{k}$ for $S=0.6$ , 0.8 and 1, for the first six modes. The stream function inside the channel is obtained by integrating expression (5.4) subject to ${\it\psi}_{2}(x\rightarrow -\infty ,z)=0$ , leading to

(5.6) $$\begin{eqnarray}{\it\psi}_{2}(x\leqslant 0,z)=\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}\text{e}^{{\it\beta}_{k}x}V_{k}(z).\end{eqnarray}$$

The pressure inside the channel is determined by integrating equation (5.1b ) subject to the pressure condition at the wall, which is available upon setting $z=0$ in (5.1a ). In addition, since (4.10) suggests that $p_{2}(x=0,z=0)=0$ , then we obtain the desired boundary condition for pressure as $p_{2}(x\leqslant 0,z=0)=-4S\sum _{k=1}^{\infty }(B_{k}/{\it\beta}_{k})(\text{e}^{{\it\beta}_{k}x}-1)-8x$ . Inserting (5.6) into (5.1b ) and using (5.5) gives

(5.7) $$\begin{eqnarray}p_{2z}(x\leqslant 0,z)=u_{0}\mathop{\sum }_{k=1}^{\infty }{\it\beta}_{k}B_{k}\text{e}^{{\it\beta}_{k}x}V_{k}=-\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}\text{e}^{{\it\beta}_{k}x}[(u_{0}V_{k}^{\prime })^{\prime }-(u_{0}^{\prime }V_{k})^{\prime }].\end{eqnarray}$$

Integrating subject to the pressure at the wall finally yields

(5.8) $$\begin{eqnarray}p_{2}(x\leqslant 0,z)=-8x-\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}\{\text{e}^{{\it\beta}_{k}x}[u_{0}(z)V_{k}^{\prime }(z)-u_{0}^{\prime }(z)V_{k}(z)]-4S\}.\end{eqnarray}$$

Table 1.

Eigenvalues and expansion coefficients for different slip lengths, $S=0.6$ , 0.8 and 1.

To this order, the flow field inside the channel is given by

(5.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}(x\leqslant 0,z)={\it\psi}_{0}(z)+Re^{-1}\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}\text{e}^{{\it\beta}_{k}x}V_{k}(z), & \displaystyle\end{eqnarray}$$

(5.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle u(x\leqslant 0,z)=u_{0}(z)+Re^{-1}\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}\text{e}^{{\it\beta}_{k}x}V_{k}^{\prime }(z), & \displaystyle\end{eqnarray}$$

(5.9c ) $$\begin{eqnarray}\displaystyle & \displaystyle w(x\leqslant 0,z)=-Re^{-1}\mathop{\sum }_{k=1}^{\infty }B_{k}\text{e}^{{\it\beta}_{k}x}V_{k}(z), & \displaystyle\end{eqnarray}$$

(5.9d ) $$\begin{eqnarray}\displaystyle & \displaystyle p(x\leqslant 0,z)=-Re^{-1}8x-Re^{-1}\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}\{\text{e}^{{\it\beta}_{k}x}[u_{0}(z)V_{k}^{\prime }(z)-u_{0}^{\prime }(z)V_{k}(z)]-4S\}, & \displaystyle\end{eqnarray}$$

(5.9e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{xz}(x\leqslant 0,z)=u_{0}^{\prime }(z)-Re^{-1}\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}\text{e}^{{\it\beta}_{k}x}\left(2{\it\beta}_{k}^{2}-\frac{u_{0}^{\prime \prime }}{u_{0}}\right)V_{k}(z), & \displaystyle\end{eqnarray}$$

Figure 7.

Upstream flow for $S=1$ at different positions: (a) velocity profiles at $Re=80$ , and (b) pressure profiles at any Reynolds number.

Although the coefficients $B_{k}$ will become available once the flow outside the channel is determined, it is helpful at this stage to nevertheless adopt their values from table 1 and examine the flow field as the fluid approaches the exit inside the channel. The upstream flow field is typically illustrated in figure 7 for $S=1$ , where the streamwise velocity and pressure profiles are depicted at various positions between $x=-1$ , where fully developed conditions are essentially met, and the channel exit is at $x=0$ . Figure 7(a) shows the velocity profiles for $Re=80$ , suggesting a rapid increase in slip as the fluid approaches $x=0$ . This is particularly clear from the inset. The difference between the wall and centreline velocities diminishes progressively, with gradual flattening of the velocity profile. Thus, as the fluid moves towards the channel exit, additional slippage occurs at the wall, and the flow decelerates in the core layer so as to satisfy the condition of continuity (see inset). This relative deceleration in the core layer is reduced for higher slip, and the flow retains increasingly its Poiseuille character. Figure 7(b) indicates, as expected, that the pressure profile at $x=-1$ presents no variation in the transverse direction. Closer to the channel exit, the pressure exhibits an almost antisymmetric variation: an increase in the pressure at the wall is accompanied by a decrease along the centreline.

Figure 8.

Influence of slip and inertia on exit length $x_{0}$ (upstream position for fully developed flow): (a) exit length against $Re$ for different slip lengths, (b) exit length against $S$ for different Reynolds numbers.

Although Poiseuille conditions are theoretically recovered in the limit $x\rightarrow -\infty$ , calculations based on expressions (5.9), such as shown in figure 7 above, indicate that these conditions prevail essentially at a distance equal to one channel width. This is also the case in the absence of slip (Khayat Reference Khayat2014). As to the influence of slip, consider the exit length, the upstream position $x_{0}$ where fully developed Poiseuille flow is recovered inside the channel. Rather than seeking an accurate numerical prediction, an estimate of the influence of inertia and slip on the exit length is obtained analytically. Since the exit length is relatively large, only the contribution of the leading eigenmode is significant. In this case, by evaluating the relative departure in the velocity from (5.9b ) at the centreline, for relatively large distance, the exit length satisfies approximately: $(B_{1}/Re(4S+1){\it\beta}_{1})\text{e}^{{\it\beta}_{1}x_{0}}V_{1}^{\prime }(1/2)\simeq -10^{-c}$ , where $c$ is an imposed tolerance exponent. Thus,

(5.10) $$\begin{eqnarray}x_{0}(Re,S)=\frac{1}{{\it\beta}_{1}}\ln \left[-10^{-c}\frac{Re(4S+1){\it\beta}_{1}}{B_{1}}V_{1}^{\prime }\left(\frac{1}{2}\right)\right].\end{eqnarray}$$

Figure 8 depicts the influence of slip and inertia on the exit length for $c=4$ , which is plotted against $Re$ for different slip lengths (figure 8 a), and against $S$ for different Reynolds numbers (figure 8 b). The logarithmic dependence on $Re$ , which also corresponds to the behaviour predicted by Poole & Ridley (Reference Poole and Ridley2007) for the entry length in a duct at large Reynolds number (see their figure 2), is not entirely obvious from figure 8(a) over the $Re$ range in the figure. Figure 8 suggests that the exit length decreases both with inertia and, not surprisingly, with slip. In both cases, the viscous effect responsible for flow development is diminished; the Poiseuille character is retained up to a smaller upstream distance. In contrast, for a fluid entering a channel or a pipe, the entry length increases with slip (Chakraborty & Anand Reference Chakraborty and Anand2008; Duan & Muzychka Reference Duan and Muzychka2010). The two findings are not contradictory. For both entry and exit flows, slip (and inertia) helps retain upstream plug and parabolic flow conditions, respectively, over a longer distance.

Perhaps the most remarkable feature in figure 8 is the significant jump in exit length as $S$ increases, especially between adhering and slipping fluids (not shown). A similar phenomenon was also noted by Chakraborty & Anand (Reference Chakraborty and Anand2008) who examined the dependence of the entry length on the thickness of a thin gas layer at the wall. They observed that ‘even with a trace amount of nanobubble formation due to hydrophobic interactions, there is a rapid increment in the entrance length’. They also point out that a thickness (in unit of channel width) as small as $10^{-7}$ of the fluid layer can result in more than 50 % increase in the entry length, which was attributed to the presence of nanobubble layers covering the solid surface. Consequently, ‘the effect of wall shear penetrates into the bulk fluid only after the core flow has axially traversed a significantly greater distance than for the case devoid of any hydrophobic interactions’. The same mechanism also applies in the case of the jet exiting a hydrophobic channel, with the gas layer delaying the transition from Poiseuille conditions. In other words, and paraphrasing Chakraborty & Anand (Reference Chakraborty and Anand2008), the microscopic roughness of the solid surface impedes the motion of the fluid, ensuring adherence. In contrast, the gas layer acts as a shield, and prevents the liquid from direct contact with the surface irregularities and roughness.

5.2 Flow in the core layer outside the channel

At this stage, the solution process undergoes a major departure from the case of an adhering jet. Indeed, the flow outside the channel cannot be expressed in terms of the eigenfunctions $V_{k}$ as previously done (Tillett Reference Tillett1968; Khayat Reference Khayat2014). This is the result of the slip effect in the boundary condition for small $z$ in the jet region. This boundary condition is obtained by eliminating $h_{1}(x)$ between (4.6) and (4.7) to give

(5.11) $$\begin{eqnarray}{\it\psi}_{2}(x\geqslant 0,z=0)-S{\it\psi}_{2z}(x\geqslant 0,z=0)=2x.\end{eqnarray}$$

Consequently, in addition to matching with (5.4) at $x=0$ , the boundary conditions needed for the solution of (5.2) are

(5.12a,b ) $$\begin{eqnarray}w_{2}\left(x\geqslant 0,z={\textstyle \frac{1}{2}}\right)=0,\quad w_{2}(x\geqslant 0,z=0)-Sw_{2z}(x\geqslant 0,z=0)=-2.\end{eqnarray}$$

In this case, the solution takes the form

(5.13) $$\begin{eqnarray}w_{2}(x\geqslant 0,z)=-{\it\psi}_{2x}(x\geqslant 0,z)=-2U_{0}(z)+\mathop{\sum }_{k=1}^{\infty }A_{k}\text{e}^{-{\it\alpha}_{k}x}U_{k}(z),\end{eqnarray}$$

where $U_{0}(z)$ satisfies the following boundary-value problem:

(5.14a−c ) $$\begin{eqnarray}U_{0}^{\prime \prime }(z)-\frac{u_{0}^{\prime \prime }(z)}{u_{0}(z)}U_{0}(z)=0,\quad U_{0}(0)-SU_{0}^{\prime }(0)=1,\quad U_{0}\left(\frac{1}{2}\right)=0.\end{eqnarray}$$

Problem (5.14) possesses an analytical solution. The equation can be recast as $(U_{0}^{\prime }u_{0})^{\prime }-(U_{0}u_{0}^{\prime })^{\prime }=0,$ which, upon integrating, applying the mixed condition at $z=0$ and recalling (2.1), lead to $U_{0}^{\prime }u_{0}-U_{0}u_{0}^{\prime }=-4$ . Solving gives

(5.15) $$\begin{eqnarray}U_{0}(z)=\frac{S+z-z^{2}}{4S+1}\left[\frac{1-2z}{S+z-z^{2}}+\frac{4}{\sqrt{4S+1}}\tanh ^{-1}\left(\frac{1-2z}{\sqrt{4S+1}}\right)\right].\end{eqnarray}$$

Figure 9 illustrates the influence of slip on the $U_{0}$ profiles. The profiles also give the behaviour of the depthwise flow in the core region far from the exit, dominated by the first term in (5.13). In particular, the dependence of $U_{0}(0)$ on $S$ reflects the increasingly contracting jet as $S$ decreases. However, note that the behaviour of $w$ near the jet surface is fully established when the composite flow is determined (see § 6).

Figure 9.

Influence of slip on $U_{0}(z)$ .

The functions $U_{k}\,(z)$ must satisfy the condition $U_{k}(0)-SU_{k}^{\prime }(0)=0$ . Without loss of generality, one then has the following eigenvalue problem:

(5.16a−d ) $$\begin{eqnarray}U_{k}^{\prime \prime }(z)+\left({\it\alpha}_{k}^{2}-\frac{u_{0}^{\prime \prime }(z)}{u_{0}(z)}\right)U_{k}(z)=0,\quad U_{k}(0)=S,\quad U_{k}^{\prime }(0)=1,\quad U_{k}\left(\frac{1}{2}\right)=0.\end{eqnarray}$$

Like $V_{k}$ , the $U_{k}$ eigenfunctions can be shown to be orthogonal (see appendix D). Note, however, that $V_{k}$ and $U_{k}$ are not mutually orthogonal. In fact, by adopting similar steps as in appendix B, it can be shown that

(5.17) $$\begin{eqnarray}\langle U_{j}V_{k}\rangle =\frac{S}{{\it\beta}_{k}^{2}-{\it\alpha}_{j}^{2}},\end{eqnarray}$$

where $\langle ~\rangle$ denotes integration over the interval $z\in [0,1/2]$ . Table 1 displays the dependence of ${\it\alpha}_{k}$ on $S$ for the first six eigenvalues.

The stream function is determined by integrating (5.13) subject to condition (5.11), suggesting that

(5.18) $$\begin{eqnarray}{\it\psi}_{2}(x\geqslant 0,z)=2U_{0}(z)x+\mathop{\sum }_{k=1}^{\infty }\left(\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}+C_{k}\right)U_{k}(z),\end{eqnarray}$$

where the coefficients $C_{k}$ are determined by matching the flow at the channel exit (see below). Note that (5.18) also satisfies ${\it\psi}_{2}(x\geqslant 0,z=1/2)=0$ as (4.3) dictates.

The pressure outside the channel is obtained by first inserting ${\it\psi}_{2}(x\geqslant 0,z)$ from (5.18) into (5.1b ), and recalling (5.16) to yield

(5.19) $$\begin{eqnarray}p_{2z}(x\geqslant 0,z)=u_{0}\mathop{\sum }_{k=1}^{\infty }{\it\alpha}_{k}\text{e}^{-{\it\alpha}_{k}x}A_{k}U_{k}=-\mathop{\sum }_{k=1}^{\infty }\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}[(U_{k}^{\prime }u_{0})^{\prime }-(U_{k}u_{0}^{\prime })^{\prime }].\end{eqnarray}$$

Integrating (5.19), subject to condition (4.10), and using the boundary conditions in (5.16), give

(5.20) $$\begin{eqnarray}p_{2}(x\geqslant 0,z)=-\mathop{\sum }_{k=1}^{\infty }\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}[u_{0}(z)U_{k}^{\prime }(z)-u_{0}^{\prime }(z)U_{k}(z)].\end{eqnarray}$$

To this order, the flow outside the channel is summarized as

(5.21a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\psi}(x\geqslant 0,z)={\it\psi}_{0}(z)+Re^{-1}\left[2U_{0}(z)x+\mathop{\sum }_{k=1}^{\infty }\left(\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}+C_{k}\right)U_{k}(z)\right], & \displaystyle\end{eqnarray}$$

(5.21b ) $$\begin{eqnarray}\displaystyle & \displaystyle u(x\geqslant 0,z)=u_{0}(z)+Re^{-1}\left\{2U_{0}^{\prime }(z)x+\mathop{\sum }_{k=1}^{\infty }\left(\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}+C_{k}\right)U_{k}^{\prime }(z)\right\}, & \displaystyle\end{eqnarray}$$

(5.21c ) $$\begin{eqnarray}\displaystyle & \displaystyle w(x\geqslant 0,z)=-Re^{-1}\left[2U_{0}(z)-\mathop{\sum }_{k=1}^{\infty }\text{e}^{-{\it\alpha}_{k}x}A_{k}U_{k}(z)\right], & \displaystyle\end{eqnarray}$$

(5.21d ) $$\begin{eqnarray}\displaystyle & \displaystyle p(x\geqslant 0,z)=-Re^{-1}\mathop{\sum }_{k=1}^{\infty }\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}[u_{0}(z)U_{k}^{\prime }(z)-u_{0}^{\prime }(z)U_{k}(z)], & \displaystyle\end{eqnarray}$$

(5.21e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{xz}(x\geqslant 0,z)=u_{0}^{\prime }(z)+Re^{-1}\left\{2U_{0}^{\prime \prime }x+\mathop{\sum }_{k=1}^{\infty }\left[\left(\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}+C_{k}\right)\frac{u_{0}^{\prime \prime }}{u_{0}}-C_{k}{\it\alpha}_{k}^{2}\right]U_{k}\right\}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

$u$

$w(x,z)$

$w(x\geqslant 0,z)\sim -2Re^{-1}U_{0}(z)$

$x$

5.3 Matching the flow at the channel exit

The coefficients, $A_{k},B_{k}$ and $C_{k}$ are now determined by matching the transverse velocity component, stream function and pressure at the channel exit. It is emphasized that although the flow is singular at the channel exit in the inner layer, it is continuous in the core region. Thus, upon evaluating $w(x,z)$ from (5.9c ) and (5.21c ) at $x=0$ , and matching, one obtains the following relation:

(5.22) $$\begin{eqnarray}2U_{0}(z)=\mathop{\sum }_{k=1}^{\infty }[B_{k}V_{k}(z)+A_{k}U_{k}(z)].\end{eqnarray}$$

This relation is a spectral representation of $U_{0}\,(z)$ in terms of the orthogonal functions $V_{k}\,(z)$ and $U_{k}\,(z)$ . Matching the pressure from (5.9d ) and (5.21d ) gives

(5.23) $$\begin{eqnarray}\mathop{\sum }_{k=1}^{\infty }\left\{\frac{B_{k}}{{\it\beta}_{k}}[u_{0}(z)V_{k}^{\prime }(z)-u_{0}^{\prime }(z)V_{k}(z)-4S]-\frac{A_{k}}{{\it\alpha}_{k}}[u_{0}(z)U_{k}^{\prime }(z)-u_{0}^{\prime }(z)U_{k}(z)]\right\}=0.\end{eqnarray}$$

A more suitable relation is obtained upon differentiating (5.23) and using the equations in (5.5) and (5.16), namely

(5.24) $$\begin{eqnarray}\mathop{\sum }_{k=1}^{\infty }[{\it\beta}_{k}B_{k}V_{k}(z)-{\it\alpha}_{k}A_{k}U_{k}(z)]=0.\end{eqnarray}$$

Note that this relation ensures the marching of the shear stress. Projecting both (5.22) and (5.24) onto $V_{k}\,(z)$ , and using (5.17), allow the expression of $B_{k}$ in terms of $A_{k}$ as

(5.25) $$\begin{eqnarray}B_{k}=\frac{S}{{\it\beta}_{k}\langle V_{k}V_{k}\rangle }\mathop{\sum }_{j=1}^{\infty }\frac{{\it\alpha}_{j}A_{j}}{{\it\beta}_{k}^{2}-{\it\alpha}_{j}^{2}},\end{eqnarray}$$

and the evaluation of the coefficients $A_{k}$ through the system of equations

(5.26) $$\begin{eqnarray}\mathop{\sum }_{j=1}^{\infty }\frac{A_{j}}{{\it\beta}_{k}-{\it\alpha}_{j}}=\frac{2}{S}{\it\beta}_{k}\langle V_{0}V_{k}\rangle .\end{eqnarray}$$

Finally, the third relation ensues from evaluating the stream functions in (5.9a ) and (5.21a ) at $x=0$ , and comparing:

(5.27) $$\begin{eqnarray}\mathop{\sum }_{k=1}^{\infty }\frac{B_{k}}{{\it\beta}_{k}}V_{k}(z)=\mathop{\sum }_{k=1}^{\infty }\left(\frac{A_{k}}{{\it\alpha}_{k}}+C_{k}\right)U_{k}(z).\end{eqnarray}$$

Upon projection and use of relations (5.17), (5.22) and (5.24), we obtain the following set of equations for $C_{k}$ :

(5.28) $$\begin{eqnarray}{\it\beta}_{k}\mathop{\sum }_{j=1}^{\infty }\left(\frac{C_{j}}{{\it\beta}_{k}^{2}-{\it\alpha}_{j}^{2}}\right)=\frac{2}{S}\left\langle V_{0}V_{k}\right\rangle -\mathop{\sum }_{j=1}^{\infty }\frac{A_{j}}{{\it\alpha}_{j}}\left(\frac{1}{{\it\beta}_{k}-{\it\alpha}_{j}}\right).\end{eqnarray}$$

Now that the coefficients have been determined, the flow field becomes available at any position in the core region, inside and outside the channel. Incidentally, expressions (5.9d ) and (5.21d ) are valid throughout the entire flow region as ${\it\varepsilon}^{2}p_{2}(x,z)$ turns out to be equal to the composite pressure (see § 6). In other words, the pressure is everywhere readily available and is given by $p(x,z)={\it\varepsilon}^{2}p_{2}(x,z)+O({\it\varepsilon}^{3})$ .

5.4 Influence of slip on the flow in the core layer and jet profile

Figure 10.

Influence of slip on flow along centreline: (a) deviation of streamwise from Poiseuille level. Inset shows actual streamwise velocity component at $Re=80$ , and (b) pressure, for different slip parameters. Inset shows magnification of pressure curves.

The character of the flow in the core region is reflected in the behaviour of velocity and pressure along the centreline. Figure 10 depicts the influence of slip on the streamwise velocity component $u(x,z=1/2)$ , and pressure $Re\,p(x,z=1/2)$ at any Reynolds number inside and outside the channel as functions of $x$ . Figure 10(a) shows the deviation from the Poiseuille level for each $S$ . The streamwise flow (inset in figure 10 a) retains its Poiseuille level, $4S$ $+$ 1, up to an upstream distance from the channel exit that decreases with slip (see figure 8). At that point $u$ decreases as a result of flow weakening downstream for any slip level, accompanied by the vanishing of the pressure (figure 10 b). There is an essentially linear growth in velocity departure, reflected in the $U_{0}^{\prime }(z)x$ term in (5.21b ), which weakens the flow and eventually halts it, as illustrated in the $S=0$ profile. However, at this point, the current formulation ceases to be valid. In reality, the velocity begins to flatten as the jet approaches uniform flow conditions (see Philippe & Dumargue Reference Philippe and Dumargue1991). The slope of the velocity at the centreline far downstream is determined from (5.21b ) at $z=1/2$ . In addition, recalling that $U_{0}^{\prime }(z)u_{0}(z)-U_{0}(z)u_{0}^{\prime }(z)=-4$ , then we have

(5.29) $$\begin{eqnarray}u_{x}\left(x\geqslant 0,\frac{1}{2}\right)=2Re^{-1}U_{0}^{\prime }\left(\frac{1}{2}\right)=-\frac{2}{(4S+1)Re}.\end{eqnarray}$$

In the presence of strong slip, inertia remains dominant, with the flow retaining its Poiseuille character over longer distance, which is also reflected in the larger (streamwise) pressure gradient (see the case for $S=1$ , for instance). Note that the upstream pressure gradient is the same for any slip level. A slipping jet experiences a milder departure from the Poiseuille level in velocity and pressure. In particular, for $S=1$ , figure 10 suggests that Poiseuille flow is essentially prevalent everywhere inside and, to a lesser extent, outside the channel; the pressure gradient remains constant for a relatively large downstream distance from the channel exit, at which point the pressure levels off and vanishes. For smaller $S$ , the contraction is relatively significant, resulting in stronger streamwise flow as figure 10 indicates. In this case, inertia becomes essentially the only driving force to balance viscous force as the jet exits the channel, without any driving pressure. In contrast, a slipping jet experiences less friction, which relatively enhances the effect of inertia. Consequently, the jet behaves closer to inviscid than viscous flow, with inertia being predominantly balanced by a driving pressure. In fact, figure 10 suggests that for large $S$ , the driving pressure force remains essentially the same as for Poiseuille flow However, there is a saturation behaviour as $S$ increases, and we may not be able to infer the large $S$ behaviour given that $S=O(1)$ in the present formulation. In general, the more slipping fluid tends to retain further (and farther) its original (upstream) Poiseuille character in pressure and velocity, with reduced contraction.

The change of $u$ along the centreline reflects the presence of normal (elongational) stress, and is, evidently, accompanied by a change in $w$ in the transverse direction, starting from the zero level at the centreline. Figure 11 displays the influence of slip on the transverse flow at $z=0.25$ . The effect of flow discontinuity remains felt at this height, but lessens with slip. Far from the exit, $w$ tends to level off at a value equal to $-\!2Re^{-1}U_{0}(0.25;S)$ . This value increases with $S$ (see figure 10 a). Figure 11 shows that $w$ is negative, as expected, since $w_{z}(x,z=1/2)=-u_{x}(x,z=1/2)>0$ , suggesting a fountain flow character of the jet in the core layer, which does not prevail necessarily over the entire (half) width of the jet unless the jet is expanding (see below).

Figure 11.

Influence of slip on transverse velocity profiles in the core layer $(x=0.25)$ .

Figure 12.

Influence of slip and inertia on relaxation length $x_{\infty }$ : (a) relaxation length against $Re$ for different slip lengths, (b) relaxation length against $S$ for different Reynolds numbers.

We consider next the relaxation length $x_{\infty }$ or the downstream distance from the channel exit where the pressure gradient vanishes as suggested by figure 10(b). Like the exit length, an estimate of the influence of inertia and slip on the relaxation lengths is obtained analytically, accounting only for the contribution of the leading eigenmode. In this case, evaluating the pressure gradient from (5.21d ) at the centreline, the relaxation length becomes

(5.30) $$\begin{eqnarray}x_{\infty }(Re,S)=-\frac{1}{{\it\alpha}_{1}}\ln \left[-\frac{10^{-c}Re}{(4S+1)A_{1}U_{1}^{\prime }\left({\textstyle \frac{1}{2}}\right)}\right].\end{eqnarray}$$

The resulting profiles are displayed in figure 12 for $c=3$ against the Reynolds number, and should be contrasted with those reported earlier in figure 8 for the exit length. The relaxation length increases with $S$ but decreases with $Re$ . Interestingly, in comparison to an adhering jet, a slipping jet tends to recover more rapidly its Poiseuille profile upstream but takes longer to relax downstream from the channel exit. In general, $x_{\infty }(S>0)>-x_{0}(S<0)$ .

Finally, the jet profile is now determined by evaluating ${\it\psi}_{2}(x,z)$ from (5.18) at $z=0$ , and substituting into (4.6). We obtain $h_{1}(x)$ , and, consequently, the jet height as

(5.31) $$\begin{eqnarray}{\it\zeta}(x;Re,S)=\frac{Re^{-1}}{4S}\left\{\left[\frac{1}{S}-2U_{0}(0;S)\right]x-S\mathop{\sum }_{k=1}^{\infty }\left(\frac{A_{k}}{{\it\alpha}_{k}}\text{e}^{-{\it\alpha}_{k}x}+C_{k}\right)\right\},\end{eqnarray}$$

where (5.15) is evaluated at $z=0$ to give

(5.32) $$\begin{eqnarray}U_{0}(0;S)=\frac{S}{4S+1}\left[\frac{1}{S}+\frac{4}{\sqrt{4S+1}}\tanh ^{-1}\left(\frac{1}{\sqrt{4S+1}}\right)\right].\end{eqnarray}$$

At the channel exit,

(5.33) $$\begin{eqnarray}{\it\zeta}(x=0;Re,S)=-\frac{Re^{-1}}{4}\mathop{\sum }_{k=1}^{\infty }\left(\frac{A_{k}}{{\it\alpha}_{k}}+C_{k}\right)=0,\end{eqnarray}$$

which follows from evaluating (5.18) at $z=0$ and using the boundary conditions in (5.5) and (5.16). Thus, the jet height vanishes at the channel exit. Figure 13 illustrates the influence of slip on the jet profile for any Reynolds number. As expected, the jet height decreases with increasing slip effect. The jet height grows monotonically with distance. Like an adhering jet (Tillett Reference Tillett1968; Khayat Reference Khayat2014), a slipping jet contracts indefinitely (until uniform conditions are reached far downstream). This was predicted and verified experimentally by Philippe & Dumargue (Reference Philippe and Dumargue1991) for an adhering jet. Expression (5.31) indicates that the jet behaves linearly close to the exit as well as far from the exit; with a decrease in slope equal to $(Re^{-1}/4)\sum _{k=1}^{\infty }A_{k}$ . Consequently, more inertial jets tend to be flatter.

Figure 13.

Jet profiles for different slip parameter values.