3.1. Dead zones at the stagnation points

The height of the yield surface in figure 1(d) illustrates that ‘dead’ zones, in which the material is entirely rigid and static, develop upstream and downstream of the obstruction. Although the yield stress is regularised somewhat in our numerical scheme (see Appendix A), the log scale enables the distinction of the dead zone to be clearly seen as black regions in figure 1(d) (here we have chosen the dead zone to correspond to $Y \lesssim 2 \times 10^{-4}$ though this arbitrary threshold has little influence on the interpretation of the flow structure).

For $B>0$, there is always a dead zone in a neighbourhood of the two stagnation points at $(\pm 1,0)$; see, e.g., figure 4. This is easily argued by contradiction. Suppose that the material is yielded at $z=0$ at $(\pm 1,0)$ with $Y>0$. Then, by symmetry, the transverse flux $\boldsymbol {q}\boldsymbol {\cdot } \boldsymbol {e}_y$ vanishes, and by the boundary condition on the wall, $\boldsymbol {q}\boldsymbol {\cdot } \boldsymbol {e}_x$ vanishes (where $\boldsymbol {e}_x$ and $\boldsymbol {e}_y$ are unit vectors in the $x$ and $y$ directions, respectively). With $Y>0$, (2.3) furnishes $\partial h/\partial y=0$ and $\partial h/\partial x= \mathcal {L}$. The yield stress term in (2.4) is then singular implying that $Y=0$, a contradiction. The material is always rigid at these locations because the flow stagnates there and there is no hydrostatic pressure gradient to induce any shear stress in the layer.

Figure 4.

(a) Steady flow thickness, $h(x,y)$, and (b) yield surface height, $Y(x,y)$, for $B=1$ and $\mathcal {L}=4$ (calculated numerically); (c) $h(x,y)$ and (d) $Y(x,y)$ for $B=1$ and $\mathcal {L}=16$.

The size of the two dead zones increases with the dimensionless yield stress $B$. The upstream dead zone becomes smaller for wider obstructions (greater $\mathcal {L}$) whilst the downstream dead zone becomes larger (e.g. figure 4). This is because for greater $\mathcal {L}$, the magnitude of the ‘smoothing’ term, $-\mathcal {L}^{-1} \boldsymbol {\nabla } h$ in (2.3) is diminished. Hence, the cross-slope gradient of the free surface at the upstream boundary is larger in order to divert the flow (see § 4), and the flow reconnects downstream over a greater distance (see Hinton & Hogg Reference Hinton and Hogg2022).

A simple corollary is that there can be a ‘cusp’ at the upstream stagnation point, i.e. $\partial h/ \partial y$ is discontinuous. This is a common feature of plugged viscoplastic material (Balmforth et al. Reference Balmforth, Craster and Sassi2002). The cusp is borne out by our asymptotic analysis for relatively wide obstructions. It is slightly smoothed in the numerical results owing to the regularisation of the yield stress.

3.2. Non-uniqueness of the steady flow

Throughout this paper, we primarily focus on the upstream flow, noting that for $\mathcal {L} \gg 1$ there is a long dead zone downstream of the obstruction and the flow will reconnect over a length scale proportional to $\mathcal {L}$ (Hinton & Hogg Reference Hinton and Hogg2022). Details of the flow structure at the edge of the downstream dead zone and the possibility of a non-unique steady state were investigated in Hinton & Hogg (Reference Hinton and Hogg2022) for flow around large mounds corresponding to $\mathcal {L} \gg 1$. The steady state downstream will depend on the initial condition prior to the supply of constant flux from upstream. For example, the plane may have been entirely dry with $h=0$ everywhere in which case the downstream dead zone mostly corresponds to a dry zone with $h=0$ (provided that $\mathcal {L} \gg 1$). If instead, the plane was coated with a uniform, rigid and stationary layer of fluid with thickness $h=B$ prior to the line source initiation, then the downstream dead zone would mostly be a region in which $h=B$. In the numerical results presented in this paper, we have assumed that this is the case; see, for example, figure 4. It should also be noted that any $0\leqslant h \leqslant B$ is allowed in the downstream dead zone provided that the layer is entirely rigid. Importantly, the downstream behaviour has negligible effect on the upstream thickness.

4.1. Relatively low yield stress ($\lambda \ll 1$)

In terms of the two original dimensionless groups, $B$ and $\mathcal {L}$, the present $\lambda \ll 1$ regime is (see figure 3)

(4.10a,b)\begin{equation} {B \ll \mathcal{L}^{-1/5}, \quad \mathcal{L} \gg 1.} \end{equation}

This regime corresponds to quasi-Newtonian flow; see figure 6. We seek a regular asymptotic expansion for the flow thickness at the wall of the form

(4.11)\begin{equation} \bar{G}(y) = \bar{G}_0(y) + \lambda \bar{G}_1(y) + \dots. \end{equation}

The terms are obtained from the ODE (4.9). The first term is given by the Newtonian solution (corresponding to $B=0$, see Hinton et al. Reference Hinton, Hogg and Huppert2020)

(4.12)\begin{equation} \bar{G}_0(y) = 10^{1/5} (1-y^2)^{1/5}. \end{equation}

The next term accounts for the yield stress (for details, see Appendix B.1)

(4.13)\begin{equation} \bar{G}_1(y)=\frac{0.9549-p(y)}{(1-y^2)^{4/5}}, \quad \text{where} \ p(y) = \frac{10^{4/5}}{5} \int_0^{|y|} (1-s^2)^{3/5} \, \mathrm{d} s. \end{equation}

The asymptotic prediction for $\lambda \ll 1$ is compared with the numerical results in figures 5(a) and 6(c), where good agreement is observed. The height of the yield surface in the deep region (4.3) is given by

(4.14)\begin{equation} Y=\max\left(0, \mathcal{L} (x+1) +\mathcal{L}^{2/5} \bar{G}_0(y) +B \mathcal{L}^{3/5}\left(\bar{G}_1(y)- \frac{1}{ |\bar{G}_0'(y)|}\right) + \dots \right), \end{equation}

which is compared with the numerical result along $y=0.6$ in figure 6(d). In this regime, the flow thickness, $h$, and the height of the yield surface, $Y$, are of the same order of magnitude in the deep ponded region, and the lateral extent of the deep region is similar to the lateral extent of the flowing region (where $Y>0$); see figure 6. Equation (4.14) can be used to obtain the boundary of the region in which $Y=0$, up to and including second-order terms,

(4.15)\begin{equation} x_B(y) =-1+ \min \left(0, \mathcal{L}^{-3/5} \bar{G}_0(y) + B \mathcal{L}^{-2/5} \left(\bar{G}_1(y)- \frac{1}{ |\bar{G}_0'(y)|}\right) \right), \end{equation}

which is shown as a blue dashed line in figure 6(b). This boundary intersects the obstruction wall near the centreline where $\bar {G}_0'(y)$ becomes small, and a different asymptotic expansion is needed in $|y| \sim \lambda$. This inner region arises because the yield stress plays a more significant role near the centreline owing to the relatively smaller gradients of flow thickness there. Indeed, as argued above, there is always a dead zone at the stagnation point. Details of this region and the matching with the outer expansion are given in Appendix B.1. Importantly, the flow thickness at $y=0$ is unchanged at first and second order from the outer expansion (4.11). Hence the maximum flow thickness, which occurs at $x=-1$, $y=0$, is given by

(4.16)\begin{equation} h_{max} = 10^{1/5} \mathcal{L}^{2/5} + 0.9549 B \mathcal{L}^{3/5}+ \dots, \end{equation}

which is shown as a red dashed line in figure 5(b). The present analysis is based upon $\lambda \ll 1$ and $h \gg h_{\infty }$ so that the expansion (4.11) remains asymptotic. In the next subsection, we analyse the other regime, of large $\lambda$.

Figure 6.

Results for a wide obstruction and relatively low yield stress; $\mathcal {L}=40$ and $B=0.1$. (a) Steady flow thickness and (b) yield surface from the full numerical method. The boundary of the flowing region predicted by (4.15) is shown as blue dashed lines. (c) Flow thickness along $y=0$ (blue) and $y=0.6$ (black). The red dashed lines show the asymptotic prediction (4.2) with $\bar {G}(y)$ given by (4.11). (d) Corresponding height of the yield surface; red dashed line is (4.14).

Figure 7.

Results for a wide obstruction and relatively high yield stress; $\mathcal {L}=40$ and $B=2$. (a) Steady flow thickness and (b) yield surface from the full numerical method. (c) Flow thickness along $y=0$ and $y=0.6$. The red dashed lines show the asymptotic prediction (4.2) with $\bar {G}(y)$ given by (4.18). (d) Flow thickness at the wall comparing the large yield stress prediction (red dashed line) with the result from the ODE (4.7) (blue dot-dashed line) and the full numerical result (black line).

4.2. Relatively large yield stress ($\lambda \gg 1$)

In terms of the two original dimensionless groups, $B$ and $\mathcal {L}$, the present $\lambda \gg 1$ regime is (see figure 3)

(4.17)\begin{equation} {\mathcal{L}^{-1/5}\ll B \ll \mathcal{L}.} \end{equation}

In this regime, the role of the yield stress is non-negligible and appears in the leading order of the expansion for $\bar {G}$. The fluid is plugged to leading order in the deep region upstream of the obstruction; see figure 7 and note the qualitatively different contour structure to the low yield stress case in figure 6(a). Hence, the first term in brackets in (4.7) becomes small. This leading-order behaviour provides no transverse flux. This motivates the following expansion

(4.18)\begin{equation} {\bar{G} = \lambda^{1/2}\left( \tilde{G}_0(y) + \lambda^{-5/6} \tilde{G}_1(y) + \dots \right),} \end{equation}

where the second term provides the transverse flux to balance the incoming flow from upstream. The leading-order plugged shape is then calculated from (4.9)

(4.19)\begin{equation} {\tilde{G}_0(y) = \left[2(1-|y|) \right]^{1/2}.} \end{equation}

The next term, which accounts for the flux balance (and hence the left-hand side of (4.7)), is (for details, see Appendix B.2)

(4.20)\begin{equation} {\tilde{G}_1(y) = \frac{1.0600 - r(y)}{(1-|y|)^{1/2}}} \quad \text{where} \ r(y) = \int_0^{|y|} \frac{s^{1/3}}{2^{2/3} (1-s)^{1/2}} \, \mathrm{d} s. \end{equation}

This prediction is shown as red dashed lines in figures 7(c) and 7(d) with $B=2$ and $\mathcal {L}=40$; the numerical results are shown as continuous lines and the blue dot-dashed line is from integrating (4.7). The height of the yield surface in the deep region is found to be (from (4.3))

(4.21)\begin{equation} Y=\mathcal{L}(1+ x) +\mathcal{L}^{1/3} B^{-1/3} 2^{1/3} |y|^{1/3} + \dots. \end{equation}

As expected, the height of the yield surface ($\sim \mathcal {L}^{1/3} B^{-1/3}$) is asymptotically smaller than the height of the free surface ($\sim B^{1/2} \mathcal {L}^{1/2}$) in this regime; see the colourscales in figures 7(a) and 7(b). The extent of the deep region is $x \sim \mathcal {L}^{-1/2} B^{1/2}$, owing to the quasi-plugged term, $\tilde {G}_0$. The extent of the region over which there is appreciable transverse flux is asymptotically smaller (in contrast to the case of relatively small yield stress) and is given by

(4.22)\begin{equation} x_B(y) =-1 -\mathcal{L}^{-2/3} B^{-1/3} 2^{1/3} |y|^{1/3}. \end{equation}

The maximum flow thickness is

(4.23)\begin{equation} h_{max} = 2^{1/2} B^{1/2} \mathcal{L}^{1/2} + 1.0600 B^{-1/3} \mathcal{L}^{1/3}+ \dots, \end{equation}

which is shown as a blue dot-dashed line in figure 5(b). The second term in (4.23) is smaller than the first provided that $\lambda \gg 1$, as expected.

For large fixed $\mathcal {L}$, the present analysis will, in fact, break down for very large $B$. This is because the far-field flow thickness is $h_{\infty } \approx B$ for large $B$, and this exceeds the pond flow thickness, ($\sim B^{1/2} \mathcal {L}^{1/2}$) for $B \gg \mathcal {L}$, which invalidates the assumption of a deep pond. Hence the expansion (4.18) is valid when (4.17) applies. The case $B \gg \mathcal {L} \gg 1$ instead corresponds to a ‘deep’ oncoming flow relative to the obstruction width, which is analysed in § 5 (see also the parameter space in figure 3). For example, in figure 7(d), there is excellent agreement as $B=2$, $\mathcal {L}=40$ and $\lambda = 4.183$, whereas the agreement in figure 5(a) for $\lambda =10$ is not so good, especially near the edges of the upstream wall, because $\mathcal {L}=20$ and $B=5.493$ with a far-field flow thickness of $h_{\infty }=5.834$.

5.1. Analogy with flow in a Hele-Shaw cell ($\mathcal {L} \ll h_{\infty }$)

To dissect the flow physics in the narrow obstruction regime, we seek an asymptotic expansion of the form

(5.3)\begin{equation} h=h_{\infty} + \mathcal{L} \left[1+x+p(x,y) \right] + O\left( \mathcal{L}^2 \right), \end{equation}

where $p(x,y)$ is associated with the hydrostatic pressure that drives the flow. To leading order in $\mathcal {L}$, the flux (2.3) is

(5.4)\begin{equation} \boldsymbol{q} =- \frac{1}{2} Y^2 \left(3h_{\infty}-Y \right) \boldsymbol{\nabla} p, \quad \text{where} \ Y=\max\left(0, h_{\infty}-\frac{B}{|\boldsymbol{\nabla} p|} \right). \end{equation}

This form of the flux is identical to that for viscoplastic flow in a Hele-Shaw cell of fixed gap width, $h_{\infty }$, with $p(x,y)$ representing the pressure that drives the flow with constant flux (which is supplied far upstream) (Hewitt et al. Reference Hewitt, Daneshi, Balmforth and Martinez2016). Hence, to leading order, the streamlines and the pressure gradient in free-surface flow around a relatively narrow obstruction are given by those in a Hele-Shaw cell with a blockage of the same cross-section. The similarity arises because in this narrow obstruction regime, the thickness of the free-surface flow is very minorly perturbed and the flux is approximately independent of this thickness. The leading-order flux depends only on the gradients of the free surface (see (5.4)) and so the system behaves in an analogous fashion to flow in a Hele-Shaw cell of fixed gap width. This equivalence was previously observed for viscous Newtonian flows (Hinton et al. Reference Hinton, Hogg and Huppert2020).

These observations conveniently allow us to exploit previous research on obstructed viscoplastic flow in a Hele-Shaw cell (Hewitt et al. Reference Hewitt, Daneshi, Balmforth and Martinez2016; Daneshi et al. Reference Daneshi, MacKenzie, Balmforth, Martinez and Hewitt2020). For example, the symmetry of the two dead zones (figure 8) is rationalised by the reversibility of steady flow in a Hele-Shaw cell. There is a slight difference in that here the far-field flow thickness (corresponding to the gap width) varies with $B$ but this merely leads to some extra factors of $h_\infty$ in the results (this alters some of the scalings, particularly for large $B$).

To analyse the flow, we use a hodograph transformation. The important results are quoted in the following, with full details given by Neuber (Reference Neuber1961), Entov (Reference Entov1970) and Hewitt et al. (Reference Hewitt, Daneshi, Balmforth and Martinez2016). We transform $(x,y)$ space to the hodograph plane of $(Q, \theta )$, where $Q=|\boldsymbol {q}|$ is the magnitude of the flux and $\theta$ is the angle that the streamlines make with the downslope direction. In this way using the results from Hewitt et al. (Reference Hewitt, Daneshi, Balmforth and Martinez2016) and correcting a sign error, we find

(5.5)\begin{equation} \mathrm{d}\kern0.7pt x + \mathrm{i}\, \mathrm{d}y=-{\rm e}^{\mathrm{i} \theta} \left(\frac{\mathrm{d} p}{S} + \mathrm{i} S^{-2}\frac{\partial S}{\partial Q} \frac{\partial p}{\partial \theta} \,\mathrm{d} Q -\mathrm{i} \frac{Q}{S} \frac{\partial p}{\partial Q}\, \mathrm{d} \theta \right) \quad \text{where} \ S=|\boldsymbol{\nabla} p|. \end{equation}

The magnitude of (5.4)a relates $Q$ and $S$ via

(5.6)\begin{equation} Q=\left(h_{\infty}-\frac{B}{S} \right)^2 \left(h_{\infty} + \frac{B}{2 S} \right) S. \end{equation}

Local mass conservation, $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {q}=0$, is recast as a linear elliptic partial differential equation for $p$,

(5.7)\begin{equation} \frac{S^2}{Q({\partial S}/{\partial Q})}\frac{\partial }{\partial Q} \left( \frac{Q^2}{S} \frac{\partial p}{\partial Q} \right) + \frac{\partial^2 p}{\partial \theta^2} =0, \end{equation}

where $S=S(Q)$ (see (5.6)). Some separable solutions to (5.7) exist, which enables analytical progress in § 5.2. The flow behaviour is qualitatively different in the regimes of a relatively low and relatively high yield stress, which are analysed in §§ 5.2–5.3, respectively.

5.2. Relatively low yield stress ($B \ll 1, \mathcal {L} \ll 1$)

For $0< B \ll 1$, $h_\infty \approx 1$ and so (5.1) becomes $\mathcal {L} \ll 1$; the present regime occupies the bottom-left of parameter space in figure 3. In this regime, viscoplastic stagnation point flow arises; see figure 9(a). A key feature is the appearance of a dead zone in the vicinity of the stagnation point; see figures 8(a) and 8(b).

Figure 9.

Direction and magnitude of the flux $\boldsymbol {q}$ obtained from the numerical method with $\mathcal {L}=0.1$ and (a) $B=0.5$ and (b) $B=8$.

The shape of the dead zone can be determined analytically by noting that $Q=0$ on the boundary of the dead zone and there is stagnation point flow further away (with larger values of $Q$) with

(5.8)\begin{equation} p \sim (1+x)^2 -y^2 = \frac{Q^2}{4} \cos 2 \theta. \end{equation}

Note that this stagnation flow solution for $p$ is defined up to an arbitrary multiplicative and additive constant (and, hence, the same applies to (5.10)). Equation (5.8) provides the boundary condition for the partial differential equation in the hodograph plane (5.7). A separable solution is sought for (5.7) and the behaviour for $Q \ll 1$ is obtained by shooting numerically. The leading-order shape of the dead zone boundary and the pressure at the boundary are then parameterised by (see Hewitt et al. Reference Hewitt, Daneshi, Balmforth and Martinez2016)

(5.9)$$\begin{gather} (x+1,y) = 0.242 B h_{\infty}^2 (-\cos^3 \theta, \pm \sin^3 \theta), \end{gather}$$

(5.10)$$\begin{gather}p \sim 0.181 B^2 h_{\infty} \cos 2 \theta, \end{gather}$$

where $\theta \in [0, {\rm \pi}/2]$. The dead zone boundary (5.9) is compared with the numerical result for $B=0.5$ in figure 8(b). It extends a distance $0.242 B h_{\infty }^2$ away from the origin along both the $x$ and $y$ axes. An identical dead zone occurs at the downstream boundary.

The dead-zone shape (5.9) is valid provided that there is stagnation point flow at the upstream wall. This requires that the dead zone occupies only a small fraction of the wall. That is, $B h_{\infty }^2 \ll 1$ or, equivalently, $B \ll 1$ since $h_{\infty } \sim 1$ for small $B$. For larger values of $B$, the dead zone encompasses the corners of the square obstruction invalidating the assumption of stagnation point flow; see figures 8 and 9. Different analysis is needed in this case; see § 5.3.

For $0< B\ll 1$, the presence of the dead zone only alters the flow thickness at order $B^2$ (see (5.10)). Away from the dead zone, there is an outer flow that is quasi-Newtonian for which the yield stress influences the flow thickness at order $B$. In this outer flow, we seek the following expansion for the hydrostatic pressure

(5.11)\begin{equation} p(x,y) = p_N(x,y) + B p_1(x,y) + O(B^2), \end{equation}

where $p_N(x,y)$ is Newtonian solution. Using (5.3) and (5.4), we find that the pressure terms satisfy

(5.12a,b)\begin{equation} \nabla^2 p_N =0, \quad \nabla^2 p_1 = \frac{3}{2} \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\frac{\boldsymbol{\nabla} p_N}{|\boldsymbol{\nabla} p_N|}\right), \end{equation}

with boundary conditions,

(5.13)$$\begin{gather} \frac{\partial p_N}{\partial n} =0, \quad \frac{\partial p_1}{\partial n} =0, \quad \text{for}\ (x,y) \in \partial \varOmega, \end{gather}$$

(5.14)$$\begin{gather}p_N \to -x-1, \quad p_1 \to 0, \quad \text{as}\ x^2+y^2 \to \infty, \end{gather}$$

where $\partial \varOmega$ is the boundary of the square and $\partial /\partial n$ denotes the normal derivative. The linear system for $p_N(x,y)$ and $p_1(x,y)$ is integrated numerically using the same finite-difference method as described in Appendix A. The solutions are shown in figures 10(a) and 10(b).

Figure 10.

Steady flow around a relatively narrow obstruction with $B \ll 1$. (a) Leading-order pressure, $p_N(x,y)$ for Newtonian flow (see (5.12a,b)). (b) The pressure perturbation $p_1(x,y)$ associated with $B \ll 1$. Flow thickness for $\mathcal {L}=0.05$ (c) along the centreline, $y=0$, upstream of the obstruction, and (d) at the wall, $x=-1$ showing numerical results (black lines) and prediction (5.11) (dot-dashed lines).

The prediction for the flow thickness from (5.11) is compared with the full numerical results along the centreline ($y=0$) upstream of the obstruction, and at the wall ($x=-1$) in figures 10(c) and 10(d), respectively. The form of $p_1(x,y)$ predicted by the full numerical result for $\mathcal {L}=0.025$ and $B=0.5$ is shown in figure 11(a), whilst the difference between the numerical and asymptotic thickness is shown in figure 11(b); the error is at most $3.5 \times 10^{-3}$.

Figure 11.

Comparison between the full numerical results and the asymptotic expansion given by (5.3) and (5.11) with $\mathcal {L}=0.025$ and $B=0.5$. (a) Pressure perturbation $p_1(x,y)$ inferred from the full numerical results (cf. figure 10b). (b) Difference in predicted flow thickness; $\mathrm {error}=h_{numerical} - h_{asymptotic}$.

As the stagnation point, $(-1,0)$, is approached the functions take the following limits

(5.15a,b)\begin{equation} p_N \to 1.353, \quad p_1 \to -0.679. \end{equation}

Although both $p_N(x,y)$ and $p_1(x,y)$ are finite everywhere, the present asymptotic analysis breaks down near the stagnation point because $|\boldsymbol {\nabla } p_N|$ vanishes there. The rigid region encompasses the entire thickness of the layer (see (5.4)) leading to the dead zone analysed previously. However, the pressure (and, hence, the flow thickness) is accurately approximated by $p=p_N(x,y) + B p_1(x,y)$ everywhere because the effect of the dead zone only appears at $O(B^2)$ in the pressure.

The maximum flow thickness is

(5.16)\begin{equation} h_{max}=h_{\infty}(B) + \mathcal{L}[1.353 - 0.679 B] + O(\mathcal{L}^2,\mathcal{L}B^2). \end{equation}

This is compared favourably to the numerical results in figure 12. The perturbation to the flow thickness reduces with increasing yield stress (this also occurs for the large $B$ regime; see § 5.3 and figure 13a). This contrasts with relatively wide obstructions for which the flow thickness perturbation increased with greater yield stress. For wide obstructions, the flow is primarily diverted around the obstruction by gradients in the free surface inducing transverse flux and for a fixed flux, greater gradients are needed at higher yield stresses.

Figure 12.

The maximum flow thickness, $h_{max}$, as a function of $\mathcal {L}$ for $B=0$, $B=0.25$ and $B=0.5$. The lines show the small-$B$ prediction (5.16), and the plus signs are from the full numerical method; see Appendix A.

Figure 13.

Maximum perturbation to the flow thickness, $h_{max}-h_{\infty }$ for $\mathcal {L}=0.1$. Analytic scaling (black line, $h_{max}-h_{\infty } \sim B^{-3/4}$ (5.25)) is compared with numerical results (blue crosses). (b) Flow thickness along the centreline for $\mathcal {L}=0.1$ and $B=8$. Numerical result (black line; see also figure 8e) is compared with the analytic prediction for the gradient (red dashed line; (5.24)).

The decrease in the flow thickness perturbation with $B$ for narrow obstructions can be interpreted through the relationship (5.6) between the volume flux, $Q$ and the pressure gradient $S=|\boldsymbol {\nabla } p|$. For $B \ll 1$, (5.6) becomes

(5.17)\begin{equation} Q = S + \tfrac{3}{2} B (S-1) + O(B^2). \end{equation}

Hence, the volume flux (relative to the far-field volume flux) of a Bingham fluid varies more rapidly with the pressure gradient, $S$, than a Newtonian fluid (this is because the pressure gradient alters the proportion of the layer that is yielded). Smaller variations in pressure are required to divert the volume flux around the obstruction for a Bingham fluid and so there is a smaller variation in the flow thickness.

5.3. Relatively high yield stress ($B \gg 1$, $\mathcal {L} \ll B$)

For large yield stresses, $h_\infty \approx B$ and (5.1) becomes $\mathcal {L} \ll B$; this case is depicted in the top-left of parameter space in figure 3. In this regime, the dead zone encompasses the entire upstream and downstream boundaries of the obstruction (see figure 8f). The streamwise extent of the dead zone increases with yield stress. Although it is not possible to use the stagnation point flow to obtain an analytic solution for the shape of the dead zone (see figure 9b), we can find scalings for the extent of the dead zone and the flow thickness. Fluid from upstream in $-1< y<1$ must be diverted around the obstruction via boundary layers at the edge of the dead zone in which $Q \sim 1$ (Hewitt et al. Reference Hewitt, Daneshi, Balmforth and Martinez2016). Since $h_{\infty } \sim B \gg 1$, (5.6) may be written as

(5.18)\begin{equation} S(Q) = \frac{B}{h_{\infty}} + \frac{1}{h_{\infty}^2} \left( \frac{2 B Q}{3} \right)^{1/2} + \dots. \end{equation}

Equation (5.7) then becomes, to leading order,

(5.19)\begin{equation} 6^{1/2} B^{1/2} h_{\infty} Q^{-1/2} \frac{\partial }{\partial Q} \left(Q^2 \frac{\partial p}{\partial Q} \right) + \frac{\partial^2 p}{\partial \theta^2} =0. \end{equation}

Since the flux from upstream is diverted by the flow in a neighbourhood of the dead zone, we have $Q=O(1)$ and balancing the two terms in (5.19) furnishes a scaling for the angle of the dead zone boundary (cf. Hewitt et al. Reference Hewitt, Daneshi, Balmforth and Martinez2016),

(5.20)\begin{equation} \theta \sim B^{-1/4} h_{\infty}^{-1/2}. \end{equation}

Since the edge of the dead zone meets the obstruction at $y=1$ and $\theta \sim y/x$, the dead zone extent is given by

(5.21)\begin{equation} x \sim B^{1/4} h_{\infty}^{1/2} \sim B^{3/4}. \end{equation}

These scalings are different to those in a Hele-Shaw cell (Hewitt et al. Reference Hewitt, Daneshi, Balmforth and Martinez2016) because the effective gap width, $h_{\infty }$ varies with $B$. Within the dead zone, $Y=0$ and using both (2.4) and (5.21) furnishes the scalings $x=B^{3/4} \chi$ and $h=h_{\infty } + \mathcal {L} B^{-3/4} h_1$. Upon substituting these into $Y=0$, we find that

(5.22)\begin{equation} h_{\infty} - \frac{B}{\sqrt{1- 2 B^{-3/2} \dfrac{\partial h_1}{\partial \chi} + B^{-3/2} \left(\dfrac{\partial h_1}{\partial y}\right)^2}} + O\left(\mathcal{L} B^{-3/4}, B^{-2}\right)=0. \end{equation}

Recalling that $h_{\infty }=B+[2/(3B)]^{1/2}+\dots,$ the thickness in the dead zone satisfies

(5.23)\begin{equation} \frac{\partial h_1}{\partial \chi} - \frac{1}{2} \left(\frac{\partial h_1}{\partial y} \right)^2 = \left(\frac{2}{3} \right)^{1/2}, \end{equation}

to leading order. With analytical boundary data at the edge of the dead zone, this equation could be solved via Charpit's method. Unfortunately, such data are not available (see Hewitt et al. Reference Hewitt, Daneshi, Balmforth and Martinez2016). However, we can make some progress by noting that along the centreline $\partial h_1/\partial y=0$ (there is no cusp to leading order in the $\mathcal {L} \ll 1$ regime) and, hence,

(5.24)\begin{equation} \frac{\partial h}{ \partial x}=B^{-3/2} \mathcal{L} \sqrt{2/3}, \end{equation}

which shows good agreement with the full numerical result (see figure 13b). The perturbation to the maximum flow thickness has the scaling

(5.25)\begin{equation} h_{max}-h_{\infty} \sim \mathcal{L} B^{-3/4}, \end{equation}

which is compared favourably to the numerical results in figure 13(a).