2. Unsteady model and switching off the source

We analyse the unsteady flow that arises when the upstream source is terminated. The dimensionless variables are given by (1.1), and dimensionless time is given by

(2.1) \begin{equation} { t = \frac {\rho g \hat {H}_\infty ^2 \sin \beta }{3\mu L}\hat {t}}, \end{equation}

where

(2.2) \begin{align} {\mu =k^{1/n} \left (\frac {\rho g \hat {H}_\infty \sin \beta }{3}\right )^{1-1/n}} \end{align}

is a viscosity scale, and $t=0$ corresponds to the switch-off time. The evolution of the flow thickness $H(x,y,t)$ is governed by

(2.3) \begin{equation} \frac {\partial {\textit{H}}}{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{q} = 0. \end{equation}

The initial condition is given by a steady flowing solution computed using the method described by Hinton et al. (Reference Hinton, Hewitt and Hogg2023), with examples plotted in figure 2(a–c).

Figure 2. Steady flowing solutions for $b=1$ and: (a) Bingham fluid, $H_\infty =2.53$ (corresponding to $q_\infty =7.1$ ); (b) Bingham fluid, $H_\infty =1.68$ ( $q_\infty =1$ ); (c) Herschel–Bulkley fluid, $n=0.5$ , $H_\infty =1.30$ ( $q_\infty =0.03$ ); see (1.3). When the flux source is switched off, all of these converge to the same final stationary slump, shown in (d). Note the different scale bars used across the panels.

Equation (2.3) is numerically integrated in time; for details of the numerical method, see Appendix A. In the computation, the source is located relatively far upstream at $x=-6$ , so at $t=0$ , the boundary condition at the left-hand end of the domain is adjusted to

(2.4) \begin{equation} H =1 \quad \text{at} \ x =-6. \end{equation}

Note that a different location for the upstream source simply leads to a time offset in the ensuing dynamics after the switch off (the reduction in $H$ simply takes longer to propagate downstream towards the barrier). The no-flux boundary condition on the obstruction (1.8) is unchanged.

Fluid drains downstream, and the flow thickness converges to a rigid stationary shape everywhere; see figures 2 and 3.The final rigid shape is approached asymptotically with the following late-time scalings when $t \gg 1$ (Matson & Hogg Reference Matson and Hogg2007; Hogg & Matson Reference Hogg and Matson2009):

(2.5) \begin{align} {H (x,y,t) = H_{\textit{final}}(x,y) + t^{-n}\, H_1(x,y) +\cdots , \quad Y(x,y,t)= t^{-n}\, Y_1(x,y) +\cdots ,} \end{align}

which are verified numerically for our problem in figure 4. Here, $H_{\textit{final}}(x,y)$ is the final height of the free surface, while $H_1(x,y)$ and $Y_1(x,y)$ dictate the late-time approach of the height of the free surface and yield surface, respectively, to their final static shapes. Henceforth, we drop the subscript ‘final’ for brevity. Figure 4 shows that fluid near the upstream corners of the barrier takes longer to drain away and approach the arrested height than elsewhere. The yield surface exhibits a maximum at the upstream corners, and is also significant on the sides of the barrier and at the downstream corners. This is because fluid draining from upstream is diverted around the obstruction and thus flows past the upstream corners, giving rise to a large flux at these locations (see figure 5). The increased flux at the corners requires a higher yield surface. A similar phenomenon occurs at the downstream corners, where fluid flowing around the sharp corner induces a large flux, and therefore a relatively high yield surface, but to a lesser extent.

Figure 3. Late-time slumping of a Bingham fluid. (a,b) The height of the free surface (dashed black lines) at the upstream wall of the obstruction ( $x=-1$ ) at times $t\in \{\textrm{e}^1,\textrm{e}^2,\textrm{e}^3,\textrm{e}^4,\textrm{e}^5,\textrm{e}^6\}$ after the upstream flux (at $x=-6$ ) is terminated. (c,d) Corresponding results along the centreline ( $y=0$ ). In (a) and (c), parameter values are $b=1,\ q_\infty =1$ , whilst for (b) and (d), $b=1,\ q_{\infty }=1/2$ . The final shape is shown with a blue line (from § 3).

Figure 4. Late-time decay of (a,b,c) a Bingham fluid to the final state, and (d,e,f) a Herschel–Bulkley fluid with $n=1/4$ . (a,d) Transient evolution of the relative flow thickness at four points along the centreline ( $y=0$ ); these show the late-time decay rate $\sim 1/t^n$ . (b,e) The relative difference between the fluid thickness at $t=20$ and the final rigid thickness ( $t \to \infty$ ). (c, f) The relative height of the yield surface at $t=20$ . Parameter values are $b=1$ and $H_\infty =1.68$ .

Figure 5. Streamlines of the flow (white lines) and magnitude of the flux (colour bar) for a Bingham fluid at $t=20$ . Parameter values are $b=1$ and $H_\infty =1.68$ .

3. Static shape

Figure 2(a–c) show the thickness of the steady flow for different values of $q_\infty$ and $n$ . When the source is terminated, these three cases all converge to the same rigid shape, suggesting that it depends only on $b$ , and specifically, is independent of the power-law exponent $n$ and the upstream source flux $q_\infty$ . In this section, we derive an exact expression for the rigid shape.

In the far field, the flow thickness converges to the critical rigid height of viscoplastic material on an inclined plane, which furnishes

(3.1) \begin{align} H \to 1 \quad \text{as} \ x^2 +y^2 \to \infty . \end{align}

The rigid shape is found by evaluating (2.3) at late times with $Y \to 0$ . Thus the maximal rigid shape is found to satisfy (Balmforth et al. Reference Balmforth, Craster and Sassi2002)

(3.2) \begin{align} \left (1-b\frac {\partial {\textit{H}}}{\partial x}\right )^2 + \left (b \frac {\partial {\textit{H}}}{\partial y}\right )^2 = \frac {1}{H^2} , \end{align}

where $b$ is the single dimensionless parameter that characterises the shape of the rigid slump. The no-flux boundary condition on the obstruction walls, which applies at all times, imposes

(3.3) \begin{align} \left (1-b\frac {\partial {\textit{H}}}{\partial x}, -b\frac {\partial {\textit{H}}}{\partial y}\right ) \boldsymbol{\cdot }\boldsymbol{n} = 0, \end{align}

which is applied at $x=\pm 1,\ {-1}\leqslant y\leqslant 1$ and $y=\pm 1,\ {-1}\leqslant x\leqslant 1$ . The slump is determined maximal when it is ‘on the verge’ of yielding; any further addition of material to the slump will induce a flow that carries the added fluid downhill. The governing partial differential equation (3.2) and boundary conditions (3.1), (3.3) feature only $b$ , and this is corroborated by the different flows in figure 2(a,b,c) all adjusting to the same arrested state.

3.1. Shape of the free surface at the upstream wall

We parametrise the upstream wall of the obstruction by $y=s$ with $-1 \leqslant s \leqslant 1$ , $x=-1$ and $H(-1,s) = H_0(s)$ , which is to be determined. Combining (3.2) and (3.3) on the upstream wall furnishes

(3.4) \begin{align} {\textit{bH}}_0 \frac {\mathrm{d} H_0}{\mathrm{d}s} = -\text{sgn}(s). \end{align}

The boundary condition in the far field, (3.1), imposes the following condition at the corners of the obstruction:

(3.5) \begin{align} H_0(\pm 1) = 1 . \end{align}

This condition is required for a self-consistent solution. Specifically, one cannot satisfy (3.1) if the height at the obstruction corners exceeds the far-field height. Equation (3.4) is integrated to obtain

(3.6) \begin{align} H_0(s)&=\sqrt {2b^{-1}(1-|s|)+1}. \end{align}

This shape is shown as a blue line in figures 3(a) and 3(b), where it is compared to the numerical results for the unsteady flow. The gradient of $H_0$ is discontinuous at $s=0$ .

3.2. Charpit’s method

Equation (3.6) provides the boundary conditions to solve (3.2) in the region directly upstream of the obstruction where the rigid material is supported. The solution is obtained using Charpit’s method for first-order nonlinear partial differential equations. Defining

(3.7) \begin{align} p=\frac {\partial {\textit{H}}}{\partial x}, \quad q=\frac {\partial {\textit{H}}}{\partial y}, \end{align}

Charpit’s equations transform (3.2) into five ordinary differential equations along the characteristics (Ockendon et al. Reference Ockendon, Howison, Lacey and Movchan2003):

(3.8a) \begin{align} \dot {x} &= -2b(1-bp) , \end{align}

(3.8b) \begin{align} \dot {y} &= 2b^2 q , \end{align}

(3.8c) \begin{align} \dot {p} &= -2p H^{-3} , \end{align}

(3.8d) \begin{align} \dot {q} &= -2q H^{-3} , \end{align}

(3.8e) \begin{align} \dot {H} &= 2b^2q^2-2bp(1-bp) , \end{align}

where a dot indicates differentiation with respect to $\tau$ , which parametrises the characteristics. The characteristics (3.8a )–(3.8e ) emanate from the upstream wall of the obstruction where the boundary data from (3.6) are specified. The forms of the characteristics are qualitatively different depending on whether they emanate from the face of the upstream wall ( $-1\lt y\lt 1$ ) or from the corners ( $y=\pm\, 1$ ), so the analysis is separated accordingly.

3.2.1. Characteristics emanating from the upstream wall ( $-1\lt y\lt 1$ )

By combining (3.6) and (3.2), we obtain the following boundary data on the upstream wall for the characteristics:

(3.9a) \begin{align} x_0(s) &=-1 , \end{align}

(3.9b) \begin{align} y_0(s) &= s , \end{align}

(3.9c) \begin{align} p_0(s) &= \frac {1}{b} , \end{align}

(3.9d) \begin{align} q_0(s) &= -\frac {\text{sgn}(s)}{b \sqrt {2b^{-1}(1-|s|)+1}} , \end{align}

(3.9e) \begin{align} H_0(s) &= \sqrt {2b^{-1}(1-|s|)+1} , \end{align}

where subscript ‘0’ indicates evaluation at $\tau =0$ . Equations (3.8a )–(3.8e ) are integrated subject to the boundary data (3.9a )–(3.9e ). This furnishes part of the exact solution, given by (full details are reported in Appendix B)

(3.10a) \begin{align} x+1 &= b \left [H-\sqrt {2b^{-1}(1-|s|)+1}\right ] \nonumber \\&\quad -\, b \log \left [\frac {\pm b\left (\sqrt {2b^{-1}(1-|s|)+1}-1\right )-y+s}{(H-1)b}\right ]\! , \end{align}

(3.10b) \begin{align} s &= y\pm b\sqrt {2b^{-1}(1-|y|)+1-H^2} . \end{align}

This solution covers the region of the $(x,y)$ plane penetrated by characteristics emanating from the upstream wall, which is given by

(3.11) \begin{equation} -1\lt \, y\lt 1 \quad \text{and} \quad b \left (A(y)-1 \right ) -b \log \left (\frac {1-|y|}{b(A(y)-1)}\right ) \leqslant \, x+1\leqslant 0, \end{equation}

with

(3.12) \begin{align} A(y) \equiv H(y,s=1)= \sqrt {1+2b^{-1}(1-|y|) - b^{-2}(1-|y|)^2}. \end{align}

The characteristics and the region penetrated are shown by black lines in figure 6. We note that $A(0) \geqslant 1$ if $b\gt 1/2$ . Thus for $b \leqslant 1/2$ , the characteristics encompass the entirety of the centreline, $y=0$ , $x \in (-\infty ,-1]$ . The solution beyond this region is obtained by considering characteristics that emanate from the upstream corners.

Figure 6. (a) Characteristics for the thickness of the rigid shape supported by the obstruction for $b=1/2$ . (b) Characteristic projections in the $(x,y)$ plane corresponding to (a). (c) Characteristic projections for $b=1$ . (d) Characteristic projections for $b=2$ . The solid black lines represent characteristics emanating from the upstream wall, whilst the red dashed lines represent those emanating from the corners. See also figure 7.

3.2.2. Characteristics from the upstream corners

At the upstream corners, the no-flux condition (3.3) is applied at all incoming angles to the corner. That is, we parametrise the boundary normal vector at the upstream corners by

(3.13) \begin{align} \boldsymbol{n}=(\sin \theta , \cos \theta ), \end{align}

with $\theta \in [-\unicode{x03C0} /2,0]$ and $\theta \in [0, \unicode{x03C0} /2]$ for the corners at $y=1$ and $-1$ , respectively, where $\theta =0$ corresponds to the positive $x$ axis. Combining (3.3) and (3.2) gives the following boundary data for characteristics emanating from the corners:

(3.14a) \begin{align} x_0(\theta ) &= -1 , \end{align}

(3.14b) \begin{align} y_0(\theta ) &= \pm 1 , \end{align}

(3.14c) \begin{align} p_0(\theta ) &= \frac {1+\sin \theta }{b} , \end{align}

(3.14d) \begin{align} q_0(\theta ) &= \mp \frac {\cos \theta }{b} , \end{align}

(3.14e) \begin{align} H_0(\theta ) &= 1 , \end{align}

where the plus and minus signs are used to distinguish between the two upstream corners. The solution for the shape covered by these characteristics is (see Appendix B)

(3.15) \begin{equation} x+1 = b (H-1) - b \log \left [ \frac {H^2 -1 + b^{-2}(1-|y|)^2}{2(H-1)} \right ]\! , \end{equation}

valid in the domain

(3.16) \begin{equation} -1 \leqslant y \leqslant 1 \quad \text{and} \quad x+1 \lt b \left (A(y)-1 \right )-b \log \left (\frac {1-|y|}{b(A(y)-1)}\right )\!. \end{equation}

The characteristics and the region penetrated are shown by red dashed lines in figure 6.

Outside the domains dictated by (3.11) and (3.16), i.e. for $|y|\gt 1$ , the solution for the fluid thickness is $H=1$ .

Figure 7. The exact solution for the stationary slump ((3.10a ), (3.10b ) and (3.15)) for (a) $b=1/4$ , (b) $b=1/2$ , (c) $b=1$ , (d) $b=2$ .

Figure 8. (a) The dimensionless maximum slump height as function of $b$ . (b) The dimensional maximum height as a function of the yield stress $\tau _Y$ for $\rho g = 10^4$ $\mathrm{kg\,m^{-2}\,s^{-2}}$ , $L=0.1$ $\mathrm{m}$ and $\beta =5 ^{\circ }$ . (c) Dimensionless excess volume of the slump ((4.3), solid blue line), compared with the small and large $b$ asymptotic solutions (see (4.4a )–(4.4b )). (d) Dimensional volume as a function of $\tau _Y$ for the above-mentioned parameter values. The scaled maximum height and volume obtained in experiments 1 and 3 (see § 5) are shown in (a) and (c), where the error bars are smaller than the extent of the markers.

4. Discussion

Equations (3.10a ), (3.10b ) and (3.15) constitute complete and exact expressions for the final slump shape in terms of $b$ . Examples of the shape for four values of $b$ are shown in figure 7. It should be noted that a steeper slope corresponds to smaller $b$ , which is associated with a thicker and more localised slump. From the exact solution, many physical aspects of the slump can be computed. The maximum thickness of the slump, attained at $(x,y)=(-1,0)$ , is given by

(4.1) \begin{equation} H_{\textit{max}} = \sqrt {1+2/b}, \end{equation}

which is shown in figure 8(a). In dimensional terms, the maximum thickness is

(4.2) \begin{equation} \hat {H}_{\textit{max}} = \sqrt {\frac {\tau _Y^2}{(\rho g \sin \beta )^2}+\frac {2 \tau _Y L}{\rho g \cos \beta }}, \end{equation}

which is plotted in figure 8(b) for an example case with $\rho g = 10^4$ $\mathrm{kg\,m^{-2}\,s^{-2}}$ , $L=0.1$ m and $\beta =5 ^{\circ }$ , which is relevant to our laboratory experiments; see § 5. The dimensionless maximum thickness is a monotonically decreasing function of $b$ , whilst the dimensional maximum thickness is a monotonically increasing function of the yield stress $\tau _Y$ . This difference arises because the dimensionless thickness has been scaled by the thickness in the far field, which is linearly proportional to the yield stress. Thus although the slump thickness is larger for a fluid with a higher yield stress, the ratio of the slump thickness to the far-field thickness is smaller. A similar feature occurs with the excess volume, discussed below.

The excess volume of the slump, $V$ , is defined as the additional volume that is supported by the obstruction as compared to the case with no obstruction for which $H\equiv 1$ , i.e.

(4.3) \begin{align} V = \int _{-\infty }^{-1} \int _{-1}^1 \left ( H(x,y) -1 \right )\, \mathrm{d}y\, \mathrm{d}x. \end{align}

The excess volume is plotted as a function of $b$ in figure 8(c). The dimensional excess volume is shown in figure 8(d) (with the same parameter values as in figure 8 b) .

The parameter $b$ , which is proportional to the yield stress, represents the ratio of the far-field rigid thickness to the obstruction width and the slope gradient (see (1.6)). The regime of small $b$ corresponds to a relatively small far-field thickness (or wide obstruction), and large $b$ corresponds to a relatively large far-field thickness. For $b \gg 1$ , the governing equations away from the margins of the arrested slump become identical to those of a static granular pile on an empty plane (see § 7.1 of Tregaskis et al. Reference Tregaskis, Johnson, Cui and Gray2022); see also the calculations in § C.2 (though it should be noted that the static granular pile on an empty plane has $H \to 0$ away from the barrier, whereas here we have $H\to 1$ ). In this regime of a relatively large Bingham number, the static solution calculated here is also a good approximation for the height of the free surface of steady flowing material from a constant flux source. This steady flow driven by a constant-flux line source was analysed by Hinton et al. (Reference Hinton, Hewitt and Hogg2023), and in their notation this regime corresponds to $B\gg 1$ and $\mathcal{L}=O(1)$ .

For small and large $b$ , (4.3) admits the respective asymptotic limits

(4.4a) \begin{align} V &\sim 1 -\left (\frac {8-2\unicode{x03C0} }{3\sqrt {2}}\right ) \sqrt {b} , & b \ll 1 , \end{align}

(4.4b) \begin{align} V &\sim \frac {1}{18b} \left (5 + 6 \log (2b) \right )\!, & b \gg 1 ; \end{align}

for details, see Appendix C. In dimensional form, this is given by

(4.5a) \begin{align} \hat {V} &\sim \frac {L^2 \tau _Y}{\rho g \sin \beta } -\left (\frac {8-2\unicode{x03C0} }{3\sqrt {2}}\right ) \left (\frac {L \tau _Y}{\rho g \sin \beta } \right )^{3/2} (\tan \beta )^{-1/2} ,& b \ll 1 , \end{align}

(4.5b) \begin{align} \hat {V} &\sim \frac {L^3 \tan \beta }{18} \left (5 + 6 \log (2b) \right )\!, & b \gg 1. \end{align}

Interestingly, the dimensional excess volume exhibits different scaling laws with respect to the yield stress $\tau _Y$ in the regimes of small and large $b$ . In the relatively high yield stress regime ( $b\gg 1$ ), the excess volume is only weakly dependent on the yield stress through the $\log (b)$ term, and instead the volume depends primarily on the geometry of the slope and the barrier. Similar dependencies arise for a cylindrical obstruction; see Appendix D.

One other interesting observation concerning the rigid solution is that the characteristic projections (figure 6) are exactly the streamlines of the late-time fluid flow as it approaches the rigid state; this is shown in figure 9 for $b=1$ . The analogy arises because the characteristics follow $(\dot {x},\dot {y}) \propto (1-b\, \partial \textit{H}/\partial x, - b\, \partial \textit{H}/\partial y)$ , which is the flow direction. The characteristics thus emanate tangentially to the obstacle, and data for the rigid solution are carried in the opposite direction to the flow.

Figure 9. The streamlines of the numerically calculated late-time flow (solid blue lines) compared with the characteristic projections from Charpit’s method (dashed black lines), for $b=1$ . The obstruction is shown in grey.

The stationary free-surface thickness exhibits a seam along the centreline, $y=0$ . Here, the characteristics of the rigid solution collide and terminate. This corresponds to the late-time flow of the slumping fluid bifurcating at the centreline.

4.1. Force on the barrier

The dimensional hydrostatic force $(\hat {F},0)$ exerted on the barrier by the arrested slump is given by the integral of the pressure over the upstream face,

(4.6) \begin{equation} \hat {F}=\int _{-L}^L \int _0^{\hat {h}(-1,\hat {y})} \rho g \big(\hat {h}(-1,\hat {y})-\hat {z} \big) \cos \beta \, \mathrm{d} \hat {z}\, \mathrm{d} \hat {y}, \end{equation}

and using the solution for the shape at this face, (3.6), we obtain

(4.7) \begin{equation} \hat {F}= \left (\frac {\tau _Y}{\rho g \sin \beta } + L \tan \beta \right ) \tau _Y L \cot \beta . \end{equation}

4.2. Insensitivity to the upstream source

The stationary slump shape calculated in § 3 is insensitive to the upstream flux source provided that the flow exceeded the slump height. Here, we consider the flow generated by an off-centre vent source located at $-2\leqslant y\leqslant -1/2$ , $x=-4$ supplying constant flux (figure 10 a). Using the techniques described in § 2 and initialising the fluid at zero height i.e. $H(x,y,t=0)=0$ , the steady flow and the flow following the switch off are calculated; see figure 10. The steady flow exhibits an asymmetric profile. However, when the source is switched off, the stationary shape upstream of the obstruction matches the analytical calculations of § 3.

Figure 10. (a,c) Flowing and (b,d) stationary solutions for Bingham fluid ( $n=1$ ) with $b=1/4,\ q_\infty =1$ , for (a,b) off-centre vent source, (c,d) infinite line source.

4.3. Downstream behaviour

The shape of the free surface of rigid material downstream of the obstruction depends on the initial conditions of the flow prior to switch off as well as the upstream flux source and the viscous rheology of the fluid. This is in contrast to the upstream shape, which is independent of these parameters (see discussion above). The dependence of the downstream shape arises because parts of the downstream region are inaccessible to the flow. Thus if the fluid is initialised in this region to exhibit a static slump, then this shape will be maintained independently of the fluid flow impinging from the flux source; e.g. see unwetted region in figures 10(b) and 10(d). Moreover, the region inaccessible to the upstream flow depends on the strength of the flux source. The stationary downstream shape is altered at higher values of the upstream flux because more fluid invades the region directly downstream, and subsequently cannot flow away; see also Hinton & Hogg (Reference Hinton and Hogg2022), who observed similar behaviour upstream and downstream of a topographic mound. The non-universality of the downstream shape is exemplified by the different static shapes obtained for an infinite line-flux source and a vent source that are initialised to zero height prior to the the flux source being turned on; compare figures 10(b) and 10(d). We illustrate this explicitly by plotting the shape of the free surface at $x=1$ (i.e. at the downstream face of the obstruction) in figure 11, at different times after the flux source is terminated for both an infinite line flux source and a vent. This shows that for the vent source (figure 11 b), less fluid invades the downstream region near $y=1$ than near $y=-1$ , as compared with the line source (figure 11 a), where the downstream region invaded by the fluid is symmetric in $y$ . This exemplifies the non-universality of the downstream rigid shape.

Figure 11. Fluid thickness at $x=1$ (i.e. the downstream face of the obstruction), at various times after the source has been switched off: (a) infinite line source, (b) vent source, as described in figure 10. The dashed black lines indicate the edges of the obstruction’s downstream face.

5. Laboratory experiments

We report experiments on an inclined plane of width 40 cm and length 50 cm held at a shallow angle to the horizontal. A cuboid obstruction of total width 20 cm, or a cylindrical obstruction with diameter 17 cm, was held fixed on the inclined plane. Hair gel was poured onto the plane far upstream from the obstacle, with sufficient flux and quantity to ensure that the plane was entirely coated by fluid, and that the thickness exceeded the rigid shape. Excess fluid was allowed to drain off the end of the plane. The pouring was then stopped, and the free surface approached its arrested shape as excess material drained around the obstruction.

To quantify the thickness of fluid, we projected an array of 76 parallel blue lines (e.g. figure 1 d) using a video projector (Epson EH-TW5200) mounted perpendicularly to the sloping surface. Similar methods have been employed elsewhere for free-surface flow of complex fluids (Hewitt & Balmforth Reference Hewitt and Balmforth2013), free-surface flow with an obstruction (Hinton, Hogg & Huppert Reference Hinton, Hogg and Huppert2020), and elastic membranes (Pihler-Puzović et al. Reference Pihler-Puzović, Juel, Peng, Lister and Heil2015). However, we extended the method here to determine the full profile of the surface in two dimensions, $\hat {h}(\hat {x},\hat {y})$ . This technique was verified by employing it to measure the thickness of a 4 mm calibration plate, which exhibited an error of less than 2 %.

Hair gel was used as a representative viscoplastic material, owing to its established rheological properties (Taylor-West & Hogg Reference Taylor-West and Hogg2024), and the fact that it does not separate over the time scales required for these experiments. We used one brand of hair gel throughout the study (Woolworths Essentials ‘Firm Hold’). A small quantity of white acrylic paint was added (approximately 2 % v/v) to introduce opacity, resulting in strong diffuse scattering of light from the surface of the hair gel. Although this addition modified the yield stress, it preserved the viscoplastic nature of the material (this was verified using a cup and bob test with the Anton Paar MCR 702 rheometer; see § 5.2). The difference in the yield stress across the three experiments is a result of small variations in the proportion of paint.

The hair gel had density approximately $1000\,\mathrm{kg\,m^{-3}}$ , and the surface tension is approximately given by that of water, $\sigma = 0.07\, \mathrm{N\,m^{-1}}$ (Taylor-West & Hogg Reference Taylor-West and Hogg2024), which furnishes a Bond number

(5.1) \begin{equation} {{Bo} = \frac {\rho g L^2 }{\sigma }=1400}, \end{equation}

suggesting that surface tension effects are negligible in the experiments. The Reynolds number is given by

(5.2) \begin{equation} {{Re} = \frac {\rho U L}{\mu }=\frac {\rho L^2}{\mu \left ( \dfrac {3 \mu L}{\rho g \hat {H}_\infty ^2 \sin \beta }\right )}=0.015,} \end{equation}

where we have used the time scale from (2.1) and viscosity scale from (2.2), and the consistency is determined with a rheometer; see § 5.2. The role of inertia is negligible.

Figure 12. (a) The fluid thickness at $\hat {t}=60$ minutes from experiment 1 (square obstruction located in $(\hat {x},\hat {y}) \in [-100 \text{ mm},100\text{ mm}]^2$ ), which is compared with the lubrication prediction for the arrested state in (b). (c,d) The fluid thickness for experiment 2 (circular obstruction) and the arrested state prediction, respectively. Parameter values are given in table 1. White in (c) denotes a region with no data due to the circle obstructing the camera’s view.

In previous experimental studies that use hair gels (or similar carbomer fluids), the elastic storage modulus $G'$ at low strain is in the range $200 {-} 800 \text{ Pa}$ (Tokpavi et al. Reference Tokpavi, Jay, Magnin and Jossic2009; van der Kolk, Tieman & Jalaal Reference van der Kolk, Tieman and Jalaal2023; Taylor-West & Hogg Reference Taylor-West and Hogg2024). For our experiments, this furnishes an elastic time scale $\lambda =(k/G')^{1/n}$ in the range $ 10^{-5}{-}10^{-3} \text{ s}$ (the consistency is determined with a rheometer; see § 5.2). Two dimensionless groups quantify the relative importance of elasticity: the Weissenberg number and the Deborah number. The former is $\textit{Wi}=2 \lambda \dot {\gamma }$ , which vanishes as the slump becomes arrested, and is always below $10^{-3}$ during the motion. The Deborah number is the ratio of the elastic time scale to the flow time scale (which is of order hours); ${De}=\lambda /T_{\textit{flo}w}$ , so ${De}=10^{-8}{-}10^{-6}$ . These values suggest that elastic effects are negligible in our experiments.

Images were recorded without and with the fluid present, using a DSLR camera (Canon 5D Mark III) equipped with a telephoto lens (Canon 70–200 mm f/4 L IS USM) at focal length 89 mm. An aperture f/22 was chosen with a large depth of field to ensure that the full fluid surface was in focus. For each of the photographs, the projected lines were identified using cubic spline fitting of points identified as exceeding two standard deviations above the mean pixel intensity. The dimensional thickness of the material was then determined using a spatially dependent pixel calibration, which considers the variable distance to the camera, and accounts for the angle of the camera relative to the inclined plane. The 76 one-dimensional slices were interpolated to determine the dimensional thickness of the entire two-dimensional fluid upstream of the obstacle (e.g. see figures 12 a,c).

5.1. Experiments 1 and 2

In experiments 1 and 2, after stopping the upstream supply of fluid, we waited 60 minutes for fluid to drain and the slump to converge towards the arrested state. Experiment 1 was conducted with the square obstruction, and experiment 2 with the cylindrical obstruction; results for the fluid thickness at $\hat {t}=60$ minutes are shown in figure 12.

Figure 13. Fluid thickness along the centreline measured in (a) experiment 1 and (b) experiment 2 after 60 minutes, and the final arrested state predicted by (3.10a )–(3.15).

Table 1. Parameters for the three experiments conducted. For experiments 1 and 2, the upstream height $\hat {H}_\infty$ was measured at $\hat {x}=-370$ mm, with $b$ calculated using (1.6), and the yield stress using (1.2). For experiment 3, the yield stress was measured using a rheometer, and $b$ and $\hat {H}_\infty$ were calculated using (1.6) and (1.2), respectively.

The measured shape of the free surface along the centreline is shown in figure 13 along with the predictions of (3.10a ), (3.10b ) and (3.15). Parameters for these two experiments are reported in table 1. The value of $b$ is obtained by measuring the thickness of the layer far upstream and then using (1.6); this does not require direct knowledge of the yield stress. The yield stress is inferred from the height in the far field (at $\hat {x}=-370$ mm) using (1.2).

Figure 13 indicates that the free surface along the centreline after 60 minutes is substantially larger than the rigid shape predicted by our lubrication model. The measured free surface height is also higher than the predicted rigid shape away from the centreline, as seen in figure 12. This is further seen in the maximum height and excess volume of the fluid slump for experiment 1 (plotted as a diamond in figure 8) exceeding the predictions of the theory. This volume prediction is calculated for $-300\;\text{mm}\leqslant \hat {x}\leqslant -100\;\text{mm}$ to avoid edge effects. We also note that the measured shape does not exhibit a sharp cusp at the centreline ( $\hat {y}=0$ ).

The transient evolution of nine points along the centreline with $\hat {t} \in [0,60]$ minutes for experiment 1 is shown in figure 14. This indicates that the convergence to the rigid shape is algebraic in time, as suggested by our theoretical results, and that the decay is very slow (cf. Ancey & Cochard Reference Ancey and Cochard2009). Results from the Anton Paar MCR 702 rheometer found that the hair gel mixture’s rheology was accurately fitted by a Herschel–Bulkley model with $n=0.3$ , which suggests convergence with $\hat {t}^{-0.3}$ . Our results suggest that in the first 60 minutes, the decay rate is even slower than this, especially near the barrier. The slower time variation could be due to a residual slip velocity, or because the late-time behaviour has not yet been reached at $\hat {t}=1$ hour. These findings motivated an experiment with a much longer time scale.

Figure 14. Transient evolution of the measured fluid thickness at nine points along the centreline from experiment 1. The pouring of gel upstream stopped at $\hat {t}=0$ , but it takes approximately 100 s for this to substantially alter the fluid thickness in the vicinity of the barrier. The time evolution follows power-law behaviour; note the log-log scale.

Figure 15. Steady-state flow curve for the hair gel and paint mixture. Data shown for up- and down-stepped tests, and the best-fit Herschel–Bulkley constitutive law (yellow line).

Figure 16. (a) The fluid thickness at $\hat {t}=17$ hours from experiment 3. (b) Theoretical prediction for the static shape with no fitting parameters.

5.2. Experiment 3

We attribute the excess material in experiments 1 and 2 to the slump not reaching its final stationary state within the time of the experiment. In experiment 3, the fluid was allowed to drain and approach the arrested state for a significantly longer time period of 17 hours.

The constitutive relation for the hair gel–paint mixture from this experiment was measured using a rheometer, which gave $\tau _Y=57.05$ Pa, consistency $k=27.57$ Pa $\,{\rm s}^{0.3}$ , and Herschel–Bulkley power-law exponent $n=0.3$ ; see figure 15. Apart from at low strain rates, there is good agreement between the up- and down-stepped tests. There was a thirty-second rest period between the up and down data, suggesting that thixotropic effects are negligible.

The measured height of the free surface after 17 hours is shown in figure 16, with transects shown in figure 17. This is compared with our analytical predictions from (3.10a ), (3.10b ) and (3.15). For this experiment, $b=1.81$ (see table 1), which is obtained directly from the Anton Paar MCR 702 rheometer; there are no fitting parameters here (unlike in experiments 1 and 2).

The comparison in figures 16 and 17 shows reasonable agreement, with the theory under-predicting the fluid thickness closer to the obstruction, and over-predicting the thickness further away from the obstruction by up to 11 %. The maximum height and excess volume of the fluid slump are shown in figure 8 by a black $\times$ , and are in excellent agreement with the lubrication model prediction. Again, the measured slump does not exhibit sharp cusps at the centreline, in contrast with the theoretical prediction. This is consistent with past experiments (see e.g. Balmforth et al. Reference Balmforth, Craster, Rust and Sassi2006), and could be due to the finite time of the experiments, since the kink is only realised asymptotically as $\hat {t} \to \infty$ , surface wrinkling obscuring this phenomenon, or perhaps surface tension effects localised to the seam at $\hat {y}=0$ .

Figure 17. The measured height of the free surface after 17 hours (solid blue line), and height predicted by the theory (dotted black line) for experiment 3 along (a) the centreline, and along the transects (b) $\hat {x}=-150$ mm and (c) $\hat {x}=-200$ mm. (d) The relative height of the yield surface after 17 hours, as predicted by (1.5), where white regions denote missing data due to the free surface being obscured by the fluid.

We briefly comment on the expected approach of the fluid to its stationary state. The time scale defined in (2.1) is approximately $10$ seconds for experiment 3. Thus 17 hours corresponds to a dimensionless time $t\approx 6\times 10^3$ ; compare to figure 4(d). The algebraic decay to the stationary state dictated by (2.5) and our numerical results predict that the fluid thickness after 17 hours is approximately 8 % thicker than the final stationary state, in agreement with the observed discrepancy. To improve on this, an even longer experiment (say over a week) would be required. However, this is impractical as other effects such as evaporation would begin to become significant over such time scales. Indeed, dam-break experiments using Carbopol have found that evaporation becomes significant after time scales of 10 hours (Cochard Reference Cochard2007; Dubash et al. Reference Dubash, Balmforth, Slim and Cochard2009).

We note that the lubrication parameter for experiment 3, being the ratio of the far-field height to the half-width of the obstruction, is $\epsilon =\hat {H}_\infty /L=0.34$ . This indicates significant departure from the lubrication approximation, with non-lubrication effects contributing errors of order $\epsilon ^2$ , being $\approx 12\,\%$ . We also note that the experimental behaviour for $\hat {x}\lt -300$ mm is attributed to edge effects from the pouring of the fluid onto the board.

Figure 17(d) shows the height of the yield surface, $\hat {Y}$ after 17 hours. This was obtained from the measured height of the free surface by smoothing $\hat {h}$ in the $\hat {x}$ direction using a smoothing spline, and the height of the yield surface, $\hat {Y}$ , is then inferred from (1.5). Figure 17(d) suggests that the height of the yield surface does not exceed 20 % of the free surface height in the bulk of the domain. However, near the upstream obstruction corner (at $\hat {y}=\hat {x}=-100$ mm), the inferred height of the yield surface is non-negligible, which indicates that the lubrication approximation may not be accurate in this region. In particular, the requirement that shear stresses far exceed deviatoric normal stresses ( $|\tau _{zx}|,|\tau _{zy}| \gg |\tau _{xx}|, |\tau _{zz}|$ ), underpinning the lubrication approximation, likely does not hold near the corners, where the stress balance requires a full three-dimensional treatment. A similar observation has been made for viscoplastic material accumulating near the wall of a moving squeegee (Ribinskas et al. Reference Ribinskas, Ball, Penney and Neufeld2024; Taylor-West & Hogg Reference Taylor-West and Hogg2024).