2. Converging flow in a channel of variable cross-section

A viscous gravity current of a Ostwald–de Waele fluid (PL) (Ostwald Reference Ostwald1929; Morrell & De Waele Reference Morrell and De Waele1920) propagates in the negative $x$ direction within a channel (or fracture or Hele-Shaw cell) of variable cross-section (see figure 1), starting at $x_0$ and reaching the origin $x=0$ at a finite touch-down time $t_c$.

Figure 1. Horizontal channel with varying gap thickness $b_1x^k$: a converging gravity current in viscous-buoyancy balance propagates toward the origin. Here $x_f$ is the instantaneous front position and $x_0$ is the front position at time $t=0$.

The PL model for a shear-thinning/thickening fluid reads in one dimension

(2.1)\begin{equation} \tau = \mu_0|\dot{\gamma}|^{n-1}\dot{\gamma} \end{equation}

in terms of the tangential stress $\tau$ and of the strain rate $\dot {\gamma }$. The consistency index $\mu _0$ represents a viscosity-like parameter, and the fluid behaviour index $n$ controls the extent of shear-thinning ($n<1$) or shear-thickening ($n>1$); $n=1$ corresponds to the Newtonian case. A slightly more complicated description using tensors is required for three-dimensional flows; this general formulation is not reported here as a one-dimensional problem is considered. The fluid advances in a horizontal channel with a gap thickness varying as $b(x)=b_1x^k$ ($[b_1]=L^{1-k}$, where L is a length scale and $0<k<1$), see figure 1, under the hypotheses of (i) hydrostatic pressure distribution; (ii) $\tau _{xy}\gg \tau _{xz}$ and negligible $\tau _{yz}$, i.e. tangential stress acting on the plane of normal $x$ along the cross direction $y$, is dominant with respect to tangential stress acting on the same plane in vertical direction $z$; (iii) no slip at the side wall at $y=\pm b/2$ and symmetry at $y=0$; (iv) negligible surface tension and no fingering at the interface with the ambient fluid; and (v) inviscid ambient fluid. The lubrication approximation holds and the flow is one-dimensional along the $x$ axis except near the origin, where also the capillary length is comparable to the gap thickness and $\tau _{xz}$ is comparable to $\tau _{xy}$. The streamwise horizontal velocity of the current averaged over the cross-section is

(2.2)\begin{equation} u(x,t)={-}\mathrm{sgn}\left(\dfrac{\partial h}{\partial x}\right)\left(\dfrac{b_1x^k}{2}\right)^{(n+1)/n}\dfrac{n}{2n+1}\left(\dfrac{\Delta\rho g}{\mu_0}\right)^{1/n} \left|\dfrac{\partial h}{\partial x}\right|^{1/n}, \end{equation}

where $h(x,t)$ is the current depth, $\Delta \rho \equiv \rho _c-\rho _a$ is the density difference between the intruding current and the ambient fluid and $g$ is gravity. The mass conservation reads

(2.3)\begin{equation} \dfrac{\partial h}{\partial t}+\dfrac{1}{x^k}\dfrac{\partial (x^khu)}{\partial x}=0. \end{equation}

The boundary conditions are motivated by the singular points in the phase plane described later. We anticipate that the boundary conditions allowing a solution are as follows: for $t<t_c$ are $h=0$ for $x=x_f$ and $u=0$ for $x\to \infty$, where $t_c$ is the touch-down time required by the front of the current to reach the origin of the channel; for $t> t_c$ are $u=0$ and $\partial h/\partial x=0$ for $x=0$; for $t=t_c$ a singularity is reached, which is then removed upon a proper reformulation of the dependent variables. The integral mass conservation, although not specifically involved in the sought self-similar solution of the second kind, are important in numerical integration schemes; the most common representation corresponds to $V\propto t^{\sigma }$, where $\sigma =0$ is the lock release of a finite and constant volume of fluid, $\sigma =1$ is the constant inflow rate, $0<\sigma < 1$ is a waning and $\sigma >1$ is a waxing inflow rate, respectively.

A first-kind self-similar solution cannot be obtained and a second-kind self-similar approach is pursued introducing the phase formalism with the adoption of the independent variables, $x$ and $t$, as length and time scales, respectively, to render the governing equations dimensionless. We let

(2.4a)\begin{gather} u(x,t)=\dfrac{x}{t_r}U(x,t_r), \end{gather}

(2.4b)\begin{gather}h(x,t)=\left(\dfrac{2}{b_1}\right)^{n+1}\left(\dfrac{2n+1}{n}\right)^n\left( \dfrac{\mu_0}{\Delta\rho g}\right)\dfrac{x^{(n+1)(1-k)}}{t_r|t_r|^{n-1}}H(x,t_r), \end{gather}

where $H$ and $U$ are dimensionless and $t_r=t_c-t$. Here $t_r>0$ represents the pre-closure phase, with the fluid advancing towards the origin of the channel; $t_r<0$ identifies the post-closure phase, with the fluid occupying the entire channel length, the fluid surface undergoing a progressive levelling and the front position being $x_f=0$. In the pre-closure phase $U$ is negative and $H$ is positive, and change sign during the post-closure phase because $t_r$ becomes negative.

Substituting (2.4a) and (2.4b) into (2.2) and (2.3) yields

(2.5a)\begin{gather} U\left|U\right|^{n-1}+(n+1)(1-k)H+x\dfrac{\partial H}{\partial x}=0, \end{gather}

(2.5b)\begin{gather}t_r\dfrac{\partial H}{\partial t_r}-nH-HU(n+2-nk)-x\dfrac{\partial HU}{\partial x}=0. \end{gather}

We assume a similarity variable $\xi =xt_r^{-1}|t_r|^{1-\delta }$, where the exponent $\delta$ cannot be determined by using dimensional arguments and must be determined in a different way. Inserting the similarity variable into (2.5a) and (2.5b) gives

(2.6a)\begin{gather} U\left|U\right|^{n-1}+\xi H'+(n+1)(1-k)H=0, \end{gather}

(2.6b)\begin{gather}\delta\xi H'+nH+\xi (HU)'+(n+2-nk)HU=0, \end{gather}

where the prime indicates the derivative with respect to $\xi$ and where the variable $t_r$ does not appear. Eliminating $\xi$ from the two equations results in

(2.7a)\begin{align} &\dfrac{\mathrm{d}H}{\mathrm{d}U}=\dfrac{H[(n+1)(1-k)H+U\left|U\right|^{n-1}]}{H[(k+1)U-(n+1)(1-k)\delta+n]-(U+\delta)U\left|U\right|^{n-1}}, \end{align}

(2.7b)\begin{align} &\dfrac{\mathrm{d}\ln\xi}{\mathrm{d}H}={-}\dfrac{1}{U\left|U\right|^{n-1}+(n+1)(1-k)H}. \end{align}

These two equations are a set of autonomous planar ODEs, with boundary conditions represented by points in the phase space. We note that the similarity variable has been embedded in its logarithm, with $\textrm {d}\xi /\xi \equiv \textrm {d}(\ln \xi )$. We define as singular points the simultaneous zeros of numerator and denominator of (2.7a), with a further singular point obtained by setting the denominator to infinity. Hence, there are four singular points, namely

(2.8)\begin{equation} \left.\begin{gathered} \mathrm{O}:(H,U) \equiv (0,0),\\ \mathrm{A}:(H,U) \equiv(0,-\delta),\\ \mathrm{B}:(H,U) \equiv\left(\left[\dfrac{n}{2+n(1-k)}\right]^n\dfrac{1}{(n+1)(1-k)},-\dfrac{n}{2+n(1-k)}\right),\\ \mathrm{C}:(H,U) \equiv\left(-\infty,\dfrac{(n+1)(1-k)\delta-n}{k+1}\right). \end{gathered}\right\} \end{equation}

In the phase plane a solution connects two points representing the boundary conditions. In this sense, the singular points are crucial because they are the candidates for the role of representatives of the boundary conditions, and the behaviour of the solution must be sought in the neighbourhood of these singularities.

Point O corresponds to null velocity and height at $t_r=0$ (and to $\xi \to \infty$); point A corresponds to the moving front where $h(x_f(t),t)=0$, with $x_f$ the front position and $x_f(t=0)=x_0$ (and to $\xi =\xi _f$); point B has no specific clear meaning, and from an analytic point of view corresponds to the condition $\textrm {d}^2H/\textrm {d}U^2\equiv \textrm {d}H/\textrm {d}U=0$; point C is active during the post-closure levelling of the current, and represents the asymptotic flow condition (it also corresponds to $\xi =0$). For $n=1$ point B corresponds to the value given in Zheng et al. (Reference Zheng, Christov and Stone2014) (where $n$ in Zheng et al. (Reference Zheng, Christov and Stone2014) corresponds to our $k$).

The expansion about point O, computed by assuming $H=\varpi U^\nu$, substituting in (2.7a) and balancing the smaller-order terms to calculate $\varpi$ and $\nu$, gives

(2.9)\begin{equation} U\approx{-}\left[\dfrac{\delta(1-k)(n+1)-n}{\delta}\right]^{1/n} H^{1/n}; \end{equation}

the expansion about point A, the front of the current, computed by assuming $H=\varpi (U+\delta )^\nu$, substituting in (2.7a) and balancing the smaller-order terms, yields

(2.10)\begin{equation} U\approx\left[\dfrac{(2+n-kn)\delta-n}{2\delta^n}\right]^{1/n} H-\delta; \end{equation}

the asymptotic expression for $H\rightarrow -\infty$ (point C) is computed by assuming $1/H=\varpi (1-U/U_C)^\nu$, substituting in (2.7a) and balancing the smaller-order terms, and gives

(2.11)\begin{equation} U\approx U_C+\dfrac{U_C^{n+1}(U_C+\delta)}{U_C(k+1)+(n+1)(1-k)}\dfrac{1}{H}, \end{equation}

where $U_C$ is the coordinate of the singular point C.

Given these properties of the singular points, we expect that the pre-closure phase is described by an integral curve joining A and O, the post-closure phase by a curve joining O and C.

Numerical integration with Mathematica (Wolfram Research, Inc. 2017) has been performed in the pre-closure time interval ($t<t_c$) by adopting (2.7a), starting nearby the origin O and assuming that $U=-\epsilon$ with the local expansion for $H$ given by (2.9) in the form

(2.12)\begin{equation} \left.H\right|_{-\epsilon}\approx\dfrac{\delta}{\delta(1-k)(n+1)-n} \epsilon^n, \end{equation}

with $\epsilon =10^{-5}$, and computing $H(U)$ in the interval $U \in [-\delta ,-\epsilon ]$. An initial value of $\delta$ was chosen, with iterative modifications of this value and stop criterion when $H(-\delta )<10^{-3}$. The computed eigenvalues, named critical eigenvalues $\delta _c$, are listed in table 2 of Appendix A.

Numerical integration in the post-closure interval ($t>t_c$) has been performed with the same equation (2.7a) used for the pre-closure interval, but starting near point C where $U=U_C-\epsilon$, with the expansion for $H$ given by (2.11) in the form

(2.13)\begin{equation} \left.H\right|_{(U_C-\epsilon)}\approx{-}\dfrac{U_C^{n+1}(U_C+\delta_c)}{U_C(k+1)+(n+1)(1-k)} \epsilon^{{-}1}, \end{equation}

again with $\epsilon =10^{-5}$, and computing $H(U)$ in the interval $U \in [\epsilon ,U_C-\epsilon ]$. No iteration was required because the value of $\delta _c$ was already known.

Figure 2 shows the phase portrait of (2.7a) with the trajectories for the critical eigenvalue $\delta _c=1.5836$ for $n=0.5$ (a typical value for a shear-thinning fluid) and $k=0.5$ (a fracture enlarging with the square root of the abscissa). Continuous and dashed curves describe the pre-closure and post-closure phases, respectively. Figure 3(a) shows the heteroclinic trajectories connecting: (i) point A and point O (continuous curves, pre-closure) and (ii) point O and point C (dashed curves, post-closure or levelling) for fixed $n=0.5$ and increasing $k$ values. For $k\to 1$ the variation of the current depth with $U$ tends to zero near the front and to $-\infty$ near the origin, and $U_C\to 0$. Figure 3(b) shows the same curves but for fixed $k=0.5$ and increasing values of the flow-behaviour index $n$. Figure 4 shows the eigenvalues as a function of $k$ for different $n$. The asymptotic value for $k\to 1$ is $\delta _c\approx n/[(n+1)(1-k)]$ and corresponds to $U_C\to 0$. The Newtonian fluid ($n=1$) shows the minimum eigenvalues for a given value of the width coefficient $k$. Relative differences between eigenvalues corresponding to different values of $n$ are modest while $k\leqslant 0.7$, then increase rapidly, especially for very shear-thinning fluids. We observe that the eigenvalues are not monotonic with $n$, and are decreasing with $n$ if $n<1$, increasing if $n>1$, see also the numerical values of $\delta _c$ in table 2. An increase in the eigenvalue results in a higher front velocity ($U_f=-\delta _c$). This seems to be the result of an interplay between rheology and channel geometry. Observing figure 5(a), it appears that the slope of the free surface current (i.e. the gradient pressure) for $\xi \to \xi _f$ increases with $n$. The front speed results from a balance between gradient pressure and flow resistance. For shear thinning fluids the reduction of resistance with $n$ is more than the reduction of gradient pressure, and the nose of the current is faster for decreasing $n$; for shear-thickening fluids the increase of resistance is less than the increase of the gradient pressure, and again the nose of the current is faster for increasing $n$.

Figure 2. Converging gravity current in a fracture of gap thickness $b=b_1x^k$. Phase portrait of (2.7a) for $n=0.5$ (shear-thinning fluid) and $k=0.5$, with $\delta _c=1.5836$. The singular points are $\mathrm {O}:(0,0)$, $\mathrm {A}:(0,-1.5836)$, $\mathrm {B}:(0.6285, -0.2222)$ and $\mathrm {C}:(-\infty , 0.4585)$. The continuous curve refers to the pre-closure phase, the dashed curve refers to the post-closure (levelling) phase, the thin red vertical line indicates the asymptote in the levelling phase, the dash-dot blue curves are the approximate solutions about points O and A.

Figure 3. Converging gravity current in a channel of gap thickness $b=b_1x^k$. Shape of the heteroclinic trajectories in rescaled coordinates for (a) $n=0.5$ (shear-thinning fluid) as $k\rightarrow 1$; (b) $k=0.5$ (a fracture enlarging with the square root of the abscissa) as $n$ varies from 0.5 (shear-thinning) to 1 (Newtonian) and to 1.5 (shear-thickening). Continuous curves refer to pre-closure, dashed curves to post-closure (levelling) phase.

Figure 4. Converging gravity current in a channel of gap thickness $b=b_1x^k$. Eigenvalues representing the exponent of the similarity variable $\xi =x/(t_c-t)^{\delta _c}$ for fluids with different flow behaviour index $n$. The dashed pink curve refers to a shear-thickening fluid with $n=1.5$, the hatched area is limited by the upper boundary, $1/(1-k)$, and by the lower boundary, $n/[(n+1)(1-k)]$, for $n=0.3$, of the critical eigenvalues.

Figure 5. Converging gravity current in a channel of gap thickness $b=b_1x^k$. Self-similar current shape profiles and velocity profiles for $k=0.5$ (a fracture enlarging with the square root of the abscissa) and for $n=0.3,0.5,0.7,1.0,1.5$ (a) in the pre-closure phase and (b) in the post-closure phase.

Once the function $H(U)$ has been computed, in order to evaluate the independent similarity variable $\xi$ it is convenient to rewrite the left-hand side of (2.7b) as $\textrm {d}\ln (\xi )/\textrm {d}U$ via the chain rule and the inverse function theorem using (2.7a), and also to map the domain of varying size $\xi \in [\xi _f,\infty ]$ into $\xi /\xi _f\in [1,\infty ]$. The resulting differential equation

(2.14)\begin{equation} \dfrac{\mathrm{d}\ln\xi/\xi_f}{\mathrm{d}U}={-}\dfrac{H}{H[(k+1)U-(n+1)(1-k)\delta_c+n]-(U+\delta_c)U\left|U\right|^{n-1}} \end{equation}

can be integrated in the range $U \in [-\delta _c,-\epsilon ]$ starting from point A, the front of the current, with the boundary condition $\left .\xi /\xi _f\right |_{U=-\delta _c}=1$, equivalent to $\left .\textrm {d}\ln (\xi /\xi _f)\right |_{U=-\delta _c}=0$. The origin O is reached for $U\to 0$ at $\xi /\xi _f\to \infty$.

Figure 5(a) shows the self-similar current shape profiles and the velocity profiles in the pre-closure phase, for $k=0.5$ and different values of $n$. We note that velocity profiles are almost coincident, whereas the shape of the current is markedly influenced by the flow behaviour index $n$, with a crest near the front more evident the more shear-thickening is the fluid.

A similar approach is adopted for the post-closure phase, when the front of the current has reached the origin and a progressive levelling occurs. In this phase it is convenient to integrate (2.7b) from point O, with $\left .\ln \xi \right |_{H=-\epsilon }=-(\delta _c/n)\ln (\epsilon )$. For increasing $H$ (in absolute value), the similarity variable $\xi \to 0$, the neighbourhood of point C where $U\to U_C$. Figure 5(b) shows the current depth and velocity for different values of the fluid behaviour index $n$ and for $k=0.5$. The results for other values of the width exponent $k$ are similar (not shown).

Inserting the expansion (2.9) near the origin O into (2.7b) and solving the resulting differential equation, yields

(2.15)\begin{gather} H\approx K\xi^{{-}n/\delta_c}, \end{gather}

(2.16)\begin{gather}U\approx{-}\left[\dfrac{\delta_c(1-k)(n+1)-n}{\delta_c}\right]^{1/n} K^{1/n}\xi^{{-}1/\delta_c}. \end{gather}

Switching to dimensional variables gives

(2.17)\begin{gather} h(x)=\left(\dfrac{2}{b_1}\right)^{n+1}\left(\dfrac{2n+1}{n}\right)^n\left(\dfrac{\mu_0}{\Delta\rho g}\right)Kx^{(n+1)(1-k)-n/\delta_c}, \end{gather}

(2.18)\begin{gather}u(x)\approx{-}\left[\dfrac{\delta_c(1-k)(n+1)-n}{\delta_c}\right]^{1/n} K^{1/n}x^{1-1/\delta_c}, \end{gather}

where $K$ is a constant. Notably, (2.17) and (2.18) are time independent.

For $x\to \infty$ the depth of the current grows in the positive direction, hence $\textrm {d} h/\textrm {d} x>0$ that, in turn, implies

(2.19)\begin{equation} \delta_c>\dfrac{n}{(n+1)(1-k)}, \end{equation}

which also satisfies the condition of a negative velocity. The minimum value of $\delta _c$ grows with $n$ and $k$, and tends to infinity for $k\to 1$ (a linearly expanding fracture).

In a similar manner, inserting the expansion (2.11) near point C into (2.7b) and solving the differential equation, yields

(2.20)\begin{equation} H\propto \xi|\xi|^{-(n+1)(1-k)-1}, \end{equation}

or

(2.21)\begin{equation} h\propto t_r|t_r|^{\delta_c(n+1)(1-k)-n-1}, \end{equation}

in dimensional variables. Equation (2.21) is independent on $x$ and represents the levelling process near $x=0$.

The condition of a time increasing $h$ during levelling, $\textrm {d}h/\textrm {d}t>0$, equivalent to $\textrm {d}h/\textrm {d}t_r<0$, requires that

(2.22)\begin{equation} \dfrac{\textrm{d}h}{\textrm{d}t_r}\propto [\delta_c(n+1)(1-k)-n]t_r|t_r|^{\delta_c(n+1)(1-k)-n-2}<0\rightarrow \delta_c>\dfrac{n}{(n+1)(1-k)}, \end{equation}

coincident with the condition (2.19). Imposing that the levelling process vanishes in time, i.e.

(2.23)\begin{equation} \lim_{t_r\to -\infty}\dfrac{\textrm{d}h}{\textrm{d}t_r}=0, \end{equation}

requires that

(2.24)\begin{equation} \delta_c<\dfrac{1}{1-k}. \end{equation}

In conclusion, the critical eigenvalue is bounded above and below, i.e.

(2.25)\begin{equation} \dfrac{n}{(n+1)(1-k)}<\delta_c<\dfrac{1}{1-k}, \end{equation}

with the two bounds collapsing to $1/(1-k)=\delta _c$ for an infinitely shear-thickening fluid or $n\to \infty$. Note that the upper bound is independent on the fluid nature. The lower bound for $n=1$ coincides with the result by Zheng et al. (Reference Zheng, Christov and Stone2014) (their $n$ is our $k$). For $k\to 0$ (constant gap thickness) and $n\to \infty$, it results $\delta _c=1$.

Owing to these constraints on the eigenvalues, the time exponent of the depth growth rate at the origin during levelling, defined in (2.21), varies in the range $[0,1]$, see figure 6 where its value is depicted versus $k$ for different values of the flow behaviour index $n$. As the parameter $k$ describing the growth rate of the fracture gap increases, the same exponent decreases up to a minimum value depending on the fluid rheology, then modestly increases; this is so only for shear-thinning fluids, whereas the exponent for shear-thickening and Newtonian fluids reaches a minimum and remains approximately constant. The minimum value of the exponent in (2.21) is reached for $k=0.75 \text {--}1$. Finally, the exponent increases as the flow behaviour index decreases from shear-thickening to shear-thinning rheology.

Figure 6. Converging gravity current in a channel of gap thickness $b=b_1x^k$. Time exponent of the growth rate of the depth of the current at the origin during levelling (see (2.21)) as a function of $k$ for fluids with different flow behaviour index $n$ ranging from shear-thinning to shear-thickening. The dashed thin curve indicates the minimum for each $n$ value.

In Appendix B we consider a related problem: the propagation of a plane gravity current of profile $h(x,t)$ toward the origin of the coordinate system in an heterogeneous porous medium with spatially variable permeability and porosity. If both quantities exhibit a PL variation in the streamwise direction $x$, a formal analogy can be established between two domains, the porous medium and the Hele-Shaw cell or channel of variable gap thickness. The analogy is subject to constraints in the values of parameters describing the heterogeneity as detailed in Appendix B.

3. Converging axisymmetric flow

We consider the converging flow of a cylindrical gravity current as shown in figure 7: the current propagates towards the origin of a cylindrical coordinate system. We assume again: (i) the current to be thin, allowing a hydrostatic pressure distribution; (ii) $\tau _{rz}$ to be the dominant shear, with negligible contributions by $\tau _{r\theta }$ and $\tau _{\theta z}$; and (iii) no surface tension effects and no fingering at the interface with the ambient fluid; (iv) inviscid ambient fluid. Gravity is the driving force balanced by viscous forces. In the limits of lubrication model, the vertically averaged horizontal velocity in the radial direction, calculated by imposing the no slip condition $u=0$ for $z=0$ and a zero tangential stress at the interface with the ambient fluid, corresponding to $\partial u/\partial x=0$ for $z=h$, is

(3.1)\begin{equation} u(r,t)={-}\textrm{sgn}\left(\dfrac{\partial h}{\partial r}\right)h^{(n+1)/n}\dfrac{n}{2n+1}\left(\dfrac{\Delta\rho g}{\mu_0}\right)^{1/n}\left|\dfrac{\partial h}{\partial r}\right|^{1/n}, \end{equation}

and mass conservation reads

(3.2)\begin{equation} \dfrac{\partial h}{\partial t}+\dfrac{1}{r}\dfrac{\partial (ruh)}{\partial r}=0. \end{equation}

Again, in a general approach we should specify the initial and boundary conditions, but as already discussed in § 2, the boundary conditions are motivated by the singular points in the phase plane.

Figure 7. Radial converging flow: a gravity current in viscous-buoyancy balance propagates towards the origin. Here $r_f$ is the instantaneous front position and $r_0$ is the front position at time $t=0$.

The dependent variables $u$ and $h$ are rendered non-dimensional as follows:

(3.3a)\begin{gather} u(r,t)=\dfrac{r}{t_r}U(r,t_r), \end{gather}

(3.3b)\begin{gather}h(r,t)=\left(\dfrac{2n+1}{n}\right)^{n/(n+2)}\left(\dfrac{\mu_0}{\Delta\rho g}\right)^{1/(n+2)} \dfrac{r^{(n+1)/(n+2)}}{t_r|t_r|^{{-}2/(n+2)}}H(r,t_r). \end{gather}

Then the similarity independent variable $\xi =r t_r^{-1}|t_r|^{1-\delta }$ is introduced, where $\delta$ is the yet unknown eigenvalue. With the same approach described for converging flow in a horizontal fracture, equations (3.1) and (3.2) are rearranged to obtain

(3.4a)\begin{align} &\dfrac{\mathrm{d}U}{\mathrm{d}H}=\dfrac{H|H|^{n+1}[2(n+2)U-(n+1)\delta+n]-(U+\delta)(n+2)U\left|U\right|^{n-1}}{H[(n+1)H|H|^{n+1}+(n+2)U\left|U\right|^{n-1}]}, \end{align}

(3.4b)\begin{align} &\dfrac{\mathrm{d}\ln\xi}{\mathrm{d}H}={-}\dfrac{n+2}{(n+2)H^{{-}1}|H|^{{-}n}U\left|U\right|^{n-1}+(n+1)H}. \end{align}

The singular points are

(3.5)\begin{equation} \left.\begin{gathered} \mathrm{O}:(H,U) \equiv (0,0),\\ \mathrm{A}:(H,U) \equiv(0,-\delta),\\ \mathrm{B}:(H,U) \equiv\left(\left[\dfrac{n+2}{n+1}\right]^{1/(n+2)}\left[\dfrac{n}{5+3n}\right]^{n/(n+2)},-\dfrac{n}{5+3n}\right),\\ \mathrm{C}:(H,U) \equiv\left(-\infty,\dfrac{(n+1)\delta-n}{2(n+2)}\right), \end{gathered}\right\} \end{equation}

and have the same meaning of the points described for converging fracture flow.

The case $n=1$ refers to a Newtonian fluid, and (3.4a)–(3.4b) become

(3.6a)\begin{align} &\dfrac{\mathrm{d}U}{\mathrm{d}H}=\dfrac{H^{3}[6U-2\delta+1]-3U(U+\delta)}{H[2H^{3}+3U]}, \end{align}

(3.6b)\begin{align} &\dfrac{\mathrm{d}\ln\xi}{\mathrm{d}H}={-}\dfrac{3H^2}{3U+2H^3}, \end{align}

and the singular points are

(3.7)\begin{equation} \left.\begin{gathered} \mathrm{O}:(H,U) \equiv (0,0),\\ \mathrm{A}:(H,U) \equiv(0,-\delta),\\ \mathrm{B}:(H,U) \equiv\left(\left[\dfrac{3}{16}\right]^{1/3},-\dfrac{1}{8}\right),\\ \mathrm{C}:(H,U) \equiv\left(-\infty,\dfrac{2\delta-1}{6}\right), \end{gathered}\right\} \end{equation}

corresponding to the differential problem in Diez et al. (Reference Diez, Gratton and Gratton1992b) with a different definition of the variables. The results are equivalent to those reported in McCue et al. (Reference McCue, Jin, Moroney, Lo, Chou and Simpson2019), where a slightly different approach and a broader perspective is adopted, with an improved clarity of the various steps. See also Gratton & Perazzo (Reference Gratton and Perazzo2010).

Numerical integration of the pre-closure phase starts with a first tentative value of $\delta$ and moving from the origin O ($\xi \to \infty$), where $H\to 0$ and the solution for $U$ is approximated by

(3.8)\begin{equation} U\approx{-}H|H|^{2/n}\left[\dfrac{(n+1)\delta-n}{(n+2)\delta}\right]^{1/n}. \end{equation}

Upon successive iterations with changes of the $\delta$ value, the integral curve reaches exactly point A ($\xi =\xi _f$), the front of the gravity current, where $H\to 0$ and the approximate solution for $U$ is

(3.9)\begin{equation} U\approx{-}\delta+\dfrac{2}{(n+3)\delta^{n-1}}H|H|^{n+1}. \end{equation}

The value of $\delta$ that allows the integral curve to reach point A from O is the critical eigenvalue $\delta _c$.

Numerical integration of the post-closure phase starts about point C, where $H\to -\infty$ (and $\xi \to 0$) and where the approximate solution for $U$ is

(3.10)\begin{equation} U\approx U_C+\dfrac{(3+n)(n+2)}{U_C^n\left[U_C+(n+2)\delta_c\right]}\dfrac{1}{H|H|^{n+1}}, \end{equation}

and the integral curve reaches point O ($\xi \to \infty$), where the approximate solution is (3.8). In this second integration, iterations are not necessary because $\delta _c$ is already known. The expansion about the singular points O, A and C has been obtained in the same way as that reported for the differential problem describing the gravity current in a converging channels, see § 2.

Figure 8 shows the phase portrait for a shear-thinning fluid with $n=0.5$, with the integral curves for the pre- and post-closure phases, corresponding to an eigenvalue $\delta _c=0.78261$. The eigenvalues for different values of $n$ are listed in table 3 of Appendix A and are shown in figure 9, where a shear-thinning behaviour is accompanied by higher $\delta _c$, although the variability is quite modest; the corresponding integral curves are depicted in figure 10(a), followed by the shape profiles and velocity profiles in the pre-closure phase in figure 10(b). No specific trend is observed, although the thickness of the current immediately behind the front is larger for more shear-thinning fluids. The horizontal velocity is almost indistinguishable for currents with different value of the fluid behaviour index. The results for the eigenvalues coincide with those shown in Figure 1 in Gratton & Perazzo (Reference Gratton and Perazzo2010), after converting the horizontal axis and with their $\lambda$ equal to $1/n$ in the present notation.

Figure 8. Converging radial gravity current. Phase portrait of (3.4a) for $n=0.5$ (shear-thinning fluid), $\delta _c=0.78261$. The continuous curve refers to the pre-closure phase, the dashed curve refers to the post-closure (levelling) phase, the thin red vertical line indicates the asymptote in the levelling phase and the dash-dot blue curves are the approximate solutions about points O and A.

Figure 9. Converging radial gravity current. Eigenvalues representing the exponent of the similarity variable $\xi =x/(t_c-t)^{\delta }$ for axisymmetric flow towards the origin for fluids with different flow behaviour index $n$.

Figure 10. Converging radial gravity current: (a) shape of the heteroclinic trajectories in the pre-closure (continuous curves) and post-closure (dashed curves) phase; and (b) shape profiles and velocity profiles in the pre-closure phase as $n$ varies from 0.5 (shear-thinning) to 1 (Newtonian) and to 1.5 (shear-thickening).

The behaviour of the integral curves approaching O can be obtained by expanding $H=a_0\xi ^b+a_1\xi ^{b+1}+\ldots\, $. Then introducing the approximation (3.8) in (3.4b) and balancing yields

(3.11)\begin{gather} H=K\xi^{{-}n/[\delta_c(n+2)]}, \end{gather}

(3.12)\begin{gather}U={-}K^{(2+n)/n}\left[\dfrac{(n+1)\delta_c-n}{(n+2)\delta_c}\right]^{1/n}\xi^{{-}1/\delta_c}, \end{gather}

where $K$ is a constant. Switching to dimensional variables, one obtains for the current depth and velocity

(3.13)\begin{gather} h(r)=\left(\dfrac{2n+1}{n}\right)^{{n}/({n+2})}\left(\dfrac{\mu_0}{\delta_c\rho g}\right)^{{1}/({n+2})} Kr^{({(n+1)\delta_c-n})/({(n+2)\delta_c})}, \end{gather}

(3.14)\begin{gather}u(r)={-}K^{(2+n)/n}\left[\dfrac{(n+1)\delta_c-n}{(n+2)\delta_c}\right]^{1/n}r^{(\delta_c-1)/\delta_c}, \end{gather}

that are time-independent. For $r\to \infty$, the depth of the current grows in the positive direction, hence $\textrm {d} h/\textrm {d} r>0$ that in turn implies $\delta _c>n/(n+1)$, which also ensures a negative velocity.

A similar analysis can be conducted to describe the post-closure phase, with results similar to those obtained for converging flow in a horizontal fracture.

4. Experimental validation

4.1. Experimental set-up

Three different experimental set-ups were developed in the Laboratory of Hydraulics of the University of Parma in order to validate the theoretical models.

The first set-up consists of a tank to reproduce the propagation of gravity currents in converging channel flow. The channel consists of two rectangular plates, 150 cm long and 20 cm high, with a front wall made of transparent plastic for an easy visualization, while the back wall is made of yellow polyvinyl chloride (PVC) machined with a computerized numerical control machine; the gap thickness is $b(x)=b_1\,x^{0.6}$, where $x$ is the abscissa with origin at the corner, as shown in figure 11; $b_1=0.01176\ \textrm {m}^{0.4}$ and the width of the channel at $x=150\ \textrm {cm}$ equals $1.5\ \textrm {cm}$. Lock release gravity currents were generated with different lengths of the lock, equal to $12, 14, 20, 30 \ \mathrm {cm}$. One experiment was performed with a different boundary condition of given flux, in order to check the sensitivity of the solution to the inflow mode. A similar set-up was adopted in Zheng et al. (Reference Zheng, Christov and Stone2014) and also documented in Ghodgaonkar (Reference Ghodgaonkar2019), with a length of $75\ \textrm {cm}$. With this set-up, the heavy fluid was Newtonian or non-Newtonian PL shear-thinning, whereas the ambient fluid was always air.

Figure 11. Experimental set-up $\#1$, a tank with a gap thickness of $b_1 \, x^{0.6}$: (a) front view; (b) top view; (c) a photo during one of the tests; the lock is almost empty but a thin layer of fluid still adheres to the transparent wall, marking the volume occupied by the fluid in the lock before lifting the gate.

A second series of experiments reproduced radial converging gravity currents and was carried out in a transparent plastic cell shaped as a circular sector (a wedge) with an angle at the centre $\beta = 12^\circ$, a radius $R=75\ \textrm {cm}$ and a height of $h=18\ \textrm {cm}$. The front, side and top view are shown in figure 12(a–c), respectively. The front vertical wall is rigidly attached to the bottom, while the back wall is removable. At a distance $r_0=60\ \textrm {cm}$ a vertical gate is inserted in order to separate the heavy fluid in the upstream lock from the light ambient fluid in the downstream chamber. Three clamps are applied on top of the tank in order to push the walls against the bottom and prevent any leakage. Figure 12(d) is a photograph of the experimental apparatus.

Figure 12. Experimental set-up $\#2$, a circular sector cell with an angle at the centre $\beta = 12^\circ$: (a) front view; (b) lateral view; (c) top view; and (d) snapshot of the tank.

A third set of experiments for radial converging gravity currents was conducted in a fully metallic cylinder with an inner radius of $R_2 = 30 \ \mathrm {cm}$ (figure 13). A smaller cylinder with an external radius $R_1 = 19.5 \ \mathrm {cm}$ is positioned concentrically to the prior one and pushed against the glass bottom (a soft rubber sealing gasket is present) in order to create an annular lock with a gap $\Delta R = R_2 - R_1 = 10.5 \ \mathrm {cm}$. Particular attention was paid to the levelling of the base of the apparatus, so as to prevent asymmetries due to the gravitational force. An electronic level is used for this purpose, with an accuracy of $0.1^{\circ }$. The annulus is filled with dense fluid up to a depth of $h_0 = 3.3 \text {--} 4.5 \ \mathrm {cm}$. The inner cylinder is filled with light ambient fluid. At the beginning of the experiments, the inner cylinder is lifted up to a known height $h < h_0$. A similar set-up was used in Diez et al. (Reference Diez, Gratton and Gratton1992b), with radius of the external/internal cylinder of 20 and 5 cm, respectively.

Figure 13. Experimental set-up $\#3$, a radial symmetric cell: (a) front view; (b) top view at $t = 0$, before the lift of the inner cylinder (sketch), and for $t>0$, after the lift of the inner cylinder (photo).

For the currents in converging channel flow (set-up $\#1$), the Newtonian fluids were prepared with glycerol, while the non-Newtonian fluids were prepared with (i) a mixture of water (95 % vol.), glycerol (5 % vol.) and carboxymethyl cellulose (CMC, 2 % by weight), or a (ii) a mixture of water (40 % vol.), glycerol (60 % vol.) and E415 (0.1 % by weight). All the non-Newtonian fluids have PL shear-thinning rheology.

For the radial converging currents (set-ups #2 and #3), the Newtonian fluids were prepared with either glycerol or a mixture of glycerol, salt and water, and non-Newtonian fluids with a mixture of water (60 % vol.), glycerol (40 % vol.) and xanthan gum (E415, 0.25 % by weight). The light ambient fluid is water or a mixture of water, salt and glycerol with a density slightly less than the current fluid. The combination of different ambient fluids and intruding fluids was required in order to guarantee the viscous regime, with negligible inertial effects, and to increase the duration of the experiments for an accurate estimation of $t_c$. Aniline water color was finally added to the denser fluid for an easy visualization of the propagating current.

The rheology was measured by the parallel plate twin-driver MCR702 rheometer by Anton Paar, at a temperature of $\varTheta = 25\text {--}27^{\circ }$, equal to the ambient temperature during the experiments. Several different tests were performed in order to evaluate the fluid behaviour index and consistency index for non-Newtonian fluids, and the dynamic viscosity of Newtonian fluids. Figure 14 shows the experimental shear-stress/shear-rate curves for two different shear-thinning fluids. The limited range of shear rate is dictated by the evidence that, except at the early stage of the current propagation, after the gate lift, the average shear rate is quite modest. The estimated accuracy is $\Delta n/n\leqslant 4\,\%$ and $\Delta \mu _0/\mu _0\le 6\,\%$. The mass density of the fluids was measured by a hydrometer (STV350023 Salmoraghi), and by the DMA 5000 by Anton Paar, with an accuracy of $\Delta \rho /\rho _0 \leqslant 0.1\,\%$.

Figure 14. Experimental rheometry of two shear-thinning fluids: (a) mixture of water, NaCl and xanthan gum, $\varTheta =27\,^{\circ }\textrm {C}$; (b) mixture of water, NaCl and xanthan gum, with two series of measurements to check repeatability, $\varTheta =25\,^{\circ }\textrm {C}$. The solid curves are the PL interpolation and the dashed curves are the 95 % confidence limits.

The profiles of the advancing current after the lift of the gate, are recorded by either a high-resolution video camera (Canon Legria $1920 \times 1080$ pixels) operating at 25 frames per second (fps), or an iPhone 4k video camera working at 30 fps. For some experiments both videos were used in order to capture the whole profile of the current with an adequate resolution, see figure 11(c) where the panoramic image results from two frames extracted from two synoptic videos. The frames extracted from the video are post-processed using a Matlab code that converts pixels into metric coordinates. This conversion is possible thanks to a square grid attached onto the inner side of the front vertical wall, which allows compensation of the optical distortion. The overall accuracy in detecting the position of the nose and the profile of the current is approximately 0.1 cm, while the accuracy in measuring time is negligible (1/50–1/60 s).

Figure 15 shows a sequence of snapshots for one of the experiments in lock-release, experiment 11, with both pre- and post-closure instants. After closure, the effect of capillarity near the origin is evident, with an increase in the level compared with the average value observed far from the origin. Local level for post-closure analysis was obtained correcting data for capillarity uprise.

Figure 15. Time lapse of the current profile for experiment 11 in a channel of varying gap thickness. Closure is at $t_c=16.4\ \textrm {s}$, then levelling starts.

4.2. Experiments

The parameters of the experiments are listed in table 1; 10 experiments were performed in the converging channel and 11 experiments in radial geometry, either in the $12^\circ$ sector or in the full circle configuration. The theory assumes the dominance of viscous and buoyancy forces over inertia. The Reynolds number for channel flow can be expressed as ratio between the inertial and viscous forces per unit volume,

(4.1a–c)\begin{equation} F_i\sim \Delta\rho\dfrac{u^2}{L},\quad F_v\sim\mu_0\dfrac{u^n}{L^{n+1}},\quad \rightarrow {Re}=\dfrac{\Delta\rho u^{2-n}L^n}{\mu_0}, \end{equation}

where $L$ is taken to be equal to the channel maximum width $L=b_1x_0^k$. The velocity scale can be assumed equal to $x_0/t_c$, hence

(4.2)\begin{equation} {Re}= \dfrac{\Delta\rho x_0^{2-n+k}b_1}{\mu_0t_c^{2-n}}. \end{equation}

Table 1. Parameters for the experiments on converging flow in a channel of variable thickness and in radial geometry. The second column indicates the geometry of the flow field, where ‘c’ denotes channel and ‘r12’ and ‘r360’ denote radial geometry with an angle at the centre of $12^\circ$ and $360^\circ$, respectively. We use ‘c-Q’, ‘l-r’ and ‘c-h’ to denote constant inflow rate, lock-release and constant head; $h$ and $h_0$ are the constant head or the initial depth in the lock; $a$ is the height of the slot at the bottom of the gate; $r_0,x_0$ is the initial position of the front of the current in radial geometry and in the channel; $n$ and $\mu _0$ are the fluid behaviour index and the consistency index (the viscosity for Newtonian fluids) of the current fluid, respectively; $\rho _c,\rho _a$ are the current/ambient fluid density; $t_c$ is the touch-down time; and Re is the Reynolds number according to the expressions (4.2)–(4.4), where an average value $h=2\ \textrm {cm}$ is assumed for radial gravity currents.

For radial gravity currents a similar analysis brings to a Reynolds number defined as

(4.3)\begin{equation} {Re}=\dfrac{\Delta\rho u^{2-n}h^{n+1}}{\mu_oR},\quad\textrm{with}\ u\sim\left(\dfrac{\Delta\rho g}{\mu_0}\right)^{1/n}\dfrac{h^{(n+2)/n}}{R^{1/n}}, \end{equation}

hence

(4.4)\begin{equation} {Re}=\left(\dfrac{\Delta\rho}{\mu_0}\right)^{2/n}g^{(2-n)/n}\dfrac{h^{(n+4)/n}}{R^{2/n}}. \end{equation}

The values of Reynolds number listed in the last column of table 1 refer to the initial stage of the current propagation; for channel flow ${Re}<1$ and for radial flow ${Re} \ll 1$, with only one test initially characterized by ${Re}\approx 0.8$. In the other tests, inertial effects are negligible from the very beginning, and the viscous balance dominates the advancement of the currents at all stages.

Figure 16 shows the time series of the front position for the experiments in converging channel flow. The experiments are separated in two groups, namely Newtonian and shear-thinning gravity currents. For the former tests the theoretical eigenvalue is $\delta _c=1.787$ and the corresponding trajectory is fairly well followed by the experiments. Similar experiments were conducted in Zheng et al. (Reference Zheng, Christov and Stone2014) in channels with different width exponent $k$, although only in lock-release mode. In the present Newtonian subset, four tests are in lock-release mode, whereas experiment 1 is a constant influx experiment. The results indicate that in lock-release mode the current requires a much longer time to forget the initial condition and to reach self-similarity, whereas with a constant influx the current evolves rather quickly to reach the intermediate asymptotic regime of self-similarity. The different lengths of the locks do not have a significant effect on the current propagation in the far field. The grey symbols are considered outliers attributed to several disturbances and interferences, such as capillarity and three dimensionality of the flow field not modelled in the theory. Experiments 3 and 5 are a repetition with the same fluid and under the same conditions, to obtain an estimate of the repeatability and uncertainty.

Figure 16. Front position of gravity currents in converging channel flow with $k=0.6$. Experiments 1–5 refer to Newtonian fluid (left vertical axis) and experiments 6–11 refer to shear-thinning fluids (right vertical axis). The straight lines correspond to the theoretical curves for $n=1$ and $n=0.54\text {--}0.90$, respectively, with eigenvalues $\delta _c=1.787$ and $\delta _c=1.852\text {--}1.794$. The parameters of the experiments are listed in table 1.

For the non-Newtonian fluid experiments the fluid behaviour index ranges from 0.54 to 0.90, and the consistency index ranges from 0.42 to $2.63\ \textrm {Pa}\ \textrm {s}^n$. The two straight lines in figure 16 correspond to extreme values of $\delta _c\approx 1.79$ and $\approx 1.85$. Again the experimental results, all in lock-release mode, adequately follow the theory, reaching the self-similar regime although the differences with the value of $n$ cannot be appreciated. Essentially, the long tank favours the evolution of the current with enough space to forget the inlet and initial conditions and before reaching the proximity of the origin, where capillarity renders the theoretical model invalid.

A comparison of the shape of the current with the experiments is reported for two experiments, with Newtonian and PL fluids, respectively. A main variable in checking the validity of the model is the time position of the front of the current. We expect that $x_f(t)\propto t_r^{\delta _c}$, hence

(4.5)\begin{equation} \dfrac{x_f(t)}{x_0}=\alpha\left(\dfrac{t_r}{t_c}\right)^{\delta_c}, \end{equation}

where the dimensionless coefficient $\alpha$ is the intercept of the straight interpolating line in a diagram $x_f/x_0-t_r/t_c$ drawn in logarithmic scale. We also have the result

(4.6)\begin{equation} \xi_f=\dfrac{\alpha x_0}{t_c^{\delta_c}}, \end{equation}

which is a constant for each experiment. The value of $\alpha$ is obtained by interpolating the experimental front position of the advancing current in pre-closure phase.

Figure 17(a,b) refers to the pre-closure phase for experiments 5 and 11, with experimental profiles sampled in several sections at different times and the continuous curve representing the theoretical model. For the Newtonian fluid case, the early time profiles show a significant discrepancy with theory, which however decreases after approximately $190\ \textrm {s}$. That means that at least for approximately $35\ \textrm {s}$ the current is in self-similar regime, ended by the insurgence of capillarity effects near the origin. For the PL fluid, experiment 11, the profile at $t=6\ \textrm {s}$ differs from the rest of the time snapshots which collapse onto one curve, so they are self-similar; all profiles differ from theory at the peak around $\xi /\xi _f\approx 2.5$ and are in excess to the same degree that the profile at $t=6\ \textrm {s}$ differs around $\xi /\xi _f\approx 10$, and is in defect. We observe that experiment 11 has the highest initial Reynolds number and shows an initial stage when inertial effects are comparable to viscous effects. However, the current forgets its original regime and evolves towards self-similarity. This is another validation of the memoryless behaviour of the gravity currents, which not only forget the type of influx (lock-release or constant influx), but also their initial regime (inertial or in transition). The analysis of the time requested by self-similar solutions to approximate within a given accuracy the (numerical) solutions was discussed by Ball et al. (Reference Ball, Huppert, Lister and Neufeld2017) and Ball & Huppert (Reference Ball and Huppert2019) in the context of gravity currents, also considering different shapes of the lock or inflow rate variations. The results obtained for first-kind self-similar solutions can be extended to experiments and to second-kind self-similar processes: the approximation improves with time, unless disturbances effects (such as capillarity) render the model invalid. It explains the different agreement in times between experiments and theory, which improves for $t>6\ \textrm {s}$ although the peak value of $H$ is underestimated by approximately $15\,\%$. The numerous approximations of the model are responsible for this discrepancy, but the self-similar nature of the profiles is adequately confirmed.

Figure 17. Experimental profiles of the current in converging channel flow with $k=0.6$ during pre-closure (symbols) compared with the theoretical self-similar solution (bold line): (a) experiment 5, Newtonian fluid, $t_c=229\ \textrm {s}$, $\delta _c=1.7874$, $\alpha =0.30$; (b) experiment 11, PL fluid, $t_c=16.4\ \textrm {s}$, $\delta _c=1.8534$, $\alpha =0.19$.

A comparison between experiments and theory has been conducted also for the post-closure stage, with the current levelling near the origin. In this case the pre-factor refers to the time evolution of the current depth in the origin, where, in the self-similar stage, we have the result

(4.7)\begin{equation} \dfrac{h(0,t)}{h_\infty}=\alpha'\left(\dfrac{-t_r}{t_c}\right)^{\delta_c(n+1)(1-k)-n}. \end{equation}

Figure 18 shows the comparison between theory and experiments for experiments 5 and 11. During post-closure, the pre-factor $\alpha '$ is computed by considering the experimental time evolution of the depth $h(0,t)$, which is initially varying according to (4.7), see the insets. The theoretical exponent of the current depth growth in the origin is $\delta _c(n+1)(1-k)-n$, equal to 0.429 and to 0.602 for experiments 5 and 11, respectively; the experimental values are 0.42 and 0.51, with a fairly good agreement for the first, an acceptable agreement for the second. The overall adherence between experiments and theory is striking, with a better collapse of the experimental data and of the theoretical curve for increasing time after closure.

Figure 18. Experimental profiles of the current in converging channel flow with $k=0.6$ during levelling (symbols) compared with the theoretical self-similar solution (bold line): (a) experiment 5, Newtonian fluid $n=1$, $t_c=229\ \textrm {s}$, $\delta _c=1.7874$, $\alpha '=1.29$; (b) experiment 11, PL fluid $n=0.54$, $t_c=16.4\ \textrm {s}$, $\delta _c=1.8534$, $\alpha '=0.41$. The insets are the time level series at the origin adopted to estimate the pre-factor $\alpha '$.

The theoretical formulation for converging radial currents was also experimentally validated. Figure 19 shows the time position of the front of the current in radial geometry: experiments are divided in two groups with Newtonian and shear-thinning fluids, respectively. The Newtonian fluid case was already validated by Diez et al. (Reference Diez, Gratton and Gratton1992b). Experiments 19 and 20 have very different closure times and different time evolution of the front position, although the characteristics of the fluid are practically similar, the only difference being the density, for both fluids a few per cent higher than water, the ambient fluid. As an explanation of this anomaly, we remind that the difference in density is small, but the reduced gravity appearing in buoyancy is significantly different. In addition, both tests are partially influenced by diffusive processes which further reduce the density difference between the current and the ambient fluid, to a greater extent for experiment 19 (less-dense current fluid of the two) than for experiment 20 (denser fluid of the current between the two). Ultimately, under such experimental conditions small variations in the parameters can amplify differences.

Figure 19. Front position of gravity currents in radial converging flow. Experiments 12–16 refer to Newtonian fluid (left vertical axis), experiments 17–21 refer to shear-thinning fluids (right vertical axis). The straight lines correspond to the theoretical curves for $n=1$ and $n=0.29\text {--}0.63$, respectively, with eigenvalues $\delta _c=0.762$ and $\delta _c=0.822\text {--}0.772$. The parameters of the experiments are listed in table 1.

In radial geometry the eigenvalue has an almost constant value for $n>0.8$ although a significant deviation from the average value of approximately $0.78$ can be observed only for very shear-thinning fluids. Observing the time series of the front position, there are no significant differences for experiments in full circle and partial circle configuration, and also the different mode of inflow does not apparently affect the results. However, we expect that the two different geometries, the circular sector and the full circle, can guarantee the experimental reproduction of self-similarity for different time intervals. In the case of the circular sector, when the current is approaching closure the flow field becomes three-dimensional, with comparable tangential stresses in the vertical planes; moreover, capillarity introduces, for the circular sector, disturbances that are not present in full-circle geometry. In summary, the full circle provides the conditions for the establishment of self-similarity more than the circular sector. This happens to a greater extent the smaller the angle at the centre of the circular sector.

The agreement between theory and experiments is generally good, with some dispersion of the data when the current is near the closure, owing to the dominance of interferences. The early stage adaptation to the self-similar solution has a different duration for different tests, but the duration of the intermediate self-similar regime is always quite long.