2.1. Power-law corners

We consider the flow of a liquid into a sharp corner created by two surfaces of possibly different geometries. The surfaces are in contact along the $z$-axis, which is oriented such that the acceleration due to gravity is $-g \boldsymbol {e}_z$, as depicted in figure 1(a). A schematic of the horizontal cross-sectional geometry is shown in figure 1(b). In the cross-section, the surfaces make contact at $x = y = 0$, and their locations are given by

(2.1)\begin{equation} h_i(x) = c_i x^n, \quad i = 1,2, \ n \geq 1, \end{equation}

and the geometric parameters $c_i$ have dimensions [Length]$^{-(n-1)}$. We impose the condition $c_2 > c_1$, where the inequality is necessary so that the surfaces do not coincide, but the $c_i$ may have the same sign. The thickness of the gap formed by the surfaces is

(2.2)\begin{equation} h(x) = h_2(x) - h_1(x) = c x^n, \end{equation}

where $c = c_2-c_1$.

Figure 1.

Geometry of a power-law corner. (a) Fluid column in the gap formed by two surfaces that meet in a vertical line. Grey lines indicate the liquid–solid and liquid–air interfaces. (b) Diagram of a planar cross-section. The corner is located at the origin, and it is formed by two surfaces that are located at $h_2(x)$ and $h_1(x)$. The width of the liquid column is $w(z,t)$, and the radius of the liquid–air interface is $s(z,t)$.

Before moving forward, we discuss the applicability of this simple corner geometry, i.e. corners formed by power-law surfaces with a single exponent $n$, to understanding corners with more varied and complex geometries. One may consider the case in which each surface has a different power, $h_i(x) = c_i x^{n_i}$, e.g. with $n_2 > n_1$. However, this case reduces to $h(x) = h_1(x) = c_1 x^{n_1}$ as $x \to 0$ since $x^{n_2}\ll x^{n_1}$. Furthermore, any arbitrary (smooth) surface profile may be approximated by a Taylor series expansion near $x=0$, which yields a power series. Taking the limit as we approach the corner, we may keep only the leading-order term and thus use a power law to describe the surface geometry. Therefore, we focus our investigations on power-law corners with a single exponent, $n_1 = n_2 = n$, so that we may capture the essential features of the phenomenon while maintaining brevity.

2.2. Scaling of the dimensions of the liquid cross-section

In this subsection, we examine the geometry of the liquid cross-section to identify scaling relationships between its various geometrical descriptors. The width of the column $w(z,t)$ is the minimal value of $x$ of the liquid–air interface, and it is a function of the altitude $z$ (figure 1). The liquid–air interface in the cross-section can assumed to be a circular arc with radius $s(z,t)$. The use and justification of this assumption in the context of capillary rise will be discussed in greater detail in § 4. We anticipate that the liquid column is long and narrow such that the triple line at which the liquid, solid and air meet is nearly parallel to the $z$-axis. Therefore, the angle made by the liquid–air interface and the solid surface in the horizontal cross-section is approximated to be the equilibrium contact angle $\theta$ of the liquid, which is assumed to be constant.

In the remainder of this section, we will identify the relationship between the interface radius $s(z,t)$, the width $w(z,t)$, and the equilibrium contact angle $\theta$. In particular, we show below that, in certain limits, the radius and width scale as

(2.3)\begin{align} 2\,s(z,t) \cos \theta \sim cw^n(z,t) = h(w(z,t)) \quad \text{or} \quad w(z,t) \sim [2 c^{-1}\,s(z,t) \cos\theta]^{1/n} \quad (n \geq 1) \end{align}

and $s(z,t) \ll w(z,t)$, which will enable the use of lubrication theory in § 3.

By defining the horizontal positions of the triple lines on the upper and lower surfaces as $x_2(z,t)$ and $x_1(z,t)$, respectively (see figure 1b), we find that

(2.4a)$$\begin{gather} x_2(z,t) = w(z,t) + s(z,t)[1 - \sin (\theta + \beta_2 (x_2(z,t)))], \end{gather}$$

(2.4b)$$\begin{gather}x_1(z,t) = w(z,t) + s(z,t)[1 - \sin (\theta - \beta_1 (x_1(z,t)) )], \end{gather}$$

where $\beta _i(x) = \tan ^{-1}(h_i'(x))$, with $h_i'(x)=\mathrm {d}h_i/\mathrm {d} x$, are the angles measured from the $x$-axis to the lines tangent to the solid surfaces. Then, by defining the centre of the circular arc of the interface as $(x_c(z,t), y_c(z,t))$, we can write

(2.5a)$$\begin{gather} y_c(z,t) + s(z,t) \cos (\theta+\beta_2 (x_2(z,t))) = h_2 (x_2(z,t)), \end{gather}$$

(2.5b)$$\begin{gather}y_c(z,t) - s(z,t) \cos (\theta - \beta_1(x_1(z,t))) = h_1(x_1(z,t)). \end{gather}$$

Subtracting the two preceding equations yields the geometrical relationship

(2.6)\begin{equation} s(z,t) [\cos (\theta + \beta_2(x_2(z,t)) ) + \cos (\theta-\beta_1(x_1(z,t)) )] = h_2(x_2(z,t)) - h_1(x_1(z,t)). \end{equation}

For the case $n=1$, in which case $c$ is dimensionless, we can take the $xz$-plane to be the symmetry plane of the corner without loss of generality, such that $h_2'(x)=-h_1'(x)=c/2$, $\beta _2(x) = -\beta _1(x) = \tan ^{-1}(c/2)$ and

(2.7)\begin{equation} x_1(z,t) = x_2(z,t) = w(z,t) + s(z,t)\left[1-\sin\left(\theta+\tan^{-1}\frac{c}{2}\right)\right]. \end{equation}

Then (2.6) becomes

(2.8)\begin{equation} 2\,s(z,t) \cos \left(\theta+\tan^{-1}\frac{c}{2}\right) = h\left(w(z,t) + s(z,t)\left[1-\sin\left(\theta+\tan^{-1}\frac{c}{2}\right)\right]\right). \end{equation}

By using the definition in (2.2), we obtain

(2.9)\begin{equation} 2\,s(z,t) \cos \left(\theta+\tan^{-1}\frac{c}{2}\right) = c\,w(z,t) + c\,s(z,t)\left[1-\sin\left(\theta+\tan^{-1}\frac{c}{2}\right)\right]. \end{equation}

Rearranging and considering the limit $c \ll 1$, we find

(2.10)\begin{align} 2\,s(z,t)\cos\theta \sim c\,w(z,t)= h(w(z,t)) \quad \text{or} \quad w(z,t)\sim 2c^{-1}\,s(z,t)\cos\theta \quad (n = 1). \end{align}

For the case $n>1$, the surfaces are tangent to the $xz$-plane at $x = 0$, and $\lim _{x\to 0}\beta _i(x) = 0$. As $x_i(z,t)\to 0$ for $i = 1,2$, we find that $x_2(z,t) \sim x_1(z,t) \sim w(z,t) + s(z,t)$ $(1-\sin \theta )$, and (2.6) simplifies to

(2.11)\begin{equation} 2\,s(z,t) \cos \theta \sim h(w(z,t) + s(z,t)\,(1-\sin \theta)) =c[w(z,t) + s(z,t) (1-\sin \theta)]^n, \end{equation}

where the last equality comes from applying the definition of $h(x)$ in (2.2). Since both $s(z,t) \to 0$ and $w(z,t)\to 0$, we anticipate that terms on the right-hand side of (2.11) with powers $s^2(z,t)$ or greater are insignificant compared to the $s(z,t)$ term on the left-hand side, and thus consider the balance

(2.12)\begin{equation} 2\,s(z,t) \cos \theta \sim c\,w^n(z,t) + cn\,w^{n-1}(z,t)\,s(z,t)\,(1-\sin \theta). \end{equation}

By rearranging terms, we find that in the limit $cn\,w^{n-1}(z,t) \ll 1$,

(2.13)\begin{align} 2\,s(z,t) \cos \theta \sim c\,w^n(z,t) = h(w(z,t)) \quad \text{or} \quad w(z,t) \sim [2 c^{-1}\,s(z,t) \cos\theta]^{1/n} \quad (n > 1). \end{align}

Therefore, in the limit as $x_i(z,t)\to 0$ for $i = 1,2$, $s(z,t)/w(z,t) \to 0$. Due to the results (2.10) obtained for $n=1$ and (2.13) for $n>1$, we may use the relation in (2.3) stated at the start of this subsection for $n\geq 1$ when $cn\, w^{n-1}(z,t) \ll 1$. In the $n=1$ case, certain conditions on the corner geometry (i.e. a small interior angle) must be satisfied. In the $n>1$ case, $c$ may vary, but for a given value of $c$, we require the dimension of the liquid cross-section to be sufficiently small.

The results derived in this section may be understood through the context of geometric similarity. When two shapes are geometrically similar, corresponding angles and dimensionless ratios between linear dimensions are identical, so that the shapes are scaled versions of each other. Geometric similarity of the liquid cross-sections holds only for $n=1$. When $n>1$, $s(z,t)$ and $w(z,t)$ are not directly proportional (see (2.3)), and the ratio $s(z,t)/w(z,t)$ tends to $0$ as the cross-section shrinks to a point. Geometric similarity is also closely tied to the nature of the constant parameters $c_i$ (and $c$). When $n=1$, the cross-sections are geometrically similar because the $c_i$, which are dimensionless descriptors of the geometry, are constant. However, when $n>1$, the definition of geometric similarity is no longer met because there are constant quantities $c_i^{-1/(n-1)}$ with units of [Length], which leads to dimensionless parameters that vary with the size of the cross-section. Perhaps surprisingly, the introduction of parameters $c_i$ with physical dimensions when $n>1$ does not preclude the presence of self-similarity in the phenomenon of capillary rise in sharp corners. This result will be discussed in greater detail in § 5.

Finally, we note that by assuming a cross-section of the form in figure 1(b), we implicitly assume that if the liquid does not extend to $z\to \infty$, then the location of the maximum height of the fluid is located on the $z$-axis ($x=y=0$), where the two surfaces make contact. However, this is not necessarily true at early times. Work by Higuera et al. (Reference Higuera, Medina and Linan2008) and Ponomarenko et al. (Reference Ponomarenko, Quéré and Clanet2011) shows that in the initial stages of capillary rise into a sharp corner, the maximum meniscus altitude is located at some finite $x$. Higuera et al. (Reference Higuera, Medina and Linan2008) develop a model using lubrication theory that reveals that in a linear corner, the location of maximum altitude approaches $x = 0$ as $t^{-1/3}$. Ponomarenko et al. (Reference Ponomarenko, Quéré and Clanet2011) use an organ model and find that in corners of general geometry, the effective radius of the leading meniscus decreases as $t^{-1/3}$. Given the scaling (2.3) between the interface radius and the width, the location of the maximum altitude is expected to approach the corner as $t^{-1/3n}$, as we discuss further in § 6 (see (6.26) and (6.27)). Experiments and numerical simulations performed by Higuera et al. (Reference Higuera, Medina and Linan2008) find that the maximum is effectively located at $x=0$ for long times. Therefore, the geometric description of the liquid column given in this section is appropriate when considering the dynamics of capillary rise in a sharp corner at long times when the liquid has assumed a long and narrow geometry. Capillary rise at early times is considered separately. A model based on the work of Higuera et al. (Reference Higuera, Medina and Linan2008) is developed for corners with $n\geq 1$ in § 6, where we show this $t^{-1/3n}$ power law.

5.1. Scaling

Equation (4.6) is made dimensionless by scaling lengths and time using

(5.1a,b,c)\begin{equation} \tilde{z} = \frac{z}{\ell_{cg}(\cos\theta)^{1/2}},\quad \tilde{s} = \frac{(\cos\theta)^{1/2} s}{\ell_{cg}} \quad \text{and}\quad \tilde{t} = \frac{\gamma (\cos\theta)^{1/2}t}{3\mu \ell_{cg}}. \end{equation}

Due to the absence of $c$ in (4.6), characteristic scales are used here that are independent of $c$, or alternatively, independent of the length scale $c^{-1/(n-1)}$ ($n>1$) of the corner geometry. A similar choice of characteristic scales for time and altitude was made by Ponomarenko et al. (Reference Ponomarenko, Quéré and Clanet2011), which reflects the expected universality of the phenomenon. For ease of comparison with other statements of this capillary rise problem (e.g. Higuera et al. Reference Higuera, Medina and Linan2008; Zhou & Doi Reference Zhou and Doi2020), the scaling and similarity transformation for the equivalent expression of (4.6) in terms of the width $w(z,t)$ are given in Appendix A.

Applying the scaling (5.1a,b,c) leads to the equation

(5.2)\begin{equation} \frac{\partial}{\partial \tilde{t}}( \tilde{s}^{1 + 1/n}) = \frac{n+1}{3n+1}\, \frac{\partial}{\partial \tilde{z}} \left[ \tilde{s}^{1+1/n}\left(\frac{\partial \tilde{s}}{\partial \tilde{z}}+\tilde{s}^{2}\right)\right], \end{equation}

subject to the conditions

(5.3)\begin{equation} \tilde{s}(\tilde{z},\tilde{t}) \sim \frac{1}{\tilde{z}} \quad \mathrm{as} \ \tilde{z} \to 0 \ \text{or} \ \tilde{t} \to \infty \end{equation}

and

(5.4)\begin{equation} \int_0^\infty \tilde{s}^{1 + 1/n} \,\mathrm{d} \tilde{z} < \infty \quad \mathrm{for}\ 0 \leq \tilde{t} < \infty. \end{equation}

5.2. Similarity solution

We observe that the problem described in (5.2), (5.3) and (5.4) is invariant under the one-parameter family of scaling transformations

(5.5a,b,c)\begin{equation} \hat{z} = \chi \tilde{z}, \quad \hat{t} = \chi^3 \tilde{t}, \quad \hat{s} = \chi^{-1} \tilde{s}, \end{equation}

thus we know that there is a similarity solution of the form (e.g. Bluman & Kumei Reference Bluman and Kumei2013; Debnath Reference Debnath2005)

(5.6)\begin{equation} \phi(\eta) = \tilde{t}^{1/3}\,\tilde{s}(\tilde{z},\tilde{t}), \quad \mathrm{where}\ \eta = \tilde{t}^{-1/3}\tilde{z}. \end{equation}

Applying the transformation produces the ODE

(5.7)\begin{equation} -\frac{1}{3}\,\eta\,\frac{\mathrm{d}}{\mathrm{d} \eta} (\phi^{1 + 1/n} ) = \frac{n+1}{3n+1}\,\frac{\mathrm{d} }{\mathrm{d}\eta} \left[ \phi^{1+1/n} \left(\frac{\mathrm{d} \phi}{\mathrm{d} \eta} + \phi^{2}\right)\right], \end{equation}

with the conditions

(5.8a)\begin{equation} \phi(\eta) \sim \frac{1}{\eta} \quad\mathrm{as} \ \eta \to 0 \end{equation}

and

(5.8b)\begin{equation} \int_0^\infty \phi(\eta)^{1+1/n}\,\mathrm{d} \eta < \infty. \end{equation}

Following an argument (see Appendix B for details) similar to that presented by Romero & Yost (Reference Romero and Yost1996) for capillary flow into a sharp horizontal groove, the inequality in (5.8b) can be used to show that the liquid column extends to a finite altitude $z^*$ (or $\eta ^*$ in dimensionless terms) at which the cross-section reduces to a point,

(5.9)\begin{equation} \phi(\eta^*) = 0, \end{equation}

as well as to derive a boundary condition at this location,

(5.10)\begin{equation} \phi'(\eta^*)=-\frac{3n+1}{3(n+1)}\,\eta^*. \end{equation}

Analogous conditions were also adopted by Weislogel & Lichter (Reference Weislogel and Lichter1998). Higuera et al. (Reference Higuera, Medina and Linan2008) and Zhou & Doi (Reference Zhou and Doi2020) derive an equivalent boundary condition expressed in terms of a function approximating the profile of the column near the tip. The ODE (5.7) was solved numerically using (5.8a), (5.9) and (5.10) (see Appendix C for details). The dimensionless tip altitudes $\eta ^*$ for a series of integer $n$ are plotted in figure 2(a). Figure 2(a) contains an inset table listing the $\eta ^*$ values corresponding to the first few values of $n$, and a complete table is provided in Appendix C. In the limit as $n\to \infty$, the boundary condition (5.10) implies that $\phi '(\eta ^*) \to -\eta ^*$. By solving the ODE numerically, we find that $\lim _{n\to \infty } \eta ^* \approx 1.4326$ (see figure 2a). The corresponding solutions $\phi (\eta )$ are plotted in figure 2(b).

Figure 2.

Solutions to the ODE (5.7). (a) The dimensionless tip altitude $\eta ^*$ as a function of $n$. The values of $\eta ^*$ corresponding to the first few integer $n$ are listed in the inset table. A complete table can be found in Appendix C. In the limit $n\to \infty$, $\phi '(\eta ^*)\to \eta ^* \approx 1.4326$. (b) The function $\phi (\eta )$ obtained by solving the ODE numerically. The condition for hydrostatic equilibrium (5.8a), $\phi (\eta ) = 1/\eta$, is shown in black.

5.3. Self-similarity and universality

This subsection discusses briefly the existence of self-similar and universal aspects of capillary rise in sharp corners. In the language of Barenblatt (Reference Barenblatt2003), phenomena are similar if they differ only in the values of their corresponding physical parameters while their corresponding dimensionless parameters are identical. This is the natural generalization of geometric similarity, in which dimensionless quantities like angles and length ratios are preserved while a shape is scaled. Furthermore, we say that phenomena are self-similar when there exists a time- or space-dependent scaling of dimensional quantities for which the phenomenon becomes invariant. Self-similar phenomena are characterized by power-law growth and the existence of similarity transformations through which it is possible to reduce the number independent variables and in this case obtain an ODE from a PDE. Self-similarity is often observed in the propagation of persistent singularities, such as the motion of contact lines at the edges of spreading drops (e.g. Eggers & Fontelos Reference Eggers and Fontelos2015). A common feature of self-similar phenomena is universality, since self-similarity is usually observed in the intermediate asymptotic regime in which both initial conditions and the large-scale structure of the solution are irrelevant. The term ‘universal’ is used here to refer to features of capillary rise in sharp corners that do not depend on corner geometry, or in other words that are common for all powers $n$ and geometric parameters $c$.

As discussed in § 2.1, there is an additional length scale $c^{-1/(n-1)}$ characteristic of the corner geometry when $n > 1$, and the liquid cross-section lacks geometric similarity due to the existence of dimensionless quantities that are not constant as the size of the cross-section is varied. The system also fails to satisfy the definition of physical similarity for the same reason. However, when written in terms of the interface radius, the cross-sectional area and flow rate (3.8) scale with the same power of $c$, thus $c$ does not appear in the evolution equation expressed in terms of the interface radius (4.6) even prior to non-dimensionalization. As noted previously, the fractional exponents in (4.6) and (5.2) simplify upon distribution of the derivatives, and information about the corner geometry remains only in the power $n$ in the coefficients. It becomes apparent that there is a similarity transformation (5.6) that is independent of $c$ and $n$, so we confirm the universality of $t^{1/3}$ proposed by Ponomarenko et al. (Reference Ponomarenko, Quéré and Clanet2011). That is, the tip altitude advances with the same power of $t$ for all corner geometries:

(5.11)\begin{equation} \frac{z^*}{\ell_{cg}(\cos\theta)^{1/2}} = \eta^*\left( \frac{\gamma (\cos\theta)^{1/2} t}{3\mu \ell_{cg}}\right)^{1/3} \end{equation}

or

(5.12)\begin{equation} \tilde{z}^* = \eta^*\tilde{t}^{1/3}. \end{equation}

The scaling of the tip altitude with time in (5.11) is equivalent to the result in Zhou & Doi (Reference Zhou and Doi2020) (see their (3.25)) up to factors of $n$, despite differences in the choices of characteristic lengths in the intervening scaling step and in the form of the ODE obtained by similarity transformation. In (5.11) and (5.12), the constant $c$ is absent. However, due to the dependence of the prefactor and dimensionless tip altitude $\eta ^*$ on $n$, the rate of capillary rise is not truly universal. The prefactor $\eta ^*$ of the power law varies with $n$ because $n$ appears in the coefficients of the nonlinear evolution equation (4.6) and in the boundary condition (5.10). The quantity $\eta ^*$ decreases monotonically with the power $n$, and its value is bounded between approximately 1.6718 and 1.4350 for $1\leq n < \infty$ (see figure 2a). Zhou & Doi (Reference Zhou and Doi2020) also report that variations in the prefactor are small based on values obtained up to $n = 5$, which are tabulated in their table 1.

In summary, we anticipate that capillary rise proceeds as $t^{1/3}$ for all $c$ and $n$, and that the rates of capillary rise in corners characterized by the same power $n$ are the same regardless of the values of $c$. Furthermore, we can identify another feature shared by liquid columns rising in corners. For a given $n$, the profiles of the interface radius $s$ as a function of the axial coordinate $z$ are identical regardless of the value of $c$, and they all share the same self-similar profile. Experimental data that illustrate these results are presented in § 7.

In the next section, we examine the early stages of capillary rise. Power laws for the evolution of geometric features of the meniscus are identified that depend differently on the geometry of the corner than those presented for capillary rise at late times in which the liquid geometry is long and narrow. The result presented in the next section suggests that the corner geometry affects the dynamics of capillary rise in ways that are not always straightforward.

6.1. Model

First, by applying lubrication theory and making use of the separation of scales $L_y \ll L_x$ and $L_y \ll L_z$, we write the velocities averaged over the gap height:

(6.1)\begin{equation} (u_x,u_z) =-\frac{h^2(x)}{\mu} \left( \frac{\partial p}{\partial x}, \frac{\partial p}{\partial z}\right), \quad 0 < z < Z(x,t),\ x>0, \end{equation}

where $p(x,z,t)$ is the pressure driving the flow. The pressure is approximately constant in the $y$-direction as a result of the geometry, and the velocity $u_y$ is negligible, $u_y/u_x \ll 1$ and $u_y/u_z \ll 1$. The corresponding flux in each direction is

(6.2)\begin{equation} (q_x,q_z) =-\frac{h^3(x)}{\mu} \left(\frac{\partial p}{\partial x}, \frac{\partial p}{\partial z}\right), \quad 0 < z < Z(x,t),\ x>0. \end{equation}

Using (2.2), we substitute for the gap height $h(x)$:

(6.3)$$\begin{gather} (u_x,u_z) =-\frac{c^2 x^{2n}}{\mu} \left( \frac{\partial p}{\partial x}, \frac{\partial p}{\partial z}\right), \quad 0 < z < Z(x,t), \ x>0, \end{gather}$$

(6.4)$$\begin{gather}(q_x,q_z) =-\frac{c^3 x^{3n}}{\mu} \left( \frac{\partial p}{\partial x}, \frac{\partial p}{\partial z}\right), \quad 0 < z < Z(x,t),\ x>0. \end{gather}$$

For an incompressible flow, mass conservation requires $\partial q_x/\partial x + \partial q_z/\partial z =0$, or

(6.5)\begin{equation} \frac{\partial}{\partial x}\left( x^{3n}\,\frac{\partial p}{\partial x} \right) + x^{3n}\, \frac{\partial ^2p}{\partial z^2} = 0, \quad 0 < z < Z(x,t). \end{equation}

At the edge of the corner where the surfaces make contact, we expect the velocity and flux to vanish, or equivalently,

(6.6)\begin{equation} \frac{\partial p}{\partial x}\to 0 \quad \text{as} \ x \to 0. \end{equation}

The reference pressure is

(6.7)\begin{equation} p(x,z=0,t) = 0, \end{equation}

and a pressure boundary condition holds at the interface:

(6.8)\begin{equation} p(x,z=Z(x,t),t) =-\frac{2 \gamma \cos\theta}{cx^n}+ \rho g\,Z(x,t), \end{equation}

where it has been assumed that axial contributions to mean curvature are negligible, that the interface is approximately circular in horizontal cross-sections, and that the definition (2.2) and asymptotic relation (2.3) have been applied. We assume that the meniscus is a material surface that travels with the average velocity $\text {D}Z/\text {D}t = u_z$, in which case

(6.9)\begin{equation} \frac{\partial Z}{\partial t} - \frac{c^2 x^{2n}}{\mu}\,\frac{\partial p}{\partial x}\, \frac{\partial Z}{\partial x} + \frac{c^2 x^{2n}}{\mu}\,\frac{\partial p}{\partial z} = 0 \quad \text{at}\ z = Z(x,t). \end{equation}

If we begin with the initial condition

(6.10)\begin{equation} Z(x,t=0) = 0, \end{equation}

then the characteristic scale in the vertical direction is small compared to that in the horizontal direction, $L_z \ll L_x$, and we may use lubrication theory to simplify (6.5) and (6.9) to

(6.11)\begin{equation} \frac{\partial ^2p}{\partial z^2} = 0, \quad 0 < z < Z(x,t), \ x>0, \end{equation}

and

(6.12)\begin{equation} \frac{\partial Z}{\partial t} + \frac{c^2 x^{2n}}{\mu}\,\frac{\partial p}{\partial z} = 0 \quad \text{at}\ z = Z(x,t). \end{equation}

6.2. Scaling and similarity solution

Scaling the lengths by

(6.13a,b)\begin{equation} (\tilde{z},\tilde{Z}) = \frac{(z,Z)}{\ell_{cg}(\cos\theta)^{1/2}}\quad \text{and} \quad \tilde{x} = \frac{x}{[2 c^{-1}\ell_{cg}(\cos \theta)^{1/2}]^{1/n}}, \end{equation}

and pressure and time by

(6.14a,b)\begin{equation} \tilde{p} = \frac{p}{\rho g \ell_{cg}(\cos\theta)^{1/2}}\quad \text{and} \quad \tilde{t} = \frac{\gamma (\cos\theta)^{1/2}t}{3\mu \ell_{cg}}, \end{equation}

we obtain the system

(6.15a)$$\begin{gather} \frac{\partial ^2\tilde{p}}{\partial \tilde{z}^2} = 0, \quad 0 < \tilde{z} < \tilde{Z}(\tilde{x},\tilde{t}), \ \tilde{x}>0, \end{gather}$$

(6.15b)$$\begin{gather}\tilde{p}(\tilde{x},\tilde{z}=0,\tilde{t}) = 0, \end{gather}$$

(6.15c)$$\begin{gather}\tilde{p}(\tilde{x},\tilde{z}=\tilde{Z}(\tilde{x},\tilde{t}),\tilde{t}) =-\frac{1}{\tilde{x}^n}+ \tilde{Z}(\tilde{x},\tilde{t}), \end{gather}$$

(6.15d)$$\begin{gather}\frac{\partial \tilde{Z}}{\partial \tilde{t}} + 12\tilde{x}^{2n}\,\frac{\partial \tilde{p}}{\partial \tilde{z}} = 0 \quad \text{at}\ \tilde{z} = \tilde{Z}(\tilde{x},\tilde{t}) \end{gather}$$

and

(6.15e)\begin{equation} \tilde{Z}(\tilde{x},\tilde{t}=0) = 0. \end{equation}

Solving (6.15a) with (6.15b) and (6.15c) yields the solution for the pressure:

(6.16)\begin{equation} \tilde{p}(\tilde{x},\tilde{z},\tilde{t}) = \frac{\tilde{z}}{\tilde{Z}(\tilde{x},\tilde{t})}\left( \tilde{Z}(\tilde{x},\tilde{t}) - \frac{1}{\tilde{x}^n}\right). \end{equation}

Substituting the expression for the pressure (6.16) into (6.15d), we obtain

(6.17)\begin{equation} \frac{\partial \tilde{Z}}{\partial t} = 12 \tilde{x}^n\Bigg(\frac{1-\tilde{Z}(\tilde{x},\tilde{t})\,\tilde{x}^n}{\tilde{Z}(\tilde{x},\tilde{t})}\Bigg), \end{equation}

which can be solved with the initial condition (6.15e) to find

(6.18)\begin{equation} -\tilde{x}^n\,\tilde{Z}(\tilde{x},\tilde{t}) - \ln [1- \tilde{x}^n\,\tilde{Z}(\tilde{x},\tilde{t})] = 12 \tilde{x}^{3n} \tilde{t}. \end{equation}

We observe that (6.18) is invariant under the one-parameter family of scaling transformations

(6.19a,b,c)\begin{equation} \hat{x} = \chi \tilde{x}, \quad \hat{t} = \chi^{-3n} \tilde{t}, \quad \hat{Z} = \chi^{-n} \tilde{Z}. \end{equation}

Therefore, there is a similarity transformation

(6.20)\begin{equation} \zeta(\xi) = \tilde{t}^{-1/3}\,\tilde{Z}(\tilde{x},\tilde{t}), \quad \mathrm{where}\ \xi = \tilde{t}^{1/3n}\tilde{x}, \end{equation}

through which we obtain

(6.21)\begin{equation} -\xi^n\zeta -\ln (1 - \xi^n \zeta) = 12 \xi^{3n}. \end{equation}

The solution for $\zeta (\xi )$ can be written using the Lambert $W$-function

(6.22)\begin{equation} \zeta(\xi) = \xi^{-n}[1+ W(-{\rm e}^{-1-12\xi^{3n}})]. \end{equation}

6.3. Evolution of the maximum meniscus altitude

Taking the derivative of (6.21) with respect to $\xi$, and setting $\zeta '(\xi )=0$, we find that there is a maximum $\zeta ^*$ located at $\xi ^*$ such that

(6.23)\begin{equation} (\xi^*)^{n}\zeta^*\left(\frac{1}{1-(\xi^*)^n \zeta^*}- 1 \right) =36 (\xi^*)^{3n}. \end{equation}

The location $\xi ^*$ of the maximum is shown in figure 3(a), and the corresponding solutions to (6.21) are shown in figure 3(b). Also, it is possible to examine (6.21) to determine how the location and value of the maximum depend on the power $n$.

Figure 3.

Similarity solution for capillary rise at early times. (a) Locations $\xi ^*$ of the maximum of the function $\zeta (\xi )$ as a function of $n$. (b) Profiles of $\zeta (\xi )$ for different $n$.

It can be shown that $\zeta ^* \approx 2.0200$ for all $n$, and that $\xi ^* \to 1$ as $n\to \infty$. By defining $\chi = \xi ^n\zeta$, we can write (6.21) and (6.23) as

(6.24a)\begin{equation} -3\chi^* - 3\ln (1-\chi^*) = 36(\xi^*)^{3n} \end{equation}

and

(6.24b)\begin{equation} \chi^* \left(\frac{1}{1-\chi^*} -1\right) = 36(\xi^*)^{3n}. \end{equation}

In writing (6.24a), we have used the fact that if (6.21) applies for all $\xi$ and $\zeta (\xi )$, then it must be true at $\xi ^*$ and $\zeta ^*$. Then, noting that the expressions on the right-hand sides of (6.24a) and (6.24b) are identical, we set the left-hand sides equal,

(6.25)\begin{equation} \chi^* \left(\frac{1}{1-\chi^*} -1\right) =-3 \chi^* -3 \ln (1-\chi^*), \end{equation}

and find that $\chi ^* \approx 0.6450$. Then we use (6.24a) or (6.24b) to solve for $(\xi ^*)^n$, and we conclude that $(\xi ^*)^n \approx 0.3193$. Here, $(\xi ^*)^n$ is a constant, so $\xi ^*\to 1$ as $n\to \infty$. Furthermore, by using the definition of $\chi$, we find that $\zeta ^* = (\xi ^*)^{-n}\chi ^* \approx 2.0200$, which is constant for all $n$. We observe these features in figure 3(b).

Due to (6.20), we find that the location of the maximum moves towards $\tilde {x}=0$ as

(6.26)\begin{equation} \tilde{x}^* = \xi^*\tilde{t}^{-1/3n} , \end{equation}

where the prefactor $\xi ^*$ is a function of $n$. Using physical quantities,

(6.27)\begin{equation} \frac{x^*}{[2 c^{-1}\ell_{cg}(\cos \theta)^{1/2}]^{1/n}} = \xi^*\left(\frac{\gamma (\cos\theta)^{1/2}t}{3\mu \ell_{cg}}\right)^{-1/3n}. \end{equation}

The maximum meniscus altitude increases as

(6.28)\begin{equation} \tilde{Z}^* = \zeta^*\tilde{t}^{1/3} \end{equation}

or

(6.29)\begin{equation} \frac{Z^*}{\ell_{cg}(\cos\theta)^{1/2}} = \zeta^*\left( \frac{\gamma (\cos\theta)^{1/2} t}{3\mu \ell_{cg}}\right)^{1/3}. \end{equation}

Therefore, in the early stages of capillary rise, we expect the rate at which the meniscus rises (6.29) to be universal, that is, independent of $c$ and $n$. However, the rate at which the location of the maximum altitude approaches the corner (6.27) depends on both $c$ and $n$.