Semi-Analytical Methods in the Problem of Deformation of a Fluid Strip

Abstract

A problem from a class of unsteady plane potential flows with a free boundary is considered. The entire boundary occupied by the liquid is free, and a zero pressure is maintained. There are neither external nor capillary forces. The motion is driven by inertia. The parameters prescribed at the initial time are the velocity field and the domain occupied by the fluid. The task is to determine these parameters at subsequent time instants. The solution is sought in the form of power series, which are then summed up with the use of the Pade approximation.

1. Introduction

Though studies of an incompressible ideal fluid with a free boundary have a long history and can be considered as classical research, there are still not too many results obtained in the exact nonlinear formulation. There are rather few examples of exact solutions. The first results were obtained by Dirichlet [1] who found a class of flows with a linear velocity field. Later, these studies were continued by Ovsiannikov [2] and Longuet-Higgins [3]. Interesting examples of exact solutions and a semi-Lagrangian algorithm for finding them were demonstrated by John [4]. A class of flows with zero acceleration on the free boundary was studied in [5,6,7]. Lui and Pego in their review [8] proposed to use the term flows with ballistic free boundaries.

The reason for the small number of exact solutions is the fact that there is no universal method for resolving complex nonlinear boundary conditions on an unknown boundary.

The majority of studies aimed at solving problems of this kind are based on using numerical algorithms.

We do not provide an exact solution in the present paper. However, semi-analytical methods applied here allow one to obtain much more information on the character of the solution than purely numerical calculations. It should be noted that the term “semi-analytical methods” was proposed by M. Van Dyke [9]. A semi-analytical study is usually understood as follows: the problem is first studied by tools of the classical analysis, and then numerical algorithms are involved to clarify these or those analytical characteristics, As an example, we can mention the paper [10], where semi-analytical methods were applied to solve the problem of motion of a cylindrical cavity in a fluid.

The combined use of the complex analysis, computer science, and semi-analytical methods allows obtaining unique results, which cannot be obtained by using any of these approaches separately. Though the complex analysis and semi-analytical methods have been used for a sufficiently long time, the increasing capabilities of computer science provide an impetus to applying these methods for solving new problems.

In the present study, the solution of liquid strip deformation is sought in the form of power series with respect to time. The Pade approximations are used for summation of series. The use of the Pade approximation is justified by their improved convergence as compared to power series.

The present paper is based on three ideas put forward by different researchers at different times:

• Application of conformal mappings for the description of plane flows;
• Presentation of the solution in the form of power series with respect to time;
• Application of semi-analytical methods.

Finding the coefficients of the power series is only one aspect of the problem; moreover, it is far from being a difficult aspect. Semi-analytical methods proper are actually involved since the time instant when the power series coefficients are already found. Two problems usually arise in solving problems in the form of power series. First, the radius of convergence of the power series is usually small. As a result, the power series may diverge at a point of interest for us though the function presented by this series does exist at this point. However, its value at this point cannot be obtained by means of direct summation. The power series has to be analytically continued outward the circle of its convergence. The second problem (finding singular points) is to find the coordinates of the singular points and to study their character. To resolve these issues, we used the Pade approximants, methods of convergence acceleration, Domb–Sykes test, and other techniques.

The method used in the study is applicable for solving all unsteady problems of the flow of an ideal incompressible fluid with a free boundary, where the initial velocity is defined by an analytical function.

Plane potential flows were intensely studied. The studies were mainly focused on proving integrability of equations that describe the flow of an ideal fluid with a free boundary. Studying the properties and singularities of the solution obtained in the present investigation can be useful for moving forward in this direction.

The idea of considering inertial motion is driven by our desire to simplify the initial problem as much as possible in order to obtain exact analytical characteristics of some elements of motion, e.g., singularities. An example of a practical scenario of inertial motion can be the breaking of gravitational waves.

There are several areas in hydrodynamics that describe real physical processes where capillary forces are ignored, e.g., the theory of jets and theory of waves on water. Most studies deal with steady flows. There is no similar theory for an unsteady situation considered in the paper, and we tried to fill this gap.

2. Formulation of the Problem

The main difficulty of solving problems with a free boundary is the fact that the domain of the solution changes with time. The use of conformal mappings allows the problem to be simplified by means of fixing the domain of its solution. The initial formulation of the problem can be simplified by using conformal mappings.

Conformal mappings for steady flows were used by Stokes [11] in studying periodic waves on the fluid surface. For studying unsteady flows, it is necessary to choose a fixed domain (circle, half-plane, strip, or something else) in the auxiliary plane $\zeta =\xi +\mathrm{i}\eta$ and then determine a conformal mapping that transforms this domain into the flow domain in the plane of the Cartesian coordinates $z=x+\mathrm{i}y$. This procedure allows one to fix the flow domain and solve a boundary-value problem in this domain.

The boundary conditions for finding two analytical functions: conformal mapping $z=Z(\zeta ,t)$ and complex potential $\phi +\mathrm{i}\psi =\mathrm{\Phi }(\zeta ,t)$ were derived for the first time in [12,13,14]. These boundary conditions turned out to be cubically nonlinear. It was noted [15] that the boundary conditions and the corresponding operator equations will become quadratically nonlinear if the complex potential is replaced by the complex velocity $U(\zeta ,t)={\mathrm{\Phi }}_{\zeta }/Z_{\zeta }$. The complex velocity is expressed via the Cartesian coordinates of the velocity vector $\overrightarrow{V}=(u,v)$ by the formula $U(\zeta ,t)=u-\mathrm{i}v.$

There are three approaches to studying plane unsteady flows with a free boundary. The time in these three approaches will be denoted below by different letters:

• Using the Euler coordinates $x,y$ and the time $\tau$. In this case, there arise the Cauchy–Riemann relations; therefore, one can pass to analytical functions of the complex variable $z=x+\mathrm{i}y$. Thus, the problem is simplified;
• Using the Lagrangian coordinates $a,b$ and the time T. The advantage of this approach is a fixed free boundary in the plane of the Lagrangian coordinates;
• Using the complex variable $\zeta$ and the time t. This approach combines the best features of the two previous approaches. First, there arise analytical functions of the complex variable $\zeta$. Second, the free surface is fixed in the plane $\zeta$. In other words, the domain ${\mathrm{\Omega }}_0$ filled by the fluid in the plane $\zeta$ is known.

In these three cases, the time is denoted by different letters. In this way, it is possible to avoid confusion in writing partial derivatives with respect to time. Thus, though $T=\tau =t$, the partial derivatives with respect to these variables have different meanings and do not coincide:

$$\frac{\partial }{\partial T}\ne \frac{\partial }{\partial \tau }\ne \frac{\partial }{\partial t}.$$

Let s be the length of the arc of the boundary $\partial {\mathrm{\Omega }}_0$ chosen in the plane $\zeta$. There are two boundary conditions (kinematic and dynamic conditions) at the domain boundaries for finding two unknown analytical functions $U(\zeta ,t)$ and $Z(\zeta ,t)$.

The kinematic condition means that the free boundary is a fluid line, i.e., it consists of the same particles. Hence, the projection of the velocity $\overline{U}$ of the fluid particle located on the free boundary and the projection of the velocity of the boundary itself $Z_t$ onto the normal to the free boundary coincide with each other. In other words, the complex vector $Z_t-\overline{U}$ is directed tangentially to the free boundary. Let us recall that the vector tangential to the free boundary is defined as $Z_s$. Therefore, the normal vector $\mathrm{i}{Z}_s$ is directed along the normal to the free boundary. The scalar product of two vectors $Z_1$ and $Z_2$ is defined by the formula $\mathrm{Re}\left[Z_1\overline{Z_2}\right]$; therefore, the kinematic condition can be written as the scalar product

$$\mathrm{Im}\left[Z_s(\overline{Z_t}-U)\right]=0.$$

(1)

The dynamic condition implies that acceleration has to be orthogonal to the free boundary $p=\mathrm{const}$. This statement directly follows from the Euler equation of motion

$$\frac{\mathrm{d}\overrightarrow{V}}{\mathrm{d}\tau }=-\frac{1}{\rho }\nabla p.$$

Thus, the scalar product of the acceleration vector $U_T$ and the tangent line vector $Z_s$ is equal to zero:

$$\mathrm{Re}\left[Z_s{U}_T\right]=0.$$

(2)

Now, we have to find the complex vector of acceleration $U_T=a_x-\mathrm{i}{a}_y$. Here, $a_x$ and $a_y$ are the projections of the acceleration vector onto the x and y axes, respectively.

As Z and $\tau$ are independent variables, we have

$$Z_{\tau }=Z_t+Z_{\zeta }{\zeta }_{\tau }=0.$$

Therefore,

$${\zeta }_{\tau }=-\frac{Z_t}{Z_{\zeta }}$$

and thus, we have

$$\frac{\partial }{\partial \tau }=\frac{\partial }{\partial t}+{\zeta }_{\tau }\frac{\partial }{\partial \zeta }=\frac{\partial }{\partial t}-\frac{Z_t}{Z_{\zeta }}\frac{\partial }{\partial \zeta }.$$

Moreover,

$$\frac{\partial }{\partial T}=\frac{\partial }{\partial \tau }+Z_T\frac{\partial }{\partial Z}=\frac{\partial }{\partial \tau }+\overline{U}\frac{\partial }{\partial Z};\frac{\partial }{\partial Z}=\frac{1}{Z_{\zeta }}\frac{\partial }{\partial \zeta }.$$

(3)

Substituting the complex velocity and conformal mapping into the differentiation operators (3), we obtain the following statement:

Statement 1.

If the complex velocity $u-\mathrm{i}v=U(\zeta ,t)$ and conformal mapping $x+\mathrm{i}y=Z(\zeta ,t)$ of the fixed domain ${\mathrm{\Omega }}_0$ in the plane $\zeta$ onto the domain occupied by the fluid in the plane Z are known, then the complex acceleration $U_T$ is described by the formula

$$U_T=U_t-\frac{U_{\zeta }}{Z_{\zeta }}\left(Z_t-\overline{U}\right).$$

(4)

Substituting the found acceleration (4) into the dynamic boundary condition (2), adding the kinematic boundary condition (1), and imposing some simplifying assumptions, we see that the following problem has to be solved to find an unsteady flow with a free boundary.

Problem 1.

It is necessary to find two analytical functions $Z(\zeta ,t)$ and $U(\zeta ,t)$ that are holomorphic at $\zeta \in {\mathrm{\Omega }}_0$ and satisfy two conditions on the boundary of the domain ${\mathrm{\Omega }}_0$

$$\mathrm{Im}\left[Z_s(\overline{Z_t}-U)\right]=0,\mathrm{Re}[Z_s{U}_t-U_s(Z_t-\overline{U})]=0(\mathit{at}\zeta \in \partial {\mathrm{\Omega }}_0)$$

and two initial conditions

$$Z(\zeta ,0)=Z^{\left(0\right)}\left(\zeta \right),U(\zeta ,0)=U^{\left(0\right)}\left(\zeta \right)(\mathit{at}\zeta \in {\mathrm{\Omega }}_0).$$

Thus, to make problem 1 more specific, we have to define a domain ${\mathrm{\Omega }}_0$ in an auxiliary plane $\zeta =\xi +\mathrm{i}\eta$ and choose two functions: $Z^{\left(0\right)}\left(\zeta \right)$ and $U^{\left(0\right)}\left(\zeta \right)$, which define the initial shape of the domain occupied by the fluid and the initial velocity field, respectively.

In the present study, the domain ${\mathrm{\Omega }}_0$ is taken to be a strip (Figure 1) described by the equation

$$-\frac{\pi }{4}\le \eta \le \frac{\pi }{4},-\infty \le \xi \le \infty ,$$

(5)

and the initial conformal mapping is defined by the linear function

$$Z^{\left(0\right)}\left(\zeta \right)=\zeta .$$

(6)

At the initial time, the variables z and $\zeta$ coincide; therefore, at $t=0$, the fluid in the physical plane z occupies a strip (1) similar to the strip (5).

We assume that the initial complex velocity is expressed via the hyperbolic tangent and contains the real constant A:

$$U^{\left(0\right)}\left(\zeta \right)=Atanh\zeta .$$

(7)

The choice of the hyperbolic tangent owing to the formula ${lim}_{\xi \to \pm \infty }tanh\xi =\pm 1$ ensures constant velocities at the right and left infinities, which differ only by the sign. At the line of symmetry of the strip $y=0$, one velocity component is always equal to zero: $v=0$. Figure 2 shows the second component of the velocity vector at the line of symmetry at the initial time. Owing to its asymptotic behavior at infinity, the hyperbolic tangent function provides an adequate description of the velocity of two colliding jets of an ideal fluid. As we consider an inertial flow, the shape of the domain and the velocity field at a certain time instant are completely determined by the initial velocity field. Two cases are considered below: $A=1$ and $A=-1$.

The first case ($A=1$) corresponds to strip extension. If the right part of the strip is pulled to the right and the left part is pulled to the left, then the strip thickness in the finite part of the domain occupied by the fluid decreases. A thin neck connecting two pieces of the fluid is formed. There arises a question: Will the original strip be torn into two parts within a finite time? Our semi-analytical calculations described in the present paper show that a continuous domain is not broken during a finite time.

The second case ($A=-1$) corresponds to a collision between the right and left parts of the strip. Two half-bands of identical width moving with identical velocities perform a symmetric frontal collision. The flow here is unsteady, but it tends to a known steady-state limit as $t\to \infty$.

As it follows from Equation (6), the function $Z(\zeta ,0)$ is linear at $t=0$; therefore, we separate the linear part of the conformal mapping and seek for it in the form

$$Z(\zeta ,t)=\zeta +W(\zeta ,t),$$

(8)

where the function $W(\zeta ,t)$ tends to zero as $t\to 0$.

Substituting a new function $W(\zeta ,t)$ into problem 1, considering the upper half-strip only, and replacing the derivative $\partial /\partial s$ by $\partial /\partial \zeta$, we obtain a new problem.

Problem 2.

It is necessary to find two analytical functions $U(\zeta ,t)$ and $W(\zeta ,t)$ that are holomorphic in the strip $0<\eta <\pi /4$ and satisfy two boundary conditions on the free boundary

$$\mathrm{Im}\left[(1+W_{\zeta })({\overline{W}}_t-U)\right]=0,\mathrm{Re}[(1+W_{\zeta })U_t+U_{\zeta }(\overline{U}-W_t)]=0,(\mathit{at}\zeta =\xi +\mathrm{i}\pi /4),$$

(9)

the symmetry conditions

$$\mathrm{Im}W=0,\mathrm{Im}U=0(\mathit{at}\eta =0),$$

and the initial conditions

$${W(\zeta ,t)|}_{t=0}{=0,U(\zeta ,t)|}_{t=0}=Atanh\left(\zeta \right).$$

(10)

3. Power Series with Respect to Time

A natural parameter for unsteady problems is time; therefore, we seek for the solution in the form of power series with respect to time. In addition to the variable $\zeta$, it is more convenient in what follows to use a new variable

$$\mu =tanh\zeta .$$

(11)

Formula (11) ensures a conformal mapping of the strip corresponding to the fluid in the plane $\zeta$ onto a circle in a new auxiliary plane $\mu$. The correspondence of some points is illustrated in Figure 3. Thus, the sought functions are the complex velocity $U(\mu ,t)$ and the nonlinear part of the conformal mapping $W(\mu ,t)$ of the unit circle onto the domain occupied by the fluid:

$$W(\mu ,t)=W^{\left(0\right)}\left(\mu \right)+W^{\left(1\right)}\left(\mu \right)t+W^{\left(2\right)}\left(\mu \right)t^2+\dots ,$$

(12)

$$U(\mu ,t)=U^{\left(0\right)}\left(\mu \right)+U^{\left(1\right)}\left(\mu \right)t+U^{\left(2\right)}\left(\mu \right)t^2+\dots .$$

(13)

As it follows from Equations (8) and (11), the series for the function of conformal mapping in the auxiliary plane $\mu$ has the form

$$Z(\mu ,t)=\frac{1}{2}ln\frac{1+\mu }{1-\mu }+W^{\left(0\right)}\left(\mu \right)+W^{\left(1\right)}\left(\mu \right)t+W^{\left(2\right)}\left(\mu \right)t^2+\dots .$$

As was noted above, the function W at the initial time is equal to zero; hence, $W^{\left(0\right)}\left(\mu \right)=0.$

The initial velocity is also known, which allows one to determine the zero coefficient of the series for U:

$$U(z,0)=Atanh\left(z\right)=A\mu =U^{\left(0\right)}\left(\mu \right).$$

Let the unit circle in the plane $\mu$ be described by the equation $\mu =exp\left(\mathrm{i}\theta \right)$, $0\le \theta \le 2\pi .$ The boundary conditions (9) were derived for the auxiliary plane $\zeta$, where the derivative $\frac{\partial }{\partial s}$ means the derivative in the tangential direction to the free boundary. For the plane $\mu$, $\frac{\partial }{\partial \theta }$ plays the role of this derivative. Thus, we have

$$\frac{\partial }{\partial s}=\frac{\partial }{\partial \theta }=\frac{\partial }{\partial \mu }\frac{\partial \mu }{\partial \theta }=\mathrm{i}\mu \frac{\partial }{\partial \mu }$$

and, therefore, the boundary conditions (9) take the form

$$\left\{\begin{array}{c}\mathrm{Im}\left[\mathrm{i}\mu W_{\mu }(\overline{W_t}-U)\right]=0,\hfill \\ \mathrm{Re}[\mathrm{i}\mu W_{\mu }{U}_t-\mathrm{i}\mu U_{\mu }(W_t-\overline{U})]=0.\hfill \end{array}\right.$$

(14)

We substitute series (12) and (13) into conditions (14) and equate the terms at identical powers of $t,$ thus obtaining the equality of imaginary (real) parts of some expressions from which there follows the equality of the expressions themselves with accuracy to the real (imaginary) constant. Owing to symmetry, all these constants should have zero values. Acting in this way, we obtain recurrent formulas for finding the coefficients $W^{\left(k\right)}\left(\mu \right)$, $U^{\left(k\right)}\left(\mu \right):$

$$W^{(k+1)}=\left[(1-{\mu }^2)\sum _{j=1}^k\left(j\overline{W^{\left(j\right)}}-U^{(j-1)}\right)W_{\mu }^{(k-j+1)}-U^{\left(k\right)}\right]/(k+1);$$

(15)

$$\begin{array}{cc}\hfill U^{(k+1)}& =\frac{1-{\mu }^2}{k+1}\left[(k+1)W^{(k+1)}{U}_{\mu }^{\left(0\right)}-\right.\hfill \\ \hfill & \left.-\sum _{j=1}^{k}j\left(W_{\mu }^{(k-j+1)}{U}^{\left(j\right)}-U_{\mu }^{(k-j+1)}{W}^{\left(j\right)}\right)-\sum _{j=1}^k{U}_{\mu }^{(j-1)}\overline{U^{(k-j+1)}}\right].\hfill \end{array}$$

(16)

Here, the subscript $\mu$ means differentiation with respect to $\mu$: $\frac{d{W}^{\left(j\right)}}{d\mu }=W_{\mu }^{\left(j\right)}.$ For calculating the conjunctions present in Formulas (15) and (16), we use the rationality of the functions, which allow the following calculations:

$$\overline{W^{\left(j\right)}}\left(\mu \right)=W^{\left(j\right)}\left(\overline{\mu }\right)=W^{\left(j\right)}\left(\frac{1}{\mu }\right).$$

Similar calculations can be performed for the complex velocity U. Using sequentially Equations (15) and (16), we obtain

$$W(\mu ,t)=\mu t-{\mu }^3{t}^2-\left(\frac{5}{3}\mu +\frac{1}{3}{\mu }^3-2{\mu }^5\right)t^3-\left(\frac{3}{2}\mu -\frac{31}{6}{\mu }^3-\frac{5}{3}{\mu }^5+5{\mu }^7\right)t^4+\dots$$

$$U(\mu ,t)=-\mu -\mu (1-{\mu }^2)t+(\mu +{\mu }^3-2{\mu }^5)t^2+\left(\frac{8}{3}\mu -5{\mu }^3-\frac{8}{3}{\mu }^5+5{\mu }^7\right)t^3+\dots$$

Several first coefficients of the series are presented in

Table 1. Coefficients of the power series with respect to time for the conformal mapping.

$\mathit{W}(\mathit{\mu },\mathit{t})$	${\mathit{\mu }}^1$	${\mathit{\mu }}^2$	${\mathit{\mu }}^3$	${\mathit{\mu }}^4$	${\mathit{\mu }}^5$	${\mathit{\mu }}^6$	${\mathit{\mu }}^7$	${\mathit{\mu }}^8$	${\mathit{\mu }}^9$	${\mathit{\mu }}^{10}$	${\mathit{\mu }}^{11}$	${\mathit{\mu }}^{12}$	${\mathit{\mu }}^{13}$
$t^0$	0
$t^1$	1
$t^2$	0	0	$-1$
$t^3$	$-\frac{5}{3}$	0	$-\frac{1}{3}$	0	2
$t^4$	$-\frac{3}{2}$	0	$\frac{31}{6}$	0	$\frac{5}{3}$	0	$-5$
$t^5$	$\frac{97}{30}$	0	$\frac{241}{30}$	0	$-\frac{259}{15}$	0	$-7$	0	14
$t^6$	$\frac{181}{18}$	0	$-\frac{153}{10}$	0	$-\frac{1159}{30}$	0	$\frac{593}{10}$	0	28	0	$-42$
$t^7$	$\frac{317}{45}$	0	$-\frac{13,631}{210}$	0	$\frac{20,687}{315}$	0	$\frac{11,449}{630}$	0	$-\frac{9298}{45}$	0	$-110$	0	132

and

Table 2. Coefficients of the power series with respect to time for the complex velocity.

$\mathit{U}(\mathit{\mu },\mathit{t})$	${\mathit{\mu }}^1$	${\mathit{\mu }}^2$	${\mathit{\mu }}^3$	${\mathit{\mu }}^4$	${\mathit{\mu }}^5$	${\mathit{\mu }}^6$	${\mathit{\mu }}^7$	${\mathit{\mu }}^8$	${\mathit{\mu }}^9$	${\mathit{\mu }}^{10}$	${\mathit{\mu }}^{11}$	${\mathit{\mu }}^{12}$	${\mathit{\mu }}^{13}$
$t^0$	$-1$
$t^1$	$-1$	0	1
$t^2$	1	0	1	0	$-2$
$t^3$	$\frac{8}{3}$	0	$-5$	0	$-\frac{8}{3}$	0	5
$t^4$	$-\frac{1}{2}$	0	$-\frac{35}{3}$	0	$\frac{103}{6}$	0	9	0	$-14$
$t^5$	$-\frac{259}{30}$	0	$\frac{47}{5}$	0	$\frac{1471}{30}$	0	$-\frac{887}{15}$	0	$-\frac{98}{3}$	0	42
$t^6$	$-\frac{58}{5}$	0	$\frac{1094}{15}$	0	$-\frac{2192}{45}$	0	$-\frac{9386}{45}$	0	$\frac{9268}{45}$	0	122	0	$-132$

. It is of interest to note that the absolute values of the elements on the main diagonals of the matrices obtained from these tables after elimination of zero columns are the Catalan numbers:

$$1,1,2,5,14,42,132\dots$$

The following series is obtained for the function $Z(\mu ,t)$ defining the conformal mapping:

$$Z(\mu ,t)=\frac{1}{2}ln\frac{1+\mu }{1-\mu }+\mu t-{\mu }^3{t}^2-\left(\frac{5}{3}\mu +\frac{1}{3}{\mu }^3-2{\mu }^5\right)t^3+\dots .$$

(17)

A total of 1100 terms of the series were found. The calculations were first performed in rational numbers; beginning from the 501th term, they were performed in real numbers with the initial length of the mantissa equal to 1200 decimal points. As the computation was continued, the accuracy deteriorated, and the last terms were found with accuracy approximately up to 600 decimal points.

4. Direct Summation of the Power Series with Respect to Time

Let us try to construct the free boundary using the coefficients of series (17) for $Z.$ In the plane $\mu$, the free boundary corresponds to a unit circumference $\mu =exp\left(\mathrm{i}\theta \right),$ where $\theta \in [0,2\pi ).$ Moreover, the points $\mu =\pm 1$ correspond to infinitely distant points $x=\pm \infty$ in the physical plane. Taking into account the symmetry of the flow domain with respect to the real axis, we construct the free boundary in the upper half-plane, i.e., $\theta \in (0,\pi ).$ For this purpose, we calculate the sum of series (17) at a particular point $(\theta ,t).$ Power series allow one to determine the position of the free boundary of the flow domain. Each point on this boundary is a result of summation of the corresponding series; if the points form a smooth line, we can assume that the corresponding series converge. This assumption is confirmed below by studying the radius of convergence of the corresponding series using Jentzsch theorem [16]. If some point falls out from the smooth line, this means that the corresponding series diverges. Figure 4 provides some examples of series divergence at some points of the free boundary. These sets of points were obtained by means of summation of the series with 1000 coefficients. We can assume that the time instant $t=0.2$ is the last one for which the series (17) converges at all points of the free boundary. This fact is evidenced by the smooth curve shown in Figure 4a. It is seen that the smoothness of the free boundary is violated already at $t=0.215$ (Figure 4b), which means that series (17) diverges at these points. At the time $t=0.22$ presented in Figure 4c, the remainders of the smooth free boundary survive in the regions of the maximum and minimum changes in its curvature. Figure 4d illustrates the situation at $t=0.3.$ The majority of the points have already left the domain, and only some of the markers calculated in an immediate vicinity of the jet tip and infinitely distant points still remain at their places.

Let us estimate the radius of convergence of the summed series with the Jentzsch theorem [16]. According to this theorem, the zero values of the partial sums of the power series line up along the boundary of the convergence circle. Figure 5 shows the roots of the equation

$$Z^{\left(0\right)}\left(e^{\mathrm{i}\theta }\right)+Z^{\left(1\right)}\left(e^{\mathrm{i}\theta }\right)t+Z^{\left(2\right)}\left(e^{\mathrm{i}\theta }\right)t^2+\dots +Z^{\left(100\right)}\left(e^{\mathrm{i}\theta }\right)t^{100}=0$$

(18)

in the complex time plane for several values of $\theta$. It is seen that all roots of Equation (18) are accurately arranged along the circumferences, which is a clear presentation of convergence circles. The maximum radius of convergence is reached at the jet tip at $\theta =\pi /2$, while the minimum radius is reached at $\theta =\pi /4$. We can see that the minimum radius is just over $0.2$. This means that for $t\le 0.2$ the points $(\theta ,t)$ for all $\theta \in (0,\pi )$ are inside the circle of convergence.

5. Pade Approximants

One of the most famous methods of numerical analytical continuation of the series outward of the convergence circle is the method based on using the Pade approximants. For an arbitrary function $f\left(t\right)$, the Pade approximant definition can be written as follows:

$$f\left(t\right)={\Sigma }_{n=0}^{\infty }{a}_n{t}^n=\frac{P_L\left(t\right)}{Q_M\left(t\right)}+O\left(t^{L+M+1}\right).$$

Here, $P_L\left(t\right),Q_M\left(t\right)$ — polynomials with powers L and M. The numbers L and M may be arbitrary natural numbers, but the case considered most often is $L=M.$

The efficiency of using the Pade approximation for summation of power series and the high quality of thus-obtained solutions have been confirmed by practice many times. For example, the paper [17] describes a comparison of numerical and semi-analytical methods for solving the problem of formulation of cumulative jets.

Let us apply the Pade summation for constructing the free boundary. Let us take 256 points $\mu =e^{\mathrm{i}j\pi /128},$$(j=\overline{0, 256})$ uniformly distributed over the circumference $\left|\mu \right|=1.$ A power series (12) is formed for each point. After that, we find the approximant [548/548] for each series and calculate its value for a certain value of $t.$ Figure 6 shows one half of the flow domain bounded by the axis of symmetry $y=0$ from below and by the free boundary from above. The free boundary position is presented for three time instants. Curve 1 shows the profile obtained for $t=0.2.$ Let us recall that this was the last time instant at which direct summation of the series at all points of the free boundary was possible. By using the Pade approximants, we were able to move significantly further in the study and to sum up series (12) up to $t=1.2.$ Curves 2 and 3 in Figure 6 show the free boundary profiles obtained for $t=0.6$ and $t=1.2$, respectively.

Clearly, the unsteady solution under consideration has a steady-state limit as $t\to \infty$ corresponding to the frontal collision of two jets. Such a steady-state solution has been known for a long time (see, e.g., [18]). In view of the initial velocity and the thickness of the colliding jets, the steady-state solution corresponding to our unsteady problem has the form

$$U=-\mu ,Z=\frac{1}{2}\left[ln\left(\frac{1+\mu }{1-\mu }\right)+\mathrm{i}ln\left(\frac{1-\mathrm{i}\mu }{1+\mathrm{i}\mu }\right)\right].$$

Figure 7 shows the flow domains for the steady-state solution (the boundary is shown by the dotted curve) and for the problem under consideration at $t=1.2$ (the boundary is shown by the solid curve).

6. Singularities Outside the Fluid

In plane problems of hydrodynamics, the functions characterizing the flow can have singular points. From the mathematical viewpoint, singular points are points, where the analytical character of the complex velocity function is violated. If such a point is found on the free boundary, this boundary is usually no longer smooth. From the physical viewpoint, this means that the flow cannot be further described within the framework of mechanics of continuous media. The formation of singularities on the free boundary in an ideal fluid and their physical interpretation were discussed, e.g., in the paper [19]. Singular points can be located inside the fluid region. It is possible to assign some physical meaning to such singularities: they can be vortices, sources, or dipoles. The function can be also holomorphic in the flow region, and its singular points can be located outside this region. Such virtual singularities have no physical meaning, but their significance is fairly large. The importance of studying the type, number, and location of singularities in the nonphysical region was first noted by Tanveer [20]. Such investigations are useful because the singular point approaching the free surface induces significant deformation of the latter. When the singular point reaches the free surface, solution breakdown occurs.

A certain analog of cumulative jets is also formed in our problem, which testifies to the existence of virtual singularities. Let us find them in the plane of the auxiliary variable $\mu .$ It should be noted that the first term of the series for conformal mapping

$$Z^{\left(0\right)}\left(\mu \right)=\frac{1}{2}ln\left(\frac{1+\mu }{1-\mu }\right)$$

yields two singular points $\mu =\pm 1$ corresponding to infinitely distant points in the physical plane. Let us find singularities for the auxiliary function $W(\mu ,t):$

$$Z(\mu ,t)=Z^{\left(0\right)}\left(\mu \right)+W(\mu ,t).$$

For this purpose, we need a series in powers of $\mu :$

$$W(\mu ,t)=B^{\left(1\right)}\left(t\right)\mu +B^{\left(2\right)}\left(t\right){\mu }^2+B^{\left(3\right)}\left(t\right){\mu }^3+\dots .$$

(19)

This series can be obtained by changing the order of summation in series (12). A specific features of a series (12) is the fact that all its terms contain only odd powers of $\mu$; therefore, we have $B^{\left(2k\right)}\left(t\right)\equiv 0.$ The even coefficients of series (19) are obtained in the form of power series in $t:$

$$\begin{array}{cc}\hfill B^{\left(1\right)}\left(t\right)& =t-\frac{5}{3}{t}^3-\frac{3}{2}{t}^4+\frac{97}{30}{t}^5+\dots ,\hfill \\ \hfill B^{\left(3\right)}\left(t\right)& =-t^2-\frac{1}{3}{t}^3+\frac{31}{6}{t}^4+\frac{241}{30}{t}^5+\dots ,\hfill \\ \hfill B^{\left(5\right)}\left(t\right)& =2t^3+\frac{5}{3}{t}^4-\frac{259}{15}{t}^5-\frac{1159}{30}{t}^6+\dots .\hfill \end{array}$$

(20)

Now, we have to perform the Pade summation of series (20). Using 1100 terms of series (12), we find the [450/450] Pade approximants for each $B^{\left(j\right)}\left(t\right),$$(j=\overline{1, 200}).$ Thus, we obtain a series for the function $W(\mu ,t)$ in powers of $\mu$. For some particular cases, it was proved that, if a function has poles, then the poles of its $[L/L]$ approximant with $L\to \infty$ tend to the poles of the function. If the function has more complicated singularities, e.g., branching points, it was noted that the poles and zeroes of the approximants become arranged along certain lines, each pointing to the position of the singular point of the function. Figure 8 shows the zeroes and poles of the $[95/95]$ Pade approximant of series (19) at several time instants. It should be noted that, at each time instant, they are arranged along two rays located at the imaginary axis in the plane $\mu .$ At the beginning of each ray, there is a singular point marked in Figure 8. With time, these singular points tend to the points $\mu =\pm \mathrm{i}$, which means that they approach the free boundary. Let us recall that the free surface in the plane $\mu$ corresponds to a circumference of unit radius with the center at the origin of the coordinate system.

To ensure better reliability of results, an independent checkup of the characteristics of the singular points of the function defined by series (19) is performed. According to the commonly known statement, if there exists a limit

$$\underset{n\to \infty }{lim}\frac{B^{\left(n\right)}\left(t\right)}{B^{(n+1)}\left(t\right)}=\delta \left(t\right),$$

(21)

when $\mu =\delta \left(t\right)$ is the location of the singular point for the function presented by series (19). In our case, limit (21) does not exist because series (19) contains only odd functions of $\mu$.

In the plane ${\mu }^2$, however, the situation becomes different: only one singularity out of two survives. Figure 9 illustrates the existence of the limit of the sequence $\{{\tilde{B}}^{\left(n\right)}/{\tilde{B}}^{(n+1)}\}$, where ${\tilde{B}}^{\left(n\right)}=B^{(2n-1)}$. As n increases, the curves corresponding to the ratio $\{{\tilde{B}}^{\left(n\right)}/{\tilde{B}}^{(n+1)}\}$ move closer and closer to each other and practically merge at $n=16$ and $n=32.$ The curve obtained in the limit characterizes the evolution of the singular point positions with time. It follows from Figure 9 that the singularity tends to the point ${\mu }^2=-1$ with increasing t. Thus, we obtain one more confirmation that the function of complex mapping in the plane $\mu$ has two singular points, which move along the imaginary axis and approach the free boundary with time, forming cumulative jets on the free boundary, which are observed in the physical plane. Moreover, Figure 9 also shows that singular points do not reach the free surface within a finite time.

Let us study the type of these singularities with the Domb–Sykes test [21]. It can be formulated as follows. Let a certain analytical function have a power-law singularity with the power index $\gamma$ at the point $z_0$:

$$f\left(z\right)=\sum _{n=0}^{\infty }{a}_n{z}^n={(z-z_0)}^{\gamma }.$$

(22)

Then, expanding the right-hand side of equality (22) in powers of z, it can be demonstrated that

$$\frac{a_n}{a_{n-1}}=\frac{1}{z_0}\left(1-\frac{1+\gamma }{n}\right).$$

(23)

This means that the ratio $a_n/a_{n-1}$ is a linear function of $\frac{1}{n}.$ If $a_n/a_{n-1}$ is a real quantity; then, using it on one axis and $1/n$ on the other axis, we obtain a family of points lying on one straight line. Denoting $y=a_n/a_{n-1}$, $x=1/n$, we rewrite Equation (23) in the form

$$y=kx+\frac{1}{z_0},k=-\frac{1+\gamma }{z_0}.$$

Therefore, we have $\gamma =-1-k{z}_0.$ The angular coefficient k and $z_0$ can be determined by solving the system

$$\left\{\begin{array}{c}{y}_1=k{x}_1+\frac{1}{z_0},\hfill \\ y_2=k{x}_2+\frac{1}{z_0},\hfill \end{array}\right.$$

(24)

where $(x_1,y_1)$ and $(x_2,y_2)$ are two specified points of the straight line. Thus, based on the coordinates of two arbitrary points of the straight line, we can determine the index $\gamma$ and, hence, the type of the singular point $z_0.$

In our case, the function $W(\mu ,t)$ is presented as a series

$$W(\mu ,t)=\mu \left({\tilde{B}}^{\left(1\right)}+{\tilde{B}}^{\left(2\right)}{\mu }^2+{\tilde{B}}^{\left(3\right)}{\mu }^4+\dots \right).$$

Therefore, it can be assumed that the function $W(\mu ,t)$ in the vicinity of the singular point is expanded into series (22), where ${\mu }^2$ plays the role of z and $a_n={\tilde{B}}^{\left(n\right)}.$ Therefore, Equation (23) should be satisfied asymptotically, and the points $\left(1/n,{\tilde{B}}^{(n+1)}/{\tilde{B}}^{\left(n\right)}\right)$ should line up along the straight line as $n\to \infty .$ Figure 10 shows the line consisting of the points $\left(1/n,{\tilde{B}}^{(n+1)}/{\tilde{B}}^{\left(n\right)}\right)$ at $t=0.6.$ Here, $n=\overline{1,97}.$ Let us draw a straight line through each pair of the neighboring points $\left(1/n,{\tilde{B}}^{\left(n\right)}/{\tilde{B}}^{(n-1)}\right)$ and $\left(1/(n+1),{\tilde{B}}^{(n+1)}/{\tilde{B}}^{\left(n\right)}\right)$; solving system (24), we calculate ${\gamma }_n$ and obtain the sequence $\left\{{\gamma }_n\right\},$ which is expected to tend to the sought index $\gamma$ in the limit as $n\to \infty$. This sequence $\left\{{\gamma }_n\right\}$ constructed for $t=0.3$ has 96 terms. The last five terms of this sequence are listed in the second column of

Table 3. Last five columns of the Neville table (even columns) illustrating the convergence of the sequence $\left\{{\gamma }_n\right\}$ to $\gamma =1/2$.

n	${\mathit{\gamma }}_{\mathit{n}}={\mathit{e}}_{\mathit{n}}^0$	${\mathit{e}}_{\mathit{n}}^2$	${\mathit{e}}_{\mathit{n}}^4$	${\mathit{e}}_{\mathit{n}}^6$
92	0.4598102	0.4993467	0.4999593	0.4999962
93	0.4602044	0.4993630	0.4999609	0.4999964
94	0.4605909	0.4993787	0.4999624	0.4999966
95	0.4609699	0.4993940	0.4999638	0.4999967
96	0.4613415	0.4994087	0.4999652	0.4999969

; considering this sequence, one does not see that the limit of $\left\{{\gamma }_n\right\}$ is $1/2.$

There are various algorithms aimed at accelerating the convergence of sequences. Let us take the algorithm called the Neville table [22]. The original sequence whose limit has to be found is placed to the zeroth column ${\gamma }_n=e_n^0.$ The elements of the r-th column are generated from those of the $(r-1)$-th column by the formula

$$e_n^r=\left(n{e}_n^{r-1}-(n-r)e_{n-1}^{r-1}\right)/r.$$

If $e_n^0$ is considered as a function of $1/n,$ then $e_n^r$ simply means crossing of the axis $1/n=0$ and the curve of the $(r+1)$-th order sequentially passing through the points $e_n^0,e_{n-1}^0,\dots ,e_{n-r}^0.$ Each next column yields a sequence converging to the same limit, but faster than that in the previous column. The Neville tables represents an algorithm of convergence acceleration based on increasing the order of the approximating curve. At the first step, the points are connected by segments of a straight line; at the second step, they are approximated by a second-order curve (parabola); then, they are approximated by a cubic parabola, etc.

Table 3 shows the lower part of the Neville table (even columns). The first column contains the ordinal numbers of the sequence elements, and the second column contains the original sequence $\left\{{\gamma }_n\right\}.$ To see the series convergence, it is necessary to compare the table column with the previous one. Comparing the second column, which shows the initial sequence, with the fifth column obtained by the algorithm of convergence acceleration, we can argue that the convergence is significantly improved, and the sequence limit with the accuracy of ${10}^{-5}$ is equal to $1/2$. Therefore, we have $\gamma =1/2$ in Equation (22), which means that the singularity found is a singular point of the square root type. Thus, it is seen that the series for the function $W(\mu ,t)$ has a singular point ${\mu }^2=-1$ of the square root type in the plane ${\mu }^2$. Therefore, the series for the function defining conformal mapping has two singularities of the square root type, which approach the free boundary with time and induce significant deformation of the latter.

7. Stretching of an Infinite Strip

Figure 11 shows the zeroes of the denominator of the $[300/300]$ Pade approximant for series (12) calculated for $\mu =\pm \mathrm{i}$ (point of intersection of the free boundary with the axis of symmetry $x=0$) in the time plane. It is in the vicinity of this point that the best convergence of the series of conformal mapping is observed (see Figure 5). It is seen that all singularities are grouped in the right half-plane. This fact allows us to hope that the convergence of the series at $t<0$ will be appreciably better. The negative time here means that the velocity vector changes its direction; hence, instead of the collision of two jets, we consider the case of stretching.

Indeed, the convergence of the series at $t<0$ is much better, though here again we use the Pade approximants for constructing the free boundary. Let the fluid at the initial time occupy a strip bounded by two free boundaries $x=\pm \pi /4.$ The initial complex velocity is defined by the function $U(z,0)=tanhz$. This choice of initial conditions models fluid strip stretching. Figure 12 illustrates the evolution of the domain occupied by the fluid for the case considered here. As it could be expected, the strip transforms to a thin layer with the smallest width at the axis of symmetry. The thickness of this layer tends to zero as $t\to -\infty ,$ but no breakdown occurs within a finite time.

8. Conclusions

A problem of unsteady deformation of a strip of an ideal incompressible fluid was solved by semi-analytical methods with the use of power series. Recurrent formulas for determining the coefficients of the power series were derived. It turned out that these coefficients are rational polynomials of the complex variable, which simplified the process of their calculation. As expected, the radius of convergence of the constructed series was sufficiently small. The Pade approximants were used to calculate the sought functions at points located outside the convergence circumference. As a result, the domain of convergence of the series under consideration was significantly extended.

Singular points in the plane of the complex variable located outside the fluid were considered in [23,24]. The basic case under consideration was a situation where two singularities of the square root type moved along a motionless straight line with time. To avoid the two-sheeted case, the singularities were connected by a cut lying on this straight line. It was assumed that there may be several cuts of this kind in the general case.

When we first encountered such a formulation of the problem, we thought it was a rare and exotic case. Now, we have changed our opinion. It seems that a situation with two singularities and a moving cut which changes its length but is always located on a motionless straight line is a widespread phenomenon. In our earlier studies [10,25], we had already found two flows where this situation does occur. Now, we have found one more new flow where we again obtained a straight line on which two singular points of the square root type connected by a straight cut move.