On the Necessary Conditions for Non-Equivalent Solutions of the Rotlet-Induced Stokes Flow in a Sphere: Towards a Minimal Model for Fluid Flow in the Kupffer’s Vesicle

Abstract

The emergence of left–right (LR) asymmetry in vertebrates is a prime example of a highly conserved fundamental process in developmental biology. Details of how symmetry breaking is established in different organisms are, however, still not fully understood. In the zebrafish (Danio rerio), it is known that a cilia-mediated vortical flow exists within its LR organizer, the so-called Kupffer’s vesicle (KV), and that it is directly involved in early LR determination. However, the flow exhibits spatio-temporal complexity; moreover, its conversion to asymmetric development has proved difficult to resolve despite a number of recent experimental advances and numerical efforts. In this paper, we provide further theoretical insight into the essence of flow generation by putting together a minimal biophysical model which reduces to a set of singular solutions satisfying the imposed boundary conditions; one that is informed by our current understanding of the fluid flow in the KV, that satisfies the requirements for left–right symmetry breaking, but which is also amenable to extensive parametric analysis. Our work is a step forward in this direction. By finding the general conditions for the solution to the fluid mechanics of a singular rotlet within a rigid sphere, we have enlarged the set of available solutions in a way that can be easily extended to more complex configurations. These general conditions define a suitable set for which to apply the superposition principle to the linear Stokes problem and, hence, by which to construct a continuous set of solutions that correspond to spherically constrained vortical flows generated by arbitrarily displaced infinitesimal rotations around any three-dimensional axis.

1. Introduction

In the early stages of development of the corporal patterns in multicellular organisms, cells become spatially arranged following a concrete body plan [1]. This process involves the laying down of the three decisive body axes: the anterior–posterior, dorsal–ventral and left–right axes, as seen in Figure 1A [2,3]. Although in most vertebrates, including humans, the external appearance is nearly left–right symmetric, our internal body arrangement is coordinated in a highly asymmetric fashion. In the most common disposition, known as situs solitus, the heart is on the left and the liver on the right side of the body. This orderly development and, most importantly, the many causes of its failure, have been the subject of extensive research with common fatal and benign pathologies [4,5] being associated with the abnormal (i.e., heterotaxy), random (situs ambiguus) or mirror-reversed (situs inversus) arrangement of internal organs.

In many organisms, left–right symmetry breaking occurs within a specialized group of cells forming a transient organ generically referred to as the “organizer” [2,3]. Elegant experimental and theoretical work has established the role of physics and, in particular, of fluid dynamics in this process. By using mice as a model system, this was first demonstrated by Nonaka et al. in 2002 in what has become a seminal work in the field [6]: In the node, tens of incessantly circling monocilia, tilted toward the posterior axis [7], collectively generate a leftward flow that effectively breaks the left–right symmetric development. The flow is understood to either serve as the carrier of a chemical morphogenic signal or to act as an intrinsic mechanical signal for the subsequent cascade of genetic and developmental events [3,8]. The nodal flow hypothesis is still to date one of the most exquisite examples of the constructive role of fluid mechanics in establishing fundamental patterns in early vertebrate development [6,9,10,11,12].

In the zebrafish Danio rerio, one of the best studied model organisms for vertebrate development, the corresponding left–right organizer (i.e., the Kupffer’s vesicle (KV)) has, similarly to mice, one layer of monociliated epithelial cells that line a lumen cavity filled with extracellular fluid [13] (see Figure 1B). Once again, the minute forces generated by tens of motile monocilia are enough to set in motion a KV flow [14]. However, accurately pinpointing the left–right asymmetric features of the flow has proved more difficult. Firstly, this is because the three-dimensional geometry of the KV cavity, compared to the effectively two-dimensional nodal case, leads to more complex three-dimensional fluid flow patterns [15]. Secondly, it is also because the flow exhibits spatiotemporal complexity [16]. The experimental observation of the existence of a relatively simple counter-clockwise circulation (when seen from the dorsal side) in the coronal mid-plane of the KV [17], as depicted in Figure 1B (right), is not enough to clarify the development of laterality. Such flow would remain left–right symmetric unless cilia orientation, spatial distribution or beating characteristics dictate otherwise. Despite recent progress in the modelling and experimental description of cilia beating and fluid flow within the KV [18,19], these details are not yet completely resolved.

From the modelling perspective, progress in the field has been facilitated by the use of exact and approximate analytical and numerical methods with which to tackle Stokes equations for the fluid mechanics of incompressible viscous flows. The Stokes regime is a simplification of the Navier–Stokes equations for when the inertial effects are negligible compared to viscous forces; i.e., it is a valid approximation for the very low-Reynolds-number regime typically encountered at the micro scale [20]. Research in polymer science and suspension rheology, micro-organism locomotion and other biophysical flows or large scale, but slow, geophysical flows, have all been based in one way or another in the Stokes equation. At the same time, these distinct scientific disciplines have all largely contributed to the development of appropriate mathematical methods with which to tackle its analysis. Nevertheless, finding exact solutions for arbitrary boundary conditions are still recognized as a difficult task with only a limited number of closed-form solutions known to date and only for the simplest highly symmetric geometric configurations [21].

A powerful analytical method available for finding approximate solutions to Stokes flows is the so called ‘singularity method’ (SM). Pioneered by Lorentz, Oseen and Burgers [22,23,24], it is based on a multipole expansion of the exact solution for a singular (i.e., delta-like in space) forcing term in the equation. By choosing the appropriate spatial distribution of individual terms from the expansion (i.e., the basic singularities known as Stokeslet, Rotlet (also called a Couplet by Batchelor [25]), Stresslet and others), it is possible to address complex steady and time-dependent problems. Examples abound and they include the use of SM for studies of slender-body, boundary element or mesh free methods in disparate branches of fluid mechanics [25,26,27,28,29,30,31,32,33,34,35,36].

SM has been already successfully employed to construct physical descriptions of the fluid mechanics of the fluid flow within the left–right (LR) organizers (both in the case of the mouse Hensen’s node and in the zebrafish KV) and to express, analyse and predict through mathematical and computational models the characteristics of these fluid flows [7,14,15,18]. Following a similar modelling spirit, we set here to seek for a closed-form solution to the KV flow, aiming to provide theoretical insight into the essence of flow generation by the use of conceptually simplified physics models. Specifically, we reduce the flow to a sum of rotational solutions but, instead of modelling the flow induced by each individual circling cilia as previously done in [7] for the mouse node and in [19] for the KV, we consider a single rotational singularity to account for the overall vortical structure of the KV flow. Moreover, we elucidate the required geometric constraints necessary to obtain a set of non equivalent solutions for the rotlet-induced internal Stokes flow within a simplified rigid spherical geometry. While the literature already provides one such an exact solution to the same problem [36], this is restricted to a very specific geometric configuration of the three-dimensional rotlet: Radially displaced along one of the axis of the sphere (e.g., the z-axis) while pointing orthogonally to it (e.g., parallel to the y-axis). Here, instead, we address the general conditions for existence of other independent solutions compatible with the symmetry and boundary conditions of the problem. By doing so, we will be able to address possible LR asymmetries in the established fluid flows that arise from anterior–posterior and dorsal–ventral asymmetries in the force field.

2. Results

2.1. Definition of the Problem and Preliminaries

The zebrafish KV is a roughly spheroidal structure approximately $70\times 60\times 30\mathsf{\mu }{\mathrm{m}}^3$ in size, transiently located at the tip of the developing embryo during early somitogenesis, Figure 1B. It is filled with fluid set in motion by tens of 2–4 $\mathsf{\mu }\mathrm{m}$ long cilia lining its inner cell layer [15]. These beating cilia generate a predominantly vortical anticlockwise fluid flow when viewed from the dorsal side, with a centre shifted towards the anterior end [17]. The small scale of the KV together with the slow speed of the KV flow imply that, from a fluid mechanics perspective, we are well within the realm of low Reynolds number flows, with $Re\lesssim {10}^{-3}$ [15]. In the context just described, the correct fluid mechanics within the KV is simply modelled by the Stokes equations for an incompressible fluid together with conditions for the boundary problem, namely:

$$\begin{array}{c}\hfill \nabla p-\mu {\nabla }^{2}u=f\\ \hfill \nabla ·u=0\\ \hfill {u|}_{\partial \Omega }=0\\ \hfill \Omega :sphere.\end{array}$$

(1)

Here $p=p\left(x\right)$ and $u=u\left(x\right)$ are the pressure and fluid velocity field at position x respectively, $\mu$ is the fluid viscosity, f the resultant of the body forces responsible for the fluid motion and $x\in {\Re }^3$. Moreover, the fluid is subjected to the usual no-slip (Dirichlet) boundary conditions with the flow velocity approaching zero close to the surface cell layer, an expected and experimentally confirmed fact in the KV [14,15,16,19]. A convenient representation in spherical coordinates of the general solution to the system of Equation (1) was given by Ranger [37] in the form:

$$\begin{array}{ccc}\hfill u(r,\theta ,\varphi )& =& cur{l}^2\left[\frac{\overrightarrow{r}\psi (r,\theta )}{rsin\theta }cos\varphi \right]+curl\left[\frac{\overrightarrow{r}\chi (r,\theta )}{rsin\theta }sin\varphi \right]\hfill \\ \hfill p(r,\theta ,\varphi )& =& \frac{\mu }{rsin\theta }\frac{\partial }{\partial r}\left[\frac{{\partial }^2\psi (r,\theta )}{\partial r^2}+\frac{sin\theta }{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{sin\theta }\frac{\partial \psi (r,\theta )}{\partial \theta }\right)\right]cos\varphi \hfill \end{array}$$

(2)

where the spherical coordinates $(r,\theta ,\varphi )$ are related to the Cartesian position vector x in the usual way, $r\in \left[0,1\right]$, $\theta \in \left[0,\pi \right]$ and $\varphi \in \left[0,2\pi \right)$, and the scalar functions $\psi$ and $\chi$ satisfy:

$$\begin{array}{c}\hfill L_{-1}^2\psi =L_{-1}\chi =0\end{array}$$

(3)

with the Stokes operator $L_{-1}$ defined by $L_{-1}\equiv \frac{{\partial }^2}{\partial r^2}+\frac{sin\theta }{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{sin\theta }\frac{\partial }{\partial \theta }\right)$ or, expressing the velocity field in its spherical components,

$$\begin{array}{ccc}\hfill u_r(r,\theta ,\varphi )& =& -\frac{1}{r^2}\frac{\partial }{\partial \theta }\left[\frac{1}{sin\theta }\frac{\partial \psi }{\partial \theta }\right]cos\varphi \hfill \\ \hfill u_{\theta }(r,\theta ,\varphi )& =& \left[\frac{1}{r}\frac{\partial }{\partial \theta }\left(\frac{1}{sin\theta }\frac{\partial \psi }{\partial r}\right)+\frac{\chi }{r{sin}^2\theta }\right]cos\varphi \hfill \\ \hfill u_{\varphi }(r,\theta ,\varphi )& =& -\left[\frac{1}{r{sin}^2\theta }\frac{\partial \psi }{\partial r}+\frac{1}{r}\frac{\partial }{\partial \theta }\left(\frac{\chi }{sin\theta }\right)\right]sin\varphi .\hfill \end{array}$$

(4)

As described above, the KV flow is known to be predominantly vortical, viscous, incompressible, approximately Newtonian and non-axisymmetric [14,15,18]. In order to construct a minimal model for the KV fluid flow capturing all these experimentally observed features, we decided to focus on the simplest singular solution capable of providing vortical fluid motion with the correct approximate geometry: a single rotlet embedded within a rigid spherical shell filled with a viscous and incompressible fluid (Figure 2A). This is, in principle, also a solution that is amenable to extensive parametric analysis and that can be easily extended to more complex configurations. It is important to notice, however, that, in contrast to previous work [7,15,18,19], here we are not using a rotlet (or any other singular solutions) to model the flow around individual beating cilia but rather to describe the overall vortical structure that emerges in the KV from tens of cilia simultaneously pumping the fluid.

More specifically, we consider a rotlet of torque $\mathbf{L}=(l_1,l_2,l_3)$ (with $l_1,l_2,l_3\in \Re$), centred at position ${\mathbf{x}}_{\mathbf{0}}=(x_0,y_0,z_0)$ (or $(c,{\theta }_0,{\varphi }_0)$ in spherical coordinates) within the unit sphere, with the origin of our coordinate system, O, fixed at its centre as depicted in Figure 2A. We first recall that in an unbounded three dimensional space, the solution $u_0$ to the Stokes problem defined above with the flow decaying to zero at infinity, takes the simplified form:

$$u_0=\frac{\mathbf{L}\times \mathbf{R}}{8\pi {∥R∥}^3}$$

(5)

which can be interpreted as the flow induced by the rotation of an infinitesimally small sphere in a viscous fluid [38,39], and it corresponds to the first axisymmetric term in the multi-pole expansion solution to Equation (1). Here $\mathbf{R}$ is the relative position vector $\mathbf{R}\equiv \mathbf{x}-{\mathbf{x}}_{\mathbf{0}}$, and $∥R∥$ its $l^2$-norm.

For an unbounded flow there is always a simple geometric transformation, consisting in a three-dimensional rotation plus translation, that brings ${\mathbf{x}}_{\mathbf{0}}$ to O, and $\mathbf{L}$ to $\widehat{k}$. However, with the boundary conditions on the sphere defined in Equation (1), the solution to the boundary problem becomes uniquely dependent on the position and orientation of the torque vector $\mathbf{L}$. Hackborn et al. [36] obtained a particular solution to this problem restricted to a very specific geometric configuration of the three-dimensional rotlet: radially displaced along one the z-axis, i.e., ${\mathbf{x}}_{\mathbf{0}}=(0,0,c)$, while pointing parallel to the y-axis, i.e., $\mathbf{L}=(0,1,0)$. Moreover, they provided a convenient closed-form for the scalar functions $\psi$ and $\chi$. Figure 2C,D show a few streamlines computed from this solution for $c=0.25$ and a set of random initial conditions on the $z=0$ plane. Figure 2B further compares the flow within a rigid sphere to the unbounded. As can be seen there, the major difference arises close to the $r=1$ boundary where the unbounded rotlet does not satisfy the required no-slip condition. In the next sections, we expand Hackborn’s analysis to provide necessary geometric conditions for the existence of other independent solutions compatible with the symmetry and boundary conditions of the problem.

2.2. Necessary Conditions of a Three-Dimensional Rotlet with Spherically-Symmetric Boundary Conditions

We can start by focusing on the numerator of Equation (5) for the unbounded solution. The cross product defines the three Cartesian components of this reduced velocity field, $u_0^{\ast }$, as

$$\begin{array}{cc}\hfill u_0^{\ast }\equiv \mathbf{L}\times \mathbf{R}=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ l_1& l_2& l_3\\ x-x_0& y-y_0& z-z_0\end{array}\right|& =\left(l_2\left(z-z_0\right)-l_3\left(y-y_0\right)\right)\overrightarrow{i}\\ & \hfill +\left(l_3\left(x-x_0\right)-l_1\left(z-z_0\right)\right)\overrightarrow{j}\\ & \hfill +\left(l_1\left(y-y_0\right)-l_2\left(x-x_0\right)\right)\overrightarrow{k}\end{array}$$

(6)

which, after making the transformation to spherical coordinates, read:

$$\begin{array}{ccc}\hfill \left(\begin{array}{c}{u}_{0r}^{\ast }\\ u_{0\theta }^{\ast }\\ u_{0\varphi }^{\ast }\end{array}\right)& =& \left(\begin{array}{ccc}sin\theta cos\varphi & sin\theta sin\varphi & cos\theta \\ cos\theta cos\varphi & cos\theta sin\varphi & -sin\theta \\ -sin\varphi & cos\varphi & 0\end{array}\right)\left(\begin{array}{c}{u}_{0x}^{\ast }\\ u_{0y}^{\ast }\\ u_{0z}^{\ast }\end{array}\right)\hfill \\ & =& \left(\begin{array}{ccc}sin\theta cos\varphi & sin\theta sin\varphi & cos\theta \\ cos\theta cos\varphi & cos\theta sin\varphi & -sin\theta \\ -sin\varphi & cos\varphi & 0\end{array}\right)\left(\begin{array}{c}\hfill l_2\left(z-z_0\right)-l_3\left(y-y_0\right)\\ \hfill l_3\left(x-x_0\right)-l_1\left(z-z_0\right)\\ \hfill l_1\left(y-y_0\right)-l_2\left(x-x_0\right)\end{array}\right)\hfill \end{array}$$

(7)

or, recalling that $x_0=csin{\theta }_{0}cos{\varphi }_0$, $y_0=csin{\theta }_{0}sin{\varphi }_0$, $z_0=ccos{\theta }_0$ and rearranging by spherical components:

$$\begin{array}{cc}\hfill u_{0r}^{\ast }=& \left[l_2\left(rcos\theta -z_0\right)-l_3\left(rsin\theta sin\varphi -y_0\right)\right]sin\theta cos\varphi \hfill \\ & +\left[l_3\left(rsin\theta cos\varphi -x_0\right)-l_1\left(rcos\theta -z_0\right)\right]sin\theta sin\varphi \hfill \\ & +\left[l_1\left(rsin\theta sin\varphi -y_0\right)-l_2\left(rsin\theta cos\varphi -x_0\right)\right]cos\theta \hfill \\ \hfill =& \left[l_3{y}_0-l_2{z}_0\right]sin\theta cos\varphi +\left[l_1{z}_0-l_3{x}_0\right]sin\theta sin\varphi +\left[l_2{x}_0-l_1{y}_0\right]cos\theta \hfill \\ \hfill u_{0\theta }^{\ast }=& \left[l_2\left(rcos\theta -z_0\right)-l_3\left(rsin\theta sin\varphi -y_0\right)\right]cos\theta cos\varphi \hfill \\ \hfill & +\left[l_3\left(rsin\theta cos\varphi -x_0\right)-l_1\left(rcos\theta -z_0\right)\right]cos\theta sin\varphi \hfill \\ \hfill & -\left[l_1\left(rsin\theta sin\varphi -y_0\right)-l_2\left(rsin\theta cos\varphi -x_0\right)\right]sin\theta \hfill \\ \hfill =& \left[l_{2}r-l_2{z}_{0}cos\theta +l_3{y}_{0}cos\theta \right]cos\varphi +\left[l_1{z}_{0}cos\theta -l_3{x}_{0}cos\theta -l_{1}r\right]sin\varphi \hfill \\ \hfill & +\left[l_1{y}_0-l_2{x}_0\right]sin\theta \hfill \\ \hfill u_{0\varphi }^{\ast }=& -\left[l_2\left(rcos\theta -z_0\right)-l_3\left(rsin\theta sin\varphi -y_0\right)\right]sin\varphi \hfill \\ \hfill & +\left[l_3\left(rsin\theta cos\varphi -x_0\right)-l_1\left(rcos\theta -z_0\right)\right]cos\varphi \hfill \\ \hfill =& \left[l_2{z}_0-l_{2}rcos\theta -l_3{y}_0\right]sin\varphi +l_{3}rsin\theta +\left[l_1{z}_0-l_{1}rcos\theta -l_3{x}_0\right]cos\varphi .\hfill \end{array}$$

(8)

For the spherically-symmetric boundary conditions to be satisfied, the convenient representation in spherical coordinates given in Equation (4) further constraints the reduced velocity to the form $u_0^{\ast }=\left(u_{0r}^{{}^{\prime }}cos\varphi ,u_{0\theta }^{{}^{\prime }}cos\varphi ,u_{0\varphi }^{{}^{\prime }}sin\varphi \right)$ [37]. Here $u_0^{{}^{\prime }}=(u_{0r}^{{}^{\prime }},u_{0\theta }^{{}^{\prime }},u_{0\varphi }^{{}^{\prime }})$ should in general be considered a function of the three spherical coordinates $u_0^{{}^{\prime }}(r,\theta ,\varphi )$. However, it is sometimes convenient to reduce the problem even further by only looking for solutions for which $u_0^{{}^{\prime }}(r,\theta )$ is independent of $\varphi$, as it is the case in the particular solution obtained in [36]. These two complementary approaches will be discussed in the next two sections.

2.3. On Some General Conditions

It immediately follows from Equation (8) that $u_0^{{}^{\prime }}$ must satisfy:

$$\begin{array}{cc}\hfill u_{0r}^{{}^{\prime }}=& \left(l_3{y}_0-l_2{z}_0\right)sin\theta +\left(l_1{z}_0-l_3{x}_0\right)sin\theta tan\varphi +\frac{\left(l_2{x}_0-l_1{y}_0\right)cos\theta }{cos\varphi }\hfill \\ \hfill u_{0\theta }^{{}^{\prime }}=& l_{2}r+(l_3{y}_0-l_2{z}_0)cos\theta +\left(l_1{z}_{0}cos\theta -l_3{x}_{0}cos\theta -l_{1}r\right)tan\varphi +\frac{\left(l_1{y}_0-l_2{x}_0\right)sin\theta }{cos\varphi }\hfill \\ \hfill u_{0\varphi }^{{}^{\prime }}=& (l_2{z}_0-l_3{y}_0)-l_{2}rcos\theta +\left(l_1{z}_0-l_{1}rcos\theta -l_3{x}_0\right)cot\varphi +\frac{l_{3}rsin\theta }{sin\varphi }\hfill \end{array}$$

(9)

where, for the above to be valid, $sin\varphi \ne 0\wedge cos\varphi \ne 0$. By using the representation of u given by Equation (4), we arrive at the following general necessary conditions for the corresponding potential functions ${\psi }_0$ and ${\chi }_0$:

$$\begin{array}{cc}\hfill -\frac{1}{r^2}\frac{\partial }{\partial \theta }\left[\frac{1}{sin\theta }\frac{\partial {\psi }_0}{\partial \theta }\right]& =\frac{1}{R^3}(\left(l_3{y}_0-l_2{z}_0\right)sin\theta \hfill \\ \hfill & +\left(l_1{z}_0-l_3{x}_0\right)sin\theta tan\varphi +\frac{\left(l_2{x}_0-l_1{y}_0\right)cos\theta }{cos\varphi }),\hfill \\ \hfill \frac{1}{r}\frac{\partial }{\partial \theta }\left(\frac{1}{sin\theta }\frac{\partial {\psi }_0}{\partial r}\right)+\frac{{\chi }_0}{r{sin}^2\theta }& =\frac{1}{R^3}(l_{2}r+(l_3{y}_0-l_2{z}_0)cos\theta \hfill \\ \hfill & +\left(l_1{z}_{0}cos\theta -l_3{x}_{0}cos\theta -l_{1}r\right)tan\varphi +\frac{\left(l_1{y}_0-l_2{x}_0\right)sin\theta }{cos\varphi }),\hfill \\ \hfill -\frac{1}{r{sin}^2\theta }\frac{\partial {\psi }_0}{\partial r}-\frac{1}{r}\frac{\partial }{\partial \theta }\left(\frac{{\chi }_0}{sin\theta }\right)& =\frac{1}{R^3}(l_2{z}_0-l_{2}rcos\theta -l_3{y}_0\hfill \\ \hfill & +\frac{l_{3}rsin\theta }{sin\varphi }+\left(l_1{z}_0-l_{1}rcos\theta -l_3{x}_0\right)cot\varphi ).\hfill \end{array}$$

(10)

2.4. On Some Restricted Conditions

We next restrict our analysis to the simpler case where $u_0^{{}^{\prime }}(r,\theta )$ is independent of $\varphi$, for which the system of equations in Equation (8) reduces to:

$$\begin{array}{cc}\hfill u_{0r}^{\ast }=& \left[l_3{y}_0-l_2{z}_0\right]sin\theta cos\varphi \hfill \\ \hfill u_{0\theta }^{\ast }=& \left[l_{2}r-l_2{z}_{o}cos\theta +l_3{y}_{0}cos\theta \right]cos\varphi \hfill \\ \hfill u_{0\varphi }^{\ast }=& \left[l_2{z}_0-l_{2}rcos\theta -l_3{y}_0\right]sin\varphi \hfill \end{array}$$

(11)

together with the following conditions:

$$\begin{array}{cc}\hfill & \mathrm{from}{u}_{0r}^{\ast }:\left[\left[l_1{z}_0-l_3{x}_0\right]sin\theta sin\varphi +\left[l_2{x}_0-l_1{y}_0\right]cos\theta =0\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \vee \left[l_1{z}_0-l_3{x}_0=0\wedge l_2{x}_0-l_1{y}_0=0\right],\hfill \\ \hfill & \mathrm{from}{u}_{0\theta }^{\ast }:\left[\left[l_1{z}_{0}cos\theta -l_3{x}_{0}cos\theta -l_{1}r\right]sin\varphi +\left[l_1{y}_0-l_2{x}_0\right]sin\theta =0\right]\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \vee \left[l_1{z}_{0}cos\theta -l_3{x}_{0}cos\theta -l_{1}r=0\wedge l_1{y}_0-l_2{x}_0=0\right],\hfill \\ \hfill & \mathrm{from}{u}_{0\varphi }^{\ast }:\left[l_{3}rsin\theta +\left[l_1{z}_0-l_{1}rcos\theta -l_3{x}_0\right]cos\varphi =0\right]\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \vee \left[l_3=0\wedge l_1{z}_0-l_{1}rcos\theta -l_3{x}_0=0\right].\hfill \end{array}$$

(14)

The first condition in Equations (12)–(14) for each of the coordinates of the vector field leads to the system of equations:

$$\begin{array}{cc}\hfill \left[l_1{z}_0-l_3{x}_0\right]sin\theta sin\varphi +\left[l_2{x}_0-l_1{y}_0\right]cos\theta & =0\hfill \\ \hfill \left[l_1{z}_{0}cos\theta -l_3{x}_{0}cos\theta -l_{1}r\right]sin\varphi +\left[l_1{y}_0-l_2{x}_0\right]sin\theta & =0\hfill \\ \hfill l_{3}rsin\theta +\left[l_1{z}_0-l_{1}rcos\theta -l_3{x}_0\right]cos\varphi & =0.\hfill \end{array}$$

(15)

As ($l_1,l_2,l_3$) and ($x_0,y_0,z_0$) can all, in principle, be considered unknown variables (i.e., the system is compatible but has infinite solutions). Without any loss of generality, we can rearrange the system in Equation (15) by assuming ($x_0,y_0,z_0$) are dependent on ($l_1,l_2,l_3$). Finally, we solve for the independent variables by Gaussian reduction:

$$\begin{array}{cc}& \begin{array}{ccc}{y}_0& z_0& x_0\end{array}\hfill \\ \begin{array}{c}sin\theta \\ cos\theta \end{array}& \left(\begin{array}{ccccc}-l_{1}cos\theta & l_{1}sin\theta sin\varphi & l_{2}cos\theta -l_{3}sin\theta sin\varphi & \mathrm{|}& 0\\ l_{1}sin\theta & l_{1}cos\theta sin\varphi & -l_{2}sin\theta -l_{3}cos\theta sin\varphi & \mathrm{|}& l_{1}rsin\varphi \\ 0& l_{1}cos\varphi & -l_{3}cos\varphi & \mathrm{|}& l_{1}rcos\theta cos\varphi -l_{3}rsin\theta \end{array}\right)\\ & \begin{array}{ccc}{y}_0& z_0& x_0\end{array}\hfill \\ \sim & \left(\begin{array}{ccccc}-l_{1}cos\theta & l_{1}sin\theta sin\varphi & l_{2}cos\theta -l_{3}sin\theta sin\varphi & |& 0\\ 0& l_{1}sin\varphi & -l_{3}sin\varphi & |& l_{1}rcos\theta sin\varphi \\ 0& 0& 0& |& l_{3}rsin\theta sin\varphi \end{array}\right)\hfill \end{array}$$

(16)

There is a solution for ($l_{1,2}\ne 0,l_3=0$) with the following dependent variables:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_0=l_1(y_0-rsin\theta sin\varphi )/l_2\hfill \\ y_0\ne 0\hfill \\ z_0=rcos\theta .\hfill \end{array}\right.\end{array}$$

Taking into account that ($x_0,y_0,z_0$) have to take fix values in order to correspond to the position vector for the rotlet and, hence, that they can not depend on the spatial coordinates ($r,\theta ,\varphi$), the obtained result is not a physically meaningful solution for the problem at hand. On the contrary, the second set of conditions in Equations (12)–(14) specify that:

• If the condition in Equation (14) is met, we have

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{if}{l}_3=0\to & l_1{z}_0-l_{1}rcos\theta -l_3{x}_0=0\hfill \\ & l_1\left(z_0-rcos\theta \right)=0\hfill \\ & l_1=0\vee z_0=rcos\theta .\hfill \end{array}\end{array}$$

  (17)
• If the conditions in Equations (13) and (17) are now both simultaneously satisfied then

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{if}{l}_3=0\to & l_1{z}_0-l_3{x}_0=0\wedge l_2{x}_0-l_1{y}_0=0\hfill \\ & \underset{l_1=0\vee z_0=0}{\underbrace{l_1{z}_0=0}}\wedge l_2{x}_0=l_1{y}_0\hfill \\ & \therefore l_1=0\wedge \underset{l_2=0\vee x_0=0}{\underbrace{l_2{x}_0=0}}\hfill \\ & \therefore x_0=0,\mathrm{because}{l}_2\ne 0,\mathrm{given}\mathrm{that}{l}_1=l_3=0.\hfill \end{array}\end{array}$$

  (18)

This provides us with a suitable set of conditions for a general solution to Equation (4), namely, that the position and torque vectors for the rotlet take the form ${\mathbf{x}}_{\mathbf{0}}=(0,y_0,z_0)$ and $\mathbf{L}=\left(0,l_2,0\right)$. In the particular case of $y_0=0$ (and renaming $z_0=c$) we recover the solution provided in [36] which describes the flow induced by a rotlet displaced along the z-axis and parallel to ŷ. For any other $y_0\ne 0$, we arrive to a non-equivalent (independent) solution for which Equation (11) reduces to:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{0r}^{\ast }=& -l_2{z}_{0}sin\theta cos\varphi \hfill \\ \hfill u_{0\theta }^{\ast }=& l_2\left(r-z_{0}cos\theta \right)cos\varphi \hfill \\ \hfill u_{0\varphi }^{\ast }=& l_2\left(z_0-rcos\theta \right)sin\varphi .\hfill \end{array}\end{array}$$

(19)

Finally, and within this approach, the necessary conditions for the corresponding potential functions ${\psi }_0$ and ${\chi }_0$ read:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -\frac{1}{r^2}\frac{\partial }{\partial \theta }\left[\frac{1}{sin\theta }\frac{\partial {\psi }_0}{\partial \theta }\right]& =\frac{-l_2}{R^3}ccos{\theta }_{0}sin\theta ,\hfill \\ \hfill \left[\frac{1}{r}\frac{\partial }{\partial \theta }\left(\frac{1}{sin\theta }\frac{\partial {\psi }_0}{\partial r}\right)+\frac{{\chi }_0}{r{sin}^2\theta }\right]& =\frac{l_2}{R^3}\left(r-ccos{\theta }_{0}cos\theta \right),\hfill \\ \hfill -\left[\frac{1}{r{sin}^2\theta }\frac{\partial {\psi }_0}{\partial r}+\frac{1}{r}\frac{\partial }{\partial \theta }\left(\frac{{\chi }_0}{sin\theta }\right)\right]& =\frac{l_2}{R^3}\left(ccos{\theta }_0-rcos\theta \right).\hfill \end{array}\end{array}$$

(20)

We recall that within the spherical symmetry of the defined domain: (i) any arbitrary displacement from the origin can be assigned to a displacement along the z-axis by simple 3D rotations; (ii) any 3D orientation of the rotlet torque, L, can be assigned to an in-plane 2D (yz-plane) orientation by an appropriate rotation around the newly established z-axis; and (iii) any in-plane 2D (yz-plane) orientation can be decomposed in a component parallel and a component perpendicular to the y-axis. In this sense, and together with the solution provided in [26], these general conditions define a suitable set for which to apply the superposition principle to the linear Stokes problem Equation (1) and, hence, by which to construct a continuous set of solutions that correspond to spherically constrained vortical flows generated by arbitrarily displaced infinitesimal rotations around any three-dimensional axis.

3. Conclusions

Fluids shape ontogeny [40]. Biology makes use of the physics at hand to transduce signals into forces, and forces into shapes. And in nearly all cases, fluids (and fluid motion) are involved, either being the carriers of the transduced signals, the force actuators or sometimes even playing a dual role. Examples abound: from planar cell polarity (PCP) [41] to cytoplasmic streaming [42]; from lumen fracking establishing the AP axis [43], to the nodal flow in developing vertebrates [7], and many other cases of internal and external flows.

Having the correct set of theoretical tools to address the mathematical modelling of such instances is essential. However, analytical solutions to the equations describing fluid motion, i.e., the Navier–Stokes equations for incompressible fluids in the Stokes regime, are scarce. Moreover, and whether we use fully fledged CFD numerical models or minimalist singularity solutions, we are always required to deal properly with fluid boundary conditions. All the more so in the Stokes regime, where flows are long-ranged and, hence, boundary effects are all-important. That is particularly relevant for internal fluid dynamics within confined geometries such as the case of cilia driven flows in the KV during zebrafish development.

In recent years, our understanding of the biophysics of left–right symmetry breaking in vertebrates has gain much from minimal approaches [7,12,15,19]. Following a similar strategy, here we have started putting together a minimal biophysical model which reduce the observed vortical fluid flow within in the KV to a set of singular solutions satisfying the imposed boundary conditions; one that is informed by our current understanding of the fluid flow in the KV, that satisfies the requirements for left–right symmetry breaking, but which is also amenable to extensive parametric analysis. Our work is a step forward in this direction. By finding the general conditions for the solution to the fluid mechanics of a singular rotlet within a rigid sphere, we have enlarged the set of available solutions in a way that can be easily extended to more complex configurations.