1. Introduction
In this paper we study the gravity-driven flow of a viscous fluid in a channel of rectangular cross-section, helically-wound about a vertical axis as shown in figure 1, where the fluid depth is assumed to be small. The channel is oriented such that the bottom is horizontal in the radial direction. We neglect the entrance and exit regions and assume that the flow is helically-symmetric; that is, independent of distance along any helix of the same pitch and orientation as the channel centreline. Thus the flow, which comprises a primary flow down the channel and a secondary flow in the cross-sectional plane, depends only on position in the two-dimensional cross-section. We make no assumptions of small channel torsion or slope and so extend the work of Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013) for channels of small centreline torsion, and of Lee, Stokes & Bertozzi (Reference Lee, Stokes and Bertozzi2014) for channels of small centreline slope. Using a non-orthogonal coordinate system, as in Lee et al. (Reference Lee, Stokes and Bertozzi2014), we are, remarkably, able to find an exact solution of our thin-film model. Our use of a non-orthogonal coordinate system and accounting for a channel slope that varies with position across the channel width reveal corrections to the results of Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013).
A helically-wound rectangular channel of width
$2a$
. The centreline is a helix of radius
$A$
and pitch
$2{\rm\pi}P$
.
The study of flows in curved geometries has been motivated by flows in rivers and pipes (Thomson Reference Thomson1876, Reference Thomson1877; Dean Reference Dean1927, Reference Dean1928), the circulatory system (Lynch, Waters & Pedley Reference Lynch, Waters and Pedley1996; Siggers & Waters Reference Siggers and Waters2005, Reference Siggers and Waters2008), and the cochlea (Manoussaki & Chadwick Reference Manoussaki and Chadwick2000). The impact of Dean’s work was such that these flows are often termed ‘Dean flow’.
An application of particular interest to the authors, and which has attracted considerable attention over several decades since the work of Holland-Batt (Reference Holland-Batt1975, Reference Holland-Batt1989), is the separation of minerals or coal from crushed ore in static spiral separators. These consist of a helically-wound channel, down which a slurry of crushed ore and water is poured. The resulting fluid motion serves to sort the particles across the width of the channel by size/density, and simple sectioning of the flow at the bottom of the channel allows particles of different sizes/densities to be collected separately. Whilst such devices are commonly used, their design is hampered by a lack of quantitative understanding of the fluid and particle motion, so that experimentation and empirical formulae are heavily used during design; for a summary of the design process see Holland-Batt (Reference Holland-Batt1995).
A desire for a better quantitative understanding to aid design has driven experimental, theoretical and computational research on both clear fluid and particle-laden flow in helically-wound channels. A steady-state empirical model of particle-free flow in a rectangular channel was developed by Holland-Batt (Reference Holland-Batt1989), in which the primary flow down the channel is described by a Manning law in an inner region near the central column around which the channel is wound, and a free vortex in the outer region. Experimental work followed to attempt to validate this model (Holtham Reference Holtham1990; Holland-Batt & Holtham Reference Holland-Batt and Holtham1991; Holtham Reference Holtham1992) but the small fluid depth typical of spiral separators makes measuring flow velocity and free-surface shape very difficult. While this work did confirm the existence of the predicted secondary flow, show the flow to be laminar in much of the flow domain, and indicate that the flow had reached a fully-developed profile within two spiral turns (justifying the assumption of helical symmetry), it gave only crude estimates of flow velocities with errors as high as
$\pm 30\,\%$
, precluding meaningful quantitative comparison of experiments with models. Recently Holland-Batt (Reference Holland-Batt2009) adapted the model for large-diameter rectangular spiral channels, with the strength of the free-vortex primary flow computed using laminar, Manning and Bagnold shear equations; little difference was found in the velocity profiles between the three options. Computational simulations were performed by Wang & Andrews (Reference Wang and Andrews1994), Matthews, Fletcher & Partiridge (Reference Matthews, Fletcher and Partiridge1998), Matthews et al. (Reference Matthews, Fletcher, Partiridge and Vasquez1999) and Stokes (Reference Stokes, King and Shikhmurzaev2001). Das et al. (Reference Das, Godiwakka, Panda, Bhattacharya, Singh and Mehrotra2007) used the semi-empirical model of Holland-Batt (Reference Holland-Batt1989) to investigate the behaviour of particles in such a flow, assuming that the particles do not modify the flow, while acknowledging that this model sometimes predicts larger flow depths than reported in experiments.
Flows in spiral separators are typically quite shallow. In experiments performed by Holtham (Reference Holtham1992) using two commercially-available spiral separators with an approximate width of 280 mm, the fluid depth was 1–12 mm, an aspect ratio
${\it\delta}<0.1$
. Thus, Stokes, Wilson & Duffy (Reference Stokes, Wilson, Duffy, Behnia, Lin and McBain2004), Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013) and Lee et al. (Reference Lee, Stokes and Bertozzi2014) have exploited the small depth of the flow to develop thin-film models of particle-free and particle-laden flow in helically-wound channels. These studies have considered limiting cases of channel geometry, such as small curvature, torsion or slope, and indicate that the assumption made by Holland-Batt (Reference Holland-Batt1989, Reference Holland-Batt2009) of a free-vortex primary flow in the channel is not, in general, valid in these limits. In this paper we retain the assumption of small fluid depth but consider particle-free flow in channels of arbitrary curvature and torsion. In future work, the model will be extended using the approach of Lee et al. (Reference Lee, Stokes and Bertozzi2014) to investigate particle-laden flow.
In the study of flows in curved geometries, considerable attention has been given to the choice of coordinate system. The natural
$s,r,{\it\theta}$
(or
$s,x,y$
) coordinate system, where
$s$
is arclength along the axis of the pipe or channel and
$r,{\it\theta}$
(
$x,y$
) are polar (Cartesian) coordinates in the cross-section, is non-orthogonal, with non-zero off-diagonal elements in the metric tensor. To avoid the complexity of dealing with non-orthogonal coordinates, Germano (Reference Germano1982, Reference Germano1989) obtained an orthogonal coordinate system by rotating the
${\it\theta}=0$
line with position
$s$
. Zabielski & Mestel (Reference Zabielski and Mestel1998) pointed out that, except in the zero-torsion limit, this coordinate system does not allow true helical symmetry (quantities are independent of distance
$s$
, while true helical symmetry requires that quantities do not change along any helical path with the same pitch as the channel centreline) and proposed an alternative coordinate system which does allow true helical symmetry. Unlike the problems considered by these authors, the flows of interest here have a free surface, for which neither of these coordinate systems is easy to use. Hence we adopt the more intuitive non-orthogonal coordinate system previously employed by Manoussaki & Chadwick (Reference Manoussaki and Chadwick2000) for the study of inviscid fluid flow in the cochlea (modelled as a rectangular duct), and which is similar to the natural
$(s,x,y)$
system.
The structure of this paper is as follows. In § 2 we describe the coordinate system and the mathematical model, and then solve the model exactly. Results are presented and discussed in § 3 and our summary and conclusions are given in § 4.
2. Mathematical model
We consider steady, helically-symmetric flow of an incompressible viscous fluid down a channel of constant rectangular cross-section with width
$2a$
, the centreline of which is a helix of pitch
$2{\rm\pi}P$
and radius
$A$
, as shown in figure 1. The channel is aligned so that its bottom is horizontal is the radial direction.
2.1. Non-orthogonal coordinates
Following Manoussaki & Chadwick (Reference Manoussaki and Chadwick2000) and Lee et al. (Reference Lee, Stokes and Bertozzi2014), we employ a helical coordinate system in which a point
$\boldsymbol{x}$
is specified using three variables,
$(r,{\it\beta},z)$
, as follows:
$$\begin{eqnarray}\boldsymbol{x}(r,{\it\beta},z)=r\cos {\it\beta}\boldsymbol{i}+r\sin {\it\beta}\boldsymbol{j}+(P{\it\beta}+z)\boldsymbol{k}.\end{eqnarray}$$
Here
$r$
is the radial distance from the axis of the helix,
${\it\beta}$
is an angle from a reference direction, and
$z$
is the vertical distance above the bottom surface of the channel. Using this description, the assumption of helical symmetry implies that the flow is independent of the angle
${\it\beta}$
. The free-surface profile is given as
$S(r,z)=h_{z}(r)-z=0$
, where
$h_{z}(r)$
is the fluid depth in the vertical direction at radial position
$r$
.
The basis vectors in this coordinate system,
$\boldsymbol{e}_{r}$
,
$\boldsymbol{e}_{{\it\beta}}$
and
$\boldsymbol{e}_{z}$
, are non-orthogonal and hence tensor calculus is required to determine differential operators. Once helical symmetry has been enforced by assuming that derivatives with respect to
${\it\beta}$
vanish, a new coordinate direction,
$n$
, is introduced to replace
$z$
. The new coordinate is in the direction of unit vector
$\boldsymbol{e}_{n}=\boldsymbol{e}_{r}\times \boldsymbol{e}_{{\it\beta}}$
, orthogonal to
$\boldsymbol{e}_{r}$
and
$\boldsymbol{e}_{{\it\beta}}$
, so the direction
$\boldsymbol{e}_{n}$
varies with position across the channel. Figure 2 shows the coordinate system on the channel centreline and in a vertically-cut cross-section of the channel. The normal direction is more intuitive, particularly for very steep channels, when the axial and vertical directions can nearly coincide. Velocity components in the
$r$
,
${\it\beta}$
and
$n$
directions are
$v^{r}$
,
$v^{{\it\beta}}$
and
$v^{n}$
, respectively,
$p$
is pressure, and the free-surface profile is given by
$S_{n}(r,n)=h_{n}(r)-n=0$
. The helically-symmetric Navier–Stokes equations in the
$r$
,
${\it\beta}$
,
$n$
coordinate system are given in appendix A; see Lee et al. (Reference Lee, Stokes and Bertozzi2014) for details of the derivation.
The coordinate system (a) on the helical centreline of radius
$A$
and pitch
$2{\rm\pi}P$
; and (b) in a vertically-cut cross-section of the channel.
2.2. Thin-film equations
Since the fluid depth is assumed to be small relative to the channel half-width
$a$
, we employ a thin-film approximation to simplify the Navier–Stokes equations. Specifically, if
$\bar{h}$
is a representative fluid depth in the direction normal to the channel bottom (i.e. the
$\boldsymbol{e}_{n}$
direction), then
${\it\delta}=\bar{h}/a$
is a measure of the aspect ratio of the flow domain, and we assume that
${\it\delta}\ll 1$
. We will define
${\it\delta}$
in terms of the physical parameters of the problem in § 2.4.
Next, we non-dimensionalise our governing equations using the channel half-width,
$a$
, axial velocity scale,
$U$
(to be specified in § 2.4), and fluid viscosity,
${\it\mu}$
, as follows (using primes to indicate dimensionless variables):
$$\begin{eqnarray}r=a(A^{\prime }+y^{\prime }),\quad n=a{\it\delta}n^{\prime },\quad (v^{r},v^{{\it\beta}},v^{n})=({\it\delta}Uv^{\prime },Uu^{\prime },{\it\delta}^{2}Uw^{\prime }),\quad p=\frac{{\it\mu}U}{a{\it\delta}}p^{\prime }.\end{eqnarray}$$
We also define
$n^{\prime }=h^{\prime }(y^{\prime })$
to be the free surface. At this point it is helpful to define some geometric parameters related to the curvature and torsion of the channel centreline. The curvature of the projection of the channel centreline onto a horizontal plane is given by
${\it\epsilon}=a/A=1/A^{\prime }$
, the dimensionless pitch is
$P^{\prime }=P/a$
, and the slope of the centreline of the channel is
${\it\lambda}=P^{\prime }/A^{\prime }$
. Figure 3 shows channels with different combinations of
${\it\epsilon}$
and
${\it\lambda}$
to help visualise different channel shapes.
Channels with (a) large curvature and moderate slope (
${\it\epsilon}=0.8,{\it\lambda}=1$
); (b) small curvature and moderate slope (
${\it\epsilon}=0.2,{\it\lambda}=1$
); (c) large slope and moderate curvature (
${\it\epsilon}=0.5,{\it\lambda}=5$
). Note that each channel is the same width, but the images are not shown at the same scale, in order to preserve detail.
The torsion,
${\it\tau}$
, and curvature,
${\it\kappa}$
, of the helical channel at position
$y^{\prime }$
across the channel are given by
$$\begin{eqnarray}{\it\tau}(y^{\prime })=\frac{{\it\epsilon}{\rm\Lambda}(y^{\prime })}{{\rm\Upsilon}(y^{\prime })(1+{\it\epsilon}y^{\prime })},\quad {\it\kappa}(y^{\prime })=\frac{{\it\epsilon}}{{\rm\Upsilon}(y^{\prime })(1+{\it\epsilon}y^{\prime })}=\frac{{\it\tau}(y^{\prime })}{{\rm\Lambda}(y^{\prime })},\end{eqnarray}$$
where
$$\begin{eqnarray}{\rm\Lambda}(y^{\prime })=\frac{{\it\lambda}}{1+{\it\epsilon}y^{\prime }},\end{eqnarray}$$
is the slope of the channel bottom at a point
$y^{\prime }$
, and for notational convenience we have introduced
$$\begin{eqnarray}{\rm\Upsilon}(y^{\prime })=1+{\rm\Lambda}^{2}(y^{\prime }).\end{eqnarray}$$
At leading order in
${\it\delta}$
, the Navier–Stokes and continuity equations are (dropping primes)
$$\begin{eqnarray}\displaystyle & \displaystyle -\frac{\partial p}{\partial y}+\frac{\partial ^{2}v}{\partial n^{2}}+2{\it\tau}(y)\frac{\partial u}{\partial n}+Re\,{\it\kappa}(y)u^{2}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial ^{2}u}{\partial n^{2}}-\frac{Re}{Fr^{2}}\frac{{\it\tau}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle -\frac{\partial p}{\partial n}-\frac{Re}{Fr^{2}}\frac{{\it\kappa}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial v}{\partial y}+\frac{{\it\epsilon}v}{1+{\it\epsilon}y}+\frac{\partial w}{\partial n}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}Re=\frac{{\it\rho}{\it\delta}Ua}{{\it\mu}}\quad \text{and}\quad Fr=\frac{U}{\sqrt{ga{\it\delta}}}\end{eqnarray}$$
are the Reynolds and Froude numbers respectively (note that the centrifugal term in (2.6a
) could be written in terms of the Dean number
$K=2{\it\epsilon}Re^{2}$
, as in Stokes et al.
Reference Stokes, Duffy, Wilson and Tronnolone2013). The boundary conditions are no-slip on the channel bottom, i.e.
$$\begin{eqnarray}u=v=w=0\quad \text{on }n=0,\end{eqnarray}$$
and, on the free surface, no stress, which implies
$$\begin{eqnarray}p=0,\quad \frac{\partial u}{\partial n}=0,\quad \frac{\partial v}{\partial n}=0\quad \text{on }n=h(y),\end{eqnarray}$$
plus the standard kinematic condition, which gives
$$\begin{eqnarray}v\frac{\text{d}h}{\text{d}y}=w\quad \text{on }n=h(y).\end{eqnarray}$$
Physically, there will also be no-slip on the channel sidewalls (
$y=\pm 1$
) but we cannot impose this condition on our solutions owing to the thin-film scaling. In practice, we anticipate the presence of thin boundary layers close to these walls.
2.3. Solution
Expressions for the pressure,
$p$
, axial velocity component,
$u$
, and radial velocity component,
$v$
, are obtained by integrating (2.6a
)–(2.6c
). We obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle p=-\frac{Re}{Fr^{2}}\frac{{\it\kappa}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}(n-h),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle u=\frac{Re}{2Fr^{2}}\frac{{\it\tau}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}n(n-2h),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & & \displaystyle v=\frac{Re}{2Fr^{2}}\frac{{\it\kappa}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}\frac{\text{d}h}{\text{d}y}n(n-2h)-\frac{Re}{2Fr^{2}}\frac{{\it\tau}^{2}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}\{(n-h)^{3}+h^{3}\}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{Re^{3}}{60Fr^{4}}\frac{{\it\tau}^{2}{\it\kappa}}{{\it\tau}^{2}+{\it\kappa}^{2}}n(n-2h)\{n^{3}(n-4h)+2h^{2}(n^{2}+2hn+4h^{2})\}.\end{eqnarray}$$
$$\begin{eqnarray}\frac{\partial }{\partial y}([1+{\it\epsilon}y]v)+\frac{\partial }{\partial n}([1+{\it\epsilon}y]w)=0,\end{eqnarray}$$
and an equation for the free-surface profile,
$h(y)$
, is found using the integrated form of (2.14),
$$\begin{eqnarray}\int _{0}^{h(y)}v\,\text{d}n=0,\end{eqnarray}$$
which is obtained by requiring that there is no net flux into or out of the flow domain. Substituting (2.13) into (2.15) and integrating yields the differential equation for
$h(y)$
,
$$\begin{eqnarray}\frac{\text{d}h}{\text{d}y}=\frac{6}{35}\frac{Re^{2}}{Fr^{2}}\frac{{\it\tau}^{2}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}h^{4}-\frac{9}{8}\frac{{\it\tau}^{2}}{{\it\kappa}}h,\end{eqnarray}$$
which can be used to eliminate the derivative term in (2.13).
To calculate the normal velocity component,
$w$
, we use a streamfunction to describe flow in the channel cross-section. The streamfunction
${\it\psi}(y,n)$
is defined by
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\psi}}{\partial n}=(1+{\it\epsilon}y)v, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\psi}}{\partial y}=-(1+{\it\epsilon}y)w, & \displaystyle\end{eqnarray}$$
$v$
into (2.17a
) and integrating subject to
${\it\psi}=0$
on
$n=h(y)$
, to give 
$$\begin{eqnarray}\displaystyle {\it\psi} & = & \displaystyle \frac{{\it\epsilon}Re^{3}}{840Fr^{4}}\frac{{\it\tau}{\it\kappa}}{{\it\tau}^{2}+{\it\kappa}^{2}}n^{2}(n-h)(n-2h)^{2}(2nh-n^{2}+4h^{2})\nonumber\\ \displaystyle & & \displaystyle -\frac{{\it\epsilon}Re}{16Fr^{2}}\frac{{\it\tau}^{2}{\it\kappa}}{({\it\tau}^{2}+{\it\kappa}^{2})^{3/2}}n^{2}(n-h)(2n-3h).\end{eqnarray}$$
We note that
${\it\psi}$
vanishes on the channel bottom,
$n=0$
, as well as the free surface,
$n=h$
, as indeed it must.
We rewrite the free-surface equation (2.16) as
$$\begin{eqnarray}\frac{\text{d}h}{\text{d}y}=\frac{6}{35}\frac{Re^{2}}{Fr^{2}}\frac{{\it\epsilon}{\rm\Lambda}^{2}}{{\rm\Upsilon}^{3/2}}\frac{h^{4}}{(1+{\it\epsilon}y)}-\frac{9}{8}\frac{{\it\epsilon}{\rm\Lambda}^{2}}{(1+{\it\epsilon}y){\rm\Upsilon}}h\end{eqnarray}$$
and seek a solution with boundary condition
$h(1)=h_{r}$
, where
$h_{r}$
is the fluid depth at the outside wall of the channel. Since it is a Bernoulli differential equation, we use the substitution
$$\begin{eqnarray}{\it\xi}(y)=h(y)^{-3},\end{eqnarray}$$
which transforms (2.19) into the linear first-order differential equation
$$\begin{eqnarray}\frac{\text{d}{\it\xi}}{\text{d}y}-\frac{27}{8}\frac{{\it\epsilon}{\rm\Lambda}^{2}}{(1+{\it\epsilon}y){\rm\Upsilon}}{\it\xi}=-3\frac{Re^{2}}{Fr^{2}}\frac{{\it\epsilon}{\rm\Lambda}^{2}}{(1+{\it\epsilon}y){\rm\Upsilon}^{3/2}},\end{eqnarray}$$
which may be solved using the integrating factor
${\rm\Upsilon}^{27/16}$
to give
$$\begin{eqnarray}h(y)={\rm\Upsilon}(y)^{9/16}\left\{{\rm\Upsilon}(1)^{27/16}h_{r}^{-3}-\frac{144}{665}\frac{Re^{2}}{Fr^{2}}\left({\rm\Upsilon}(1)^{19/16}-{\rm\Upsilon}(y)^{19/16}\right)\right\}^{-1/3}.\end{eqnarray}$$
Although in the above we have imposed a boundary condition in terms of the fluid depth at the outside wall of the channel, in practice this is difficult to control, and a more natural condition to specify is the fluid flux down the channel. Thus, for a prescribed flux, we determine the corresponding value of
$h_{r}$
to give the appropriate free-surface profile. The dimensional flux down the channel, denoted
$Q$
, is given by
$$\begin{eqnarray}Q=-Ua^{2}{\it\delta}\int _{-1}^{1}\int _{0}^{h}u\,\text{d}n\text{d}y=Ua^{2}{\it\delta}\hat{Q}\end{eqnarray}$$
and, on substituting for
$u$
and integrating, we find the dimensionless fluid flux
$$\begin{eqnarray}\hat{Q}=\frac{1}{3}\frac{Re}{Fr^{2}}\int _{-1}^{1}\frac{{\it\tau}}{\sqrt{{\it\tau}^{2}+{\it\kappa}^{2}}}h^{3}\text{d}y.\end{eqnarray}$$
Note that the minus sign appears in (2.23) because our axial coordinate direction,
$\boldsymbol{e}_{{\it\beta}}$
, points up the channel, and hence fluid flowing down the channel has negative axial velocity. In general, we cannot solve (2.24) exactly, and so cannot obtain the exact value of
$h_{r}$
for a chosen value,
$\hat{\mathscr{Q}}$
, of the flux. We solve numerically using an adaptive Gauss–Kronrod quadrature algorithm provided by Matlab to approximate the integral, and a bisection-type search method to find an approximate value of
$h_{r}$
such that
$\hat{Q}$
is within some specified tolerance of
$\hat{\mathscr{Q}}$
. We have required the relative error in
$\hat{Q}$
to be less than
$10^{-8}$
. Note that we can use the fluid depth at any location in the channel as the boundary condition for (2.19), but choose
$y=1$
to improve the numerical conditioning of the search algorithm. At
$y=1$
the fluid depth is most sensitive to flux, whereas at
$y=-1$
it is, in general, least sensitive, and so small changes in the depth at
$y=-1$
cause very large changes in the flux, which makes calculating the appropriate boundary condition there more difficult than at
$y=1$
.
The dimensional cross-sectional area of the fluid domain is denoted
${\rm\Omega}$
, and is defined as
$$\begin{eqnarray}{\rm\Omega}=a^{2}{\it\delta}\int _{-1}^{1}h\,\text{d}y=a^{2}{\it\delta}\hat{{\rm\Omega}},\end{eqnarray}$$
where
$\hat{{\rm\Omega}}$
is the dimensionless cross-sectional area
$$\begin{eqnarray}\hat{{\rm\Omega}}=\int _{-1}^{1}h\,\text{d}y.\end{eqnarray}$$
As with (2.24), we cannot in general solve this integral exactly, and must use numerical integration.
2.4. Scaling
Thus far, we have not specified the velocity scale,
$U$
, and depth scale,
${\it\delta}$
, in our problem. We note that the gravitational forcing term in (2.6b
) may be written
$$\begin{eqnarray}\frac{Re}{Fr^{2}}\frac{{\rm\Lambda}}{\sqrt{1+{\rm\Lambda}^{2}}}\end{eqnarray}$$
and hence we choose to set
$$\begin{eqnarray}\frac{Re}{Fr^{2}}\frac{{\it\lambda}}{\sqrt{1+{\it\lambda}^{2}}}=1\end{eqnarray}$$
so that the term has unit value at the channel centreline
$y=0$
. In addition, to ensure non-trivial free-surface profiles for any choice of
${\it\epsilon}$
and
${\it\lambda}$
, we set the coefficient of
$h^{4}$
in (2.19) to have unit value at
$y=0$
, so that
$$\begin{eqnarray}\frac{6}{35}Re\frac{{\it\epsilon}{\it\lambda}}{1+{\it\lambda}^{2}}=1.\end{eqnarray}$$
With these scalings, the velocity scale is
$$\begin{eqnarray}U=\left[\left(\frac{35}{6{\it\epsilon}}\right)^{2}\frac{(1+{\it\lambda}^{2})^{3/2}}{{\it\lambda}{\it\epsilon}^{2}}\frac{g{\it\mu}}{{\it\rho}}\right]^{1/3}\end{eqnarray}$$
and the thin-film parameter is
$$\begin{eqnarray}{\it\delta}=\frac{1}{a}\left[\frac{35}{6}\frac{(1+{\it\lambda}^{2})^{3/2}}{{\it\epsilon}{\it\lambda}^{2}}\frac{{\it\mu}^{2}}{g{\it\rho}^{2}}\right]^{1/3}.\end{eqnarray}$$
Note that our thin-film assumption requires
${\it\delta}\ll 1$
, and this limits the possible values of the parameters. For example, for a 1 m wide channel (
$a=0.5~\text{m}$
), with water (
${\it\mu}=10^{-3}~\text{kg m}^{-1}~\text{s}^{-1},{\it\rho}=10^{3}~\text{kg m}^{-3}$
) and
$g=9.81~\text{m s}^{-2}$
, we require
$$\begin{eqnarray}\frac{{\it\epsilon}{\it\lambda}^{2}}{(1+{\it\lambda}^{2})^{3/2}}\gg 5\times 10^{-12}\end{eqnarray}$$
to ensure that
${\it\delta}$
remains small. This inequality fails to hold in only a very small part of the parameter space, when the channel approaches a straight (
${\it\epsilon}\rightarrow 0$
), flat (
${\it\lambda}\rightarrow 0$
), or steep (very large
${\it\lambda}$
) geometry, which are not of interest here.
With this scaling, our free-surface equation (2.22) becomes
$$\begin{eqnarray}h(y)={\rm\Upsilon}(y)^{9/16}\left\{{\rm\Upsilon}(1)^{27/16}h_{r}^{-3}-\frac{24}{19}\frac{(1+{\it\lambda}^{2})^{3/2}}{{\it\epsilon}{\it\lambda}^{2}}\left({\rm\Upsilon}(1)^{19/16}-{\rm\Upsilon}(y)^{19/16}\right)\right\}^{-1/3}.\end{eqnarray}$$
3. Results
We now use our analytic solution to investigate the effects of
${\it\epsilon}$
,
${\it\lambda}$
and
$\hat{Q}$
on the free-surface profile, and the pressure and velocity fields. In § 3.1 we consider the free-surface profile from which we determine the fluid velocity components and pressure. In § 3.2 we present several velocity and pressure solutions, and discuss general features of the results. We give particular attention to large flux in § 3.3.
3.1. Free-surface profile
Figures 4 and 5 show some representative free-surface profiles for different choices of
${\it\lambda}$
and
${\it\epsilon}$
at a fixed flux
$\hat{Q}=1$
. These illustrate some of the qualitatively different types of free-surface profile that are possible with different channel geometry parameters
${\it\epsilon}$
and
${\it\lambda}$
. For sufficiently small slope the fluid depth at the inside (left) wall of the channel,
$h_{l}=h(-1)$
, decreases with increasing curvature
${\it\epsilon}$
and the fluid depth increases monotonically across the width of the channel from the inside to the outside wall; figure 4(a). For any
${\it\epsilon},h_{l}$
increases with
${\it\lambda}$
(figure 5), and for
${\it\epsilon}$
not too close to unity, the fluid depth increases monotonically across the width of the channel from inside to outside wall; figures 4(a) and 5(a). However, for
${\it\epsilon}$
sufficiently close to unity and sufficiently large
${\it\lambda}$
, the free-surface profile changes significantly, with the fluid depth at first decreasing with distance from the inside wall and then increasing; figures 4(b,c) and 5(b,c); there is a build-up of fluid near the inside wall where the channel slope is largest. For very large
${\it\lambda}$
and
${\it\epsilon}$
very close to unity the fluid depth is largest at the inside wall (where the channel bottom will be near vertical) and decreases monotonically across the width of the channel, figures 4(c) and 5(c).
For small
${\it\epsilon}$
, we observe that the solutions become nearly independent of
${\it\lambda}$
(figure 5
a), and, indeed, this is seen from the small-
${\it\epsilon}$
limit of (2.33) which is independent of
${\it\lambda}$
:
$$\begin{eqnarray}h(y)=[h_{r}^{-3}-3(y-1)]^{-1/3}.\end{eqnarray}$$
Note that in this limit, exact expressions can be obtained for the flux,
$\hat{Q}$
, and area,
$\hat{{\rm\Omega}}$
; see Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013) for details. As
${\it\epsilon}$
increases to near unity, the free-surface solution becomes more sensitive to changes in
${\it\lambda}$
. This is intuitive: at large
${\it\epsilon}$
the slope of the channel changes significantly across its width, and so the effects of changing
${\it\lambda}$
will be greater.
Free-surface profiles for fixed slope
${\it\lambda}$
: (a)
${\it\lambda}=0.01$
; (b)
${\it\lambda}=0.2$
; (c)
${\it\lambda}=5$
, for curvatures
${\it\epsilon}=0.01,0.255,0.5,0.745,0.99$
, and flux
$\hat{Q}=1$
. Arrows show direction of increasing
${\it\epsilon}$
.
Free-surface profiles for fixed curvature
${\it\epsilon}$
: (a)
${\it\epsilon}=0.3$
; (b)
${\it\epsilon}=0.7$
; (c)
${\it\epsilon}=0.99$
, with slopes
${\it\lambda}=0.01,0.67,1.34,2$
, and flux
$\hat{Q}=1$
. Arrows show direction of increasing
${\it\lambda}$
.
Increasing flux (with fixed
${\it\epsilon}$
and
${\it\lambda}$
) always increases the fluid depth at any point in the channel, and the fluid depth at the outer wall,
$h_{r}$
, grows without bound as flux becomes infinite; see figure 6. Increasing flux tends to increase
$h_{r}$
more than
$h_{l}$
, i.e. the depth at the outside wall is much more sensitive to changes in flux.
We can explain the free-surface profiles that we observe physically, in terms of two competing mechanisms, corresponding to the two terms on the right of the free-surface equation (2.19). Centrifugal force pushes the fluid to the outside wall of the channel, which is exacerbated by increasing the flux. This effect was also described by Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013), and is related to the first term on the right of our free-surface equation. However, the model presented in the current work also includes the effect of the varying slope across the width of the channel (neglected by Stokes et al.
Reference Stokes, Duffy, Wilson and Tronnolone2013 who assumed constant slope). When
${\it\epsilon}$
is close to unity, and the channel is tightly wound about the vertical axis, the channel slopes much more steeply near the inside wall than at the outside wall. This means that the axial flow direction in this region is more aligned with the direction of gravitational acceleration, which results in a significant gravitational effect, captured by the second term on the right of (2.19), so that the fluid effectively cascades down the inside of the channel. This gravitational effect is magnified by increasing the slope of the channel centreline,
${\it\lambda}$
. Thus there is a balance between centrifugal effects pushing the fluid to the outer wall, and gravitational effects pulling the fluid downwards and to the steepest part of the channel.
Free-surface profile for fixed slope
${\it\lambda}$
and curvature
${\it\epsilon}$
, with varying flux
$\hat{Q}$
ranging from
$10^{-3}$
to the maximum that is computationally possible. From left to right,
${\it\lambda}$
is increasing, and from top to bottom,
${\it\epsilon}$
is increasing. Increasing flux always increases fluid depth everywhere. Plot (a) is representative of plots for all
${\it\lambda}$
with
${\it\epsilon}\ll 1$
. (a)
${\it\lambda}=0.01$
,
${\it\epsilon}=0.5$
; (b)
${\it\lambda}=0.5$
,
${\it\epsilon}=0.5$
; (c)
${\it\lambda}=5$
,
${\it\epsilon}=0.5$
; (d)
${\it\lambda}=0.01$
,
${\it\epsilon}=0.99$
; (e)
${\it\lambda}=0.5$
,
${\it\epsilon}=0.99$
; ( f)
${\it\lambda}=5$
,
${\it\epsilon}=0.99$
.
3.2. Velocity and pressure profiles
A representative solution for velocity components and pressure is shown in figure 7. The parameters used in this figure are chosen for a channel with curvature and slope that roughly correspond to a Vickers FGL commercial spiral separator (
${\it\epsilon}=0.67,{\it\lambda}=0.33$
, Holtham Reference Holtham1992). Streamlines of the secondary flow and contours of the axial velocity and pressure are shown. The secondary flow shows a single rotating cell, cut off by the outer wall. The streamfunction of the secondary flow,
${\it\psi}$
, is zero on the free surface and channel bottom, corresponding to the top and bottom of the fluid domain; however, multiple streamlines meet the channel walls, violating the no-slip condition. As explained earlier, this is due to the thin-film scaling. Boundary layers will exist which are not captured by our leading-order (in
${\it\delta}$
) equations. Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013) compared numerical solutions to the full Navier–Stokes equations with their thin-film results, and found agreement everywhere away from the edges of the channel. Imposing no-slip on the sidewalls of the channel would cause streamlines to form closed curves, and we would observe a single clockwise-rotating closed cell in the channel cross-section.
Streamlines of the secondary flow
${\it\psi}$
(a), and contour plots of axial velocity
$u$
(b) and pressure
$p$
(c), for
${\it\lambda}=0.33,{\it\epsilon}=0.67$
and for
$\hat{Q}=0.4$
. Arrows indicate the direction of the secondary flow in (a). The free surface is shown as a dashed curve.
In figure 7(b), the axial velocity increases as the distance from the channel bottom increases, and the maximum axial velocity occurs at the outside wall of the channel on the free surface. For some choices of the geometry, notably as the slope increases significantly towards the inside wall, the maximum axial velocity
$u$
can move to the inside wall (figure 8), or somewhere between the two walls (figure 9), although it always occurs on the free surface,
$n=h(y)$
(the minimum axial velocity is, of course, zero, on the channel bottom). For any choice of parameters, at the free surface we always have a radial velocity
$v>0$
across the whole width of the channel, so we always have transport to the outside of the channel along the free surface. At the channel bottom, we always have
$v=0$
, but
$\partial v/\partial n<0$
so that there we have transport to the inside wall of the channel. Nevertheless, although a single rotating cell (as in figure 7
a) is the most prevalent type of cross-sectional flow, the formation of multiple rotating cells is possible. Figure 8(a) shows a case for large
${\it\lambda}$
with two clockwise-rotating cells, one close to each wall, within the outer clockwise-rotating flow. This change in the flow pattern has potentially important implications for particle segregation. Segregation requires particles of different size/density to collect in different regions of the channel cross-section and a secondary flow with multiple rotating cells may inhibit or facilitate particle segregation. A question still under investigation is whether more than two rotating cells can form. In visualising a large number of cross-sectional flow profiles we have seen no evidence of more than two cells, nor does our intuition suggest a mechanism by which they could form. However, we cannot, as yet, prove this claim.
Streamlines of the secondary flow
${\it\psi}$
(a), and contour plots of axial velocity
$u$
(b) and pressure
$p$
(c), for
${\it\lambda}=1.5,{\it\epsilon}=0.8$
and for
$\hat{Q}=0.5$
. Arrows indicate the direction of the secondary flow in (a). The free surface is shown as a dashed curve.
Streamlines of the secondary flow
${\it\psi}$
(a), and contour plots of axial velocity
$u$
(b) and pressure
$p$
(c), for
${\it\lambda}=0.15,{\it\epsilon}=0.75$
and for
$\hat{Q}=0.2$
. Arrows indicate the direction of the secondary flow in (a). The free surface is shown as a dashed curve.
The case of small slope and flux is also of interest. As shown in figure 9 we observe a free surface with negative second derivative, and the maximum axial velocity occurs away from the channel walls (we note that this is the only case consistent with a free-vortex primary flow approximation as assumed by Holland-Batt Reference Holland-Batt1989, Reference Holland-Batt2009). As flux increases, centrifugal force acts to push the fluid towards the outside wall, reversing the sign of the second derivative of the free-surface profile in the vicinity of that wall.
3.3. Large fluid flux
In Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013) it was found that as
$\hat{Q}\rightarrow \infty ,h_{l}$
, the fluid depth at the inside wall of the channel, and
$\hat{{\rm\Omega}}$
, the fluid cross-sectional area, each approached finite limiting values, whilst
$h_{r}$
, the fluid depth at the outside wall, became unbounded. We now examine if this remains true for the more general problem considered in the current paper.
Although the flux and area integrals, (2.24) and (2.26), cannot be evaluated in closed form, elementary manipulations of the free-surface equation (2.19), allow us to express the cross-sectional area as
$$\begin{eqnarray}\displaystyle \hat{{\rm\Omega}} & = & \displaystyle \frac{1}{2h_{l}^{2}}\left(\frac{{\it\lambda}^{2}+(1-{\it\epsilon})^{2}}{{\it\lambda}^{2}+1}\right)^{3/2}+\frac{1}{2}\frac{{\it\epsilon}{\it\lambda}^{2}}{(1+{\it\lambda}^{2})^{3/2}}\int _{-1}^{1}\frac{9}{4}\frac{\sqrt{{\rm\Upsilon}}}{h^{2}}+3\frac{\sqrt{{\rm\Upsilon}}}{{\rm\Lambda}^{2}}\frac{1}{h^{2}}\,\text{d}y\nonumber\\ \displaystyle & & \displaystyle \,-\frac{1}{2h_{r}^{2}}\left(\frac{{\it\lambda}^{2}+(1+{\it\epsilon})^{2}}{{\it\lambda}^{2}+1}\right)^{3/2},\end{eqnarray}$$
and the flux as
$$\begin{eqnarray}\displaystyle \hat{Q} & = & \displaystyle \frac{3}{8}\frac{{\it\lambda}^{2}}{1+{\it\lambda}^{2}}\log \left(\frac{1+{\it\epsilon}}{1-{\it\epsilon}}\right)-\frac{1}{3}\frac{{\it\lambda}^{2}+(1-{\it\epsilon})^{2}}{1+{\it\lambda}^{2}}\log \left(h_{l}\right)\nonumber\\ \displaystyle & & \displaystyle \,-\frac{2}{3}\frac{{\it\epsilon}}{1+{\it\lambda}^{2}}\int _{-1}^{1}(1+{\it\epsilon}y)\log \left(h(y)\right)\,\text{d}y+\frac{1}{3}\frac{{\it\lambda}^{2}+(1+{\it\epsilon})^{2}}{1+{\it\lambda}^{2}}\log (h_{r}).\end{eqnarray}$$
We further note that there is a maximum possible fluid depth at
$y=-1,h_{l_{M}}$
, such that
$0<h_{l}<h_{l_{M}}$
where
$$\begin{eqnarray}h_{l_{M}}=\left\{\frac{24}{19}\frac{(1-{\it\epsilon})(1+{\it\lambda}^{2})^{3/2}}{{\it\epsilon}{\it\lambda}^{2}\sqrt{{\it\lambda}^{2}+(1-{\it\epsilon})^{2}}}\left[1-\left(\frac{(1-{\it\epsilon}^{2})^{2}+{\it\lambda}^{2}(1-{\it\epsilon})^{2}}{(1-{\it\epsilon}^{2})^{2}+{\it\lambda}^{2}(1+{\it\epsilon})^{2}}\right)^{19/16}\right]\right\}^{-1/3},\end{eqnarray}$$
which is found by considering the limit as
$h_{r}\rightarrow \infty$
of (2.33) at
$y=-1$
.
We consider (3.2) and (3.3) in the limit
$h_{r}\rightarrow \infty$
. In (3.2), all three terms are bounded in this limit, so
$\hat{{\rm\Omega}}$
is bounded. In figure 10(a), the complete expression for
$\hat{{\rm\Omega}}$
(evaluated numerically) is plotted, along with the first two terms in (3.2), showing the area reaching a limiting value (approximately 2.1 for the parameter values chosen). It can be shown that the first three terms in (3.3) are bounded as
$h_{r}\rightarrow \infty$
, and hence
$$\begin{eqnarray}\hat{Q}\sim \frac{1}{3}\frac{{\it\lambda}^{2}+(1+{\it\epsilon})^{2}}{1+{\it\lambda}^{2}}\log (h_{r})\quad \text{as }h_{r}\rightarrow \infty .\end{eqnarray}$$
This behaviour is illustrated by figure 10(b), for the case
${\it\lambda}=0.5,{\it\epsilon}=0.5$
. We have thus confirmed that for large fluxes (
$\hat{Q}\rightarrow \infty$
), the qualitative behaviour of
$h_{l},h_{r}$
and
$\hat{{\rm\Omega}}$
is the same as that found in Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013).
The maximum fluid depth at the inner wall,
$h_{l_{M}}$
is plotted against
${\it\epsilon}$
for different values of
${\it\lambda}$
in figure 11. For
${\it\epsilon}<1,h_{l_{M}}$
is bounded. In the limit
${\it\epsilon}\rightarrow 0$
, (3.4) reduces to
$h_{l_{M}}=6^{-1/3}$
, which is independent of
${\it\lambda}$
. Limits of
$h_{l_{M}}$
as
${\it\lambda}\rightarrow 0$
and
${\it\lambda}\rightarrow \infty$
are also plotted (dashed curves). It can be seen from the plot, and confirmed by taking limits, that as
${\it\epsilon}\rightarrow 1,h_{l_{M}}\rightarrow \infty$
for any
${\it\lambda}>0$
.
Figure 12 shows the effect of increasing the flux,
$\hat{Q}$
, on the cross-sectional area of the flow,
$\hat{{\rm\Omega}}$
. As shown above, and in Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013),
$\hat{{\rm\Omega}}$
approaches an upper bound as
$\hat{Q}$
becomes large. Roughly speaking, a flux of around 2 to 3 is sufficient for
$\hat{{\rm\Omega}}$
to approach its limiting value, corresponding to
$h_{l}$
nearing its limiting value
$h_{l_{M}}$
, and in the specific case
${\it\epsilon}=0.5$
and
${\it\lambda}=0.5$
, equivalent to
$h_{r}\approx 10$
. This suggests that a flux of roughly 2 to 3 is representative of large flux. In this regime, increasing
${\it\epsilon}$
increases the cross-sectional area. However when flux is small a different relationship is observed. Figure 12(b) magnifies the
$0\leqslant \hat{Q}\leqslant 0.5$
region of figure 12(a), showing the difference more clearly. We see that
$\hat{{\rm\Omega}}$
does not vary significantly until
${\it\epsilon}$
nears unity, when it decreases. Thus, for small flux, cross-sectional area is weakly dependent on
${\it\epsilon}$
over much of the parameter space, but for large flux the dependence is stronger. Equivalent plots, omitted for brevity, with fixed
${\it\epsilon}$
and varying
${\it\lambda}$
show little variation between curves for different values of
${\it\lambda}$
, showing that cross-sectional area depends weakly on the slope of the channel bottom.
The dependence of (a) cross-sectional area
$\hat{{\rm\Omega}}$
, and (b) flux
$\hat{Q}$
on
$h_{r}$
with
${\it\epsilon}=0.5$
and
${\it\lambda}=0.5$
. Note the different
$x$
-axes. The solid curves are given by (3.2) and (3.3) respectively; the dashed curves are (a) the first two terms, and (b) the last term of these expressions.
4. Conclusions
We have considered flow in helically-wound channels of rectangular cross-section, assuming the flow to be helically-symmetric, and that the typical depth of the fluid is small so that a thin-film approximation is appropriate. The Navier–Stokes equations were expressed in a non-orthogonal coordinate system, then transformed to three orthogonal directions:
$\boldsymbol{e}_{{\it\beta}}$
in the direction of increasing angle
${\it\beta}$
along the helical centreline,
$\boldsymbol{e}_{r}$
in the horizontal radial direction, and
$\boldsymbol{e}_{n}$
normal to the bottom of the channel.
Maximum fluid depth at the inside wall,
$h_{l_{M}}$
, against
${\it\epsilon}$
. Different curves correspond to different values of
${\it\lambda}$
, the arrow showing the direction of increasing
${\it\lambda}$
. Dashed curves are the
${\it\lambda}\rightarrow 0$
and
${\it\lambda}\rightarrow \infty$
limits.
Fluid cross-sectional area against flux. Plotted for
${\it\epsilon}=0.01$
, 0.15, 0.29, 0.43, 0.57, 0.71, 0.85, 0.99, with
${\it\lambda}=0.5$
. The arrow in (a) shows the direction of increasing
${\it\epsilon}$
.
The most convenient parameters for describing the geometry of the channel were found to be
${\it\epsilon}$
, corresponding to the curvature of the circle obtained by projecting the helical centreline onto a horizontal plane, and
${\it\lambda}$
, the slope of the channel centreline. Effectively, these parameters correspond to the radius
$A$
and the pitch
$2{\rm\pi}P$
of the centreline. We have
${\it\epsilon}=1/A$
and
${\it\lambda}=P/A$
, and for physically meaningful channels we must have
$0<{\it\epsilon}<1$
. For a right-hand helix we take
${\it\lambda}>0$
. The commonly used curvature and torsion of the channel centreline may be expressed in terms of
${\it\epsilon}$
and
${\it\lambda}$
. The general solution we obtain is valid for channels with centrelines of arbitrary curvature and torsion. The case where torsion is large at the inside wall but small at the channel centreline, corresponding to
${\it\epsilon}$
very near 1, was identified in Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013) as requiring further investigation. In that paper, torsion and curvature were considered constant across the width of the channel, but here we account for their variation. This explains differences observed between our results and those of Stokes et al. (Reference Stokes, Duffy, Wilson and Tronnolone2013).
The free-surface profile is given by the solution of the ordinary differential equation (2.16). The equation has an analytic solution, (2.22). Three parameters govern our solutions:
${\it\epsilon}$
, the modified curvature;
${\it\lambda}$
, the slope of the channel; and
$\hat{Q}$
, the dimensionless fluid flux down the channel. We use scalings to relate the Reynolds and Froude numbers to the geometric parameters
${\it\epsilon}$
and
${\it\lambda}$
in order to plot results. The three parameters are all important, and their effects are complicated, but we can make general statements about their effects on the solutions, and some conclusions hold independently of geometry. We find that the fluid depth at the inside wall (
$h_{l}$
) is bounded for all curvature
${\it\epsilon}<1$
, and that the fluid depth at the outside wall becomes unbounded (
$h_{r}\rightarrow \infty$
) as
$\hat{Q}\rightarrow \infty$
for any geometry. We always have transport to the outside of the channel at the free surface, and transport to the inside of the channel near the bottom. Rotating cells always rotate clockwise. Most other conclusions, however, are dependent on geometry.
The effects of geometry and flux can be characterised by considering the balance of gravitational and centrifugal effects. The steeper the channel, the more aligned the channel bottom with the vertical direction, and hence the stronger the gravitational effects on the flow. We have steep channels for large
${\it\lambda}$
, and at the inside wall when
${\it\epsilon}$
is large, and in these cases the free surface has a negative gradient at the inside wall. Centrifugal effects are dominant when the fluid flux is large, when
${\it\epsilon}$
is small, or
${\it\lambda}$
is not large. They drive fluid to the outside wall of the channel which results in a free surface with positive gradient near this wall. For moderate parameter values we see these two effects competing.
For a channel with small curvature (large centreline radius), the solutions depend weakly on the slope of the centreline. We see the fluid depth increasing monotonically across the channel from inside to outside wall, a single clockwise-rotating cell in the cross-section, and the maximum axial velocity at the outside wall. For a channel with large curvature (small centreline radius), the solution depends strongly on the slope of the centreline and the flux, and solutions change qualitatively as these parameters vary. For example, in place of a single rotating cell in the cross-section, two cells may develop, which has not been previously seen from analyses assuming small torsion or small slope.
The impact of flux and geometry on the flow solution, especially for channels with large curvature, has some practical implications. Whilst the shape of a channel (and thus
${\it\lambda}$
and
${\it\epsilon}$
) would not change during operation of a spiral separator, the flux may vary, and in some regimes this would significantly affect the flow. This, in turn, might alter the particle separation capabilities of the spiral. Again we point out that our analysis does not, in general, support the approximation of the primary flow by a free-vortex flow as done by Holland-Batt (Reference Holland-Batt1989, Reference Holland-Batt2009).
Future work will investigate channels with arbitrarily-shaped cross-sections, rather than the simple rectangular channels we have considered here. Spiral particle separators typically feature curved cross-sections and our equations can be generalised for such geometries. We will also extend our model to particle-laden flow in channels of arbitrary curvature and slope (torsion) by coupling to a particle transport model as in the work of Lee et al. (Reference Lee, Stokes and Bertozzi2014), who considered monodisperse particulate flow in channels of small slope.
Acknowledgements
We gratefully acknowledge funding from an Australian Postgraduate Award to D.J.A. and an Australian Research Council Discovery Early Career Researcher Award (DE130100031) to J.E.F.G. We also thank an anonymous referee for several insightful comments that have resulted in substantial improvements to this paper.
Appendix A. Orthogonal equations
The Navier–Stokes equations in the
$r,{\it\beta},n$
coordinate system are presented in Lee et al. (Reference Lee, Stokes and Bertozzi2014), which can be consulted for full details. For completeness, we summarise them below. Note that whilst these equations are valid for a fluid with spatially-varying viscosity, in the current work we consider fluids of constant viscosity, so derivatives of
${\it\mu}$
vanish.
The continuity equation is
$$\begin{eqnarray}\frac{1}{r}\frac{\partial }{\partial r}(rv^{r})+\frac{\partial v^{n}}{\partial n}=0\quad \text{or}\quad \frac{\partial v^{r}}{\partial r}+\frac{v^{r}}{r}+\frac{\partial v^{n}}{\partial n}=0,\end{eqnarray}$$
while the momentum equations are, in the
$\boldsymbol{e}_{r}$
direction,
$$\begin{eqnarray}\displaystyle & & \displaystyle {\it\rho}\left[v^{r}v_{,r}^{r}+v^{n}v_{,n}^{r}-\frac{1}{r{\rm\Upsilon}}(v^{{\it\beta}}-{\rm\Lambda}v^{n})^{2}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =-p_{r}+{\it\mu}\left[v_{,rr}^{r}+v_{,nn}^{r}+\frac{1}{r}\left(v_{,r}^{r}-\frac{v^{r}}{r}\right)\right.\left.+\frac{2{\rm\Lambda}}{r{\rm\Upsilon}}(v_{,n}^{{\it\beta}}-{\rm\Lambda}v_{,n}^{n})\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2{\it\mu}_{,r}v_{,r}^{r}+{\it\mu}_{,n}\left[v_{,n}^{r}+v_{,r}^{n}-\frac{{\rm\Lambda}^{2}}{r{\rm\Upsilon}}v^{n}\right],\end{eqnarray}$$
in the
$\boldsymbol{e}_{{\it\beta}}$
direction,
$$\begin{eqnarray}\displaystyle & & \displaystyle {\it\rho}\left(v^{r}v_{,r}^{{\it\beta}}+v^{n}v_{,n}^{{\it\beta}}+\frac{1}{r{\rm\Upsilon}}v^{r}v^{{\it\beta}}\right)={\it\mu}\left[v_{,rr}^{{\it\beta}}+v_{,nn}^{{\it\beta}}+\frac{1}{r}v_{,r}^{{\it\beta}}-\frac{2{\rm\Lambda}}{r{\rm\Upsilon}}(v_{,n}^{r}-v_{,r}^{n})\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.-\left(\frac{1+2{\rm\Lambda}^{2}}{r^{2}{\rm\Upsilon}^{2}}\right)v^{{\it\beta}}+\frac{2{\rm\Lambda}^{3}}{r^{2}{\rm\Upsilon}^{2}}v^{n}\right]+{\it\mu}_{,r}\left[v_{,r}^{{\it\beta}}-\frac{1}{r{\rm\Upsilon}}(v^{{\it\beta}}-2{\rm\Lambda}v^{n})\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,{\it\mu}_{,n}\left[v_{,n}^{{\it\beta}}-\frac{2{\rm\Lambda}}{r{\rm\Upsilon}}v^{r}\right]-\frac{{\rm\Lambda}}{\sqrt{{\rm\Upsilon}}}{\it\rho}g,\end{eqnarray}$$
and in the
$\boldsymbol{e}_{n}$
direction,
$$\begin{eqnarray}\displaystyle & & \displaystyle {\it\rho}\left(v^{r}v_{,r}^{n}+v^{n}v_{,n}^{n}-v^{r}\frac{{\rm\Lambda}}{r{\rm\Upsilon}}(2v^{{\it\beta}}-{\rm\Lambda}v^{n})\right)=-p_{n}+{\it\mu}\left[v_{,rr}^{n}+v_{,nn}^{n}-\frac{2{\rm\Lambda}}{r{\rm\Upsilon}}(v_{,r}^{{\it\beta}}-{\rm\Lambda}v_{,n}^{r})\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.+\,\frac{1}{r}\left(v_{,r}^{n}-\frac{v^{n}}{r}\right)+\frac{1}{r^{2}{\rm\Upsilon}^{2}}(v^{n}+2{\rm\Lambda}v^{{\it\beta}})\right]+{\it\mu}_{,r}\left[v_{,n}^{r}+v_{,r}^{n}-\frac{{\rm\Lambda}^{2}}{r{\rm\Upsilon}}v^{n}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2{\it\mu}_{,n}\left[\frac{{\rm\Lambda}^{2}}{r{\rm\Upsilon}}v^{r}+v_{,n}^{n}\right]-\frac{1}{\sqrt{{\rm\Upsilon}}}{\it\rho}g.\end{eqnarray}$$
