# Fluid Drag Reduction by Magnetic Confinement

Arvind Arun Dev<EMAIL_ADDRESS>Institut de Physique et Chimie
des Matériaux de Strasbourg, UMR 7504 CNRS-UdS, 67034 Strasbourg, France
Peter Dunne  Institut de Physique et Chimie des Matériaux de Strasbourg, UMR
7504 CNRS-UdS, 67034 Strasbourg, France  Thomas M. Hermans  Université de
Strasbourg, CNRS, UMR 7140, 67000 Strasbourg, France  Bernard Doudin
<EMAIL_ADDRESS>Institut de Physique et Chimie des Matériaux
de Strasbourg, UMR 7504 CNRS-UdS, 67034 Strasbourg, France

###### Abstract

The frictional forces of a viscous liquid flow are a major energy loss issue
and severely limit microfluidics practical use. Reducing this drag by more
than a few tens of percent remain elusive. Here, we show how cylindrical
liquid–in–liquid flow leads to drag reduction of 60–99% for sub-mm and mm-
sized channels, regardless of whether the viscosity of the transported liquid
is larger or smaller than that of the confining one. In contrast to
lubrication or sheath flow, we do not require a continuous flow of the
confining lubricant, here made of a ferrofluid held in place by magnetic
forces. In a laminar flow model with appropriate boundary conditions, we
introduce a modified Reynolds number with a scaling that depends on
geometrical factors and viscosity ratio of the two liquids. It explains our
whole range of data and reveal the key design parameters for optimizing the
drag reduction values. Our approach promises a new route for microfluidics
designs with pressure gradient reduced by orders of magnitudes.

###### keywords:

American Chemical Society, LaTeX

Université de Strasbourg, CNRS, UMR 7140, 67000 Strasbourg, France IR,NMR,UV

##  1 Introduction

Friction is a multifaceted problem existing in most physical processes and
accounts for almost 25% of energy loss in the world 1, 2. Fluid friction or
hydrodynamic viscous drag is decisive when designing energy efficient large
scale flow systems 3. At smaller, microfluidic sizes, channels provide unique
advantages for handling small volumes with reduced waste and manufacturing
costs, ideal for drug delivery and discovery 4, 5. Flow-focusing using
microfluidic channels for drug delivery and chaotic mixing of reactants forms
an essential part of pharmaceutical research 6. However, delivery of highly
concentrated drugs through sub-mm diameter medical needles is difficult
because of large drag due to the nonlinear increase in viscosity with
increasing concentration 7. This results in pumping forces beyond the range of
manual injection 8. Furthermore, to limit shear damage, aggregation, and
sedimentation in medically significant flows, like blood through tubes and
arteries 9, reducing viscous drag is essential. Enabling drag control is also
key in studying the response of cancer cells 10 and viruses 11. Reducing the
hydrodynamic viscous drag and shear are key design issues and have led to many
solutions, like mixing with additives 12, surface chemical treatment 13, or
thermal creation of two-phase systems14; while Nature’s way, the Lotus
effect15, has steered research towards engineered (super)hydrophobic
surfaces16, 17, 18 with stabilized19, 20 liquid/gas interfaces 21, 22, 23. To
overcome the drawback of the limited time stability of interstitial gas,
oil/liquid infused surfaces have been proposed 24, 25. Establishing liquid
walls by means of an interstitial liquid lubricant is of particular interest
for microfluidic applications, with several strategies documented in the
literature26, 27, 28, 29, 30, 31. Nanostructured surfaces are shown to enhance
lubricant retention thereby extending the lifetime of planar lubricated
surfaces32. Hydrodynamic fluid focusing and microdroplets microfluidics are
designed for surrounding the transported liquid material with an immiscible
liquid envelope33, 34. Core annular flows, where the transported liquid is
lubricated coaxially using a liquid of lower viscosity is proposed to handle
highly viscous flows in microfluidics 8. However, forming a stable annular
flow is difficult, and the risk of draining the lubricating liquid35, 36
limits the maximum achievable drag reduction. The use of ferrofluids as
confining liquid can improve the stability and draining issues, taking
advantage of magnets to generate a force field that hold the confining
ferrofluid in place.

Figure 1:  a) Experimental setup for differential pressure measurement (P1,
P2). An antitube of diameter $d$ inside a rigid cavity of diameter $D$ is
shown with a velocity profile following the hypothesis of a 2-fluid, 3-region
model. The red line between the ferrofluid and antitube depicts the liquid-
liquid interface. b) X-ray absorption contrast image of the fluidic circuit,
inside a cavity surrounded by magnets. The brighter central flowing liquid
(antitube) is surrounded by a darker immiscible ferrofluid. c) Pressure-drop
versus flow rate for an antitube of diameter 200 $\mu$m, with inset showing
the X-ray image near the outlet (bright central part is antitube). Markers are
experiments, the continuous line is the expectation from a Poiseuille flow of
diameter $d$, and the dashed line is the prediction of the model detailed in
the text below. d) Corresponding drag reduction with respect to Poiseuille
flow. The viscosity ratio between the transported liquid and the ferrofluid is
$\eta_{r}$=11.

Recent results using ferrofluid-infused surfaces showed superhydrophobic
behaviour 37, and early pioneers 38 showed how pressure gradients can be
reduced by an intermediate ferrofluid layer in large pipes39, 40. Here we show
how drag reduction, defined as the percentage change in friction factor13, 41,
23, 42, 43, 44, is remarkably enhanced in milli- and micro-fluidic circuits
using magnetic confinement of ferrofluid lubricants with appropriate magnet
assemblies. Its key element is the implementation of our recent design of
quadrupolar confining magnetic field45, capable of stabilizing the cylindrical
flow of a diamagnetic liquid (called ‘antitube46’) inside a ferrofluid
envelope attracted towards the magnets. Our experimental system forms an ideal
liquid–in–liquid tube system where the lubricating liquid forms a perfect
concentric confinement for the flowing liquid, using a magnetic force design
best suited to preserve the cylindrical geometry and providing optimum
robustness of the system. This allows us to compare and verify a simple,
axially-symmetric theory with experiments for a significant range of
hydrodynamic parameters. The study goes beyond the assumption of
unidirectional flow of the involved liquids, ideal for maximum drag reduction
29, 25, 8. We previously discussed the expected and measured equilibrium
diameter $d_{eq}$ of the antitube under static conditions (no flow),
stabilized by the equilibrium between magnetic pressure and surface tension
between the liquids45. Here we investigate the dynamic case, where the fluid
viscosities are key properties, and show how antitube circuits can exhibit
drastically reduced pressure drop for significant ranges of viscosity and flow
conditions.

##  2 Experiments

The fluidic cell design follows our previous work 45 and is sketched in Figure
1a. To perform the flow experiments, the four-magnet assembly were housed in a
3D-printed support with built-in fluidic connectors for pressure measurements
(cross-section Fig. 1a). For larger diameter antitubes ($>$ 1 mm) we used a
support with a magnet spacing $w$ = 6 mm, internal cavity with diameter $D$ =
4.4 mm and inlet-outlet separation $L$ = 52 mm, and for sub-mm antitube $w$ =
0.9 mm, $D$ = 600 µm, and $L$ = 12 mm. The 3D printed cavity ($D$, Fig. 1a) is
first filled with the non-magnetic liquid, then slowly replaced by injecting
the ferrofluid with resulting formation of the antitube.
We control the diameter d, which is always $>$ $d_{eq}$, by varying the
injected trapped volume of ferrofluid. We determined the antitube diameter
from X-ray absorption contrast images (see supporting information S.I. S1),
either with 20 $\mu$m resolution setup45 (Fig. 1b), or a micro computed
tomography system (RX-Solutions EasyTom 150/160) with 6 $\mu$m resolution,
Fig. 1c inset. The antitube absorbs X-rays less than the ferrofluid, resulting
in an image with a bright central region surrounded by two darker bands Fig.
1b). At the top and bottom of the image are the magnets which are opaque to
X-rays at this thickness, and appear as two black regions, separated from the
ferrofluid by the nearly-invisible 3D printed support. Near the ends of the
magnets, the antitubes widen due to the fringe magnetic fields; fortuitously,
these curved inlets and outlets improve the trapping of the ferrofluid (see
S.I. S1).
We measured the pressure drop, $\Delta P$, between inlet and outlet, with
Honeywell pressure transducers (HSCDLND001PG2A3, HSCDLNN400MGSA5) under a
constant flow set by a syringe pump (Harvard apparatus PHD 2000). Fig. 1c
shows the measured pressure drop as a function of flow rate of glycerol (Sigma
Aldrich) flowing through EMG900 (FerroTech) ferrofluid (markers). This
pressure drop is more than one order of magnitude lower than a Poiseuille flow
in a solid channel with equivalent diameter (200 $\pm$ 6 $\mu$m). This
illustrates how it is beneficial to have a cylindrical flowing liquid confined
by a liquid interface rather than by solid walls.
The reduction in pressure drop is better described by the dimensionless
friction factor, $f$. For a fluid with density $\rho$, the measured pressure
drop $\Delta P$ resulting from a flow rate $Q$ through a tube of diameter $d$
and length $L$ is related to the experimental friction factor by 23

$f_{exp}=\frac{\pi^{2}\Delta Pd^{5}}{8\rho Q^{2}L}$ (1)

The drag reduction factor $DR$ is then defined as13

$DR=\frac{f_{P}-f_{exp}}{f_{P}}\times 100$ (2)

which is the percentage change of the measured friction factor $f_{exp}$
compared to the factor $f_{p}$ under the same flow rate that follows
Poiseuille’s law. At a solid wall boundary, $f_{p}$ = 64/Re , where the
Reynolds number Re = $\rho$Ud/$\eta$ is defined for diameter $d$, density
$\rho$ , viscosity $\eta$ and average velocity $U$ corresponding to the flow
rate $Q$. Due to the liquid-liquid interface, the zero-velocity boundary
condition at the transported liquid wall (antitube–ferrofluid interface) does
not apply, and the deviation from Poiseuille’s law results in hydrodynamic
drag reduction. Fig. 1d) shows the experimental drag reduction (markers) based
on Eq. 1 and Eq. 2 and from data in Fig. 1c), with values reaching 95.5 %.

Figure 2:  Drag reduction, $DR$, for three antitube diameters as a function of
flow rate for a) honey as the transported liquid with APG314 ferrofluid (F1)
as the confining liquid, b) honey with APGE32 ferrofluid (F2), c) glycerol
with F1 and d) glycerol with F2. $\eta_{r}$ is the ratio between the antitube
and ferrofluid viscosities. Legend shows the antitube diameter in mm. Lines,
markers and filled markers compare model, simulations, and experiments,
respectively.

A more complete insight into drag reduction possible values is gained by
testing how it evolves under flow when varying the viscosity of both
transported and confining liquids as well as the antitube diameter. We limit
ourselves to viscous liquids, for measurable pressure differences and
investigate near mm-sized channels for straightforward X-ray imaging. The
transported viscous liquids were glycerol (Sigma Aldrich) and honey (Famille
Michaud), and the confining ferrofluids used were APG314 (F1) and APGE32 (F2)
(FerroTech), resulting in experiments on four combinations of magnetic and
non–magnetic liquids.

Figure 3:  Pressure drop reduction. a) Honey-APG314 ($\eta_{r}$=52), b) Honey-
APGE32 ($\eta_{r}$=7.0), c) Glycerol-APG314 ($\eta_{r}$=4.8), d) Glycerol-
APGE32 ($\eta_{r}$=0.65). $\alpha_{p}$ = $\Delta P_{exp}$ / $\Delta P_{p}$ is
the ratio of pressure drop for solid wall tube to the antitube with identical
diameter. The error bars are smaller than the size of markers where not
visible.

The viscosities of these liquids were measured using a viscometer (Anton Paar
MCR 502, see SI S2); for glycerol it is constant at 1.1 Pa.s and for honey
11.99 Pa.s under strain rate of 100 s${}^{\text{-}1}$ (maximum strain rate in
our experiments with honey in antitube is below 10 s${}^{\text{-}1}$). Our
measurements span a viscosity ratio $\eta_{r}$ between transported and
confining liquids both larger ($\leq$52) and smaller ($\geq$0.65) than one,
with data for three different antitube diameters and four different viscosity
ratios summarized in Fig. 2 and Fig.3.
All 4 combinations of transported and confined liquids show remarkably high
drag reductions ranging from 60 % to 99.3 % (Fig. 2). Drag reduction increases
with increasing viscosity ratio, with a maximum for Honey-F1 with $\eta_{r}$ =
52 (Fig. 2a). On the contrary, no drag reduction is expected when the ratio
goes to zero, describing an infinitely viscous envelope, or a solid wall.
However, in contrast to prior expectations 25, large drag reduction can still
be achieved, even if the confining ferrofluid has a larger viscosity than the
transported one, such as for Glycerol-F2, where a drag reduction of up to 80%
is observed (Fig. 2d). Additionally, the drag reduction increases with
decreasing antitube diameter (Fig. 2a,b,c,d), which is beneficial when
miniaturizing the fluidic circuit.
Alternatively, the large drag reduction can be expressed as an improvement
ratio $\alpha_{p}$ = $\Delta P_{exp}$/$\Delta P_{p}$, which compares the
measured pressure drop in a liquid-walled interface, $\Delta P_{exp}$, to a
solid-wall interface, $\Delta P_{p}$, of equivalent diameter, d. Fig. 3 shows
more than two orders of magnitude of improvement ($\alpha_{p}$) can be
achieved. For $\eta_{r}$ = 52, the antitube system results in 157 times less
pressure drop than the solid walled tube (Fig. 3a, red markers). Interestingly
a viscosity ratio $\eta_{r}$ = 0.65 still results in an almost 6 times smaller
pressure drop (Fig. 3d), and the improvement ratio $\alpha_{p}$ increases with
decreasing antitube diameter (Fig. 3a,b,c,d).

##  3 Modelling

We explain our results using a two-fluid model, following the previous
pioneering works 40, based on the steady-state one dimensional Navier-Stokes
equation with velocity as a function of radius in a cylindrical geometry, $u$
= $u$($r$), under modified boundary conditions. Note that we present below
equations that differ from those presented in literature40, motivated by the
need to compare the analytical expressions to numerical simulations. We
checked that the outcomes of both analytical approaches are identical, under
the hypothesis of non-deformable interfaces detailed below. A key ingredient
of the model is the occurrence of a counter flow within the confining
ferrofluid (Fig. 1a) resulting from avoiding drainage of the ferrofluid by
means of the magnetic sources. This suppression is due to the non-uniform
magnetic fields at the inlet and outlet opposing any egress of ferrofluid. As
the ferrofluid cannot escape but noting that a) flux must be conserved and b)
that the drag reduction should result from a non-zero velocity at the
ferrofluid-antitube interface, a return path for the ferrofluid flow must
exist. The simplest hypothesis is illustrated by the velocity profile in Fig.
1a, where we define three regions: I inside the antitube, II the part of the
ferrofluid that travels alongside the antitube flow, and III where counter-
flow occurs. The non-dimensional governing equations for the three regions (i
= I, II, II) are given by

$\frac{1}{Re_{i}\thinspace r^{\star}}\frac{\partial}{\partial
r^{\star}}\left(r^{\star}\frac{\partial u^{\star}_{i}}{\partial
r^{\star}}\right)=\frac{\partial P^{\star}_{i}}{\partial z^{\star}}$ (3)

where $u^{\star}_{i}$,$r^{\star}$ are dimensionless velocity and coordinates
scaled by the average velocity $u_{m}$ and the diameter $d$ of the region I
(antitube), respectively. The dimensionless pressure is defined as
$P^{\star}_{i}$=$P_{i}$/$\rho_{i}u_{m}^{2}$ with the corresponding Reynolds
numbers for each region being $Re_{i}$=$\rho_{i}u_{m}d$/$\eta_{i}$ . Note that
the pressure gradients along the main flow in both ferrofluid regions are
equal, under the hypothesis that these two regions do not mix, resulting in
the absence of pressure gradient along $r$, and therefore $\frac{\partial
P_{II}^{\star}}{\partial z^{\star}}=\frac{\partial P_{III}^{\star}}{\partial
z^{\star}}$.

The magnetic-nonmagnetic interface is modelled as a non-deforming fixed liquid
wall. This hypothesis limits us to experiments under low-enough flow values.
The case of diameter depending on flow and position along $z$ is beyond the
scope and minimal set of hypotheses of this paper, as it is related to the
interface stability of non-miscible liquids, and the need to introduce
explicitly magnetic stress forces in the model. We checked by X-ray imaging
that we are not in this case, aware the deviations for the model will increase
with the flow rate, as deformations cannot be neglected anymore.
We assume here that the pressure gradients in regions I and II are different,
to ensure a non-deformable interface. Note that earlier research40 presented
the analytical model as a 2 fluid model with equal pressure gradient,
$\frac{\partial P_{I}^{\star}}{\partial z^{\star}}=\frac{\partial
P_{II}^{\star}}{\partial z^{\star}}$., but since the numerical simulations do
not take into account the magnetic forces field, it is essential to define the
magnetic-nonmagnetic interface as non-deformable wall resulting in
$\frac{\partial P_{I}^{\star}}{\partial z^{\star}}\neq\frac{\partial
P_{II}^{\star}}{\partial z^{\star}}$. (see section 4 and SI S3). We insist,
that so long as the shear stress and velocity boundary conditions at the
magnetic-nonmagnetic interface are satisfied, the assumption of pressure
gradients do not affect the drag reduction calculations.
Our hypothesis allows us to consider a diameter $d$, set experimentally by the
amount of ferrofluid trapped in the cavity, independent of the flow rate and
treat the problem numerically to match our experimental conditions. These
governing equations are solved for all velocities
($u^{\star}_{I}$,$u^{\star}_{II}$,$u^{\star}_{III}$), pressure gradient
$\frac{\partial P_{III}^{\star}}{\partial z^{\star}}$ and thickness $n$ of
region II, using boundary conditions depicted in the Fig. 1b: finite velocity
at the antitube centre, zero velocity at the solid wall and at interface of II
and III, continuity of velocity and shear stress at interfaces I-II and II-
III. Along with these boundary conditions, the volume conservation of
ferrofluid dictates that the flow rate in region II and III must be equal,
$Q_{II}^{\star}=Q_{III}^{\star}$. We present two models, one with no
assumption (full model) and another with assumption $\frac{\partial
P_{II}^{\star}}{\partial z^{\star}}=0$, which explicitly shows the
contribution of geometric and fluid parameters for drag reduction. In the full
model Eq. 3 expands to

$\frac{1}{Re_{III}\thinspace r^{\star}}\frac{\partial}{\partial
r^{\star}}\left(r^{\star}\frac{\partial u^{\star}_{III}}{\partial
r^{\star}}\right)=\frac{\partial P^{\star}_{III}}{\partial z^{\star}}$ (4)
$\frac{1}{Re_{II}\thinspace r^{\star}}\frac{\partial}{\partial
r^{\star}}\left(r^{\star}\frac{\partial u^{\star}_{II}}{\partial
r^{\star}}\right)=\frac{\partial P^{\star}_{II}}{\partial z^{\star}}$ (5)
$\frac{1}{Re_{I}\thinspace r^{\star}}\frac{\partial}{\partial
r^{\star}}\left(r^{\star}\frac{\partial u^{\star}_{I}}{\partial
r^{\star}}\right)=\frac{\partial P^{\star}_{I}}{\partial z^{\star}}$ (6)

Solving Eq. 4 to Eq. 6 with the boundary conditions mentioned gives the
analytical expressions of the flow rate in the antitube as:

$Q^{\star}_{I}=\frac{\pi Re_{I}}{4}\frac{\partial P^{\star}_{I}}{\partial
z^{\star}}\left[\frac{1}{32}+a_{3}-a_{4}-a_{5}\right]$ (7)

and the resulting friction factor written as

$f_{A}=64/Re_{I}\thinspace\beta$ (8)

where

$\beta=32(a_{5}+a_{4}-a_{3})-1$ (9)

and $a_{3}$, $a_{4}$, $a_{5}$ are scalar constants that can be expressed as
explicit functions of $d$, the thicknesses $n$ of the region II and $t_{f}$ of
the ferrofluid.
This ‘full model’ is therefore fully analytically solvable; however, Eq. 8 and
Eq. 9 together presents a complex expression where the contribution of fluid
and geometric properties are hidden. Simplified expressions illustrating
better the key contributions to drag reduction are obtained by neglecting the
pressure gradient in region II. We show below that this artificial hypothesis
has limited impact on the accuracy of the results. This approximation
$\frac{\partial P_{II}^{\star}}{\partial z^{\star}}=0$ results in a simplified
expression for $\beta$:

$\beta_{0}=1+4\ln\bigg{(}1+\frac{2n_{0}}{d}\bigg{)}\eta_{r}$ (10)

where $n_{0}$ is the thickness of the ferrofluid in region II under this
approximation. $\eta_{r}=\eta_{a}/\eta_{f}$ is the ratio of viscosity of
antitube liquid ($\eta_{a}$) and ferrofluid ($\eta_{f}$). Note that we take
the viscosity of the ferrofluid at saturation in a magnetic field (see SI S2).
Complete and explicit equations for the scalar constants and simplified model
are given SI 4.

###  3.1 4\. Numerical visualization of counter flow

Greater confidence in our equations results were gained by computational fluid
dynamics (CFD) simulations using ANSYS CFX 18, where detailed information on
the occurrence of counter flow and the resulting velocity vector field were
obtained.

Figure 4:  . Simulated visualization of counter flow. a) and b) Velocity
vectors near the outlet showing counter flow for Honey-F1 and Glycerol-F2
respectively. c) and d) comparison of the non-dimensional velocity profile of
the full analytical model and simulations in Fig. 4(a) and Fig. 4(b). e) and
f) comparison using simplified analytical model neglecting pressure gradient
term. Lines are model predictions and markers are simulations calculations.
Inset shows the magnified view of velocity profile in an antitube. The
antitube diameter for left and right column is 1.73 mm and 1.54 mm
respectively. $\eta_{r}$ is viscosity ratio.

In the numerical simulations, we solve the three-dimensional Navier–Stokes
equation by considering the magnetic-nonmagnetic interface as a non-deformable
fixed liquid wall of infinitesimal width. The axial velocity and shear stress
are equal on both sides of the wall, i.e., continuous across the interface.
Since the magnetic field gradient only exists in the radial direction far from
the inlet or outlet, no magnetic body force is considered on the ferrofluid.
The finite curved edges of ferrofluid near inlet and exit of the flow are
modelled as a free slip wall with zero normal velocity (no flow across the
curved edges) (See SI S3 for numerical algorithm). The numerical simulations
also consider the shear dependent viscosity of ferrofluids. As illustrated in
Fig. 4, a counter flow occurs in the ferrofluid close to the outer wall for
Honey-F1 with $d$ = 1.73 mm (Fig. 4a) and Glycerol-F2 with $d$ = 1.54 mm (Fig.
4b), and flow rate fixed at $Q$ = 300 $\mu$l min${}^{\text{-}1}$. Good
agreement between numerical and analytical velocity profiles using the full
model for both Honey-F1, $\eta_{r}$ = 52 (Fig. 4c), and Glycerol-F2,$\eta_{r}$
= 0.65, (Fig. 4d) validates our numerical algorithm. To more accurately model
the experimental system shown in Fig. 1, we extended the simulations to
consider the finite length of a device, and the effect of fringe fields on the
shape of interface at the inlet and outlet (curved inlet and outlet, Fig. 1b,
see SI S3), beyond the hypothesis of the infinite tube of the analytical
model. Numerical simulations were found to reproduce well the drag reduction
data in Fig. 2a), while systematically underestimating observed drag reduction
for $\eta_{r}$ = 0.65 (Fig. 2d). Note that the numerical drag reduction is
calculated using Eq. 1 and Eq. 2 with $\Delta P$ obtained from numerical
simulations.
The simplified model obtained by neglecting pressure gradient term in region
II is also compared with the numerical simulations shown in Fig. 4e and Fig.
4f. The difference between the two analytical models is apparent on comparing
Fig. 4c with Fig. 4e and Fig. 4d with Fig. 4f.

###  3.2 Discussion

The analytical model with a minimum set of hypotheses presented here captures
reasonably well the experimental measurements for all measurements presented
in Fig. 1 and Fig. 2 for mm and sub-mm channels. The modified Reynolds number
is the non-dimensional governing parameter and spans four orders of magnitude
in our experiments.

Figure 5:  Comparison of experimental fexp (markers) and analytical $f_{A}$
(line) friction factors. Inset gives the diameter of antitubes (mm) for each
case. Re denotes the Reynolds number and $\beta$ is the scaling factor (= 1
for solid-walled tube).

For a cylindrical tube flow $f$ = 64/Re , the liquid-in-liquid system results
in $f_{A}$=64/Re$\beta$ for the same flow rate $Q$, illustrating how the
friction factor is reduced by the scaling factor $\beta$. Fig. 5 illustrates
how the calculated friction factor $f_{A}$ which is a function of the modified
Reynolds number (from Eq. 3.6) compares with the experimental one $f_{exp}$
computed using Eq. 2.1. The friction factor $f_{A}$ is like the fit parameter
presented in literature40. We provide here full analytical expressions for
$f_{A}$, necessary for the equation of the drag reduction defined as
percentage change in friction factor $f_{A}$ with respect to friction factor
for Poiseuille flow $f_{p}$. Similar rescaling of the friction factor was also
discussed for a highly water-repellent wall system13. Although the model
cannot completely account for minor offsets observed at low viscosity ratio
values, where deformation of the antitube starts to take place, Fig. 5
nevertheless illustrates the broad range of fluidic conditions that the model
can apply to. Note that a significant variation of drag reduction with flow
rate is found for the largest viscosity ratio ($\eta_{r}$ = 52). In such
cases, the pressure drop becomes very small at small flow rates, and the
difficulties in neglecting the pressure loss related to the interconnects and
pressure indicators limit the reliability of the data, systematically
underestimating the drag reduction values. We also expect that more
complicated fluid velocity profiles along $z$ can develop, especially for high
viscosity confining liquids where the magnetic/non-magnetic interface might
deform significantly due to the high pressure drop but taking them into
account is beyond our current model.

$Case$ | d (mm) |  $t_{f}$ (mm) |  $n$ (mm) | $\beta$ |  $n_{0}$ (mm) | $\beta_{0}$
---|---|---|---|---|---|---
| 2.45 | 0.98 | 0.34 | 41.6 | 0.38 | 58.68
Honey-APG314 | 1.77 | 1.32 | 0.49 | 74 | 0.53 | 98.3
| 0.98 | 1.71 | 0.67 | 151.68 | 0.73 | 183.35
| 2.40 | 1.00 | 0.36 | 6.74 | 0.39 | 9.12
Honey-APGE32 | 1.85 | 1.28 | 0.47 | 10.21 | 0.51 | 13.39
| 1.20 | 1.60 | 0.61 | 17.44 | 0.67 | 21.49
| 2.30 | 1.05 | 0.38 | 5.24 | 0.41 | 6.95
Glycerol-APG314 | 1.72 | 1.34 | 0.50 | 7.98 | 0.54 | 10.26
| 1.07 | 1.67 | 0.65 | 13.62 | 0.70 | 16.46
| 2.3 | 1.05 | 0.38 | 1.57 | 0.41 | 1.8
Glycerol-APGE32 | 1.53 | 1.44 | 0.54 | 2.11 | 0.58 | 2.44
| 1.05 | 1.68 | 0.65 | 2.74 | 0.71 | 3.12
Table 1: Comparison of the full and simplified models

The simplified expression $\beta_{0}$ in Eq. 10, illustrates the deviation
from the asymptotic $\beta_{0}$ = 1 value for solid walls, and is a simple but
explicit way to quantify the reduction in the friction resulting from liquid-
in-liquid flow. Compared to $\beta$, this approximation underestimates the
frictional drag (visible in Fig. 4) and becomes more apparent when the
thickness of the ferrofluid decreases (see Table 1). However, $\beta_{0}$,
explicitly reveals the contributions of the antitube geometry and fluid
properties: the drag reduction can be tuned by the choice of viscosities
($\eta_{r}$) and the amount of ferrofluid trapped in the device cavity, i.e.
antitube diameter $d$, and ferrofluid thickness $t_{f}$.

####  3.2.1 Conclusions

We have studied the flow of viscous liquids through cylindrical liquid-in-
liquid tubes where the confining liquid is held in place by a quadrupolar
magnetic field. Our results show that drag reductions exceeding 99% can be
achieved by exploiting the non-zero velocity of a viscous liquid at its
interface with the encapsulating liquid. The friction reduction is quantified
by rescaling of the Reynolds number with a factor $\beta$ in Eq. 9. The drag
reduction improves when decreasing the diameter of an antitube relative to its
surrounding ferrofluid, or when increasing the ratio of the antitube to
ferrofluid viscosities. The former is relevant for the needs and length-scales
of microfluidics, while the latter indicates that large drag reduction is
expected when flowing highly viscous liquids. Moreover, with antitube
diameters as small as 10 $\mu$m already achievable45, antitube diameter to
ferrofluid thickness ratios of order 100 are within reach. Therefore, very
large drag reduction in microfluidic channels is possible for a broad range of
confining liquid viscosities. Downsizing or designing magnetic force gradients
along the flow direction can also further enhance the stability of the
ferrofluid against shearing, paving the way to both high velocity and low
viscosity fluidic applications in domains ranging from nanofluidics to marine
or hydrocarbon cargo transport.

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 766007. We also acknowledge the support of the University of
Strasbourg Institute for Advanced Studies (USIAS) Fellowship. This work was
also partly supported by IdEx Unistra (ANR 10 IDEX 0002), and by SFRI STRAT’US
project (ANR 20 SFRI 0012) and EUR (QMat-ANR-18-EUR-0016) under the framework
of the French Investments for the Future Program. We thank A. Zaben, Prof.
Cēbers and Dr. Kitenbergs of MMML lab, Riga, Latvia for ANSYS CFD software
facility and magnetoviscosity measurements. The authors thank Prof. Gauthier,
Antoine Egele and Damien Favier of Institut Charles Sadron, Strasbourg, France
for the use of the EasyTom X-Ray Tomography facility.

The supporting information consist of a PDF document with

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Procedures for characterising antitube diameters.

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Viscosity measurements of antitube liquids and confining ferrofluids.

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Numerical simulation procedures and flow chart.

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Details of analytical model used in this article.

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