This document describes an apparent slip enhanced magnetohydrodynamic (MHD) pump. MHD pumps use electromagnetic forces to pump liquids without moving parts, making them reliable for thermal management. However, efficiency is limited by friction losses. The document explores how creating apparent slip through micro-nano structured surfaces on pump walls could significantly reduce losses and enhance pumping rates. Initial analysis found apparent slip could more than double maximum flow rates for relatively short pumps. However, overall performance is dictated by losses from fringe magnetic fields, so slip effects only manifest in long pump sections.

1. Apparent Slip Enhanced Magnetohydrodynamic Pump
Ryan Enright1,†
, Aritra Ghosh1,2
, Anne Gallagher1,3
, Shenghui Lei1
and Tim Persoons3
1
Efficient Energy Transfer (ηET) Department, Bell Labs Ireland, Nokia, Dublin 15, Ireland
2
Mechanical Engineering Department, University of Illinois, Chicago, USA
3
School of Engineering, Trinity College Dublin, Dublin 2, Ireland
†
Corresponding author: ryan.enright@alcatel-lucent.com
Abstract
Reliable thermal management is a key aspect in the operation
of telecommunications equipment, both photonic and
electronic. In liquid-cooled thermal management, conventional
pump designs incorporate moving parts, introducing reliability
concerns. The magnetohydrodynamic (MHD) phenomenon
can be used to fabricate highly reliable, compact, solid-state
pumps that avoid moving parts and make it suitable for rack
level integration in telecommunications equipment. However,
pumping efficiency and performance is limited by pressure
losses in the pump. Here we explore the performance of an
MHD pump demonstrating apparent hydrodynamic and
thermal slip on its major walls (perpendicular to the magnetic
field) that can be achieved using micro-nano structured
surfaces to reduce pressure losses due to friction. Our initial
analysis suggests that, while the pump section performance
can be significantly enhanced by apparent slip, the overall
pump performance is strongly dictated by the nature of
pressure losses associated with the fringing magnetic field at
the pump inlet and outlet such that enhancement due to
apparent slip is only manifested for relatively long pump
sections.
Introduction
Device and component integration is critical to enable next
generations of efficient and scalable electronic and
telecommunications products. The level of integration
required has severe implications for hardware design in
general but even more considerable challenges from a thermal
perspective. The thermal challenge grows with ever-increasing
levels of integration, as the designer struggles to build more
functionality into shrinking package space. Packing more
functionality (e.g., devices and components) into smaller
package footprints leads to increased thermal densities, which
in turn, drives the development of new thermal solutions [1].
In particular, Galinstan, an electrically conductive liquid metal
(melting point ~ -19 °C, thermal conductivity ~ 16× of water,
viscosity ~ 3× of water), has recently received interest for use
as a high performance coolant for thermal management
applications [2]–[6]. While demonstrating useful
thermophysical properties, it also has the advantage that it can
be pumped reliably using the magnetohydrodynamic (MHD)
effect. Similar to other liquid metals, Galinstan also has a
large surface tension (~ 7× of water) and does not readily wet
the electrically insulating wall material of a typical MHD
pump, displaying advancing contact angles of ~150° in the
smooth surface limit [3]. This leads us to consider the
potential for introducing apparent hydrodynamic slip into the
design of an MHD pump using surfaces structured at capillary
length scales. Indeed, surface structuring and the manipulation
of surface forces in the context of heat transfer has recently
become a widely investigated area for applications including
microchannel heat transfer [3], [7], boiling [8] and
condensation [9], [10].
Here, the efficacy of creating apparent slip at the flow
boundaries is analyzed for a simple Hartmann-type MHD
pump design [11]. Of particular interest is the fact that the
modified velocity profile associated with Hartmann flow
allows relatively small slip lengths (with respect to the channel
dimension) to significantly decrease the fluid resistance [12].
Specifically, we consider a nominally parallel MHD flow
between two plates (Hartmann flow) separated by 2h (see
Figure 1) and characterized by a Hartmann number of Ha ≤
100. For pump walls demonstrating moderate scaled apparent
slip length of l/h ≤ 0.2, we calculate the resulting enhanced
maximum flow rate (based on modified Hartmann flow theory
with slip boundary conditions) of > 2×. However, assessment
of the overall pump performance when considering pressure
losses due to fringing magnetic fields shows that the apparent
slip effect only becomes significant when the relative length of
the pump section becomes large.
KEY WORDS: MHD pump, apparent slip
NOMENCLATURE
b induced magnetic field strength (T)
B0 imposed uniform magnetic field strength (T)
B magnetic field (T)
E electric field (V)
h channel half-height (m)
L channel length (m)
l slip length (m)
p pressure (Pa)
Δp pressure difference (Pa)
Q flow rate (m3
/s)
t thickness (m)
v stream-wise fluid velocity (m/s)
w channel half-width (m)
x stream-wise coordinate
y span-wise coordinate in direction of magnetic field
z span-wise coordinate in direction of electric field
Greek symbols
ϵ channel span wise aspect ratio (= h/w)
ϵ’ channel stream wise aspect ratio (= L/w)
μ fluid dynamic viscosity (Pa.s)
ρ fluid density (kg/m3
)
σ fluid electrical conductivity (S/m)
978-1-4673-8121-5/$31.00 ©2016 IEEE 1030 15th IEEE ITHERM Conference

2. Subscripts
c channel
f fringing field
m mean
ns no slip
s slip
Superscripts
* dimensionless
Figure 1. Schematic of a MHD Hartmann pump section
where flow is generated by the interaction of a fixed
magnetic flux density (Bo) transverse to a current density (j)
imposed by a electric field (E). The walls of the pump
parallel to the current flow direction demonstrate a slip
velocity (vw) characterized by a slip length (l). Adapted
from ref. [13].
Apparent Slip MHD Pump Theory
When an electrically conductive fluid in a rectangular
channel is exposed to a magnetic (B) and electrical (E) field
perpendicular to each other, the fluid experiences a force:
Lorentz force, mutually perpendicular to both the E and B,
and flows in the direction of induced force. A pump operating
with the above principle is known as a magnetohydrodynamic
(MHD) pump [13], [11]. Here, we consider the case of fully-
developed, steady, incompressible, laminar MHD flow in a
parallel plate geometry (ϵ = h/w → 0). We assume that the
flow can be described using the inductionless formulation
such that the magnetic Reynolds number, Rem ( )
1, where the fluid magnetic permeability is approximately
that of free-space μ0 (= 4π ×10-7
H.m-1
) and σ is the fluid
electrical conductivity. The competition between
electromagnetic and viscous forces in an MHD flow are
described by the Hartmann number
Eq. 1
where the geometric scale, h, is the half-channel width, Bo is
the magnitude of the imposed magnetic flux density, σ is the
fluid electrical conductivity and μ is the fluid dynamic
viscosity [13]. We choose as the
velocity scale, where dp/dx is the pressure gradient in the
stream-wise direction, and consider the major walls of the
pump to be electrically insulating such that the wall
conductance ratio c = 0.
Thus, the dimensionless governing equations for flow
momentum and induced magnetic field over the domain
are, respectively,
Eq. 2
and
Eq. 3
where v*
is the dimensionless stream-wise velocity and b*
is
the dimensionless induced magnetic field strength. The
boundary conditions applied at are
Eq. 4
where l*
is the dimensionless slip length and I*
is the
dimensionless electrical current ( ) imposed by
an external power supply.
Solution of the above system of equations gives the following
expressions for the velocity and induced magnetic field
profiles
Eq. 5
and
Eq. 6
We then integrate Eq. 5 over the channel cross-section to
obtain the volumetric flow rate, which corresponds to twice
the dimensionless mean velocity and will be negative when
the fluid is being pumped against an adverse pressure gradient,
Eq. 7
The portion of the flow rate that is independent of current is
called the Hartmann flow rate , which can be used to define
the ratio [13]
Eq. 8
Thus, in order to generate flow against an adverse pressure
gradient (pump operation) requires that . Note that
this ratio holds regardless of slip length so long as the
hydrodynamic boundary conditions are the same for both. We
are further interested in comparing the MHD pumping flow
rate between the slip and no-slip case
Eq. 9
which we find is independent of the injected current. Thus, the
enhancement due to slip in MHD pumping flow rate is
identical to that of a simple pressure driven MHD flow [12].
Next, we consider the efficiency of the pump defined
generally as
Eq. 10
where and are the pressure and
volumetric flow rate developed by the pump, respectively. The
Joule component of the electrical power dissipated by the
pump is where and are the
injected electrical current and pump section electrical
Bo
v
y
x z
E
j
vw = l(dv/dy)w
2h

3. resistance, respectively. For convenience, we return Eq. 7 to
dimensional quantities and then re-scale using the pressure
gradient scale and the velocity scale to arrive
at
Eq. 11
where is the ratio of the back
pressure to the pressure developed by the pump (stall pressure)
in the absence of end losses. Note, this form of the
dimensionless volumetric flow rate scales out the injected
current density. We can then define the pump efficiency in
terms of dimensionless quantities as
Eq. 12
Figure 2 plots the dimensionless pump curve and efficiency of
the pump section (no end losses) for several values of
dimensionless slip length and Ha. For , the
enhancement in pumping section flow rate is rather modest (<
2×) reflecting the fact that the drag reduction is defined by a
shear layer thickness set by the channel dimension. However,
at larger Ha the flow rate enhancement is more pronounced
(>4×) since the shearing layer thickness scales as Ha-1
[12].
Conversely, the pump section efficiency sees a broader and
larger peak enhancement at smaller Ha coinciding with the
fact that the viscous and Lorentz forces are closer in
magnitude.
Figure 2. MHD pump section curves and efficiencies for (a,
b) Ha = 1, (c, d) Ha = 10 and (e, f) Ha = 100. The solid,
dashed and dot-dashed curves are for l*
= 0, 0.02 and 0.2,
respectively.
Clearly, when the predominant loss in the MHD pump is
frictional apparent slip can play a significant role in improving
pump performance. However, in the presence of a spatially
varying (i.e., fringing) magnetic field strength, an MHD flow
undergoes additional three-dimensional forces that introduce a
fundamental pressure loss that has a disadvantageous scaling
with pump size, which we discuss next.
As the fluid enters and leaves the pump section, it is
subject to a large change in magnetic field strength. The
voltage induced upstream between the walls is higher than the
downstream induced voltage, resulting in a current flowing in
stream-wise and counter stream-wise direction near the walls
[13]. To close the electrical path, current also flows in
different cross-stream directions before and after the transition
region. The resulting Lorentz forces act to repel the fluid from
the core, first accelerating and then decelerating the near wall
flow, and vice versa for the core flow.
Moreau et al. (2010) proposes a new simplified model,
which includes inertia terms and the effect of current exchange
between the core of the flow and the Hartmann layers [14].
The governing 3D equations are reduced to a quasi 2D form
for vorticity, stream function and electric potential. A scaling
parameter is introduced to characterize the effect of inertia
, where , the duct aspect ratio is
and w is the pump section half width. They give the
following expression for pressure drop in the presence of an
exponentially decaying fringing magnetic field strictly valid
for , , (inertialess limit)
Eq. 13
where is the channel aspect ratio. Note, for the
remainder of the analysis, we fix the channel aspect ratio at a
maximum of ϵ = 0.1 to be reasonably consistent with the 1D
behavior of the hydrodynamics modeled. We also apply Eq.
13 for cases where Ha < 100. Considering the additional end
loss, we can show that the dimensionless back-pressure
gradient is now
Eq. 14
where is the ratio of the pump section’s length to
half-width. The first term on the right hand side of Eq. 14 is
the pressure gradient generated by the Lorentz force, the
second term is the parasitic pressure gradient due to frictional
pressure losses through the pump section and the third term is
the inlet and outlet fringing field pressure loss given by twice
Eq. 13 divided by the pump section length. Evident from the
fringing field term is the fact that, for fixed channel aspect
ratio, its relative contribution can be minimized by making the
pump section longer [15]. The maximum flow rate is then
found by setting the left hand side of Eq. 14 equal to zero and
re-arranging to give
Eq. 15
Figures 3 - 5 plot the pump curves and efficiencies including
end losses due to the fringing magnetic field with the channel
aspect ratio fixed at ϵ = 0.1. For ϵ’ = 1000 (Figure 3) and Ha =
0 0.5 1 1.5
p*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6
p*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.2
0.4
0.6
0.8
Q
0 0.2 0.4
p*
p
0
0.2
0.4
0.6
0.8
1
Q/Qmax
0 0.5 1
0
0.2
0.4
0.6
0.8
1
a b
c d
e f
ΔPΔPΔP
Q∗
Q∗
/Q∗
max

4. 1, there is a modest reduction in the flow rate and efficiency
compared to the case without end losses (Figure 2) and
apparent slip still results in a significant improvement in
performance. However, as Ha increases there is a marked
reduction in the efficacy of apparent slip and, generally, the
pump flow rate as a result of the Ha3/2
dependency in the end
loss term.
Figure 3. MHD pump section curves and efficiencies for ϵ =
0.1 and ϵ’ = 1000 with (a, b) Ha = 1, (c, d) Ha = 10 and (e, f)
Ha = 100. The solid, dashed and dot-dashed curves are for l*
=
0, 0.02 and 0.2, respectively.
Further decrease in the stream-wise aspect ratio to ϵ’ = 100
(Figure 4) results in almost no effect due to apparent slip for
Ha = 100, while some flow rate enhancement is still possible
for lower values of Ha. This is contrast to the case with no end
losses where the largest gains in pumping flow rate due to
apparent slip are observed at large Ha. Also observable is a
change in the characteristic shape of the pump efficiency
curve from linear to increasingly parabolic at larger Ha. Thus,
the shift towards a more parabolic efficiency curve at
relatively large Ha (≥ 10) is indicative of the importance of
end losses on the overall pump performance.
Figure 4. MHD pump section curves and efficiencies for ϵ =
0.1 and ϵ’ = 100 with (a, b) Ha = 1, (c, d) Ha = 10 and (e, f)
Ha = 100. The solid, dashed and dot-dashed curves are for l*
=
0, 0.02 and 0.2, respectively.
As the relative channel length is shortened further to ϵ’ = 10
(Figure 5), the performance curves for a pump section
demonstrating apparent slip become almost indistinguishable
from the no-slip case and, at the highest Ha, the flow rate is
now three orders smaller than in the case with no end losses.
Interestingly, in the range of Ha = 10 - 100, the pump peak
efficiency is between 10% and 20% and the shape is parabolic,
which compares well with the ~10% pump efficiency found at
Q/Qmax ≈ 0.5 by Ghoshal et al. for Ha ≈ 40 [5]. Thus, despite
the simplistic nature of our analysis, we replicate the salient
features of MHD pump performance in a parameter range
consistent with compact devices suitable for electronic thermal
management applications. This point also serves to highlight
the unfortunate scaling of MHD pump performance for
compact, low aspect ratio form factors most suited for
electronic thermal management applications.
Summary & Conclusions
The effect of hydrodynamic apparent slip on the
performance of a MHD Hartmann pump (quasi 1D flow) has
been analyzed. The results demonstrate that, while significant
enhancements in the MHD pump section performance can be
achieved for modest apparent slip lengths, pressure losses due
to fringing magnetic field effects significantly reduce the
impact of apparent slip on overall device performance. Future
work will explore the impact of apparent slip on a wider range
0 0.5 1
p*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.05
0.1
0.15
0.2
0 0.1 0.2 0.3
p*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Q
0 0.01 0.02
p*
p
0
0.2
0.4
0.6
0.8
1
Q/Qmax
0 0.5 1
0
0.2
0.4
0.6
0.8
a b
c d
e f
ΔPΔPΔP
Q∗
Q∗
/Q∗
max
0 0.2 0.4 0.6
p*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.05
0.1
0.15
0 0.02 0.04
p*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.1
0.2
0.3
0.4
Q 10-3
0 1 2
p*
p
0
0.2
0.4
0.6
0.8
1
Q/Q
max
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
a b
c d
e f
ΔPΔPΔP
Q∗
Q∗
/Q∗
max

5. of pump geometries by implementing the above analysis in
two dimensions to asses under what conditions apparent slip
may improve the performance of compact MHD pumps for
thermal management applications. Furthermore, detailed
studies of MHD flows at apparent slip boundaries are
required. Finally, there is the need to develop structured
surface designs that are compatible with liquid metal working
fluids and robust, in terms of wetting state stability, to ensure
reliable MHD pump operation. More generally, our analysis
serves to restate the issue of end losses due to fringing field
effects that lead to a significant performance trade-off for
compact devices of low aspect ratio. In the context of thermal
management applications, this would suggest that there is a
need to further explore MHD pump designs that maintain
reliability, but with an improved performance/form factor
trade-off.
Figure 5. MHD pump section curves and efficiencies for ϵ =
0.1 and ϵ’ = 10 with (a, b) Ha = 1, (c, d) Ha = 10 and (e, f) Ha
= 100. The solid, dashed and dot-dashed curves are for l*
= 0,
0.02 and 0.2, respectively.
Acknowledgments
Bell Labs Ireland acknowledges the financial support of the
Industrial Development Agency (IDA) Ireland.
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0 0.05 0.1 0.15
p
*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.01
0.02
0.03
10
-3
0 2 4
p*
p
0
0.2
0.4
0.6
0.8
1
0 0.5 1
0
0.02
0.04
0.06
0.08
0.1
0.12
Q 10-4
0 1 2
p*
p
0
0.2
0.4
0.6
0.8
1
Q/Qmax
0 0.5 1
0
0.05
0.1
0.15
0.2
0.25
a b
c d
e f
ΔPΔPΔP
Q∗
Q∗
/Q∗
max