Laminar flow characterization using low-fieldmagnetic resonance techniques

Cite as: Phys. Fluids 33, 103609 (2021); doi: 10.1063/5.0065986Submitted: 6 August 2021 . Accepted: 19 September 2021 .Published Online: 11 October 2021

Jiangfeng Guo (郭江峰), Michael M. B. Ross, Benedict Newling, Maggie Lawrence, and Bruce J. Balcoma)

AFFILIATIONS

UNBMRI Centre, Department of Physics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada

a)Author to whom correspondence should be addressed: bjb@unb.ca

ABSTRACT

Laminar flow velocity profiles depend heavily on fluid rheology. Developing methods of laminar flow characterization, based on low-fieldmagnetic resonance (MR), contribute to the widespread industrial application of the MR technique in rheology. In this paper, we outline thedesign of a low-cost, palm-sized permanent magnet with a 1H resonance frequency of 20.48MHz to measure the laminar flow. The magnetconsists of two disk magnets, which were each tilted at an angle of 1� from an edge separation of 1.4 cm to generate a constant gradient,65G/cm, in the direction of flow. Subsequently, a series of process methods, for MR measurements, were proposed to characterizeNewtonian and non-Newtonian fluid flows in a pipe, including phase-based method, magnitude-based method, and a velocity spectrummethod. The accuracy of the proposed methods was validated by simulations, and experiments in Poiseuille flow and shear-thinning flowwith the designed magnet. The new velocity profile methods proposed are advantageous because the MR hardware and measurement meth-ods are simple and will result in a portable instrument. Although the governing equations are complicated, the data analysis isstraightforward.

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I. INTRODUCTION

Laminar flow, in fluid dynamics, is characterized by fluid microelements flowing in parallel layers.1 The characterization of laminarflow, including measurements of average velocity and velocity profile,is of considerable value in chemical and allied processing industries.2–6

Various fluids exhibit different flow behaviors under laminar condi-tions, dependent on fluid rheological properties.7–13 Laminar flowcharacterization is therefore helpful to characterize rheologicalproperties.

Magnetic resonance/magnetic resonance imaging (MR/MRI) isattractive for flow measurements because of its noninvasive capabili-ties for measuring optically opaque objects.13–15 Multiple MR- andMRI-based methods have been reported to characterize fluid flow.MRI-based methods for measuring flows are based on the applicationof magnetic field gradients, including frequency-, phase-, and motion-encoding gradients, to yield quantitative information about velocitydistributions of the flowing fluid.14–20 There are also some modifiedMRI-based methods that only use one type of gradient.21–24 MRI-based methods, resolving flow velocity profiles, have been used tomeasure various types of flows, for example, laminar flow,19,20,25

turbulence,26–29 and flow in porous media.30–32 Unfortunately, thesemeasurements are predominantly performed on laboratory research

instruments. The chief challenges to MRI-based methods, for wide-spread industrial application, are the expense of the superconductingequipment and the demand of high-performance gradient systems.MR-based methods for flow measurements do not need complicatedequipment, a permanent magnet with a static magnetic field gradientis sufficient. MR methods therefore have the prospect of industrialapplication and great potential in characterizing fluid flow.

MR-based methods for characterizing flow are based on the effectof flow on the MR signal. Suryan measured MR signals of flowingwater in a U-tube between the pole pieces of a magnet at 20MHz, andreported the continuous wave MR signal increased as the partially sat-urated spins were replaced by unsaturated flowing spins.33 Singerexploited this principle to demonstrate in vivo flow measurements.34

Hirschel and Libello showed the steady state MR signal is a function offluid velocity in the presence of flow.35 Arnold and Burkhart employeda spin echo to study the influence of flow on MR signal under laminarflow conditions.36 Hayward et al., Stejkal, and Grover and Singerextended this work using a pulsed field gradient technique.37–39

Since the effect of flow on the MR signal was first studied, multi-ple MR-based methods for characterizing flows have been reported.These methods can be classified into two main categories: (1) netphase accumulation-based techniques38–40 and (2) magnitude-based

Phys. Fluids 33, 103609 (2021); doi: 10.1063/5.0065986 33, 103609-1

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time-of-flight techniques.41–44 Net phase accumulation-based techni-ques rely on the application of a constant or pulsed magnetic field gra-dient in the direction of flow. The phase shift of the signal detected isproportional to the average velocity component in the direction of thegradient. For example, Song et al. employed the technique of MultipleModulation Multiple Echoes (MMME) to measure fluid flow with astatic magnetic field gradient.40 A series of coherence pathways weregenerated by the MMME technique, and each of them exhibits a phaseshift dependent on average velocity. Magnitude-based time-of-flighttechniques are based on the variation of signal magnitude proportionalto the quantity of excited spins in the detector, related to flow velocity.These techniques do not require the use of any magnetic field gra-dients. It is therefore very popular to employ this technique in low-costlow-field MR spectrometers. Beyond the two basic kinds of MR-basedmethods, O'Neill et al. invented an Earth's field magnetic resonanceflow meter to measure the velocity probability distribution and T1-velocity correlation probability distribution of multiphase flows.45–47

From the descriptions of existing low-cost MR-based methods, itcan be found these methods focus on the average velocity of fluid flow,which is insufficient to support the study of fluid rheology. The flowbehavior index is an important parameter in fluid rheology, which hasa direct impact on the flow velocity profile under laminar conditions.Determination of the velocity profile is, therefore, helpful to study fluidrheology. In this paper, to make possible widespread industrial appli-cation of the MR technique in fluid rheology, we designed a low-cost,low-field, palm-sized permanent magnet with a flow-directed constantmagnetic field gradient. Furthermore, we proposed correspondingMR-based methods to characterize laminar flows in a pipe, includingaverage velocity and velocity profile, based on first odd echoCarr–Purcell–Meiboom–Gill (CPMG) MR measurements. The pro-posed methods were verified by simulations and flow experiments onthe designed magnet.

II. METHODOLOGYA. Equipment used

1. Sensor design and hardware

A magnet constructed with a separation between two disk mag-nets, as a function of distance along the symmetry axis, will intuitively

lead to a magnetic field gradient directed along the symmetry axis. Forthe purposes of flow sensitization, this idea works remarkably well.Garwin and Reich48 made a conceptually similar field modificationwith an aluminum plate added to an electromagnet for the purposes ofdiffusion sensitization in MR measurements that were published veryearly.

The optimal separation and tilt angle for a desired constant gradi-ent of 60G/cm between the two N52 NdFeB K&J Magnetics(Pipersville, PA) disk magnets of 5.1 cm diameter and 1.3 cm thicknesswas determined via CST Studio Suite (Providence, RI) simulation. Inthe simulation, each disk magnet was tilted at an angle of 1� from anedge separation of 1.4 cm between the magnets. 60G/cm with thisgeometry was judged to be near ideal for the flow measurements envis-aged. Figure 1(a) depicts two disk magnets, each rotated by 1� aboutthe y axis.

A 6� 6� 4 cm3 casing fabricated from Garolite G-10 (McMasterCarr, Elmhurst, IL) was machined to house the magnets. The casingwas divided into two separate pieces, where each piece had a slot intowhich a magnet could be placed. Each slot was machined to permitthe 1� tilt relative to the symmetry axis. The casing included a 1.2 cmdiameter cylindrical hole through the shell, along the direction of theimposed gradient, to permit the placement of glass tubing to supportthe flow. A four-turn solenoidal RF coil with 1.0 cm inner diameterwas formed around a glass pipe, and was capacitively matched to50 X. The RF coil, fabricated from 0.8mm diameter copper wire, wascentered in the Proteus (PROTon Embedded sUbmersible Sensor)magnet.49 The interior and exterior of the Proteus magnet werewrapped with 0.2mm copper tape to limit external RF interferenceand suppress acoustic ringing. Figure 1(b) shows a photo of theProteus magnet.

Magnetic field plots of the sensitive spot in the Proteus magnetwere acquired with a LakeShore 460 3-Channel Gaussmeter(Westerville, OH) connected to a BiSlide Positioning System andVXM Stepping Motor Controller (Velmex Inc., Bloomfield, NY).Magnetic field data were read and processed through a customMATLAB script (Mathworks, Natick, MA).

Figure 2 depicts the simulated two-dimensional (2D) magneticfield magnitudes of the tilted Proteus magnet in the Y-Z, X-Y, and

FIG. 1. (a) Diagram of the tilted disk magnets and (b) photo of the Proteus magnet. The two disk magnets are 5.1 cm in diameter and 1.3 cm in thickness. They were sepa-rated and tilted by 1� to generate a constant magnetic field gradient directed along the x axis in the central region of the two magnets.

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X-Z planes. The magnetic field has contributions from Bx, By, and Bz,but in all cases Bz dominates. The proposed gradient strength wasselected on the basis of the ability to observe flow rates within an aver-age velocity range of 1–5 cm/s with echo times below 1ms. Figure 3(a)is the 1D profile of the magnetic magnitude field along the central axisof the X-Z plane, taken from Fig. 2(c). The 60G/cm gradient Gx isobserved6 0.5 cm about the origin in Fig. 3(a). Simulation shows, inthe region of the RF probe, that gradient Gx is uniform to within3G/cm when displaced 3.45mm off the central axis in the X-Y planeand within 2G/cm in the X-Z plane. In order to ensure that the phaseshift measured would be observed in a region of constant gradient, anRF coil with a length of 0.32 cm and inner diameter of 1.0 cm wasplaced in the centermost region.

Figure 3(b) is the experimental field plot of 1D magnetic fieldmagnitude along the X central axis, Y¼ 0, Z¼ 0. As before, the mag-netic field has contributions from Bx, By, and Bz. The finite size of thefield sensor permitted only on axis measurement. From simulation,the region of constant gradient, on axis, was6 0.5 cm about the origin.The experimental field plot showed the region of constant gradientwas reduced to 6mm compared to simulation. The experimental field

plot yields a Gx value, near the origin, of 64G/cm. The discrepanciesin spatial extent and Gx value from simulation are likely due to non-ideal disk magnets. The RF probe was centered about the magnet ori-gin. The Proteus magnet was tuned to a 1H frequency of 20.48MHz.Average velocity measurements of known water flows were performedto confirm the Gx gradient amplitude of 65G/cm.

The RF probe was attached to a TecMag (Houston, TX) trans-coupler with a k/4 cable via BNC connectors. The transcoupler wasjoined to a Tomco Technologies (Stepney, Australia) 250W RF ampli-fier and a L3 Nard-MITEQ (Hauppauge, NY) 0.7–200MHz preampli-fier with a Mini-Circuits (Toronto, Ontario) 30MHz low-bandpassfilter. Radio frequency excitation and signal detection were accom-plished using a four-turn solenoidal RF coil driven by a TomcoTechnologies 250W RF amplifier. The system was run by a TecMagLapNMR console.

2. Flow network

The flow network was identical to the setup previously employedin Ref. 50 for time-of-flight flow experiments. In this configuration, a

FIG. 2. Simulated 2D magnetic field magnitudes of the tilted Proteus magnet in the central 2D Y-Z, X-Y, and X-Z planes. (a) The field plot in the Y-Z plane is largely uniformwithin the volume of interest. The field contour interval is 7 G. (b) and (c) illustrate the constant gradient in the region of measurement (dashed box). Field contour intervals are6 and 12G, respectively.

FIG. 3. (a) 1D field magnitude along the central line of the X-Z transverse plane, obtained from Fig. 2(c). (b) Experimental 1D field magnitude measured along X with Y¼ 0,Z¼ 0 of the 1� tilted Proteus magnet. The field strength and region of the desired constant gradient are determined coarsely compared to the simulated results, but are, never-theless, similar. Discrepancies observed in the experimental field compared to the simulated field are likely indicative of imperfections in the disk magnets.

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gravity-fed flow from a reservoir suspended several feet above theProteus magnet was refreshed via a pump from another reservoir atfloor level to establish a constant flow through the Proteus magnet. Toensure a constant fluid level in the upper reservoir, and therefore aconstant pressure head driving the flow, a submersible pump (HidomElectric, Shenzhen, China) provided more inflow to the upper reser-voir than was flowing through the magnet. An overflow was installedin the upper reservoir to return excess water to the lower reservoir. AMasterflex Variable-Area Flowmeter (Cole-Parmer model # RK-32460-34, Montreal, Canada) was used to control the average flowrate. Flexible Fisherbrand clear PVC tubing (Fisher ScientificCompany, Ottawa, Canada) with an inner diameter of 0.8 cm wasincorporated throughout the construction of the flow network exceptfor the portion running through the magnet, where a 70 cm length ofglass tubing with an ID of 0.67 cm was utilized.

B. Basic fluid dynamics model

When a power-law fluid flows in a circular pipe under laminarconditions, the shear stress is proportional to the shear rate raised tothe power n, where n is the flow behavior index. Assuming the flowdirection to lie along the x axis, the constitutive equation can beexpressed as51

rxr ¼ m _cn; (1)

where rxr is the shear stress on the radial position r;m is the fluid con-sistency coefficient; and _c is the shear rate and it can be expressed as

_c ¼ dv rð Þdr

; (2)

where vðrÞ is the flow velocity at the radial position r.The axial momentum of the fluid in a pipe can be written as

0 ¼ � dpdx

þ 1r

@ rrxrð Þ@r

; (3)

where dpdx ¼ Dp

L is the pressure gradient along the pipe. Integrating Eq.(3) with respect to r, we can obtain

rxr ¼ rDp2L

: (4)

Substituting Eqs. (2) and (4) into Eq. (1), we can obtain

rDp2L

¼ mdv rð Þdr

� �n

: (5)

Integrating Eq. (5) with respect to r, we can obtain the flow velocityprofile in a pipe51 as follows:

v rð Þ ¼ DpR2mL

� �1n nRnþ 1

1� rR

� �1nþ1

!: (6)

The volume flux Q of the pipe flow can be expressed as

Q ¼ðR02prv rð Þdr ¼ pnR3

1þ 3nDpR2mL

� �1n

¼ vavgpR2: (7)

Equation (7) shows vavg ¼ nR1þ3n ðDpR2mLÞ

1n. Substituting vavg into Eq. (6),

we obtain

v rð Þ ¼ 3nþ 1nþ 1

vavg 1� rR

� �1nþ1

!: (8)

For computational convenience, we define n0 ¼ 1n þ 1, and then Eq.

(8) can be simplified as

v rð Þ ¼ n0 þ 2n0

vavg 1� rR

� �n0 !

: (9)

Equation (9) shows the maximum flow velocity vmax at the centerof the pipe, under laminar conditions, is related to vavg , described as

vmax ¼ n0 þ 2n0

vavg : (10)

Different fluids exhibit different n0 for pipe flow. For n0 < 2.0(n > 1), the fluid exhibits shear-thickening behavior. For n0 ¼ 2.0(n ¼ 1), the fluid shows Newtonian behavior. For n0 > 2.0 (n < 1),the fluid shows shear-thinning behavior. We plot three typical 1Dvelocity profiles (n0 ¼ 1.5, 2.0, and 5.0) of laminar flow at the samevavg¼ 5 cm/s, in Fig. 4. The velocity profile shape depends on n0, andthe larger the n0, the lower the maximum flow velocity at the samevavg. The velocity profile becomes increasingly blunt (more plug-like)as n0 increases. Therefore, n0 and vavg are the two necessary parametersfor determining the flow velocity profile.

C. Flow parameter determination from CPMGmeasurement

1. Phase-basedmethod

The CPMGMRmethod is composed of a 90x� pulse followed bya series of 180y� pulses with 2s time spacing. This measurement canbe described as

900x � s� 1800y � s� echoh i

j: (11)

FIG. 4. Three typical 1D velocity profiles of laminar flows at the same vavg¼ 5 cm/swith n0 ¼ 1.5 (Blue dashed line), 2.0 (Red solid line), and 5.0 (Green dashed line).The velocity profile shape varies from n0, and the larger the n0, the lower the maxi-mum flow velocity. The velocity profile becomes increasingly blunt (more plug-like)as n0 increases.

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Each 180y� pulse refocuses the magnetization to generate an echo atthe time t ¼ 2js, where j denotes the jth spin echo during the CPMGmeasurement. When the CPMG measurement is performed for fluidflow with a constant magnetic field gradient (G) in the direction offlow, a phase shift for all odd echoes will occur. For a constant velocityvc, the net phase accumulation /c of any odd echoes can be expressedas16,52

/c ¼ cGvcs2; (12)

where c is the gyromagnetic ratio.For a general steady flow, where flow velocity depends on posi-

tion, the net phase accumulation /odd of odd echoes can be expressedin discrete form as follows:

/odd ¼1PcGs2

XPp¼1

vp

0@

1A ¼ cGs2

1P

XPp¼1

vp

0@

1A ¼ cGvavgs

2; (13)

where vavg is the average velocity. For a general laminar flow in a pipe,we can also calculate the net phase accumulation /odd of odd echoesby integration as follows:

/odd ¼

ð ð/ rð Þrdrdhð ðrdrdh

¼

ðR0/ rð ÞrdrðR0rdr

¼

ðR0cGv rð Þs2� �

rdrðR0rdr

¼ cGs2

ðR0

vmax 1� rn0

Rn0

� �� �rdrðR

0rdr

¼ n0

n0 þ 2vmaxcGs

2

¼ cGvavgs2; (14)

where R is the pipe radius.It can be seen from Eqs. (13) and (14) that the /odd depends

on G, vavg, and s, and therefore vavg can be determined fromvavg ¼ /odd= cGs

2. The vavg, determined from net phase accumula-tion of an echo, suffers from the signal-to-noise ratio (SNR) of theecho measured. To obtain vavg more reliably, multiple first oddecho phase accumulations at different s were employed in thispaper. The odd echo net phase accumulations at different s2 were

fitted by Eq. (13), and the slope k can be used to determinevavg ¼ k=cG.

2. Magnitude-based method

Under laminar flow conditions, the flow velocity profile is a dis-tribution of velocities, which results in a distribution of accumulatedphases at the odd echoes, leading to a change in signal magnitude.50

Employing odd echo magnitudes is therefore a workable strategy todetermine flow parameters. Assuming complete polarization, the oddecho magnitude Modd detected with a flow-directed gradient can beexpressed as

Modd ¼ M0MRM/; (15)

where M0 is the equilibrium magnetization value, which dependson the detected fluid type and quantity. MR is the normalizedmagnitude caused by the decay of NMR signal through the so-called spin–spin relaxation process, which has a time constant T2

and depends on the relaxation property of the detected fluid. M/

is the normalized magnitude resulting from phase accumulation,related to the velocity distribution. MR and M/ are not magni-tudes but factors reducing the magnitude M0 in a strict sense. M0

and MR are independent of the fluid field, and thus M/ can beobtained from dividing the acquired magnitude for stationarysolution by the acquired magnitude for flowing fluid with thesame acquisition parameters.

The normalized signal S/ of all odd echoes due to phase accumu-lation is the same, and can be expressed as

S/ ¼

ð ðexp �i/ rð Þð Þrdrdhð ð

ds¼

ðcos / rð Þð Þrdrð

rdr� i

ðsin / rð Þð Þrdrð

rdr;

(16)

where i is the imaginary unit and ds ¼ rdrdh is the differential ofcross-sectional area. From Eq. (16), the normalized real signal SRe

¼Ð

cos ð/ðrÞÞrdrÐrdr

and the normalized imaginary signal SIm

¼ �Ð

sin ð/ðrÞÞrdrÐrdr

due to phase accumulation for all odd echoes. For a

circular pipe with a radius R, they can be modified as follows:

SRe ¼

ðR0cos / rð Þð ÞrdrðR

0rdr

¼

ðR0cos X 1� rn

0

Rn0

� �� �rdrðR

0rdr

¼eXi �ið Þ 2

n0 C2n0

� �� C

2n0;Xi

� �� �þ e�Xii

2n0 C

2n0

� �� C

2n0;�Xi

� �� �n0X

2n0

; (17)

and

SIm ¼ �

ðR0sin / rð Þð ÞrdrðR

0rdr

¼ �

ðR0sin X 1� rn

0

Rn0

� �� �rdrðR

0rdr

¼ ieXi �ið Þ 2

n0 C2n0

� �� C

2n0;Xi

� �� �� e�Xii

2n0 C

2n0

� �� C

2n0;�Xi

� �� �

n0X2n0

; (18)

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where X ¼ n0þ2n0 cGvavgs2 and Cða; xÞ ¼

Ð1x wa�1e�wdw. Detailed der-

ivations of Eqs. (17) and (18) are given in the Appendix. The normal-ized magnitude M/ of odd echoes due to phase accumulation can becalculated from

M/ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSReð Þ2 þ SImð Þ2

q

¼ 2

n0X2n0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC

2n0

� �� C

2n0;Xi

� �� �C

2n0

� �� C

2n0;�Xi

� �� �s:

(19)

It can be seen from Eq. (19) thatM/ is not only related to instru-ment and acquisition parameters, G and s, but also to laminar flowparameters, n0 and vavg. Based on Eqs. (14) and (19), we present severalschemes for determining the laminar flow parameters, as follows:

Scheme 1: We use only one first odd echo to calculate laminarflow parameters. The vavg is determined from the echo net phaseaccumulation using Eq. (14), and then n0 is solved by Eq. (19) withthe calculated vavg. This scheme, involving one odd echo datapoint, suffers from noise, and thus the results have a poor reliabil-ity for realistic flow measurements.Scheme 2: Based on the magnitude data of first odd echoes at dif-ferent s, the vavg and n0 are directly fitted by Eq. (19). Due to thecomplexity of Eq. (19), insufficient data detected might affect thefitting accuracy for this scheme.Scheme 3: Based on the net phase accumulation of first odd echoesat different s, the vavg is fitted by Eq. (14). Subsequently, the n0 isfitted by Eq. (19) based on the magnitude data of first odd echoesat different s and the fitted vavg.

Schemes 1–3 all employ first odd echo signals detected withCPMG measurement with a flow-directed constant G, to solve for thelaminar flow parameters, n0 and vavg. We note it would be feasible touse vavg determined from a known volumetric flow rate and pipediameter as an alternative to the vavg determined by the net phaseaccumulation of odd echoes in Schemes 1 and 3. We note in schemes1–3 above the use of the first odd echo in a CPMG acquisition meansthat subsequent data points result from additional first odd echoes innew CPMG acquisitions with different s.

3. Velocity spectrummethod

The complex signal SðqÞ of all odd echoes in a CPMG measure-ment after removing the diffusion effect can be expressed as21,53

S qð Þ ¼ðþ1

�1p vð Þ exp �iqvð Þdv; (20)

where q ¼ cGs2, and pðvÞ is the velocity spectrum. Equation (20)shows that SðqÞ is the Fourier transformation of pðvÞ with respect to v.pðvÞ can therefore be determined by the inverse Fourier transforma-tion of SðqÞ with respect to q, described as

p vð Þ ¼ðþ1

�1S qð Þ exp iqvð Þdq: (21)

For an acquisition system with a constant magnetic field gradient,we can only change s during the measurement. Note that this method

does not involve the use of phase-, frequency-, and motion-encodingmagnetic resonance gradients. When the magnetic field gradient isparallel to the flow direction, q is a positive number, and thus Eq. (21)can be modified as

p vð Þ ¼ðþ1

0S qð Þ exp iqvð Þdq: (22)

To meet the uniform sampling requirement of q with a Fouriertransformation, we increase s2 with a constant step size. The Field ofFlow (FOF) is determined by 2p=Dq, where Dq ¼ cGDðs2Þ is the stepsize of q. Since FOF should be no less than the maximum velocity offlow, a short step size of s2 is required. To obtain a velocity spectrumwith an adequate resolution, a large number of different s2 values maybe indicated. A compromise may be required for each measurement toconstrain the total measurement time. Velocities greater than the max-imum velocity in the flow field should ideally have zero amplitude inthe velocity spectrum. One can therefore determine the maximumvelocity based on the break point in the velocity spectrum. Combinedwith the average velocity from the net phase accumulation of odd ech-oes or volumetric flow rate and pipe diameter, n0 can be solved forwith a known maximum velocity.

For a Poiseuille flow, one can directly use the velocity spectrumto calculate the flow profile by20,21

r2 vð Þ ¼ R2 1�ðvvmin

p vð Þdv" #

; (23)

where rðvÞ is the radial position associated with a flow velocity, andvmin is the minimum velocity at r¼R.

III. NUMERICAL SIMULATIONS AND ANALYSES

To assess the presented methods in Sec. IIC for determining theflow parameters in a circular pipe, a few numerical simulation testswere performed. Owing to the use of normalized data during the sim-ulations, we can replace the simulations on the whole circular pipewith those on a circular cross section. The cross section was discretizedusing a 500� 500 grid. A diagram of the discretized cross section via a10� 10 grid is shown in Fig. 5. We note a larger grid (>5002) did notshow appreciable changes for the normalized signal simulated. Theintersections on the grid in the circular cross section were consideredduring the simulations. The velocity at each intersection can be calcu-lated based on the flow velocity profile [Eq. (9)]. The normalized realand imaginary signal due to a phase accumulation, for odd echoes dur-ing CPMG measurement, can be written in the discrete form asfollows:

SRe ¼ 1N

XNi¼1

cos cGvis2

� �; (24)

and

SIm ¼ � 1N

XNi¼1

sin cGvis2

� �; (25)

where N is the number of intersections in the circular cross sectionand vi is the velocity at the ith intersection. Therefore, the net phaseaccumulation /odd of odd echoes can be calculated by

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/odd ¼ arctan � SImSRe

� �¼ arctan

PNi¼1

sin cGvis2ð ÞPNi¼1

cos cGvis2ð Þ

26664

37775: (26)

The normalized magnitudeM/ of odd echoes due to phase accu-mulation can be determined from

M/ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSReð Þ2 þ SImð Þ2

q

¼ 1N

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNi¼1

cos cGvis2ð Þ" #2

þXNi¼1

sin cGvis2ð Þ" #2vuut : (27)

The radius of the cross section was set to 1 cm and the magneticfield gradient was set to 65G/cm during the simulations. To match theexperimental data, Gaussian noise with a SNR of 50 was added tothe real and imaginary signal. We note that the effects of B1

inhomogeneity, resonance offset, and pulse imperfection are not con-sidered in the simulations. We note that all simulations employed firstodd echoes in model CPMG echo trains. We verified the effectivenessof the three methods in Sec. II C to determine the laminar flow param-eters via processing the simulated data, as follows.

A. Phase-basedmethod verification

The normalized real and imaginary signals of first odd echoes forthe three types of laminar flows shown in Fig. 4, at seven different sranging from 100 to 400 ls with a step size of 50 ls, were calculatedbased on Eqs. (24) and (25). After adding noise, the net phase accumu-lations can be determined by Eq. (26). Figure 6 shows the relation ofthe net phase accumulation /odd of first odd echoes to s

2 for the lami-nar flows, where /odd were the mean of ten separate simulations anderror bars were determined by their standard deviations. From Fig. 6,we can see that the net phase accumulation of first odd echoes, at thesame vavg, are very close, independent of the flow type.

The simulated data were fitted employing Eq. (13), and the vavgwere determined to be 5.016 0.01, 4.946 0.02, and 5.046 0.02 cm/sfor Poiseuille flow, shear-thickening flow, and shear-thinning flow,respectively, which are similar to the model vavg¼ 5 cm/s, within 1%.These indicated that the net phase accumulation of first odd echoescan be used to determine the average velocity of any type of laminarflow.

B. Magnitude-based method verification

To ensure the accuracy of the magnitude-based method, it isdesirable to employ a broader range of echo times. Here, the normal-ized magnitudes M/ of first odd echoes due to phase accumulation at19 different s ranging from 100 to 1000 ls with a step size of 50 lswere employed. Figure 7 shows the relation of M/ to s2 for laminarflows, where M/ are the mean of ten separate simulations and errorbars are determined by their standard deviations. The trends of M/

significantly differ with the type of laminar flows at the same vavg.Schemes 2 and 3 were both employed to process the simulated magni-tude data to obtain laminar flow parameters. For scheme 3, the fittedvavg from the phase-based method in Sec. IIIA was used. The fittedresults of the two schemes are shown in Fig. 7.

For the Poiseuille flow, n0 ¼ 1.926 0.03 and vavg¼ 4.936 0.06 cm/s by scheme 2, and n0 ¼ 1.986 0.02 by scheme 3. For theshear-thickening flow, n0 ¼ 1.546 0.03 and vavg¼ 5.056 0.10 cm/s by

FIG. 5. Diagram of the pipe cross section discretized via a 10� 10 grid. Weassumed all fluids were positioned on the intersections of the grid in the circularcross section, and there is nothing inside the single squares. We can calculate the

flow velocity at each intersection, for example, vE ¼ n0þ2n0 vavgð1� rn

01

Rn0 Þ at the inter-section E, where r1 is the distance of the intersection E from the center O.

FIG. 6. Fitted results of /odd of first odd echoes at different s2, for Poiseuille flow (a), shear-thickening flow (b), and shear-thinning flow (c). The fitted relationships between/odd of first odd echoes and s

2 were (a) /odd ¼ ð5:016 0:01ÞcGs2, (b) /odd ¼ ð4:946 0:02ÞcGs2, and (c) /odd ¼ ð5:046 0:02ÞcGs2.

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scheme 2, and n0 ¼ 1.486 0.02 by scheme 3. For the shear-thinningflow, n0 ¼ 4.996 0.03 and vavg¼ 4.976 0.03 cm/s by scheme 2, andn0 ¼ 5.046 0.03 by scheme 3. The fitted n0 and vavg by scheme 2 agreewith the model, within 4% and 1%. Similarly, the fitted n0 by scheme 3are close to the model, within 1%, for the three types of laminar flows.The fitted results indicated that schemes 2 and 3 can both be used todetermine the laminar flow parameters by processing the normalizedmagnitude of odd echoes during CPMG measurement with a flow-directed gradient. The error of the fitted n0 by scheme 3 is slightlylower than those by scheme 2, due to there being fewer parameters ofscheme 3, revealing that scheme 3 is somewhat superior to scheme 2.

Based on the fitted flow parameters from schemes 2 and 3, theflow velocity profiles were reconstructed and then compared with themodel, as shown in Fig. 8. From Fig. 8, we can see that the recon-structed 1D flow velocity profiles for the three laminar flows are veryclose to the model. The error plots show that the velocity errors in thepipe are less than 0.2 cm/s, revealing the effectiveness of themagnitude-based method to determine flow parameters.

C. Velocity spectrummethod verification

To meet the requirements of the velocity spectrum method, thedata must be sampled with a fixed increment of s2. During the simula-tions, the normalized signals at 128 different s2 ranging from

6.25� 10−4 to 31.75 ms2 with a step size of 0.25 ms2, for the threeflows, were calculated and then Gaussian noise was added. The FOFwas therefore 14.45 cm/s. Before undertaking Fourier transformationof simulated data, the exponential filtering method54 was employed toimprove the resolution of the velocity spectrum. The velocity spectrafor Poiseuille flow, shear-thickening flow, and shear-thinning flow areshown in Fig. 9. The increase in amplitude at the maximum displayedvelocity in the real velocity spectra is an artifact.

It can be seen from Fig. 9 that the characteristics of the velocityspectrum vary depending upon the laminar flow type. Based on theircharacteristics, the flow type can be identified qualitatively. Moreimportantly, the real, imaginary, and magnitude velocity spectra allhave a break point at the same v, and the vmax of the laminar flows canbe determined from the break point. The discontinuity is most readilyobserved for Poiseuille flow and shear-thinning flow, panels (a) and(c) in Fig. 9. The elevated baseline in panels (a)–(c) of Fig. 9 is due toFourier transformation of solely +q data points. The vmax of the threeflows were 10.056 0.11 cm/s, 11.636 0.11 cm/s, and 7.116 0.11 cm/s,respectively. Combined with their vavg from net phase accumulation inSec. IIIA, their n0 were determined to be 1.99, 1.48, and 4.87, respec-tively, by Eq. (10). Their calculated n0 values are similar to the models,within 3%.

For Poiseuille flow, the 1D flow velocity profile was reconstructedby Eq. (23) based on the real velocity spectrum, as shown in Fig. 10.

FIG. 7. Fitted results of M/ of first odd echoes at different s2 using scheme 2 (Green solid line) and scheme 3 (Blue dashed line) for Poiseuille flow (a), shear-thickening flow(b), and shear-thinning flow (c). All the M/ for different laminar flows decrease with oscillations as s2 increases. The trends of M/ significantly differ with the type of laminarflows at the same vavg. The fitted curves by schemes 2 and 3 are both in agreement with the simulated data.

FIG. 8. Comparisons of 1D flow velocity profile models (Red solid line) with those reconstructed by scheme 2 (Green dashed line) and scheme 3 (Blue dashed line) forPoiseuille flow (a), shear-thickening flow (b), and shear-thinning flow (c). The bottom subplots represent the absolute error between the reconstructed profile and model, show-ing that the velocity errors for any laminar flows in the pipe are less than 0.2 cm/s.

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The reconstructed velocity profile coincides with the model, and theabsolute error is no more than 0.1 cm/s, which verifies the feasibility ofthe velocity spectrum method to reconstruct the Poiseuille flowprofile.

This paper emphasizes laminar flow characterization by simplenumerical fitting methods, rather than the Fourier transformationmethod. We focused on the experimental verifications of the phase-based method and the magnitude-based method and did not performexperiments to validate the velocity spectrum method. In future, wewill experimentally examine the velocity spectrummethod.

IV. EXPERIMENTS

Two solutions, one Newtonian fluid and one shear-thinningfluid, were prepared for flow experiments. Distilled water and glycerolwere mixed in a ratio of 6:1 to prepare the Newtonian fluid. Xanthangum solution, an example of a shear-thinning fluid,23,55 was preparedin concentration of 0.42wt.% using distilled water. For xanthan gum,complete dissolution was achieved by stirring for 10 h using a low gearmixer (Mastercraft, Toronto, Canada). The two solutions were dopedwith 0.33wt.% copper sulfate to reduce their T1 lifetimes to ensure themeasured fluid was completely polarized. T1 lifetimes of the glycerol/

distilled water solution and the xanthan gum solution were 42 and39ms.

Glycerol/distilled water solution flow experiments were per-formed at flow rates of 406 1 and 786 1ml/min to produce averagevelocities of 1.896 0.05 and 3.696 0.05 cm/s. Reynolds numbers were82 and 160 for the two flows, and thus the flows are laminar. Xanthangum solution flow experiments were performed at flow rates of 356 1and 666 1ml/min to produce average velocities of 1.656 0.05 and3.126 0.05 cm/s. The flowing fluid at the two average velocities arewithin the laminar flow regime which is typically observed forReynolds numbers up to 2000.56 All the flow rates were determinedfrom outflow with a measuring cylinder and timer.

The CPMG measurement was employed to measure the twotypes of flows. Echo CPMG measurement using a single echo timerequired approximately 2.5min with a repetition time of 300ms and512 averages. The 90x� pulse RF amplitude was set to half the 180y�

pulse RF amplitude, and the quadrature echo method57 was used to setthe common 90x� and 180y� pulse durations. Each pulse was 3.2 ls.

The measured magnitude data for flowing fluid were divided bycorresponding data collected for a stationary solution with the samemeasurement parameters, to obtain the normalized magnitude M/ ofodd echoes due to phase accumulation. Only the phase andM/ of thefirst odd echo were processed employing the phase-based method andthe magnitude-based method presented in Sec. IIC. This avoids anypossible effects due to pulse imperfection and resonance offset.

Based on the experimental real and imaginary signals, phaseaccumulations of the first odd echo at different s2, for the two types offlows, were calculated. The phase accumulations / were plotted withrespect to s2, as shown in Fig. 11. Since the experimental data had asystem phase /0, the phase accumulation / of odd echoes can be writ-ten as

/ ¼ /0 þ /odd ¼ /0 þ cGvavgs2: (28)

Based on Eq. (28), the phase accumulations / of the first oddechoes at different s2 were fitted for each flow by a linear fittingmethod, and the fitted results are shown as solid lines in Fig. 11. Thefitted /0 ¼ �0:656 0:01rad and vavg¼ 1.886 0.02 cm/s for thePoiseuille flow at vavg¼ 1.896 0.05 cm/s; /0 ¼ �0:646 0:01radand vavg¼ 3.576 0.04 cm/s for the Poiseuille flow at vavg¼ 3.696 0.05 cm/s; /0 ¼ �0:586 0:01rad and vavg¼ 1.626 0.02 cm/s for

FIG. 9. Real (Red solid line), imaginary (Blue dashed line), and magnitude (Green dashed line) velocity spectra obtained by Fourier transformation of the simulated signals ofodd echoes for Poiseuille flow (a), shear-thickening flow (b), and shear-thinning flow (c). Based on the break point in any of the real, imaginary, and magnitude velocity spectra,the maximum velocities of the three flows were 10.056 0.11, 11.636 0.11, and 7.116 0.11 cm/s, respectively.

FIG. 10. Comparison of Poiseuille flow velocity profile model (Red solid line) withthat reconstructed by the velocity spectrum (Blue dashed line). The bottom subplotrepresents the absolute error between the reconstructed profile and model, showingthat the velocity errors for the Poiseuille flow in the pipe are less than 0.1 cm/s.

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the shear-thinning flow at vavg¼ 1.656 0.05 cm/s; and /0 ¼ �0:686 0:01rad and vavg¼ 3.156 0.02 cm/s for the shear-thinning flow atvavg¼ 3.126 0.05 cm/s.

Figure 11 shows that the fitted phase accumulations of the firstodd echoes agree with the measured phase accumulation, indicatingthe reliability of the fitted parameters. The fitted vavg is similar to thevavg from flow rate data of each flow, within 3%. The processed resultsof experimental data reveal that the phase-based method is feasibleand practical in determining the average velocity of laminar flow.

The normalized magnitude M/ of the first odd echo at differents2, for the two types of flows, is displayed as red dots in Fig. 12. Aftervavg was determined from the phase-based method, scheme 3 wasemployed to process the experimental data to obtain the flow parame-ter n0. The fittedM/ of different flows with respect to s2 are shown assolid lines in Fig. 12.

The fitted n0 were 2.116 0.06 and 1.976 0.09 for the glycerol/distilled water flows at vavg¼ 1.896 0.05 and vavg¼ 3.696 0.05 cm/s,respectively. The fitted n0 ¼ 5.386 0.19 and n0 ¼ 5.376 0.17 forthe xanthan gum solution flows at vavg¼ 1.656 0.05 and 3.126 0.05 cm/s, respectively. A comparison of the fitted n0 for the

glycerol/distilled water flows with theoretical n0 of Poiseuille flowshows that the fitted n0 is very close to the theoretical n0, within 6%,revealing the effectiveness and practicality of scheme 3 for laminarflows. Both fitted n0 are more than 2.0 for the xanthan gum solutionflow. These results confirm that the flows are shear-thinning flows,which is as anticipated. The flow behavior index n¼ 0.23 was deter-mined from the fitted n0 at two flow velocities for the xanthan gumsolution flow. The calculated n is very similar to that from Blytheet al.22 for similar solution concentrations under laminar conditions,verifying the reliability of scheme 3 for non-Newtonian flows.

Based on the fitted flow parameters n0 and vavg by the phase-based method and the magnitude-based method, flow velocity profileswere reconstructed, as shown in Fig. 13. For Poiseuille flow, the flowvelocity profiles can be predicted by flow rates due to known n0 ¼ 2.0.The theoretical predictions are shown as solid lines in Fig. 13(a), wherethe bottom subplot represents the absolute error between the recon-structed profile and model. These results show that the velocity errorsbetween reconstructed profiles and theoretical predictions in the pipeare less than 0.2 cm/s, which indicates the reliable accuracy of thephase-based method and the magnitude-based method.

FIG. 11. Processed results of the phase-based method for the glycerol/distilled water flows at vavg¼ 1.896 0.05 (a) and 3.696 0.05 cm/s (b) and for the xanthan gum solu-tion flows at vavg¼ 1.656 0.05 (c) and 3.126 0.05 cm/s (d). Red dots show the calculated phase accumulation data of the first odd echo, and the solid line shows the fittedresults based on Eq. (28). The fitted /0 ¼ �0:656 0:01rad and vavg¼ 1.886 0.02 cm/s for (a); /0 ¼ �0:646 0:01rad and vavg¼ 3.576 0.04 cm/s for (b); /0¼ �0:586 0:01rad and vavg¼ 1.626 0.02 cm/s for (c); and /0 ¼ �0:686 0:01rad and vavg¼ 3.156 0.02 cm/s for (d).

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FIG. 12. Processed results of the magnitude-based method (scheme 3) for the glycerol/distilled water flows at vavg¼ 1.896 0.05 (a) and 3.696 0.05 cm/s (b) and for the xan-than gum solution flows at vavg¼ 1.656 0.05 (c) and 3.126 0.05 cm/s (d). Red dots show the M/ data of the first odd echo, and the solid line shows the fitted results basedon Eq. (19). The fitted n0 ¼ 2.116 0.06 for (a), n0 ¼ 1.976 0.09 for (b), n0 ¼ 5.386 0.19 for (c), and n0 ¼ 5.376 0.17 cm/s for (d).

FIG. 13. (a) Comparisons of 1D flow velocity profiles for the glycerol/distilled water flows from the magnitude-based method (dashed line) with the theoretical prediction (solidline), where the bottom subplot represents the absolute error between the reconstructed profile and theoretical profile, showing that the velocity errors between reconstructedprofiles and theoretical predictions in the pipe are less than 0.2 cm/s. (b) 1D flow velocity profiles for the xanthan gum solution flows based on the fitted n0 and vavg.

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V. CONCLUSIONS AND FUTUREWORK

In this work, a palm-sized Proteus permanent magnet, with aconstant magnetic field gradient, was designed to be used for the mea-surement of laminar flows. The Proteus magnet consists of two porta-ble disk magnets tilted at an angle of 1� from an edge separation of1.4 cm. Furthermore, we proposed a phase-based method, amagnitude-based method, and a velocity spectrum method to charac-terize laminar flow in a pipe, including average velocity and velocityprofile, from the CPMG measurement. The proposed methods wereverified by simulations and flow experiments on the designed magnet.The following conclusions can be drawn:

(1) A phase shift occurs on the odd echoes during CPMG measure-ment for flow measurement with a flow-directed constant gra-dient. The phase accumulation is related to gradient, echo time,and average velocity. The phase-based method employs multi-ple first odd echo phase accumulations at different echo timesto fit the average velocity of flow.

(2) The normalized magnitude M/ of odd echoes, due to phaseaccumulation, was derived, dependent on gradient, echo time,average velocity, and the flow behavior index. Themagnitude-based method obtains average velocity and flowbehavior index based on the fitting by M/ of first odd echoesat different echo times. With a modest number of first oddecho data points at different echo times, we can obtain theaverage velocity from the phase-based method, and then fit-ted flow behavior index, with known average velocity, fromM/.

(3) The velocity spectrum method is based on a Fourier transfor-mation approach. Due to the fixed gradient of the Proteus mag-net, this method requires change of s during the measurement.The maximum flow velocity can be determined based on thebreak point in the velocity spectrum. Combined with the aver-age velocity from the phase-based method, the flow behaviorindex can be deduced, and in turn the flow profile isdetermined.

CPMG measurement on low-field MR equipment with a flow-oriented gradient can be directly used for the determination of flowvelocity profile based on the proposed methods. Our methods inthis paper are developed to process data with complete polarization.The flow measurement, based on our equipment, requires a shortT1 of measured fluid, usually ensured using a contrast agent, tomake the detected fluid completely polarized. In future work, wewill consider new magnet designs amenable to pre-polarizationwhich will permit incorporation of incomplete sample magnetiza-tion into the flow profile analysis. We will also work to employadditional odd echoes, and even echoes, in CPMG data acquisitionsfor flow profile measurement.

ACKNOWLEDGMENTS

B.J. Balcom thanks NSERC of Canada for a Discovery Grantand the Canada Chairs program for a Research Chair in MRI ofMaterials. B. Newling thanks NSERC of Canada for a DiscoveryGrant (No. 2017-04564). This study was funded by an NSERCDiscovery Grant (No. 2015-6122) and CRC Grant (No. 950-230894) held by B.J. Balcom.

APPENDIX: NORMALIZED SIGNAL OF ODD ECHODUE TO PHASE ACCUMULATION

A. Derivation of the normalized real signal SRe of oddechoes due to phase accumulation

The normalized real signal SRe of odd echoes due to phaseaccumulation can be expressed as

SRe ¼

ðcos / rð Þð Þrdrð

rdr¼

ðR0cos X 1� rn

0

Rn0

� �� �rdrðR

0rdr

; (A1)

where X ¼ n0þ2n0 cGvavgs2.

The term cos ðXð1� rn0

Rn0 ÞÞ can be rewritten as

cos X 1� rn0

Rn0

� �� �

¼ 12

exp iX 1� rR

� �n0 ! !

þ exp �iX 1� rR

� �n0 ! !" #

¼ 12

eXi exp �Xið Þ 1n0rR

� �n0 !

þ e�Xi exp Xið Þ 1n0rR

� �n0 !" #

:

(A2)

Thus,ðR0cos X 1� rn

0

Rn0

� �� �rdr

¼12

ðR0

eXiexp �Xið Þ 1n0rR

� �n0 !

rþe�Xiexp Xið Þ 1n0rR

� �n0 !

r

" #dr:

(A3)

Equation (A3) can be regarded as half of the sum of two integrals,

namely,Ð R0 eXi exp ððð�XiÞ 1

n0 rRÞn

0 Þrdr and Ð R0 e�Xi exp ðððXiÞ 1n0 rRÞn

0 Þrdr.We define u ¼ ðXiÞ

1n0 r

R ; and then obtain

ðR0eXi exp �Xið Þ 1

n0rR

� �n0 !

rdr ¼ð Xið Þ 1n0

0eXie�un

0 uR2

Xið Þ 2n0du

¼ R2eXi

Xið Þ 2n0

ð Xið Þ 1n0

0e�un

0udu: (A4)

Next, we define w ¼ un0, and Eq. (A4) can be written as

ðR0eXi exp �Xið Þ 1

n0rR

� �n0 !

rdr ¼ R2eXi

Xið Þ 2n0

ðXi0w

1n0e�w w

1n0�1

n0dw

¼ R2eXi

n0 Xið Þ 2n0

ðXi0w

2n0�1e�wdw: (A5)

The gamma function CðaÞ ¼ Ð10 wa�1e�wdw and the incompletegamma function Cða; xÞ ¼ Ð1x wa�1e�wdw. Thus, Eq. (A5) can bewritten as

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ðR0eXi exp �Xið Þ 1

n0rR

� �n0 !

rdr

¼ R2eXi

n0 Xið Þ 2n0

ð10w

2n0�1e�wdw� R2eXi

n0 Xið Þ 2n0

ð1Xiw

2n0�1e�wdw

¼ R2eXi

n0 Xið Þ 2n0

C2n0

� �� C

2n0;Xi

� �� �: (A6)

Similarly, we can obtainðR0e�Xi exp Xið Þ 1

n0rR

� �n0 !

rdr¼ R2e�Xi

n0 �Xið Þ 2n0

C2n0

� ��C

2n0;�Xi

� �� �:

(A7)

We substitute Eqs. (A6) and (A7) into Eq. (A3), and then Eq. (A1)can be rewritten as

SRe¼

ðR0cos X 1� rn

0

Rn0

� �� �rdrðR

0rdr

¼eXi �ið Þ 2

n0 C2n0

� ��C

2n0;Xi

� �� �þe�Xii

2n0 C

2n0

� ��C

2n0;�Xi

� �� �n0X

2n0

:

(A8)

B. Derivation of the normalized imaginary signal SIm ofodd echoes due to phase accumulation

The normalized imaginary signal SIm of odd echoes due tophase accumulation can be expressed as

SIm ¼ �

ðR0sin / rð Þð ÞrdrðR

0rdr

¼ �

ðR0sin X 1� rn

0

Rn0

� �� �rdrðR

0rdr

; (A9)

where X ¼ n0þ2n0 cGvavgs2.

The term sin ðXð1� rn0

Rn0 ÞÞ can be rewritten as

sin X 1� rn0

Rn0

� �� �

¼� i2

exp iX 1� rR

� �n0 ! !

� exp �iX 1� rR

� �n0 ! !" #

¼� i2

eXi exp �Xið Þ1n0 rR

!n00@

1A� e�Xi exp Xið Þ

1n0 rR

!n00@

1A

24

35:

(A10)

Thus,

ðR0sin X 1� rn

0

Rn0

� �� �rdr ¼ � i

2

ðR0

eXi exp �Xið Þ 1n0rR

� �n0 !

r

"

�e�Xi exp Xið Þ 1n0rR

� �n0 !

r

#dr:ðA11Þ

Equation (A11) can be regard as � i2 times the difference of two

integralsÐ R0 e

Xi expððð�XiÞ 1n0 rRÞn

0 Þrdr and Ð R0 e�Xi expðððXiÞ 1n0 rRÞn

0 Þrdr.Equations (A6) and (A7) give the two integral expressions. Thus,Eq. (A9) can be rewritten as

SIm¼�

ðR0sin X 1� rn

0

Rn0

� �� �rdrРR

0 rdr

¼ ieXi �ið Þ 2

n0 C2n0

� ��C

2n0;Xi

� �� ��e�Xii

2n0 C

2n0

� ��C

2n0;�Xi

� �� �n0X

2n0

:

(A12)

DATA AVAILABILITY

The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.

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Phys. Fluids 33, 103609 (2021); doi: 10.1063/5.0065986 33, 103609-14

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