V
⁡
(
r
)

=

Ar
+

B
r

(
4
)

with coefficients A and B dependent on the rotation rates of the inner and outer cylinders, Ω₁and Ω₂, and their radii R₁and R₂. In an embodiment of the present invention, the outer cylinder (stator) is fixed and the inner cylinder (rotor) is rotated at a constant rate. In this simplification, A and B are defined as follows,

A
=

-

η
2

⁢

Ω
1

1
-

η
2

(
5
)

B
=

R
1
2

⁢

Ω
1

1
-

η
2

where η is the radius ratio,

R
1

R
2

.

Components of the velocity distribution are obtained by solving the linearized Navier-Stokes equation in cylindrical coordinates. For laminar flow—often referred to as circular Couette flow (CCF)—the flow is axisymmetric and the velocity distribution can be described analytically. For a power-law fluid with non-slip boundary conditions, the azimuthal component of the velocity distribution is given by [28],

υ
θ

(
r
)

=

r
⁢

Ω
1

[

1
-

(

r

R
2

)

-
2

n

(

1
-

η

-
2

n

)

]

(
6
)

For a Newtonian fluid (n=1), the above expression is reduced to,

υ
θ

(
r
)

=

Ω
1

⁢

r
-

R
2
2

r

1
-

η

-
2

(
7
)

As the shear rate increases, axisymmetric disturbances grow exponentially and the flow becomes unstable, resulting in the formation of vortices in the axial direction (Taylor Vortex Flow). The onset of instabilities can be estimated using the Taylor number, defined in the narrow gap approximation

as,

T
=

Ω
1
2

⁢

r
1
2

⁢

d
3

v
2

(
8
)

where d is the gap width, R₂−R₁, and v is the kinematic viscosity. In the narrow gap approximation, the critical Taylor number, T_cis approx. 1700. In an embodiment of the present invention, an object is to maintain laminar flow, as the instrument is only sensitive to flow in the direction of the magnetic field gradient. T_ccan be verified experimentally by observing the shear rate at which the measured magnitudes begin to deviate from theoretical predictions for laminar flow. A more sophisticated analysis can be used to describe the Taylor number for non-Newtonian fluids [23]; however, previous work has shown that for dilute solutions of xanthan gum in water, the critical Taylor number is close to that of water [29].

Magnetic Resonance Methods for Measuring Flow

The most common and straightforward method in motion-sensitized NMR encodes the velocity in measurements of phase. In the presence of a gradient, the NMR signal phase accumulates according to [30],

ϕ
=

∫

γ
⁢

G
→

(
t
)

·

r
→

(
t
)

⁢
dt

(
9
)

where γ is the gyromagnetic ratio. Assuming that the velocity remains constant over the time of the echo, the position {right arrow over (r)}(t) can be approximated using a Taylor expansion.

Expanding to first order, the phase at the first echo is given by,

ϕ
⁡
(

2
⁢
τ

)

=

ϕ
0

+

γ
⁢

G
→

·

υ
→

⁢

τ
2

(
10
)

where φ₀is the phase associated with a stationary sample. After normalizing to the stationary sample signal to eliminate relaxation/diffusion effects, the motion-sensitized NMR signal is given by,

S
=

∫

R
1

R
2

ρ
⁡
(
r
)

⁢

∫

θ
1

θ
2

e

-
i

⁢

ϕ
⁡
(
r
)

⁢
rdrd
⁢
θ

(
11
)

In many cases, direct measurements of phase can be used to extract information on the flow velocity. However, as shown in FIG. 1, the symmetry of the Couette flow system leads to a net cancellation of phase, preventing the use of phase-sensitive methods for measuring flow velocity. Instead, the present inventors can consider velocity induced phase-interference which leads to an overall attenuation of signal magnitude, similarly to past conventional MR measurements of rotational flow [31, 32].

Taking the component of the flow velocity in the direction of the gradient, the signal is given by,

S
=

∫

R
1

R
2

ρ
⁡
(
r
)

⁢

∫

θ
1

θ
2

e

-
i

⁢

(

γ
⁢
G
⁢

υ
θ

(
r
)

⁢
cos
⁢

θτ
2

)

⁢
rdrd
⁢
θ

(
12
)

Assuming the spin density distribution ρ(r), is uniform within the excited slice, the present inventors normalize it as ρ(r)=1.

The above integral expression can be simplified using one of two approximations. First, the integration over θ can be eliminated by assuming a very thin excited volume of the sample. In this approximation, the present inventors can separate the contributions from spins with opposing velocities on either side of the sample and the signal at the first echo is given by,

S
=

∫

R
1

R
2

e

(

-
i

⁢
γ
⁢
G
⁢

υ
θ

(
r
)

⁢
cos
⁢

θτ
2

)

⁢
rdr

+

∫

R
1

R
2

e

(

i
⁢
γ
⁢
G
⁢

υ
θ

(
r
)

⁢
cos
⁢

θτ
2

)

⁢
rdr

=

∫

R
1

R
2

cos
(

γ
⁢
G
⁢

υ
θ

(
r
)

⁢
cos
⁢

θτ
2

)

⁢
rdr

(
13
)

The signal is most sensitive to flow when the velocity is parallel/anti-parallel to the gradient (θ=0). Practically this is not always possible and measurement of the angle, θ, between the velocity and the gradient is required.

The present inventors can measure θ indirectly using a Newtonian reference sample, or directly using a method described below in section “DANTE pulses for calibration”.

The present inventors bring attention to the similarity of Equation 13 and the following equation described in [33],

S
⁡
(
t
)

=

∫
0
∞

dv
⁢

ρ
⁡
(
υ
)

⁢

cos
⁡
(

1
2

⁢
γ
⁢
G
⁢
υ
⁢
τ
⁢
t

)

(
14
)

In Equation 13, a factor describing the spin distribution of velocity, ρ(v_θ), is absent, and signal integrated over the spatial dimensions of the flow cell. Given that flow is laminar, it is reasonable to assume that the velocity distribution is stationary. Therefore, each position corresponds to a single velocity, and the distribution of velocities can be described as a delta function, ρ(v_θ)=δ(v−v_θ(r)).

In an alternative approximation, one can assume that the excited slice completely covers the sample. In this case, the integration over θ leads to a Bessel function of the first kind and the signal is given by,

S
=

2
⁢
π
⁢

∫

R
1

R
2

J
0

(

γ
⁢
G
⁢

υ
θ

(
r
)

⁢

τ
2

)

⁢
rdr

(
15
)

In both approximations, the radial part was integrated numerically. Although current instrumentation does not allow for complete excitation of the sample, it is still useful to investigate the expected behaviour in simulation.

In theory, the above expressions are valid for all echoes; however, imperfect RF pulses in conjunction with an inhomogeneous magnetic field can generate coherence pathways that interfere with later echoes. This makes quantitative characterization the flow behavior index using multiple echoes in a Carr-Purcell-Meiboom-Gill (CPMG) echo train difficult. Therefore, flow effects are encoded in the first echo.

Effects of T₂and Diffusion

In addition to flow effects, the signal will also be attenuated further by the transverse relaxation time, T₂, and diffusion as follows [35],

S
=

exp
[

-
2

⁢

τ

T
2

-

2
3

⁢

γ
2

⁢

G
2

⁢
D
⁢

τ

3

]

(
16
)

where D is the self-diffusion coefficient. Normalizing the signal to that of a stationary sample allows for diffusion and T₂effects to be neglected, assuming that shear flow has a negligible effect on the “apparent” self-diffusion coefficient. Echo times and the strength of the magnetic field must be carefully considered to ensure that the remaining signal after diffusive attenuation and T₂relaxation is sufficient to normalize and resolve flow effects.

Effects of Taylor Dispersion

Taylor dispersion is a process by which fluid elements are dispersed around the average direction of motion in a diffusion-like manner. Dispersion manifests as an apparent increase in diffusive attenuation of signal and can lead to incorrect measurements of flow parameters. Dispersion of fluid elements in the direction of flow is typically negligible for laminar flow [36]; therefore, when observing Couette flow, one must only consider perturbations to the apparent self-diffusion coefficient due to dispersion in the radial direction. This contribution is difficult to quantify using portable NMR; however, conventional PFG techniques have previously been employed to investigate the effects of Taylor dispersion in various flow geometries [37, 38]. Modifying the expression to account for a static gradient, the increase in apparent diffusivity in the radial direction due to Taylor dispersion, D_Flow, is given by,

D
Flow

=

3

1
⁢
6

⁢

〈

Δυ
r

(
r
)

2

〉

⁢
τ

(
17
)

where [Image Omitted] Δv_r(r) is the average change in the radial component of velocity over the time of the acquisition. For sufficiently low shear rates and gap sizes, Δv_r(r) is small and D_Flowcan be neglected.

DANTE Pulses for Calibration

Practically, it can be difficult to ensure perfect alignment of thin slices with the centre of the Couette cell. As described above, it is often necessary to calibrate the measurement by determining the angle, θ, between the velocity and the magnetic field gradient. One possibility is calibrating to a Newtonian reference sample (n=1); however, this requires changing the sample prior to the measurement. An alternative and more practical approach is to apply a composite 90° DANTE pulse. DANTE (Delays Alternating with Nutation for Tailored Excitation) was developed as an alternative method for selective excitation [39] and has since been optimized for use in inhomogeneous fields [40]. DANTE calibration does not require a Newtonian reference sample and can be used to quickly determine the offset of the excited slice from the center of the Couette cell and by extension, the component of velocity in the direction of the magnetic field gradient. Rather than exciting the sample with a “hard” 90° pulse, N short pulses separated by a delay, δ, are applied resulting in transverse magnetization within N small slices of the sample separated by a distance, Δx, given by,

Δ
⁢
x

=

2
⁢
π

γ
⁢
G
⁢
δ

(
18
)

Following the DANTE pulse, “hard” 180° refocusing pulses can be applied to generate spin echoes. Taking the Fourier transform of the echo reveals a 1D profile with N distinct peaks. If the sample is perfectly aligned, the profile should be symmetric. If the profile is asymmetric, a simple geometric argument, described in appendix A can be used to determine the offset from centre, and by extension the angle, θ, between the velocity and magnetic field gradient. This procedure can become complicated if a slice is positioned outside of the inner cylinder, or on the boundary between the two cylinders. Therefore, the present inventors provide a two-step calibration procedure. First, a DANTE measurement on a continuous sample to correct for larger offsets if necessary, followed by a second measurement with the rotor in position to make fine adjustments.

Simulations

Current Gradient Simulations

The current magnetic field gradient (254 G/cm) provides sufficiently close agreement with the thin slice approximation of the signal equation. Simulations were used to select acquisition parameters in this approximation and compared with simulations of the complete coverage approximation. FIG. 2 shows the dependence of signal magnitude (normalized to that of a stationary sample) on echo time (2τ). Flow behaviour indices of; n=0.2 (solid line), n=0.4 (dashed line), and n=1 (dash-dot line) are shown, each at a shear rate of 4.85 s⁻¹. Normalization of signal to that of a stationary sample leads to decreased sensitivity to flow effects at longer echo times due to T₂relaxation and more prominently, diffusive attenuation. To investigate the effects of diffusion, the present inventors have determined the τ values at which diffusion leads to a reduction in the stationary sample signal by a factor of e. These values are indicated by vertical dotted lines for diffusion coefficients of: I) water (2 μm²/ms), II) 1/10 of water (0.2 μm²/ms), and III) 1/100 of water (0.02 μm²/ms). Note that it becomes difficult to resolve the complete decay of signal for samples with diffusion coefficients that are close to that of water.

Alternatively, the effects of diffusion can be held constant by acquiring signal at a fixed echo time and varying the shear rate. However, the longer echo times needed to resolve significant flow effects result in decreased initial magnitude and a correspondingly decreased SNR for the entire measurement. This makes variable shear rate methods impractical for the current samples and magnet. Furthermore, multiple measurements of the shear rate can add to the uncertainty of the measured flow behaviour index. The simplest way to overcome the need for longer echo times for a given gradient is to use a sample with a smaller diffusion coefficient. Although SNR would be improved for such samples, this is not a viable solution as it would restrict the technique to a limited set of fluids.

Weaker Gradient Simulations

In both of the proposed measurements—variable echo time and variable shear rate—the present inventors see that their effectiveness can be improved by reducing the magnetic field gradient. The advantage of using a weaker gradient is two-fold. Most importantly, lowering the gradient significantly reduces the effects of diffusive attenuation and improves sensitivity. Furthermore, a weaker gradient allows for excitation of larger slices of the sample, and possible verification of the complete coverage approximation (Equation 15). For the same bandwidth as the current RF probe (approx. 200 kHz), the present inventors can determine the gradient needed to excite a 1 cm slice that completely covers the sample. Using the following equation, the gradient needed is approx. 50 G/cm,

Δ
⁢
x

=

2
⁢
πΔ
⁢
f

γ
⁢
G

(
19
)

FIG. 3 shows the same simulations as FIG. 2, apart from a weaker gradient (50 G/cm) and a change in scale. In comparing with FIG. 2, the present inventors can expect to resolve an added 10% of flow-induced attenuation before the stationary sample signal is attenuated by a factor of e due to diffusion alone.

FIG. 4 shows a comparison between simulations of first echo magnitude at variable shear rates (0 s⁻¹−6 s⁻¹) after diffusive attenuation (D=2 μm²/ms) for gradients of 254 G/cm and 50 G/cm. For a gradient of 254 G/cm, the τ of 0.75 ms needed to resolve flow effects maintains <60% of the initial magnitude after diffusive attenuation. Reducing the gradient to 50 G/cm requires a longer τ (1.5 ms); however, >80% of the initial magnitude remains after diffusive attenuation.

Experimental

Magnet and RF Coil

The experimental setup, depicted in FIG. 5, employed a unilateral three-magnet array (A) to provide a static magnetic field, B₀. The magnet has a constant gradient of 254 G/cm directed perpendicular to its surface [41].

To excite transverse magnetization a radiofrequency with five turns of copper wire (B) generated an oscillating magnetic field, B₁, at a frequency of 5.73 MHz (the Larmor frequency approx. 1.5 cm from the surface of the magnet array).The coil consisted of 5 turns of copper wire wound around a 3D printed former with a diameter of 1.1 cm and height of 1.5 cm. Using the reflection mode method [42], the RF probe quality factor was determined. The quality factor of 60 corresponds to a bandwidth of approx. 191 kHz, which limits the slice thickness to a maximum of 1.76 mm [43]. Custom 3D printed hardware (C) was fabricated to position the coil at a fixed distance away from the magnet. Copper tape was wrapped around the coil mount to prevent corruption of echoes by acoustic ringing interactions between the B₁field and the magnet array casing.

Couette Flow Setup

Referring to FIG. 5, a portable NMR Couette flow setup used in an embodiment of the present invention includes a three-magnet array (A), radiofrequency coil (B), 3D printed radiofrequency coil mount (C), motor mount (D), 12 V DC motor (E), garolite inner cylinder (rotor (F)), aluminium shaft (G), plastic sample tube (stator) (H), and fotonic gap sensor (I). An NMR console (J) is operably connected to radiofrequency coil(B) and fotonic gap sensor I) is operably connected to an oscilloscope (L). A computer system (K) is operably connected to NMR console (J) and is programmed with method steps according to embodiments of the present invention.

A separate mount (D) was 3D-printed to suspend a 12 V, 150 RPM DC motor (E) in place above the Couette cell. The mount was designed to provide a safe spacing of 10 cm between the motor and magnet array. The present inventors connected the rotor (a 6 mm diameter Garolite rod (F)) to the motor using an aluminum shaft (G) and set screws. Garolite was selected for several reasons; it produces no MR signal, the surface is smooth with no significant irregularities, and it was readily available in the desired size. The present inventors fitted the opening of the stator (a 10 mm diameter plastic tube (H)) with rotary shaft seals (lubricated with Lithium grease) to maintain alignment of the rotor with the central axis of the stator. The gap size between the rotor (R₁=3 mm) and the stator (R₂=5 mm) was 2 mm. The motor was driven by the variable DC output of an SDG1025 Arbitrary Waveform-Function Generator (Siglent, CN). For measurement of rotation speeds, a KD-300 Fotonic gap sensor (I) (MTI Instruments, NY) was slotted into the motor mount and connected to a digital oscilloscope.

Data Acquisition and Processing

To acquire MR signal, the RF probe was connected to a duplexer with the transmit end connected to an RF pulse amplifier (Tomco, AU) and the receiving end connected to the input of a pre-amplifier (Miteq, N.Y). The output of the pre-amplifier was connected to a LapNMR console (Tecmag, TX), with a 10.7 MHz low-pass filter (Mini Circuits, N.Y) in-line to reduce noise.

Pulse sequences were configured using NTNMR (Tecmag, TX). For validity of the thin slice approximation, 9 us pulses were attenuated by 23 and 29 dB to generate 90° and 180° pulses, respectively. Pulse widths were set by comparing the amplitudes of the first three echoes in a quadrature echo pulse sequence [44], a perfect 90° pulse should result in a ratio of 1:1.5:1.5. The 9 us pulse results in excitation of a 0.8 mm slice of the sample. 2D tables were included in the pulse sequence to sweep through a range of echo times.

Low sensitivity typically encountered when working with low-field NMR is further exacerbated by the requirement of a thin slice across the narrow gap that contains the sample. Slight improvement in SNR was achieved by sizing the coil former specifically around the sample tube to increase the fill factor. Alternatively, the coil could be wound directly around the sample tube; however this introduces potential influences from vibrations.

SNR can be improved drastically by adding the echoes in a CPMG echo train. The first echo, acquired with a variable echo time, encodes flow effects. Subsequent echoes are acquired with the shortest possible echo time with the intent of maximizing the number of added echoes while minimizing further attenuation due to flow. The dead-time of the RF probe places a lower limit on the echo time; thus to avoid corruption of echoes due to probe ringing, these echoes were acquired with an echo time of 0.2 ms. Including CPMG enhancement, the pulse sequence can be described in two steps. First, a preparation step consisting of a 90° pulse followed by a delay of length τ₁before applying a 180° pulse and acquiring the first echo after a total time TE=2τ₁. Immediately following the preparation step, a CPMG echo train is acquired with an echo time of 0.2 ms.

In testing DANTE pulses as a calibration method, the present inventors decreased the attenuation values to 17 and 23 dB to allow for shorter pulses. The composite 90° pulse consisted of three 1.5 us pulses, separated by a delay, δ, of 36 us, corresponding to a nominal slice separation, Δx, of 0.257 mm. The dwell time was increased to 8 us to provide a narrower field of view of approx. 0.12 cm. Following a delay, τ, of 0.2 ms, a “hard” 180° pulse refocused transverse magnetization, generating an echo at 0.4 ms. The built-in FFT algorithm was then used to obtain the 1D profile and intensities of the peaks were compared to estimate the offset between the excited slice and center of the Couette cell.

Sample Preparation

Measurements were performed on two different Newtonian fluids, doped water, and corn oil. Distilled water was doped with 30 mM of CuSO₄to reduce the longitudinal relaxation time and improve sensitivity. Corn oil was used to investigate the limitations of diffusive attenuation and the effects of adding echoes. For shear-thinning fluids, xanthan gum (Bob's Red Mill Natural Foods Inc, OR) was mixed with distilled water in concentrations of 0.2 and 0.4 percent weight using a low-gear mixer (Mastercraft, TN) over a period of ten hours. Xanthan gum samples are a common choice of shear thinning fluid in NMR rheology measurements, and values of the flow behavior index can be found in the literature for a range of concentrations [45].

Results and Discussion

Verification of DANTE Calibration

The corn oil sample was used to test the DANTE calibration procedure, and the 1D profile containing three DANTE slices spaced by approx. 0.24 mm is shown in FIG. 7. The intensity of the left peak is approx. 1.32 times larger than that on the right. By comparing intensities and using the analysis described in Appendix A., the offset, Δs is estimated to be approx. 2.69 mm. The agreement is remarkably close compared to a calculation using water as a reference sample (2.68 mm), confirming that DANTE pulses provide a faster and more direct calibration method.

Calibration is a very simple use of DANTE excitation; however, the same techniques could potentially be used to acquire spatially resolved flow behavior information. Although promising, this measurement is not without limitations. Acquiring more slices requires smaller separations, and hence, longer delays between pulses and larger acquisition windows. Therefore, due to the longer echo times required, this measurement is limited by diffusion even more so than others. It can be used on samples with slower diffusion, such as oil, but for water-based samples, a weaker gradient is needed.

Verification of CPMG Enhancement

To verify that flow effects can be neglected when using CPMG enhancement, the present inventors acquired a series of CPMG echo trains with varying echo times. Signal was acquired for a rotating sample of corn oil at a shear rate of 4.85 s⁻¹and normalized to the stationary sample signal. FIG. 8 shows the twice normalized magnitude (first to a stationary sample, and then to the normalized first echo magnitude) for the rotating sample after adding 32, 64, 96, and 128 echoes. The present inventors observe that for values of τ2 below 0.15 ms, up to 128 echoes can be added while maintaining >95% agreement with measurements of only the first echo. In addition to verifying the enhancement procedure, FIG. 8 demonstrates that for sufficiently short echo times, the CPMG echo train carries information on T₂and diffusion that is unaffected by flow. This opens the possibility of combining measurement of rheological properties with conventional MR parameters.

Measurements of T₂and Diffusion

Measurements of T₁, T₂, and the self-diffusion coefficient (D) for doped water and xanthan gum solutions are shown in Table 1. The present inventors observe that the differences between water and xanthan gum solutions are negligible, providing verification for the use of a Newtonian reference sample for calibration.

TABLE 1

Measurements of T₂, T₁, and the self diffusion coefficient

(D) for samples of; distilled water doped, 0.2% wt.

xanthan gum, and 0.4% wt. xanthan gum. All samples are doped

with approx. 30 mM of CuSO₄.

Sample
T₂(ms)
T₁(ms)
D (μm²/ms)

Water
7.9 ± 0.6
14.6 ± 0.5
2.15 ± 0.05

0.2% Xanthan Gum
7.8 ± 0.6
17.1 ± 0.4
2.21 ± 0.05

0.4% Xanthan Gum
7.7 ± 0.5
14.4 ± 0.3
2.01 ± 0.05

Taylor Instabilities

FIG. 9 provides verification for assuming laminar flow, showing measurements of a doped water sample at a fixed r of 0.25 ms over shear rates ranging from approx. 7 s⁻¹to 35 s⁻¹. The present inventors observe significant deviation from theoretical predictions at a shear rate of approx. 10 s⁻¹. Furthermore, the shear rate of 4.85 s⁻¹used in all other measurements falls well below the shear rate of 7.2 s⁻¹at which the onset of instabilities is predicted by the critical Taylor number (approx. 1700).

Measurements of the Flow Behavior Index

FIG. 10 show measurements of signal magnitude (after adding 16 echoes), normalized to the stationary sample signal for solutions of water, 0.2% wt. xanthan gum in water, and 0.4% wt. xanthan gum in water (all doped with 30 mM CuSO₄). The data of Equation 13, using the Newtonian reference calibration procedure to measure the angle, θ, between the velocity and the magnetic field gradient. Fitted values of the flow behavior index are included in Table 2 and compared with literature values [45].

TABLE 2

Table of measured and expected values [45] of

the flow behavior index, n, for samples of 0.2% wt.

and 0.4% wt. xanthan gum in distilled water.

Sample
Measured n
Expected n

0.2% wt. Xanthan Gum
0.40 ± 0.02
0.39 ± 0.01

0.4% wt. Xanthan Gum
0.20 ± 0.02
0.23 ± 0.01

Measurements With Slower Diffusion

FIG. 11 shows measurements of magnitude (after adding echoes and normalizing to a stationary sample) for a sample of corn oil at 21 different echo times ranging from 0.2 ms to 4.6 ms, equally spaced by 0.2 ms. This measurement is useful to investigate the behaviour at longer echo times. The data maintains a close agreement with the predicted behaviour up to τ=1.0 ms. After this point, the present inventors see oscillations in magnitude as predicted by theory; however, they do not occur with the predicted amplitude or frequency. It is possible that the decreased SNR makes these oscillations difficult to resolve, or that the thin slice approximation is less effective at these points.

Appendix A

Consider a DANTE excitation consisting of three pulses separated by a delay, δ, resulting in generation of transverse magnetization within three slices separated by

Δ
⁢
x

=

2
⁢
π

γ
⁢
G
⁢
δ

.

Assuming a cylindrical volume of sample centred about the origin and defining the offset of the central slice from the origin as Δs, the ratio of intensities, I₋and I₊, is equivalent to the ratio of the chord lengths, c− and c+, of the circular cross section at Δs−Δx and Δs+Δx, respectively. Therefore, the offset, Δs, can be found be by solving the following equation,

I
-

I
+

=

c
-

c
+

=

R
2
2

-

(

Δ
⁢
s

-

Δ
⁢
s

)

2

R
2
2

-

(

Δ
⁢
s

+

Δ
⁢
x

)

2

(
20
)

Note that the above equation describes the case of a continuous cylindrical sample. In the presence of an inner cylinder, it becomes,

I
-

I
+

=

c
-

c
+

=

R
2
2

-

(

Δ
⁢
s

-

Δ
⁢
x

)

2

-

R
1
2

-

(

Δ
⁢
s

-

Δ
⁢
x

)

2

R
2
2

-

(

Δ
⁢
s

+

Δ
⁢
x

)

2

-

R
1
2

-

(

Δ
⁢
s

+

Δ
⁢
x

)

2

(
21
)

In both cases, it is easiest to solve for the offset, Δs, graphically. Knowing Δs, the angle between the velocity and the magnetic field gradient can be determined,

θ
=

arcsin