Consider a perfectly elastic material, such as a spring. When subjected to a sinusoidally oscillating stress, it responds instantaneously with a sinusoidal strain (completely in phase with the applied stress). By contrast, a completely viscous material, such as a dashpot, dissipates energy when subjected to stress and responds with a strain that is 90° out of phase from the applied stress (as stress is proportional to the strain rate). No physical material is free from viscous losses; therefore, all materials are viscoelastic and can be described by combining viscous and elastic models (dashpots and springs). Viscoelastic materials are commonly described using the complex (or dynamic) modulus, E* [68], E * = E ′ + i E ″ = | E | e i δ (6) where the real component (storage modulus), E′, and the imaginary component (loss modulus), E″ represent the elastic and viscous characteristics, respectively. Therefore, when a viscoelastic material is subjected to stress, it responds with a phase difference, or loss-angle, δ, between stress and strain.

The simplest case to consider in elastometry is a sample under compressive stress where only the longitudinal component of the wave is significant in the sensitive region. Consider a sample excited by a sinusoidal force of amplitude, F ₀, and frequency, ω, F ⃗ t = − F 0 ⁡ sin ω t x ̂ (7)

the sample responds with a sinusoidal displacement of amplitude, A, and phase offset, δ, relative to the applied stress x ⃗ t = A ⁡ sin ω t − δ x ̂ (8) where δ is the loss-angle defined in Equation 6. This simplest approach for calculating the phase accumulated in the MR signal is to Taylor expand the position, x t ≈ x 0 + v 0 t + 1 2 a 0 t 2 + ⋯ (9)

the relative phase of the MR signal is then given by, ϕ = γ x 0 ∫ G t d t + v 0 ∫ G t t d t + a 0 ∫ G t t 2 d t + ⋯ (10) These integrals are referred to as “moments” of the magnetic field gradient and by varying G(t), one can manipulate the effects of motion on the MR signal [69]. In the case of a constant magnetic field gradient, the phase at the first echo is influenced by the average acceleration and velocity within the sensitive region and is given by, ϕ 1 = ϕ 0 + γ G v avg τ 2 + γ G a avg τ 3 (11) The phase at the second spin-echo is only influenced by the average acceleration and is given by, ϕ 2 = ϕ 0 + 2 γ G a avg τ 3 (12)

When applying this approximation to an oscillating sample, it is assumed that velocity (and acceleration) remain constant over the time of acquisition. Stated another way, the echo time must be significantly shorter than the period of oscillation. Complications can arise when working with high frequencies, or low amplitudes when longer echo times are needed to maintain phase sensitivity. Alternatively, if the echo time (twice the time between pulses) is significant compared to the period, then the phase of any Nth echo can be determined by integrating x(t) directly, ϕ N = γ G A ω − cos 2 N ω τ + δ + − 1 N cos δ + 2 ∑ k = 0 N − 1 cos 2 k + 1 ω τ + δ − 1 k (13)

Encoding motion in the first spin-echo is trivial and can be done using the first odd echo in a Carr-Purcell-Meiboom-Gill sequence (CPMG) [70] (more details on MR pulse sequences can be found in Section 4.1). The second spin-echo can be used if the stimulated echo resulting from imperfect excitation is cancelled with a phase cycle [71]. Subsequent echoes require a more detailed analysis of coherence pathways and thus, were not considered in the research presented in this review, but are a potential area of focus for future investigation.

Measurements of velocity and acceleration at various points throughout the vibration period can be used to characterize the time-varying response of the sample to applied stress. Viscoelastic properties can then be estimated through comparison with independent measurements of the applied stress.

Analysis of the MR signal can become more complicated if the shear component of a wave is dominant within the sensitive region. In addition to the potential effects of velocities distributed in time (finite echo time considerations), spatial velocity distributions can lead to significant phase interference, especially if the wavelength covers a significant fraction of the sensor size. If not properly accounted for, the presence of shear waves can lead to detrimental effects on the MR signal; however, there is also potential for exploiting this dependence to characterize viscoelastic properties.

If viscous effects are significant, there can be a significant reduction in vibration amplitude within the sensitive volume that must be considered. This can be easily seen by considering a plane wave that propagates in the z-direction in a homogeneous, isotropic, linear viscoelastic medium. Particles are displaced in the x-direction with amplitude, A, and frequency, ω, x z , t = A e − α z ⁡ sin k z sin ω t + x 0 (14) where k is the wave-number and α is the attenuation coefficient.

Both the attenuation and shear wave speed are influenced by the viscoelastic properties (storage and loss moduli) of the material [72]. The attenuation coefficient, α, is given by, α ω = ρ ω 2 E ′ 2 + E ′ ′ 2 − E ′ 2 E ′ 2 + E ′ ′ 2 (15)

And the shear wave speed, c, is given by, c ω = ω k ω = 1 ρ 2 E ′ 2 + E ′ ′ 2 E ′ + E ′ 2 + E ′ ′ 2 (16)

Refocusing pulses used to generate echoes in a CPMG sequence result in effective square-wave modulation of the magnetic field gradient [41, 73]. Thus, the gradient can effectively be “synchronized” with oscillations applying pulses separated by integer multiples of the vibration period to generate an oscillating “apparent gradient” (as seen by spins in the sensitive region). In the case of a sinusoidally varying gradient, G t = G 0 ⁡ sin ω t + ϕ (17) To simplify the analysis, it can be assumed that attenuation is negligible within the sensitive volume. The MR signal phase is then given by integrating over integer multiples of the vibration period, ϕ z = ∫ 0 n T γ A G 0 ⁡ cos ω t + δ cos k z − ω t = γ G 0 A n T 2 sin k z cos ϕ (18) In the case of constant gradient portable MR, refocusing pulses results in an effective gradient in the form of a square wave; hence, the coefficient of 1 2 is replaced by 2 π [33].

Given the finite dimensions of the MR sensor, the distribution of velocities resulting from the propagating shear wave leads to phase interference that depends on the ratio between the wavelength and sensor size. As this ratio increases, so does the amount of phase interference, and effects on signal magnitude become more apparent.

Consider a simple approximation, a 1D sensitive region, through which the shear wave propagates in the z-direction, with particle displacements in the direction of the gradient (x). If the sensor size covers a complete wavelength, there exists a closed-form analytical solution for the MR signal [33], S = ∫ 0 λ ρ e − i 2 γ G 0 A n T ⁡ cos ϕ sin k z d z ∝ λ J 0 2 γ G A n T ⁡ cos ⁡ ϕ π (19) where J _n is the Bessel function of the first kind of order n defined as follows, J n z = 1 2 π i n ∫ 0 2 π e i z ⁡ cos ⁡ ϕ e i n ϕ d ϕ (20)

The “Bessel-like” behaviour of the signal is expected even in cases where the sensor size is smaller than the wavelength. Although closed-form analytical solutions cannot be obtained, the signal can be expressed as a series solution using the Jacobi-Anger expansion [74].

Depending on the physical situation, the analysis can be modified. In this review, we describe the implementation of this simple case; however, the general analysis forms a basis for extensions to more complex situations such as accounting for the presence of 2D harmonics, prominent viscous effects within the sensitive region, and incomplete coverage of the wavelength.

The flow measurements outlined in this review describe fluids using a power-law model; however, alternative models can be used as long as it is possible to describe the velocity distribution analytically. There may also be potential for incorporating non-analytical numerical modelling of the velocity distribution into the MR signal analysis, but this has not yet been attempted. Using the power-law model, the shear stress, σ can be related to the shear rate, γ ̇ as follows [82], σ = m γ ̇ n (21) where n is the flow behaviour index and m is the flow consistency index. If n < 1, the fluid is shear thinning and the apparent viscosity decreases with increasing shear rate, while the opposite occurs for shear thickening fluids (n > 1). Different values of n result in changes to the shape of the velocity distributions and the amount of phase interference. In some cases, such as laminar flow, closed-form analytical solutions for the velocity distribution can be determined and used to predict the behaviour of the MR signal. Figure 2 shows laminar pipe flow (Figure 2A) and circular Couette flow (Figure 2B) geometries that are characterized in the experiments discussed in this review.

In the case of laminar pipe flow of power-law fluids, the radial distribution of the axial velocity is given by, v r = 3 n + 1 n + 1 v avg 1 − r R 1 n + 1 (22) where R is the radius of the pipe. In the original work [35], the parameter n ′ = 1 n + 1 was defined for mathematical convenience; however, in most cases in this review, we will use the variable n to describe the flow behaviour index.

For circular Couette flow, more specifically, with a static outer cylinder of radius, R ₂, and inner cylinder of radius, R ₁ with angular velocity Ω, the radial distribution of the azimuthal velocity component is given by [83], v θ r = r Ω 1 − r R 2 − 2 n 1 − η − 2 n (23) where r is the radial position inside the pipe and η is the radius ratio, R 1 R 2 .

Figure 3 shows example velocity profiles for pipe flow (Figure 3A) and circular Couette flow (Figure 3B) of fluids with different values of the flow behaviour index, n, describing a shear-thinning fluid (dashed line), a Newtonian fluid (solid line), and a shear-thickening fluid (dotted line).

These equations can be used to predict the dependence of MR signal on parameters such as the flow velocity and flow behaviour index, often involving integrals with non-trivial solutions. In the following sections, we will review integration techniques and approximations that have been used to generate analytical solutions and simplify the integration.

In laminar flow, the velocity distribution is stable and remains constant in time; therefore, the MR signal phase can be approximated using a first-order Taylor approximation, ϕ r = ϕ 0 + γ G v r τ 2 (24)

Integrating over the dimensions of the pipe, the odd-echo phase can be related to the average velocity as follows, ϕ odd = ϕ 0 + γ G v avg τ 2 (25) Thus, the average flow velocity in pipe flow can be determined through measurements of phase at varying echo times while measurement of the average flow velocity directly at a single echo time requires a stationary reference.

Determination of the flow behaviour index, n, requires consideration of the MR signal magnitude. The normalized signal of all odd echoes due to phase accumulation can be expressed as, S ϕ = ∬ e − i ϕ r r d r d θ ∬ d s = ∫ cos ⁡ ϕ r r d r ∫ r d r − i ∫ sin ⁡ ϕ r r d r ∫ r d r (26) A thorough analytical treatment of these integrals can be found in [35]. Here, we provide the final solution for the normalized magnitude of odd echoes due to phase accumulation, M ϕ = S R e 2 + S I m 2 = 2 n ′ X 2 n ′ Γ 2 n ′ − Γ 2 n ′ , X i Γ 2 n ′ − Γ 2 n ′ , − X i (27) where X = n ′ + 2 n ′ γ G v avg τ 2 , n ′ = 1 n + 1 , and Γ ( a , x ) = ∫ x ∞ w a − 1 e − w d w .

Several measurement schemes can be used to extract the flow parameters using Eqs 25, 27.

• Use only the first echo to calculate laminar flow parameters. The v _avg is determined from the echo net-phase accumulation using Eq. 25 and used in Equation 27 to determine n′. Using only one odd echo data point suffers from noise, and is not reliable in realistic flow measurements.

The equations defined above are valid if the fluid within the sensitive volume is completely polarized. The following equations describing incomplete polarization have been verified on a 4.7 T vertical bore instrument [84], the ultimate goal is to translate this work to a portable MR instrument. ϕ odd ′ = arctan D t = 1 c − 2 n ′ + 1 0 1 − e − 1 t 1 C − t sin A t D t = 1 c − 2 n ′ + 1 0 1 − e − 1 t 1 C − t cos A t , 1 < n ′ < 2 , arctan N t = 1 c 0 e − 1 t ⁡ sin A t N t = 1 c 0 e − 1 t ⁡ cos A t , n ′ = 2 , arctan D t = 1 c − 2 n ′ 0 1 − e − 1 t sin A t D t = 1 c − 2 n ′ 0 1 − e − 1 t cos A t , n ′ > 2 , (28) M ϕ ′ = Γ 2 n ′ − 1 2 C 2 n ′ n ′ D t = 1 c − 2 n ′ + 1 0 1 − e − 1 t 1 C − t e − i A t D t = 1 c − 2 n ′ + 1 0 1 − e − 1 t 1 C − t e iAt , 1 < n ′ < 2 , C N t = 1 c 0 e − 1 t e − i A t N t = 1 c 0 e − 1 t e iAt , n ′ = 2 , Γ 2 n ′ 2 C 2 n ′ n ′ D t = 1 c − 2 n ′ 0 1 − e − 1 t e − i A t D t = 1 c − 2 n ′ 0 1 − e − 1 t e iAt , n ′ > 2 , (29) where Γ(x) is the gamma function Γ ( x ) = ∫ 0 ∞ w x − 1 e − w d w ; D x − p a f ( x ) is the Riemann-Liouville fractional integral D x − p a f ( x ) = 1 Γ ( p ) ∫ a x ( x − w ) p − 1 f ( w ) d w ; N x a f ( x ) is defined as N x a f ( x ) = ∫ a x f ( w ) d w ; C = L pol T 1 V max = n ′ L pol ( n ′ + 2 ) T 1 V avg ; and A = γ G v max τ 2 C = γ G L pol τ 2 T 1 . γ is the gyromagnetic ratio, G the gradient in the flow direction, and 2τ the CPMG echo time.

For circular Couette flow, the symmetry of the velocity distribution leads to a net cancellation of phase; however, the above approach can be used to extract the flow behaviour index from the signal magnitude. Taking the component of the azimuthal velocity distribution (Eq. 23) in the direction of the magnetic field gradient, the MR signal magnitude is given by, S = ∫ R 1 R 2 ρ r ∫ θ 1 θ 2 e − i γ G v θ r cos ⁡ θ τ 2 r d r d θ (30) Assuming the spin density distribution ρ(r), is uniform within the excited slice, ρ(r) = 1.

Given the geometry of the flow system, the magnitude can change significantly depending on the dimensions of the excited slice (a vertical cross-section of the outer cylinder). We present two approximations that can be used to simplify the MR signal analysis depending on the dimensions of the excited slice (only the thin slice approach has been verified experimentally):

• Complete coverage: If the dimensions of the excited slice are comparable to those of the Couette cell. This approximation requires small samples or weak gradients (wide bandwidths of excitation). Sample alignment problems are less likely; however, there is a greater possibility for complications due to B ₁ field homogeneity.

In the complete coverage case, a Bessel function comes from integration over angular coordinate, θ. The remaining integral over the radial coordinate is non-trivial and can be performed numerically. S = 2 π ∫ R 1 R 2 J 0 γ G v θ r τ 2 r d r (31)

In the thin-slice approximation, the angular coordinate is fixed, and the integration over the radial coordinate can be performed numerically. S = ∫ R 1 R 2 ⁡ cos γ G v θ r cos ⁡ θ τ 2 r d r (32)

We bring attention to two important consequences of the thin-slice approximation. The first is the need for precise knowledge of the slice position relative to the center of the Couette cell (or the angle between the velocity and magnetic field gradient, θ). Second is the reduction in sample volume and by extension, SNR. Practical considerations that can be used to account for these limitations will be described in Section 4.1.2 on experimental procedures.

Synchronization is a shared feature of both elastometry experiments and can be easily achieved through the use of an external trigger in the acquisition sequence. In the longitudinal case, synchronization of vibrations with the MR acquisition allows for velocity measurements at several points along the vibration period through the introduction of a series of delays before the MR excitation (the effective gradient is constant over the echo acquisition). Measured velocity waveforms can then be compared to obtain relative measurements of viscoelastic properties.

In shear wave experiments, synchronization provides a well-defined square wave modulation of the effective magnetic field gradient with a constant phase offset. It will be shown however, that synchronization is not completely necessary to observe the effects of vibration on the MR signal, and that averaging of random phase offsets still leads to an overall reduction in signal magnitude. Label (2) in Figure 5 indicates synchronization of the CPMG acquisition with the harmonic excitation for τ values that are integer multiples of the vibration period, leading to an effective square wave magnetic field gradient.

Given the sinusoidal motion, another modification allows for the use of the second echo which encodes acceleration. In the general spin echo sequence shown above, the second echo is a superposition of a spin echo (due to the refocusing of the first echo) and a stimulated echo (a result of imperfect RF pulses). The stimulated echo can be cancelled upon even numbers of acquisitions with a slight modification of the CPMG sequence described above where the phase of the first refocusing pulse is cycled between + and − y [71]. Label (1) in Figure 5 indicates the acquisition of the first and second echo with the same τ value.

In compression experiments, the effects of phase interference are minimal, and the motion can be completely characterized by measuring the net-phase accumulation. In shear wave experiments, phase interference depends on the wavelength and phase offset of the propagating shear wave; thus, both magnitude and phase can be used to gain insight into the dynamic properties that influence the shape of the velocity distribution.

Pipe flow measurements require very few modifications to the general sequence described above. Simply acquiring the spin-echo at a variety of echo times is sufficient to characterize both the average flow velocity (through the phase) and flow behaviour index (through the magnitude). Acceleration is negligible in laminar flow; therefore, the second echo carries no useful information regarding motion. Label (3) in Figure 5 indicates that only the first echo is used in pipe flow measurements,

It is possible to perform Couette flow measurements similarly; however, complications arise due to the small excited volume of the sample, leading to low SNR. These complications are addressed by acquiring a CPMG echo train with a very short echo time following the first motion encoding echo. Given the short echo time, the effect of flow on these echoes is negligible, and they can be added together to enhance SNR [94]. Label (4) in Figure 5 indicates that subsequent echoes are acquired with a shorter echo time.

The other complication that arises when working with thin excited slices in Couette flow is the difficulty in knowing the exact position of the slice across the cylinder. The offset of the excited slice can be measured using another modification to the general spin-echo sequence. Rather than applying a single “hard” 90° excitation pulse, which generates magnetization within a single continuous volume. Several pulses, separated by delays of length, δ, can be applied to generate magnetization within multiple regions of the sample, separated by [95, 96] a distance, Δ x = 2 π γ G δ . The offset can be deduced by comparing intensities from each of the excited slices in the 1D profile generated by the Fourier transform of the spin echo.

In shear and longitudinal wave experiments, a dual-channel signal generator was used to synchronize the MR acquisition with mechanical vibrations [33, 34]. The first channel sent a signal to a mechanical actuator. Shear waves were transmitted into the samples by an aluminum plate which covered the surface of the sample. Excitation stingers were used to induce longitudinal waves. An excitation stinger is a thin rod that is stiff in the axial direction, and flexible in other directions, ensuring that forces are only transmitted along the axis of the sample.

Both experiments relied on independent measurements of vibration amplitudes displayed on an oscilloscope. In shear wave measurements, an optical sensor was used to scan the sample surface. These measured amplitudes were then used to predict the decay of signal magnitude due to vibrations. In longitudinal wave measurements, a piezoelectric force sensor was positioned between the sample and the excitation stinger. These measurements were compared to the velocity waveforms measured with the portable MR sensor.

Pipe flow experiments were performed using the gravity-fed flow from a reservoir suspended several feet above the MR sensor and refreshed by fluid pumped from a second reservoir at floor level [35]. A submersible pump was used to ensure a constant pressure head driving the flow, providing more inflow to the upper reservoir than was flowing through the magnet. An overflow in the upper reservoir was used to return excess fluid to the lower reservoir. A flow meter was used to control the average flow rate. Most tubing in the flow network was clear PVC tubing with an inner diameter of 0.8 cm, except for a 70 cm section of 0.67 cm diameter glass tubing that was connected at a distance (of at least 50 diameters) from the opening of the magnet to allow for a steady flow profile to develop. Flow rates were independently measured from outflow using a measuring cylinder and timer.

Couette flow experiments were performed by suspending a 12 V DC motor, attached to a 6 mm diameter inner cylinder inside of a 10 mm sample tube. Rotary seals were fitted into the opening of the sample tube to maintain the central alignment of the inner cylinder [36]. An optical sensor was directed at the rotating inner cylinder to provide independent measurements of the rotation speed, which could be used to calculate the shear rate at the inner wall.

In compression measurements [34], viscoelastic urethane polymers (Sorbothane) were analyzed due to their long-term stability and well-characterized dynamic properties. The samples varied in hardness and were rated on a durometer scale, with OO40 being the softest, and OO70 the hardest. Dynamic properties were calculated using vibration calculators provided by the manufacturer. Cylindrical samples were cut into 1.65 cm diameter cylinders from 1.27 cm thick rectangular sheets and pre-compressed by approx. 0.15 mm against the surface of the magnet array. Vibrations were induced at 100 Hz along the axis of the sample with forces ranging from 0.7 to 2.1 N peak-to-peak. The MR acquisition was synchronized with vibrations using the external trigger. Echoes were acquired with an echo time (TE = 2τ ₁) of 0.5 ms at 10 points along the vibration period (10 ms) by including a table of delays evenly spaced by 1 ms.

Phase in the MR signal was used to determine the velocity, which was plotted alongside the piezoelectric force sensor measurements, allowing for relative comparisons between samples. Velocity waveforms are shown in Figure 6. Symbols represent measured velocities, solid lines represent sinusoidal fits, and dashed lines represent piezoelectric force sensor measurements (shifted in phase by 90° for comparison with velocity waveforms). The differences in viscoelasticity between samples can be seen through the variation in phase offsets and amplitudes. The loss angle was determined through the fitted value of the phase offset and the magnitude of the complex modulus was determined by plotting the MR-measured velocities against force and taking the slope. Figure 7 shows the MR-measured velocities (symbols) plotted against the piezoelectric force sensor measurements for each sample fitted linearly to extract the magnitude of the complex modulus.

Despite clear differences in the responses that depend on the dynamic properties of each sample, an agreement between measured and tabulated values was not achieved. Measured magnitudes of the complex modulus were in closer agreement than loss-angles; however, they were still found to deviate more for softer samples.

Several factors, unrelated to the MR instrumentation, may have contributed to deviations from tabulated properties. Imperfect sample geometries may have resulted in undetected shear components in directions orthogonal to the gradient, delays that were not accounted for may have contributed to additional phase offsets, and positioning of the sensitive slice near the loaded edge of the sample (where strain is larger) may have corresponded to a physical situation not reflected by the geometric parameter.

Although the experiment was not successful in achieving absolute measurement of viscoelastic properties, the relative measurements provide evidence that a portable MR instrument can indeed be used to characterize the viscoelastic properties of samples under longitudinal excitation. Phase measurements using the second echo were also found to be in close agreement with expectations.

Cylindrical (16 mm thick, 76 mm diameter) polyurethane and square (50 mm × 50 mm × 6 mm) Sorbothane samples were analyzed in shear wave experiments [33]. CPMG echo trains with 16 echoes were acquired with echo times corresponding to half the periods of mechanical vibrations at amplitudes of 0.152, 0.38, 0.48, 0.71, 0.86, 1.01, 1.2, and 1.35 μm.

Figure 8 shows echo intensities (cross symbols) plotted against vibration amplitude for square and cylindrical samples using the first echo (Figures 8A, C) and the second echo (Figures 8B, D). The dash-dotted line shows theoretical predictions obtained using Eq. 19, assuming a constant gradient throughout the sensitive volume. The solid line shows simulations based on the measured gradient distribution throughout the sensitive volume.

For both samples, the “Bessel Integral” and simulations provide a reasonable fit to the first echo measurements; however, the quality of the fit deteriorates for the second echo, especially at larger amplitudes. Agreement with simulations was improved by accounting for a constant offset associated with incomplete refocusing.

One would expect that the agreement with theory could be further improved by employing the phase cycle described in [71]. Furthermore, the theoretical description assumes that the size of the sensitive volume is comparable to the wavelength. Deviations from this condition result in changes to the velocity distribution present within the sensitive volume, and by extension decreased phase interference and modulation of signal magnitude.

These results demonstrate that, in the presence of strong magnetic field gradients, any active vibration can completely eliminate the MR signal. Therefore, unless the vibration is of interest in the measurement, echo times should be carefully considered for significant attenuation.

A natural approach to decrease vibration effects is to eliminate synchronization; however, we can see that when CPMG measurements are performed at a range of echo times (0.4–1.5 ms, corresponding to frequencies from 1,250 Hz down to 333 Hz) without synchronizing to the vibration of frequency (500 Hz) and amplitude of (255 ± 40 nm), the signal intensity remains sensitive. Figure 9 shows normalized second and third echo intensities, with the most significant signal loss at echo times of 0.9 and 1 ms (when 1/2 TE = 500 Hz).

This type of measurement can be used in two completely different ways. In applications where vibrations are detrimental to the desired measurement, it can be used to characterize the frequency and amplitude of vibrations and select echo times that avoid significant attenuation of signal intensity. There are also clear applications in elastometry, where vibrations can be controlled, allowing measurement of a sample’s viscoelastic properties through the modulation of the signal.

Two solutions, one Newtonian fluid and one shear-thinning fluid were analyzed in pipe flow experiments [35]. The Newtonian fluid was made by mixing distilled water and glycerol in a ratio of 6:1. For the shear-thinning fluid, xanthan gum was dissolved in distilled water at a concentration of 0.42 wt%. To ensure complete polarization, distilled water and xanthan gum solutions were doped with 0.33 wt% copper sulfate to reduce T ₁ lifetimes to 42 and 39 ms, respectively.

Each fluid was measured at two different flow rates. Newtonian fluid experiments were performed at flow rates of 40 ± 1 and 78 ± 1 mL/min corresponding to Reynolds numbers of 82 and 160 (well within the laminar regime typically observed for Reynolds numbers up to 2000). Shear-thinning flow experiments were performed at flow rates of 35 ± 1 and 66 ± 1 mL/min.

Figure 10 shows measurements of the first echo phase at different τ values (dots), fitted using a linear fitting method (shown by solid lines). Close agreement between the fitted and measured average velocities (within 3%) points towards the reliability of the phase-based method for determining the average velocity of laminar flow.

Figure 11 shows the normalized magnitude of the first echo. Using the fitted values of v _avg , Eq. 27 was used to fit the data and determine the flow behaviour index.

These results show that measurements of the first odd echo in a CPMG sequence acquired using a portable, low-field MR sensor with a flow-oriented gradient can be directly used for the determination of flow velocity profiles of non-Newtonian fluids.

Four different fluids were analyzed in Couette flow measurements [36]. For a Newtonian fluid, distilled water was doped with copper sulfate at a concentration of 30 mM. A Newtonian oil sample (corn oil) was used to investigate the limitations of diffusive attenuation on the measurement due to its smaller diffusion coefficient. For shear-thinning fluids, xanthan gum was mixed with distilled water in concentrations of 0.2 wt% and 0.4 wt%, and doped with copper sulfate at a concentration of 30 mM.

Calibration was compared using the DANTE and reference sample procedures (described in Section 4.1.2). The offset of the excited slice from the centre of the Couette cell was estimated to be approx. 2.69 mm using the DANTE procedure, and 2.68 mm using distilled water as a reference sample.

Figure 12A shows measurements of signal magnitude (after adding 16 echoes) at a shear rate of 4.85 s ⁻¹ normalized to the stationary sample signal at different values of τ ₁. There is a clear dependence of signal magnitude on the flow behaviour index, with the distilled water sample (solid line) exhibiting a more rapid decay compared to the 0.2% xanthan gum fluid (dash-dot line) and 0.4% xanthan gum fluid (dashed line). To extract the flow behaviour index, the data were fitted to a numerical integration of Eq. 32. Portable MR-measured flow indices were in close agreement with conventional high-field Rheo-MR measurements on samples with identical xanthan gum concentrations [97].

Figure 12B investigates the effects of Couette flow on signal magnitude at longer echo times, taking advantage of the smaller diffusion coefficient of the corn oil sample. Again, measurements agree with theoretical predictions initially, and while oscillations in signal magnitude are observed after τ ₁ = 1 ms, they do not occur with the expected amplitude or frequency.

These results further validate the methods used to characterize laminar pipe flows of non-Newtonian fluids and provide a basis for future research on other rotating flows.