Mathematical Model
A mathematical analysis of the fluid mechanics within the IDDSI flow test was conducted. The model assumes the liquid contained is subject only to hydrostatic pressure which is proportional to the height of the liquid. The fluid dynamics inside the syringe can be described by the Navier–Stokes equations, a set of continuity and momentum equations covering both fluids in this model: liquid and air [a recent explanation is provided by Batchelor,3 for example]. In some simple cases these can be solved analytically; to investigate whether a simplification is possible for this flow test, the process involves comparing the ratio of different forces acting on the liquids and investigating the dominance of each term.
In this model, gravity creates hydrostatic pressure which in turn converts to inertia for the motion which is resisted by viscous (friction) forces exerted by the walls on the liquid. The fluid dynamics of this system can be characterized by two dimensionless numbers (for Newtonian cases): the Froude and Reynolds numbers. The Froude number is the ratio between inertia and gravity forces and is defined as:
$$ Fr = \sqrt {\frac{{U_{\text{m}}^{2} }}{gD}} $$
(1)
in which \( U_{\text{m}} \) is the mean liquid velocity, \( g \) is the gravitational acceleration and \( D \) is the diameter.
The Reynolds number is the ratio between inertia and viscous forces:
$$ Re = \frac{{\rho U_{\text{m}} D}}{\mu } $$
(2)
where \( \rho \) is the density and \( \mu \) is the dynamic viscosity of the liquid.
For this system, the barrel diameter is much larger than the nozzle and hence the flow inside the barrel is much slower than in the nozzle. Thus, the Reynolds number in the barrel is much smaller than inside the nozzle. The effect of gravity is dominant in the barrel, however, inertia is dominant in the nozzle. Using nominal values for water gives a first approximation of the range of these dimensionless numbers: the Froude number can change from 0.02 (gravity dominated) inside the barrel to 5 (inertia dominated) inside the nozzle, and the Reynolds number can change from 100 inside the barrel to 1000 in the nozzle (both inertia dominated). For thicker liquids, both Froude and Reynolds numbers will decrease.
This simple analysis indicated it is not generally possible to neglect any categories of the forces and simplify the fluid motion equations for this system to find an analytical solution. This outcome has also been found earlier by Kutter et al., modelling a similar geometry.16 In such situations, computational fluid dynamics (CFD) can be applied to find solutions by dividing the geometry into many (usually thousands) discrete, simple elements and solving the dynamic equations simultaneously.
Computational Model
Since the geometry is axially-symmetric throughout and the pressure acts purely along the axial direction it was assumed that the flow inside the syringe was also axially symmetric and hence, the flow motion equations only needed to be solved in a cut-plane of the syringe. This two-dimensional (2D) configuration requires much less computational resources compared to three-dimensions. The 2D geometry of the computational domain is shown in Fig. 1.
The Volume of Fluid (VOF) model was used to track the interface between water and air inside the syringe over time. The tracking of the interfaces between the phases is accomplished by the solution of a continuity equation for the volume fraction of one of the phases. In our case, since the two fluids (liquid and air) are incompressible and there is no mass transfer happening between them, this equation can be simplified as:
$$ \frac{\partial \alpha }{\partial t} + \nabla \cdot \alpha \vec{\varvec{U}} = 0 $$
(3)
In which \( \alpha \) is the volume fraction of the secondary phase (here the liquid) and \( \vec{\varvec{U}} \) is its velocity field. The volume fraction of the primary phase (here air) requires no additional equation because the sum of the two volume fractions is under the constraint of \( \alpha_{\text{air}} + \alpha_{\text{liquid}} = 1 \). Air has the density of \( \rho_{\text{air}} = 1.184 {\text{kg}}/{\text{m}}^{3} \) and the dynamic viscosity of \( \mu_{\text{air}} = 1.86e - 5 \;{\text{Pa}}\;{\text{s}} \) . For the liquids, the density is calculated by measuring their mass for a specific volume and the viscosity is measured using a shear rheometer (see “Materials” section and “Results” section).
Thickened liquids were created from mixing powdered starch or gum with water until homogeneous (described later in “Thickened Drink Preparation”). The liquids were therefore assumed to comprise uniform continuous media having a no-slip interface with the walls of the geometry. Fluent 17.2 software (CAE Associates, Middlebury, CT, USA) was used to solve the set of continuity, momentum and the volume fraction tracking equations. A grid-independence study was performed using three different numbers of mesh cells. Following this, a structured grid with 7000 quadrilateral cells was selected and all the simulations were carried out using the same grid. At time zero, the computational domain is assumed to be filled with the liquid and as the liquids drains from the nozzle, air replaces it from the top. The top boundary of the domain is assumed a pressure outlet boundary with atmospheric pressure. The circular contact line at the upper liquid / air / solid-wall interface does not move; as the simulated liquid drains, it leaves a film coating on the solid walls. In reality, the residue on the walls is variable in quantity and patterning, however there is not a practical, reliable, computational approach to model a moving interface more realistically.
The exit section of the nozzle is also assumed as a pressure outlet. In the cases where the liquids drain continuously from the nozzle (thinner liquids) it was assumed that as soon as they exit to ambience, the pressure reaches atmospheric. However, in some experiments with thick liquids, the liquids were observed to form a droplet at the exit section of the nozzle and drip instead of continuously flowing. This was observed at Levels 2 and 3 for Starch and Gum-thickened liquids but only at the thickest (Level 3) Glycerol-water mixture. In these cases, the pressure at this boundary is not atmospheric: surface tension and viscous stresses would apply a significant periodic outlet pressure as each droplet is formed and released. Attempting to model individual droplet formation dynamically was rejected on the basis that the additional computational complexity could not be justified by any predicted gain in fidelity: the rheological measurements of these materials could not be assumed to extrapolate to droplet-formation. However, the effect of dripping was included in the model by using a steady-state approximation of the mean boundary pressure. This model has been discussed and validated in similar dripping-mode applications10,41 where it was judged to give a reasonable accuracy especially at the final stages of droplet formation. In this application, viscous effects on surface tension and droplet curvature change are assumed negligible in comparison to the capillary flow resistance of the nozzle tube. The applicable pressure was calculated by assuming the constant presence of a hypothetical droplet at the exit of the nozzle with a diameter equal to the nozzle. The pressure inside this hypothetical droplet can be calculated using the Young–Laplace equation:
$$ P_{\text{droplet}} - P_{\text{atmosphere}} = \frac{2\sigma }{r} $$
(4)
where \( \sigma \) is the surface tension between the liquids and air and \( r \) is the radius of the nozzle at the exit section. The surface tension between the gum-thickened liquids and air decreases slightly as the concentration of the gum increases, but for these concentrations it is very close (within 2%) to the surface tension for water and air at room temperature (0.072 N/m).11 Thus, given \( r \) = 0.9 mm, the assumed droplet pressure is 160 Pa, which was used as the outlet boundary condition for cases with dripping liquids.
A time interval of 0.01 s was selected as the time step of the transient simulation for the whole 10 sec period of the test, after applying the Courant–Friedrichs–Lewy (CFL) condition and verifying that residual errors were typically < 10−13.
The simulation of a fluids’ flow is governed by its rheology, and in this case the dysphagia-management drinks are known to be non-Newtonian. Herschel-Bulkley models have been successfully applied to similar starch- and gum-thickened liquids previously and that model type was adopted here. Model parameters were identified by linear regression of experimental shear rheometry data using Matlab software (Mathworks, Natick, MA, USA); see “Results” section.