Over the last 60 years, a number of groups have investigated the morphology of filiform papillae (oftentimes motivated with associated diseases). Detailed studies exist on rats [14, 15], mice [16], porcupine [17], wild boar and pig [18], and humans [19, 20]. An overview of evolution in papilla morphology among vertebrates was detailed in Iwasaki [21]. An overview of the literature indicates the following morphological characteristics. In small vertebrates, individual papillae have a typical width ranging from 15 to 20 μm, and often taper conically at their tip. This width corresponds to the size of two to three epithelial cells, and is about the same order as the typical distance between papillae on the tongue. Their length can vary greatly between the anterior and posterior parts of the tongue and range from 50 to 100 μm. A precise set of measurements for humans were reported in Yamashita and OdDalkhsuren [22] where a distinction is made between the wide papillae body, which is anchored in the tongue, and the hairs on the papillae which do protrude upward from the tongue and deform under flow. Their size is reported to be 34 ± 16 μm in width and and 250 ± 62 μm long. Other measurements reports significantly larger sizes, with widths that can range 100–300 μm and length 200–500 μm [21, 23, 24]. Given the wide range of measured values among all the gathered papers, we will consider a half-width (radius) of 50 μm and length 250 μm in this paper as representative numbers.

No measurements of the Young's modulus of filiform papillae is available in the literature. A measure of E = 15 kPa of the pig's tongue modulus was obtain on fresh pig tongues [24]. That number in the range which would have been obtained by focusing solely on the elasticity of epithelial cells, for example E ≈ 1 − 10 kPa for monkey kidney epithelial cells [25] while E ≈ 0.1 − 300 kPa for human foreskin epithelial cells, with an average of 14 kPa for young cells and 33 kPa for old cells [26]. In light of these published data, we will use a value of E = 25 kPa in this study.

Based on the previous section, we can now put forward the model considered in this work, illustrated in Figure 1. We consider the fluid-solid interactions in a model mouth. The palate is modeled as a smooth, rigid, flat surface, at a distance H from the tongue. The tongue is assumed to be a smooth, rigid, flat surface on which filiform papillae are distributed. The total length of the flow region from the anterior to the posterior side of the mouth is denoted W. The focus of the work is on the deformation of the filiform papillae. Each papilla is modeled as a straight elastic rod of length L, radius a, clamped on the tongue. The Young's modulus of the rod is denoted E. As discussed above, we take E ≈ 25 kPa as a representative value. We also assume a = 50 μm, leading to a bending modulus B = πa⁴E/4 ≈ 1.2 × 10⁻¹³ J.m.

As stated in the introduction, the bovine tongue sensory innervation has already been partially described [10]. In addition to this, a recent study carried out in mice reported the presence of corpuscular endings innervating each filiform papillae [9], confirming a similar finding from previous work also describing innervation of filiform papillae in cats [27]. Since those endings do not project to the surface of the oral epithelium, we can hypothesize that they are mechanosensitive in nature and do not pertain to chemical stimulation. Our motivation to study the deformation mechanics of filiform papillae under flow lies in the fact that applied macroscopic strains will cascade into microscopic cell and transmembrane protein deformations and lead to somatosensory sensing as already known in the mamallian skin [28]. This should thus open a window for biophysicists regarding the levels of stresses that are likely to be applied to mechanosensory cells when mammals probe food for its texture.

As we are interested in the interactions between deforming papillae and the flow of the food product, it is important to accurately model the driving of the flow between the palate and the tongue. During tasting and swallowing the palate is stationary and the tongue moves horizontally parallel to and vertically toward the palate driven by the action of lingual muscles. The typical setup is therefore that of (1) a squeeze flow driven by the vertical motion of the bottom surface (so-called transverse motion in the literature [29]) and (2) a shear component also present during tasting (referred to as longitudinal motion [29]). The shear component of the tongue movement can play an important role when acting alone, but as we see below the stresses it creates can safely be neglected when a transverse flow is present.

The study is motivated by the measurements of textures on complex fluids such as ice cream. Rheological measurements for 38% fat cream show a viscosity of about 20 mPa.s, hence approximately 20 times that of water [30]. Ice-creams typically have a viscosity in the range 20–300 times that of water [31], with strong variation with temperature and fat content. Recent measurements for ice cream with fat, fat replacers, and sweeteners indeed confirm this showing viscosities O(100 mPa.s) [32, 33]. In this study we pick a reference viscosity of μ = 100 mPa.s, equivalent to 100 times that of water at room tenperature.

It is also necessary to account for typical velocities in mouth during food consumption. Detailed measurements of movement in the mouth during swallowing in show velocities ranging from 1 to 15 cm/s, with an average around 10 cm/s Shawker et al. [29]. This is consistent with results from computational simulations [34]. For a tongue-palate distance of H ≈ 5 mm, a papilla size L ≈ 250 μm, and a viscosity μ = 100 mPa.s, this leads to Reynolds numbers O(1 − 10) in the mouth while the flows around the papillae are characterized by Re ~ 10⁻¹ or even smaller due to the no-slip boundary condition near the papillae. Given that the relevant Reynolds number in narrow geometries is the regular Reynolds number times the channel aspect ratio, we are in the low-Reynolds number regime and can neglect the influence of inertia on the dynamics of the fluid in the mouth [35, 36].

For the flow in the mouth we consider the basic setup illustrated in Figure 2. The tongue is assumed to move both in the longitudinal (horizontal) direction, with a typical velocity U, and in the transverse (vertical) direct with typical velocity V. We use x and y to denote the directions along and perpendicular to the tongue (see Figure 2), with the tongue located at y = 0 and the palate at y = H.

The longitudinal motion leads to a shear flow in the fluid. The solution of Stokes equations in this case is simply u = u_∥e_x with a linear profile [37] (1)u∥=UH-yH, where we use the subscript ∥ to indicate that the flow is driven by motion parallel to the tongue. The typical shear rate acting on the tongue in this case is on the order of γ˙∥=O(U/H).

The transverse motion leads to a pressure-driven (squeeze) flow in the thin gap between the tongue and the palate. Assuming unidirectional flow, we have in this case u = u_⊥e_x with u_⊥ satisfying (2)μd2u⊥dy2=dpdx, and where we use the subscript ⊥ to indicate that the flow is driven by motion perpendicular to the tongue. The solution to Equation (2) is the standard parabolic flow (3)u⊥(y)=12μdpdxy(y-H). The pressure gradient is found using conservation of mass. Indeed, the flow rate along the tongue, Q, needs to the equal to the flow rate induced by the motion of the tongue, and so Q = VW. Integrating Equation (3) across the gap we obtain the scaling (4)Q~-H3μdpdx, and therefore (5)1μdpdx~-VWH3, leading to a flow profile scaling as (6)u⊥(y)~VWy(H-y)H3· The typical shear rate on the tongue in this case is therefore given by γ˙⊥=O(VW/H2) and the ratio between the typical longitudinal and transverse shear rates is thus given by (7)γ˙∥γ˙⊥~U/HVW/H2~(UV)(HW). Detailed measurements on tongue movement during swallowing shows that U is usually inferior to V and, in the rare cases where it is above, at most a factor of 2 larger [29]. Given that we are in a very high aspect-ratio geometry, H ≪ W by about one order of magnitude, and thus we obtain the result that γ˙∥≪γ˙⊥. The shear flow induced by the tongue in the anterior-posterior direction leads to stresses on the tongue which are smaller in comparison with the stresses induced by the tongue motion in the superior/inferior direction and the resulting squeeze flow. In summary, when a squeeze flow is present, the shear flow is small and can be safely ignored.

With the Newtonian flow in the tongue characterized in Equations (1) and (6), we now consider its effect on a single papilla. Before computing the shape, and the strain, of a deforming papilla, a number of questions need to be addressed in order to properly characterize their dynamic regime.

First, do papilla deform with unsteady dynamics (transient motion and relaxation) or are the time scales involved sufficiently long that we can model them as deforming in a quasi-steady fashion? This is a question in the realm of fluid-structure interactions of filaments in viscous fluids [38]. With a bending modulus B and a papilla of length L in a viscous fluid of viscosity μ, the typical time scale for the deformation to reach its steady-state shape is given by an elasto-hydrodynamics time scale, t_eh, [39] (8)teh~ξL4B, where ξ is the drag coefficient for flow near the papilla, ξ ≈ 4πμ/[ln(2L/a) + 1/2]. With μ = 100 mPa.s, L ~ 250 μm, and B = 1.2 × 10⁻¹³ J.m, we obtain a typical time scale of t_eh = 0(10 ms). How does this compare with the typical time scale, T, for the flow in the mouth? This timescale is given by H/V, it is the time scale over which the flow in the mouth is going to change. The vertical velocity is on the order of V ≈ 10 cm/s while the thickness is H ≈ 5 mm and thus T = H/V ≈ 50 ms. Since the high value for V can be considered an upper bound, the time scale we obtained of 50 ms should be viewed as a lower bound. Given that T ≳ t_eh we are thus in a regime where we expect unsteadiness to not play an important role and the deformation will be treated as quasi-static.

The second important question about the dynamics concerns the issue of small vs. large deformations. Do we expect the papillae to deform in the nonlinearly geometric regime where internal tension plays an important role and careful attention needs to be paid to the extensibility of the papilla, or do we remain safely in the linear regime? If we were to remain in the linear regime, the typical deflection of a linear beam would be scaling in the following manner. Let us call δ the typical magnitude of the papilla tip deflection. For a force of magnitude q per unit length, δ scales as δ ~ qL⁴/B [40]. Here the load is due to the fluid drag and thus q ~ ξu where we use u to denote the typical velocity of the fluid relative to that of the papilla, leading to δ~ξuL4/B=tehu.

We need to then consider separately the shear and squeeze flows. In the case of a shear flow, the typical relative velocity around the papilla is u_∥ ≈ UL/H leading to the scaling δ/L ~ t_ehU/H. With t_eh = 10 ms, H = 5 mm, and U ≲ 10 cm/s we obtain δ/L = 0(10⁻¹) and remain safely in the linear regime. In the case of a squeeze flow we have the scaling near the papilla u ~ VLW/H² and thus we obtain δ/L~tehVW/H2. Here again with t_eh = 10 ms, H = 5 mm, V ≲ 10 cm/s, and W = 5 cm, we obtain δ/L = 0(1), which is at the limit but is a regime in which the linearized equations of solid mechanics will be at least qualitatively accurate, and approximately quantitatively accurate.

In order to describe the deformation of a papilla in a flow, and the elastic strain that it applies to the tongue, we use the formulation for the dynamics of elastic filaments in viscous flows. Let us denote by r the location of a point along the filament and s the arc length along the filament (from s = 0 to s = L). We use u to denote the external flow field located at position r and τ the tension (force per unit length) enforcing inextensibility of the filament. The instantaneous balance of forces and moments on the filament leads then to [38] (9)ξ(rt-u)=-Brssss+(τrs)s. In Equation (9) we have used the shorthand notation that (…)_s means a partial derivative along the s direction and (…)_t a partial time derivative (so that the first term on the right-hand side of Equation (9) has four derivatives along s).

To simplify the mathematical approach and the interpretation of the results we proceed to non-dimensionalize the problem. We use the length of the papilla, L, as the characteristic length, t_eh as the characteristic time (Equation 8), L/t_eh as the characteristic velocity and B/L² as characteristic tension. Doing so, Equation (9) becomes (10)rt=u-rssss+(τrs)s. The equation for the tension τ is found by enforcing that r_s · r_s remains equal to one for all times, leading to (11)τss-τ(rss·rss)+4rss·rssss+3rsss·rsss+rs·us=0. In Equation (12) the term including u_s is small and can be neglected since it quantifies gradients of the flow along the length of the papilla. In the linear regime, r_s = t = e_y is the tangent to the papilla while u_s = ∂u/∂y is the y derivative of the flow, which is nonzero along the x direction only, and therefore r_s · u_s = 0. We thus obtain the further simplification (12)τss-τ(rss·rss)+4rss·rssss+3rsss·rsss=0. Furthermore, using our scaling approach from the previous section we can neglect any nonlinear terms in the shape of the papillae. As can be seen in Equation (12), the tension τ is expected to be quadratic in the filament amplitude and therefore the last term in Equation (10) is cubic, and can be neglected. We thus end up with the linear beam equation (13)u-rt=rssss, which, when simplified further due to the fast relaxation of unsteady modes (see Section 4.1), becomes the quasi-steady equation (14)u=rssss, Physically, Equation (14) indicates a balance between bending and the drag forces from the fluid.

Calling δ the typical deflection of an individual papilla and coming back to dimensional equations, we thus obtain the linearized balance (15)ξu~Bd4δdy4, leading to a typical tip deflection on the order of (16)δ~ξL4uB· We can then use the two velocity profiles, Equation (1) and Equation (6), to obtain the typical order of magnitude of the deflections.

We are now in a position to answer the main question of this work: do the papillae, through their deformation and the fact that they are anchored to the tongue, lead to deformation of the tongue which is stronger than if the papillae were not present. In other words, if the mechano-sensors are distributed beneath the surface of the tongue, could papillae amplify deformation of the tongue mediated through fluid-structure interactions? We can quantitatively address this question within our mathematical framework. In addition, since sensory neurons have also been shown to be mechanosensitive themselves (and thus not solely serving conduction of surface mechanosensory cells [41]), it is important to ask whether such neurons could be impacted by papillae strains.

The first step is to consider the Green's function for the deformation of a semi-infinite elastic substrate due to a point force. With the Green's function known, all other types of deformations can be tackled mathematically.

Consider an elastic medium located in the x₃ < 0 semi-infinite plane, as a model for the tongue. The Young's modulus of the tongue is denoted E and its Poisson's ratio ν. A constant force of magnitude F₃ is applied at the surface of the tongue, with the surface being parallel to the (x₁, x₂) plane. The force is being applied at the origin of the coordinate system. The solution to this Green's function problem is classical [40] and we summarize it here. The elastic displacements in the tongue along each direction, {u₁, u₂, u₃} at position {x₁, x₂, x₃} are given by (23a)u1=1+σ2πE[-x1x3r3-(1-2σ)x1r(r-x3)]F3, (23b)u2=1+σ2πE[-x2x3r3-(1-2σ)x2r(r-x3)]F3, (23c)u3=1+σ2πE[x32r3+2(1-σ)r]F3, where r2=∑ixi2.

For simplicity we then assume that the tongue is incompressible, meaning its Poisson's ratio is given by σ = 1/2. This simplifies the results for the displacements as

With the elastic displacements known, we can then compute the strain field, and it is given by the symmetric part of the displacement gradients. As can be seen in Equation (24), the typical displacement inside the tongue scales as u ~ F/Er at a distance r from the point where the force F is being applied. The strains in the tongue, ε, are therefore expected to take the approximate values (25)ε~ur~FEr2·

The deforming papilla is not applying a force on the tongue but instead it applies a moment at its base. If we denote by M the applied moment then we need to take one additional spatial derivative to obtain the strains in the tongue, leading to (26)ε~MEr3· The maximum tongue strain is found near the base of the papilla, r ~ a where a is the radius of the papilla leading to the relevant strain value of (27)ε~MEa3·

The result shown by Equation (27) needs now to be compared with what would happen in the absence of papillae-induced deformation.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. BL and CP are employees of the Nestlé Research Center. EL was consulting with Nestlé Research Center during the study.