On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within

Abstract

In this chapter, we analyze the steady-state microscale fluid–structure interaction (FSI) between a generalized Newtonian fluid and a hyperelastic tube. Physiological flows, especially in hemodynamics, serve as primary examples of such FSI phenomena. The small scale of the physical system renders the flow field, under the power-law rheological model, amenable to a closed-form solution using the lubrication approximation. On the other hand, negligible shear stresses on the walls of a long vessel allow the structure to be treated as a pressure vessel. The constitutive equation for the microtube is prescribed via the strain energy functional for an incompressible, isotropic Mooney–Rivlin material. We employ both the thin- and thick-walled formulations of the pressure vessel theory, and derive the static relation between the pressure load and the deformation of the structure. We harness the latter to determine the flow rate–pressure drop relationship for non-Newtonian flow in thin- and thick-walled soft hyperelastic microtubes. Through illustrative examples, we discuss how a hyperelastic tube supports the same pressure load as a linearly elastic tube with smaller deformation, thus requiring a higher pressure drop across itself to maintain a fixed flow rate.

Appendix

In this appendix, we consider the flow rate–pressure drop relationship for steady flow of a power-law fluid within a linearly elastic, thick-walled pressure vessel of thickness t, and inner radius $r_{i}=a$. The pressure vessel is subject only to an internal distributed pressure load p, with zero external pressure. Then, the state of stress evaluated at the inner radius (see [40]) is:

$$\begin{aligned} \sigma _{\theta \theta }&= \left( \frac{r_{o}^2+r_{i}^2}{r_{o}^2-r_{i}^2}\right) p,\end{aligned}$$

(46a)

$$\begin{aligned} \sigma _{rr}&= -p,\end{aligned}$$

(46b)

$$\begin{aligned} \sigma _{zz}&= \left( \frac{r_{i}^2}{r_{o}^2-r_{i}^2}\right) p. \end{aligned}$$

(46c)

The hoop strain is given by the constitutive equations of linear elasticity as:

$$\begin{aligned} \varepsilon _{\theta \theta } = \frac{u_r}{r_i}=\frac{1}{E}\big [\sigma _{\theta \theta }-\nu (\sigma _{zz}+\sigma _{rr})\big ]. \end{aligned}$$

(47)

Using Eqs. (46) and (47) yields

$$\begin{aligned} \frac{u_r}{r_i}= \frac{1}{E}\left[ \left( \frac{r_o^2+r_i^2}{r_o^2-r_i^2}\right) p - \nu \left( \frac{r_i^2}{r_o^2-r_i^2}-1\right) p\right] , \end{aligned}$$

(48)

which, upon using Eqs. (18) and (21), becomes

$$\begin{aligned} {\frac{u_r}{r_i} = \bar{t}\left[ \frac{(1+\bar{t})^2(1+\nu )+(1-2\nu )}{(1+\bar{t})^2-1}\right] \beta \bar{p}; \qquad \beta =\frac{\mathcal {P}_c}{E \bar{t}}.} \end{aligned}$$

(49)

After deformation, the inner radius is ${R}_{i}=r_i + u_{r}$ (where, initially, $r_i=a$). Thus, the dimensionless inner radius is

$$\begin{aligned} \bar{R}_i = \frac{r_i+u_r}{r_i} = 1+\frac{u_r}{r_i}=1+ \left[ \frac{(1+\bar{t})^2(1+\nu )+(1-2\nu )}{2 + \bar{t}}\right] \beta \bar{p}. \end{aligned}$$

(50)

Substituting the expression for $\bar{R}_i$ from Eq. (50) into Eq. (33), we obtain an ODE for the dimensionless pressure $\bar{p}$:

$$\begin{aligned} \frac{\mathrm {d}\bar{p}}{\mathrm {d}\bar{z}} = -2[(3+1/n)\bar{q}]^n\left( 1+ \mathfrak {K} \beta \bar{p}\right) ^{-(1+3n)}, \end{aligned}$$

(51)

where we have defined $\mathfrak {K} := [(1+\bar{t})^2(1+\nu )+(1-2\nu )]/(2+\bar{t})$ for convenience. As usual, the ODE (51) is separable and subject to a pressure outlet BC [i.e., $\bar{p}(1)=0$], thus we obtain:

$$\begin{aligned} \bar{p}(\bar{z}) = \frac{1}{\mathfrak {K}\beta } \left( \left\{ 1 + 2(2+3n) \mathfrak {K}\beta [(3+1/n)\bar{q}]^n (1-\bar{z})\right\} ^{1/(2+3n)} - 1\right) . \end{aligned}$$

(52)

Then, the full pressure drop is simply $\varDelta \bar{p}=\bar{p}(\bar{z}=0)$. Note that ${\mathfrak {K} = (1-\nu /2) + \mathcal {O}(\bar{t})}$, thus the expression for $\varDelta \bar{p}$ based on Eq. (52) [i.e., Eq. (40) above] reduces to Eq. (39) (based on [3]) identically for thin shells ($\bar{t}\ll 1$).