2.3.2. Mathematical Problem Statement

The mass and momentum conservation equations (Equations (12)–(13)) for an incompressible fluid (Equation (14)) can be expressed as

$$
\rho\_\mathbf{f} \left( \frac{\partial \mathbf{u}}{\partial \mathbf{t}} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \sigma \tag{12}
$$

$$
\nabla \cdot \mathbf{u} = \mathbf{0},
\tag{13}
$$

$$
\sigma = -\mathbf{p}\mathbf{I} + \tau \tag{14}
$$

$$
\pi = \mathfrak{n}(\dot{\chi}) \mathbf{D}\_{\prime} \tag{15}
$$

where ρ<sup>f</sup> is the fluid density, p is the pressure, u is the fluid velocity vector, and u<sup>g</sup> is the moving coordinate velocity. In the arbitrary Lagrangian–Eulerian (ALE) formulation, (u−ug) is the relative velocity of the fluid with respect to the moving coordinate velocity. Here, τ is the deviatoric shear stress tensor (Equation (15)). This tensor is related to the velocity through the strain rate tensor; in Cartesian coordinates it can be represented as follows:

$$\mathbf{D} = \frac{1}{2} (+\nabla \mathbf{u}^{\mathsf{T}}).\tag{16}$$

The momentum conservation equation for the solid body can be written as follows:

$$
\nabla \cdot \sigma\_{\sf s} = \varrho\_{\sf s} \ddot{\sf u}\_{\sf s} \tag{17}
$$

where ρ<sup>s</sup> , <sup>σ</sup>s, and .. u<sup>s</sup> are the density, stress tensor, and local acceleration of the solid, respectively.

It is known that blood vessels can be described as hyperelastic materials [18–20]. Because of a similar anatomical composition, the bile ducts can also be considered hyperelastic materials. For hyperelastic materials, the stress–strain relationship is written as follows:

$$
\sigma\_{\rm s} = \frac{\partial \mathbf{W}}{\partial \varepsilon} \,, \tag{18}
$$

where ε is the strain tensor and W is the strain energy density function. The Mooney–Rivlin hyperelastic potential is shown in Equation (1).

The mathematical statement of blood flow in the aorta–shunt–pulmonary artery system is governed by Equations (12)–(18)).

The FSI interface should satisfy the following conditions:

$$\mathbf{x}\_{\mathbf{g}} = \mathbf{x}\_{\mathbf{s}} \tag{19}$$

$$\mathbf{u}\_{\mathbf{g}} = \mathbf{u}\_{\mathbf{s}} \tag{20}$$

$$
\mathfrak{o}\_{\mathfrak{F}} \mathfrak{h}\_{\mathfrak{F}} = \mathfrak{o}\_{\mathfrak{s}} \mathfrak{h}\_{\mathfrak{s}}.\tag{21}
$$

The displacements of the fluid and solid domain should be compatible, as in Equation (19). The tractions at this boundary must be at equilibrium (Equation (21)). The no-slip condition for the fluid should satisfy Equation (20). In the above conditions, Equations (19)–(21) give the displacement, stress tensor, and boundary normal, respectively. The subscripts f and s indicate fluid and solid parts, respectively. Blood is assumed to be a Newtonian fluid. The blood density is equal to ρ = 1060 kg/m<sup>3</sup> ; the dynamic viscosity is constant and equal to µ = 0.0035 Pa·s. The velocity profile during the systolic and diastolic phases of the left ventricle was applied at the aortic root inlet (Figure 1). The left ventricular systole period is t = 0.22 s. The period of ventricular diastole is t = 0.28 s. The total cardiac

cycle is t = 0.5 s. The peak velocity is 1.4 m/s. A time-dependent pressure profile was used as the boundary conditions at the aortic outlets. Constant pressure of P = 20 mm Hg was applied at the pulmonary artery outlets.
