In previous section, the membrane driven force of vesicle is obtained by minimizing total vesicle energy. The next step is to couple the motion of equation for fluid to the membrane force. The framework of fluid vesicle mechanism system will be clearly stated in this section.
2.4.1 Incompressible Navier-Stokes equations
Firstly, we begin by explaining mathematical formulation of fluid incompressibility.
Consider a specific system of fluid with control volume V (shape can change with time), but the total mass does not change. Denote density by ρ and velocity field by u, under Lagrangian framework, by tracking total mass of this control volume, the mass conservation law is stated as
D Dt
Z
V (t)
ρ dV = 0, (2.8)
where DtD = ∂t∂ +u·∇ denotes for material derivative. By Reynolds transport theorem, the above equation can be written by
Z
V
∂ρ
∂t + ∇ · (ρu) dV = 0. (2.9)
Since control volume is arbitrary chosen, the integrand in Eq. (2.9) is always zero, i.e.,
∂ρ
∂t + ∇ · (ρu) = 0. (2.10)
Eq. (2.10) is called continuity equation and can be alternatively written in the form of
Dρ
Dt + ∇ · u = 0. (2.11)
For incompressible fluid, shape of control volume can change but mass and volume remain, that is, the density of fluid element remains constant. Under this assumption, we can deduce DρDt = 0, thus mass conservation law of Eq. (2.11) is reduced by
∇ · u = 0. (2.12)
The Navier-Stokes equation is govern by conservation law of momentum. In clas-sical mechanics, momentum is the product of the mass and velocity of an object.
The conservation of momentum describes the total amount of momentum within a control volume keeps as a constant. That is to say, momentum is neither created nor destroyed, but it can be changed by external forces. Actually, the conservation of momentum is an application of Newton’s second law of motion, which states the rate change of momentum of a fluid mass is equal to net external forces acting one the mass. Such external forces are consisted of two classes. One is the body force (body force per unit mass), gravitational force or electromagnetic force for instance;
the other one is surface force (surface force per unit area), such as pressure forces or viscous stress.
With basic physical property of fluid mechanics, now we begin to introduce math-ematical formulation of Navier-Stokes equation. We assume the fluid is Newtonian, based on constitutive law, the expression for stress tensor is
σij = −pδij + µ µ∂ui
∂xj +∂uj
∂xi
¶
, (2.13)
where δij presents Kronecker delta function (δij = 1 if i = j; δij = 0 if i 6= j), p is the pressure and µ is the viscosity. Throughout this dissertation, the density and viscosity are both assumed to be uniform constant. Coupling the fluid incompress-ibility constrain Eq. (2.12), the mathematical equations of motion consist of a viscous incompressible fluid in a domain Ω can be written as follows.
ρ µ∂u
∂t + (u · ∇) u
¶
= ∇ · σij = −∇p + µ∆u in Ω,
∇ · u = 0 in Ω.
Notice that there is no equation describing evolution of pressure p. In fact, pressure p plays a role as Lagrange’s multiplier to enforce the fluid incompressibility constraint.
On the other hand, for well-poseness of Navier-Stokes equation, the velocity field is equipped with boundary condition, and we name several types of boundary condition as follows.
• No-slip boundary condition: There is no fluid which penetrates the boundary, the fluid is at rest status on boundary, that is
u|∂Ω= 0.
• Inflow boundary condition: The velocity field is given specifically on the bound-ary, that is
u|∂Ω= ub.
• Outflow boundary condition: Neither velocity component changes in normal direction to the boundary, that is
∂u
∂n = 0.
2.4.2 Coupling the fluid equations to vesicle forces
In this section, the governing equation of fluid motion and force acting on vesicle membrane are combined by Immersed boundary formulation. In immersed boundary framework, the fluid variables (velocity field, pressure, etc.) are defined in Eulerian coordinate and interfacial variables (membrane position, membrane force, etc.) are defined in Lagrangian coordinate. These two types of variable are linked by inter-action equations in which the Dirac delta function plays a prominent role. Now we come to state the whole fluid system of vesicle dynamics through immersed boundary method in two-dimensional space. The mathematical equations of motion consist of a viscous incompressible fluid in a domain Ω containing an immersed, inextensible, massless vesicle boundary (or interface) Γ(t) which can be written in an immersed boundary formulation as
ρ µ∂u
∂t + (u · ∇)u
¶
= −∇p + µ∆u + Z
Γ
F(s, t)δ(x − X(s, t)) ds in Ω, (2.14)
∇ · u = 0 in Ω, (2.15)
∇s· U = 0 on Γ, (2.16)
∂X
∂t (s, t) = U(s, t) = Z
Ω
u(x, t)δ(x − X(s, t)) dx on Γ. (2.17) The interfacial force F consists of tension and bending force as
F = Fσ+ Fb = ∂
∂s(στ ) − cb∂4X
∂s4 . (2.18)
Eqs. (2.14)-(2.15) are the incompressible Navier-Stokes equations with a singular force term F arising from the vesicle membrane force. Eq. (2.16) represents the inextensibility constraint of the vesicle surface which is equivalent to the zero surface divergence of the velocity along the interface. Eq. (2.17) simply explains that the interface moves along with the local fluid velocity (the interfacial velocity). Here, the
interfacial velocity U is the interpolation of the fluid velocity at the interface defined as in traditional IB formulation. The interaction between the fluid and the interface is linked by the two-dimensional Dirac delta function δ(x) = δ(x)δ(y). One can easily extend to fully three-dimensional Navier-Stokes equation by taking three-dimensional vesicle membrane force mentioned previously.
It is worthy to mention that, in fact, the tension σ acts like a Lagrange’s multiplier function to enforce the local inextensibility constraint along the surface which is exactly the same role played by the pressure to enforce the fluid incompressibility in Navier-Stokes equations. Therefore, the development of an efficient and accurate algorithm for vesicle dynamics remains quite challenging; not to mention the vesicle surface moves along with the surrounding fluid as well.
Chapter 3
Immersed boundary method
The immersed boundary (IB) method was firstly proposed by Peskin [48] to simulate blood heart value problems. The IB method has found a wide variety of applications in biological mechanics and has been successfully applied to study fluid-structure interaction problems. The main idea of IB method is to treat Lagrangian material as a part of fluid in which arises additional interfacial forces. The fluid variables are solved in regular Cartesian domain (uniform rectangular lattice) without any modification of fluid equation; the interface is tracked by a set of discrete Lagrangian markers which can freely move in Eulerian domain and the shape of interface can has complex geometry. The interfacial forces can be computed through spatial position of Lagrangian markers and only has effect in its vicinity fluid, i.e. these forces are spread into its surrounding fluid lattices. With presence of these forces, the new fluid velocity can be obtained. The interface position is then advanced by this new fluid velocity through an interpolation.
3.1 Connection between fluids and interfaces
In this section, we introduce the linkage between fluid equation and Lagrangian in-terface. By taking advantage of Dirac delta function, a value of function can be obtained through product with a shifted delta source function, and then integrate over the whole space (thus the shifted delta function is regarded as a kernel of in-tegral). We clarify this definition by writing the spreading forces on Cartesian grid by
f(x, t) = Z
Γ
F(s, t) δ(x − X(s, t)) ds, (3.1)
and the interpolation of fluid velocity field by u(X(s, t), t) = U(s, t) =
Z
Ω
u(x, t) δ(x − X(s, t)) dx. (3.2)
Notice that Eqs. (3.1) and (3.2) represent for a line integral along the interface and an area integral over whole fluid domain respectively (in three-dimensional case, they are surface and volume integral). It is interesting to see that the total amount of singular force in Eq. (3.1), which is an integral of f(x, t) over whole domain, is equal to the total force on the immersed interface. This means although the interfacial forces (force per length) are converted to body force (force per area), the total amount of applied body force still preserves to act on the fluids. The important feature of IB method is Eq. (3.1) converts Lagrangian to Eulerian coordinate while Eq. (3.2) does in opposite direction.