3.2 Material de los sustratos
3.2.1 Procedimiento para la fabricación en autoclave de piezas de
problems and references in order to set the results of later chapters in the proper context.
2.2
Derivation of the equations
In this section we give a derivation of the Navier–Stokes equations for in- compressible flow in 3D (from which the Euler equations can be derived). The analogous equations ind-dimensions ford≥2 can be derived similarly. This is not meant to be a complete explanation of the required continuum mechanics, although we will attempt to briefly justify the arguments from the first principles of Newtonian mechanics. Our discussion here is informed by several sources; some modern (Chorin and Marsden (1993), Gonzalez and Stuart (2008), Majda and Bertozzi (2002) and Robinson et al. (2016)) and some less so (Sommerfeld (1950) and Stokes (1845)). We refer the reader to these texts and references therein for a more complete discourse. The Navier–Stokes equations can be derived by modelling fluid in the in- terior of a domain as a continuum (i.e. we assume that the nuances of molec- ular interaction have a negligible effect on the macroscopic behaviour). For the purposes of the derivation, we will only consider smooth velocity fields. As above, we denote the velocity of the fluid at timetand positionxby
u(x, t). We will assume that the flow is volume preserving, i.e. that there is no net flow across the boundary of any compact subdomain Ω0 bΩ:
Z
∂Ω0
u·dn= 0
wheren denotes the outward unit normal. This corresponds to the point- wise constraint
∇ ·u= 0.
It is natural to also require that mass is conserved, which can be formulated as
∂tρ+∇ ·(uρ) =∂tρ+ (u· ∇)ρ= 0,
whereρ(x, t) gives the distribution of density in the fluid. Equivalently d dt Z Ω0 ρ=− Z ∂Ω0 uρ·dn
2.2. Derivation of the equations
for any Ω0 bΩ.
We will only be considering the case of a homogeneous fluid, i.e. one with constant density. To save notation let us assume thatρ≡1, then the mass- conservation constraint becomes a redundant copy of the incompressibility constraint.
Setting aside the extrinsic body forcing, we assume that two effects govern the motion of the fluid, namely pressure and viscous forces (friction). We will consider the total force acting on a region of fluid Ω0 at one instant of time, which (by Newton’s third law of motion) we assume to be the integral over the boundary∂Ω0 of the forces exerted there.
The force caused by the pressure is modelled as acting in the direction of the inward normal (−n) to∂Ω0 at every point, with magnitude equal to the pressure at that point. This contributes the force
Z ∂Ω0 −pndA= Z Ω0 −∇pdx.
To model the viscous forces, we assume that they are proportional to the rate of strain (defined below) across the boundary in the outward direction. The intuition here generalises Newton’s model that friction between moving lamina is proportional to the derivative of the velocity, taken perpendicular to the lamina. Essentially, the rate of strain tensor is the component of the gradient of the velocity, that gives a first-order approximation of how the flow is“pulling apart”, relative to any rigid motion.
In more detail, at a point x∈∂Ω0, we consider a linear approximation to the velocity:
u(x+δx, t)≈u(x, t) + (∇u)δx=u(x, t) + [∂juiδxj]i,
for sufficiently smallδx ∈ R3. We therefore approximate the evolution of
the fluid relative to the motion at (x, t) by considering trajectories corre- sponding to a fixed velocity fieldv, given by:
v(y) =∇u(x, t)y.
Indeed let X and Y be Lagrangian trajectories corresponding to the flow
2.2. Derivation of the equations
X−Y evolves initially as d
dt(Y(t)−X(t))≈(∇u)δx=v(Y(t)−X(t)). (2.5)
Consider the anti-symmetric and symmetric components of∇u. That is a rigid (or rotational) part
R(x, t) := 1 2 ∇u(x, t)−(∇u(x, t))> ,
and an (elastic) strain part, (named in analogy with the theory of elastic solids)
E(x, t) := 1 2
∇u(x, t) + (∇u(x, t))>.
We see that, as a first-order approximation,X−Y evolves as
X(t+τ)−Y(t+τ) = eτ Reτ E(X(t)−Y(t))
for sufficiently small τ > 0. Note that this decomposition of a velocity field locally into a translation, a rotation, and a strain is essentially an observation of Helmholtz (1858).
The anti-symmetric componentRcorresponds to a rigid (i.e. rotational) motion. Indeed it is straightforward to check that the system
d
dtZ(s) =RZ(s)
describes an isometric evolution i.e.
|eτ Rz|=|z|
for allτ ∈Rand allz∈Rd. Hence evolution under eτ Rdoes not contribute
to the strain.
The remainderE we call the rate-of-strain tensor at (x, t). This can be further decomposed into a “rate-of-dilation” component
Ed=
1 3∇ ·uI,
2.2. Derivation of the equations
of-shear” component
Es=E−Ed.
We can now make precise our modelling assumption that the viscous force is proportional to the rate of strain accross the boundary, namely we model the contribution of these forces on the region Ω0 by
2ν Z
∂Ω0
E·dn,
whereν >0 (orν = 0 in the derivation of the Euler equations) acts as the relative weight of the viscous forces in the evolution. In other words 2ν is the constant of proportionality in the Newtonian model.
By the incompressibility constraint, the viscous force amounts to
ν Z ∂Ω0 [(∇u) + (∇u)>]·dn=ν Z Ω0 ∆u+ν Z Ω0 ∇(∇ ·u) =ν Z Ω0 ∆u.
Combining the above expressions for the two principle intrinsic forces in the model, and adding a fixed body-forcef : Ω→ R3, we arrive at the
the following equation for the evolution of the momentum of the fluid in an arbitrary fixed region Ω0bΩ:
Z Ω0 ∂2 ∂s2X(x, s) dx= Z Ω0 ν∆u− ∇p+fdx, (2.6) whereX(x,·) : [t, t+ε)→Ω, for someε >0, denotes the trajectory of the fluid that passes through the pointx at time t (we will discuss trajectory maps in more detail later). The left-hand side of (2.6) is the net acceleration of the fluid in Ω0. For all x∈Ω, X(x,·) satisfies the system
∂
∂sX(x, s) =u(X(x, s), s), X(x, t) =x,
for all s ∈ [t, t+ε). Hence the acceleration at (x, t) can be expressed in terms ofu as follows: d2 dt2X(x, t) = ∂ ∂tu(X(x, t), t) +∂xiu(X(x, t), t) ∂ ∂tXi(x, t) =∂tu+ (u· ∇)u.