TheoreticalandNumericalIssuesofIncompressibleFluidFlows .... Frey, chap1.pdf · Context Objective: describethemotionofafluidfillingadomain tˆRd,t>0,given 0. Representations: motionoffluidflowcanbemodeledintwoways:

Aspects Théoriques et Numériques pour les Fluides IncompressiblesTheoretical and Numerical Issues of Incompressible Fluid Flows

Chapter 1: Fluid Mechanics

Instructors: Pascal Frey, Yannick Privat

Sorbonne Université, CNRS4, place Jussieu, Paris

master ANEDP, 2018

Chapter IFluid Mechanics

Section 1.1

Notations, vectors, tensors

Differential operators

(x1, . . . , xd) Cartesian coordinates of point x ∈ Rd

∂iu partial derivative (in the distributional sense) of u with respect to xi∂mi u partial derivative (in the distributional sense) of order m of u with respect to

xi , m ∈ N∂i ju partial derivative (in the distributional sense) of u with respect to xi and xjα multi-index: α = (α1, . . . , αd)t ∈ Nd with αi ∈ N for every i ∈ 1, . . . , d|α| length of the multi-index: : α : |α| = α1 + · · ·+ αd

∂αu ∂α1x1. . . ∂

αdxd

∇u gradient of u: ∇u = (∂1u, . . . , ∂du)t ∈ Rd if u is a scalar function, and∇u = (∂jui)1≤i≤m,1≤j≤d ∈ Rm,d if u is a function with values in Rm

∇ · u divergence of u : ∇ · u = div u =∑di=1 ∂iui where u is a function with values

in Rd

∇× u curl of u : ∇× u = (∂2u3 − ∂3u2, ∂3u1 − ∂1u3, ∂1u2 − ∂2u1)t where u is afunction with values in R3

∆u Laplacian of u : ∆u = ∇ · (∇u) =∑di=1 ∂i iu

Aspects théoriques et numériques pour les fluides incompressibles M2 ANEDP, UPMC, 2018 4/ 41

Vectors in fluid dynamics

• Velocity gradient

∇u = (∇ux ,∇uy ,∇uz) =

∂xux ∂xuy ∂xuz∂yux ∂yuy ∂yuz∂zux ∂zuy ∂zuz

The trace of ∇u is div u = ∇ · u

• Deformation rate tensor

D(u) =1

2(∇u + (∇u)t) =

1

2

2∂xux (∂xuy + ∂yux) (∂xuz + ∂zux)

(∂yux + ∂xuy ) 2∂yuy (∂yuz + ∂zuy )

(∂zux + ∂xuz) (∂zuy + ∂yuz) 2∂zuz

• Convective derivative

u · ∇u = ux∂xu + uy∂yu + uz∂zu


Vector/tensor calculus

Let consider T = ti j ∈ R3×3, u = (u1, u2, u3) ∈ R3 and α ∈ R. We recall :

1. αT = α ti j,

2. T 1 + T2 = t1i j + t2

i j

3. u · T = (u1, u2, u3)

t11 t12 t13t21 t22 t23t31 t32 t33

=3∑i=1

ui (ti1, ti2, ti3)︸︷︷︸i-th row

4. T · u =

t11 t12 t13t21 t22 t23t31 t32 t33

u1u2u3

=3∑j=1

t1jt2jt3j

uj (j-th column)

5. T 1 · T 2 =

t111 t1

12 t113

t121 t1

22 t123

t131 t1

32 t133

t

211 t2

12 t213

t221 t2

22 t223

t231 t2

32 t233

=

3∑k=1

t1ikt

2kj

6. T 1 : T 2 = tr(T 1 · (T 2)t) =

3∑i=1

3∑k=1

t1ikt

2ik


Gauss-Green theorem

• Let Ω be a bounded, open subset of Rn and ∂Ω is C1.Then, along ∂Ω is defined the outward unit normal vector field n.

• Let u ∈ C1(Ω), the normal derivative of u is defined by:

∂u

∂n= ∇u · n =

∂u

∂x· n .

Theorem 1 (Gauss-Green theorem). Suppose u ∈ C1(Ω); then∫Ω

∂u

∂xidx =

∫∂Ωu ni ds (i = 1, . . . , n) .

Theorem 2 (Integration by parts formula). Let u, v ∈ C1(Ω); then∫Ω

∂u

∂xiv dx = −

∫Ωu∂v

∂xidx +

∫∂Ωu v ni dx (i = 1, . . . , n) .

Theorem 3 (Green’s formulas). Let u, v ∈ C2(Ω); then

1.∫

Ω∆u dx =

∫∂Ω

∂u

∂nds ;

2.∫

Ω∇v · ∇u dx = −

∫Ωu∆v dx +

∫∂Ω

∂v

∂nu ds ;

3.∫

Ωu∆v − v∆u dx =

∫∂Ωu∂v

∂n− v

∂u

∂nds .


Section 1.2

Conservation laws

Mathematical models

Partial Differential Equations (PDE) form the kernel of very many mathematical models and areused to describe physical, chemical, biological, economical phenomena, . . .

• Mathematical Modelling:

role: description of the behavior of a system using a set of PDEs endowed with initialand boundary conditions;

challenge: theoretical framework for mathematical analysis;

well-posedness: existence, uniqueness (or multiplicity) of solutions and stability w/r tothe data.

• Numerical Simulation:

target: imitating / reproducing the behavior of the complex system to analyze and possiblyimprove it;

methods: relies on mathematical analysis, numerical analysis, scientific computing.


Numerical simulation

• has become a consequent part of the mathematical modelling process and is possibly the solethat can give access to specific solutions that cannot be solved analytically;

• the choice of the approximation method is strongly related to the model and to the natureand the behavior of the phenomenon represented as well as to the desired properties of thesolutions (regularity, stability, etc.);

• the quality of the approximation method is related to mathematical properties (stability,convergence and order of convergence, error estimate) and to more practical issues (accuracy,reliability, efficiency). Both criteria are often antagonists and incompatible with one another.

• numerical analysts shall then try to obtain a good compromise between these criteria, pre-serving the accuracy of the solution at a reasonable computational effort.


From modelling to simulation

• the mathematical modelling and numerical simulation of a PDE problem is a highly multidis-ciplinary topic to obtain the numerical solution;

• process requires advanced knowledge of theoretical and applied mathematics and of computerscience.

• the numerical simulation bridges the gap between theoretical analysis and experiments.

Implementation

Computer

(4)(3)

Analysis

Numerical

(2)

Analysis

Mathematical

(1)

Model

Mathematical

and Physical

Figure 1: A continuum from modelling to simulation.


Context

• Objective: describe the motion of a fluid filling a domain Ωt ⊂ Rd , t > 0, given Ω0.

• Representations: motion of fluid flow can be modeled in two ways:

1. in the Lagrangian description, the trajectory t 7→ χ(x0, t, t0) of any individual fluidparticle, occupying position x0 at time t0, is followed during the time period.This approach is widely used in solid and particle mechanics, but yields more tediousanalysis in the case of fluid flow.

2. In the Eulerian representation, the study takes place in a fixed referential, and no particularfluid particle is followed. Each point in the domain Ωt ⊂ Rd -whose Cartesian coordinatesare denoted by x = (xi)i=1,...,d - is considered over the time period, independently ofwhich specific particle is occupying the place.


Notations

Let u(x, t) denote the velocity of the fluid particle occupying the position x in Ωt at time t. Ateach instant t, u is a vector field on Ωt , called the velocity field of the fluid.

Definition 1. 1. The flow field is said to be conservative if there exists a scalar function φ(x, t)

(called a scalar potential) such that

u = ∇φ .

2. The vorticity ω of the flow is defined as:

ω = curl(u) = ∇× u.

The flow is said irrotational (or curl-free) if ω = 0.

3. The flow field is said incompressible (or divergence-free) if the material density ρ is constantin time and space; this is equivalent to:

div u = ∇ · u = 0 .

4. The deformation tensor D = D(u) of the flow is defined as:

D(u) =1

2(∇u +∇ut).


Physical meaning

Proposition 1. Let u : U → Rd any vector field, defined over a subset U ⊂ Rd . Then thefollowing asymptotic expansions hold:

(d = 2), u(x + h) = u(x) + D(u)(x).h +1

2curl(u)(x)h⊥ +O(h2).

(d = 3), u(x + h) = u(x) + D(u)(x).h +1

2curl(u)(x)× h +O(h2).

Hence, locally in the vicinity of any point x , u is submitted to:

1. an infinitesimal deformation D(u), which admits d principal deformation directions: the prin-cipal directions of the symmetric matrix D(u).

2. an infinitesimal rotation h 7→ 12curl(u)(x)× h.

If d = 3, the axis of the rotation is the unit vector in the direction of curl(u)(x), and thespeed of the rotation is proportional to the modulus of curl(u)(x).


Vector calculus

• Exercises: prove the following identities (u vector field, f scalar function)

div(f u) = u · ∇f + f div u;

div(curl u) = div(∇× u) = 0 and curl(∇f ) = 0;

curl(curl u) = ∇(div u)− ∆u.

• The divergence theorem relates the flow of a vector field through a surface to the behaviorof the vector field in the domain.

Theorem 4 (Divergence theorem). Suppose Ω is a compact subset of R3 with a Lipschitz-continuous boundary ∂Ω. Then, if v ∈ C1(Ω) is a continuously differentiable vector fielddefined on a neighborhood of Ω, then we have∫

Ω∇ · v dx =

∫∂Ωv n ds , (1)

where n is the outward pointing unit normal field of the boundary ∂Ω.


Motion of a particle

• The motion of a fluid, filling a domain Ωt , is considered during the time I = [t0, t].

• Usually, Ωt0 (resp. Ωt) is called the reference configuration (resp. current configuration).

• In a Lagrangian perspective, the motion is described by a family of mappings and we denoteby χ(·, t, t0) the diffeomorphism:

x0 ∈ Ωt0 7→ x = χ(x0, t, t0) ∈ Ωt ,

which maps the position x0 of a considered particle at time t0 to its position x at t.

Definition 2. The trajectory of a particle of fluid is the curve χ(x0, t, t0)t∈I.The velocity of this fluid particle at position x at time t is then the vector u, independent ofthe reference time t = 0, i.e.

u = u(x, t) = u(χ(x0, t, t0), t) =∂χ

∂t(x0, t, t0) . (2)


Motion of a particle (2)

Definition 3. The acceleration γ(x, t) of the particle at time t and at x = χ(x0, t, t0) is:

γ(x, t) = γ(χ(x0, t, t0), t) =d

dtu(x, t) =

d

dtu(χ(x0, t, t0), t)

=∂u

∂t(χ(x0, t, t0), t) +

∑i

∂u

∂xi(χ(x0, t, t0), t)

∂χi∂t

(x0, t, t0)︸︷︷︸=ui(x,t)

, (3)

which also reads, using (2)

γ(χ(x0, t, t0), t) =∂u

∂t(χ(x0, t, t0), t) +

∑i

∂u

∂xi(χ(x0, t, t0), t)ui(χ(x0, t, t0), t)

=∂u

∂t(χ(x0, t, t0), t) + (u · ∇)u(χ(x0, t, t0), t) . (4)


Motion of a particle (3)

Considering the inverse mapping χ−1, for x ∈ Ωt we deduce from the previous formula

γ(x, t) =∂u

∂t+ (u · ∇) u(x, t)

which is known as the fundamental Eulerian formula for acceleration of fluids.

Definition 4. The operator

D

Dt

def=

∂

∂t+ (u · ∇)

is called the total derivative or the material derivative. It accounts for the derivative of a quantityalong the trajectory of a particle with velocity u.

For an arbitrary function f (x, t), the chain rule allows to write

f (x, t) =d

dtf (x, t)

def=

∂f

∂t(x, t) + (u(x, t) · ∇)f (x, t) =

Df

Dt(x, t) .

With this notation, we have γ(x, t) = u(x, t).


Notion of mass

• For each time t > 0, we assume that there exists a positive measure µt carried by Ωt calledthe mass distribution, which is regular with respect to the Lebesgue measure∗ dx .

• Hence, there exists a regular function ρ = ρ(x, t) (sufficiently smooth) such that:

dµt(x) = ρ(x, t) dx ,

where ρ is called the mass density of the system at point x at time t.

• For any subset Wt ⊂ Ωt , the mass of fluid contained in Wt at time t is defined by:

m(Wt) =

∫Wt

dµt(x) =

∫Wtρ(x, t) dx .

Notice that the assumption of existence for ρ is a continuum assumption that no longer holdsat molecular level.

∗Non mathematicians can simply consider dµt(x) as a notation for the measure ρ(x, t)dx .


Forces

The motion of a material system is entailed by the action of external forces of two types,represented by a regular vector measure dϕt(x) carried by a volume or a surface.

1. Contact (or stress) forces:expressed as forces across the surface ∂Wt of any subsystem Wt ⊂ Ωt . Their descriptionrelies on Cauchy’s theory of stress tensor.

Definition 5. In a d-dimensional evolving continuum Ωt , Cauchy’s stress tensor σ(x, t) isa d-dimensional tensor, defined by the property that, for any subsystem Wt ⊂ Ωt withsmooth enough boundary ∂Wt , the surface density of forces applied on ∂Wt by the rest ofthe continuum reads:

s(x, t) = σ(x, t).n(x, t),

The fact that there is no momentum of the stress efforts inside a fluid in static equilibriumhas the following fundamental mathematical translation.

Theorem 5. Cauchy’s stress tensor σ is symmetric.


Forces (2)

The total force exerted on the fluid inside Wt by means of stress on its boundary is

s∂Wt =

∫∂Wt

σ(x, t).n(x, t) ds(x) .

where ds is a regular surface measure.

Definition 6. A fluid is said to be an ideal fluid when there exists a function p(x, t), calledthe pressure, such that the stress tensor σ is of the form:

σ(x, t).n = −p(x, t) n, (5)

for any unit vector n.

This means that there is no tangential component in the contact forces. This definitionalso implies that there cannot be a rotation initiating in such fluid.


Forces (3)

Definition 7. A viscous Newtonian fluid is defined by the form of its stress tensor σ:

σ = −p Id + L(D(u)) ,

where p is the pressure of the fluid, D(u) is the deformation tensor and L is a linear mapping.

Further hypothesis (isotropic medium, invariance under a change of observer), lead to thefollowing form of the stress tensor

σ = −p Id + λ div u Id + 2µD(u) . (6)

where λ, the volume viscosity, and µ > 0, the dynamic viscosity, are the viscosity coefficientsof the fluid (also called the Lamé coefficients).

Viscosity measures the resistance of a fluid under a shear stress deformation.


Forces (4)

2. External (or body) forces:

• exert a force per unit volume on the fluid; e.g. gravity

• described by a density

dϕt(x) = f (x, t) dx , for x ∈ Ωt .

For instance, in the classical case of gravity, we have

f (x, t) = −ρg e3 ,

where g ∈ R3 is the gravitational acceleration and e3 is the vertical unit vector pointingupward.


Energy and mechanical work

Two kinds of mechanical works (energy exchange with exterior) can be distinguished, due toexternal or interior forces.

1. External work:

Definition 8. Suppose a fluid in motion, with velocity field u(x, t), experiences an externalvolume force f (x, t); the instantaneous mechanical work at time t done by f on a portionWt of Ωt is defined as: ∫

Wtf (x, t) · u(x, t) dx.

Similarly, if external loads g(x, t) are applied on the boundary of Ωt , the mechanical workdone by g reads: ∫

Ωtg(x, t) · u(x, t) ds.

When f and u have the same orientation, then f · u ≤ 0, and the mechanical work done byf is positive, in the sense that the action of the force "helps" the motion.


Energy and mechanical work (2)

2. Internal work:

Definition 9. The instantaneous power received by a portion of fluid in Wt ⊂ Ωt at time treads:

−∫Wtσ : D(u) dx

To get an intuition, assume that the considered fluid is incompressible and viscous.As seen before, one possibility for modelling the stress-strain relation reads:

σ = −p Id + 2µD(u),

where p stands for the scalar pressure field. Then, the instantaneous power received by aportion Wt of fluid is:

−∫Wtσ : D(u) dx = −

∫Wtp div(u) + 2µ|D(u)|2 dx = −2

∫Wtµ|D(u)|2 dx.

Hence viscous forces only entail energy dissipation.


Conservation principles

Establishing the partial differential equations that describe the evolution of Ωt according toNewtonian mechanics relies on three conservation laws:

1. mass conservation: mass is neither created nor destroyed;

2. balance of momentum: according to Newton’s second law, the rate of change of the linearmomentum equals the total applied force on the considered system;

3. conservation of energy: energy is neither created nor destroyed.

The derivation of the three forthcoming equations of conservation follows a common generalsketch: a small portion of fluid, evolving in time, is chosen, and the quantities attached to it(mass, momentum, and energy) should satisfy the aforementioned conservation principles.

Giving a precise mathematical meaning to this idea involves differentiation of integrals on movingdomains.


Differentiation of a volume integral

It is often required to compute the time derivative of integrals of the form:

g(t) =

∫Ωtf (x, t) dx ,

where f = f (x, t) is a given scalar function and Ωt is a bounded domain in Rd , evolving withrespect to a velocity field u(x, t).

Theorem 6 (Transport theorem (Liouville’s theorem)). Assume f = f (x, t) is a function ofclass C1 for x ∈ Ωt and t ∈ I, and that u is of class C1 with respect to x and t. Then,

dg

dt(t) =

∫Ωt

∂f

∂t(x, t) dx +

∫Ωt

div(f · u)(x, t) dx , (7)

=

∫Ωt

∂f

∂t(x, t) dx +

∫∂Ωt

f (x, t) u(x, t) · n(x) ds(x) , (8)

where ∂Ωt denotes the boundary of Ωt , n is the unit outward normal vector on ∂Ωt and ds isthe surface measure on ∂Ωt .


Conservation of mass

The rate of change of mass in a subset Wt is

d

dtm(Wt) =

d

dt

∫Wtρ(x, t) dx︸︷︷︸

=0 (conservation)

.

Using the Transport theorem 6 with f = ρ yields, for all Wt in Ωt :∫Wt

∂ρ

∂t+ div(ρu) dx = 0 ,

We obtain the so-called continuity equation:

∂ρ

∂t+ div(ρu) = 0 in Ωt , (9)

In case of an incompressible fluid, ρ is constant and equation (9) becomes

div u = 0 in Ωt .


Conservation of mass (2)

Based on this continuity equation, the following convenient form of theorem 6 is suited to dealwith mass quantities:

Corollary 1. Let f = f (x, t) is a function of class C1 for x ∈ Ωt and t ∈ I, and assume thatu is of class C1 with respect to x and t. Define the integral quantity g(t) as:

g(t) =

∫Ωtρ(x, t)f (x, t) dx.

Then,dg

dt(t) =

∫Ωtρ(x, t)

Df

Dt(x, t) dx . (10)

Proof:

dg

dt(t) =

∫Ωt

∂(ρf )

∂t(x, t) + div((ρf )u)(x, t) dx , (11)

=

∫Ωt

ρ

(∂f

∂t+ u.∇f

)(x, t) +

(∂ρ

∂t+ div(ρu)

)(x, t) dx , (12)

+ definition 4 of the material derivativeDf

dt.


Balance of momentum

The total force F acting on Wt ⊂ Ωt filled by the fluid reads:

F =

∫Wtf (x, t) dx︸︷︷︸external

+

∫∂Wt

σ(x, t).n(x, t)) ds(x)︸︷︷︸contact f orce

. (13)

Balance of momentum:

• related to Newton’s second law F = mγ;

• change of momentum related to the force acting on Ωt .

Given Wt = χ(W, t, t0), volume of fluid moving, the linear momentum in Wt writes:

ML(Wt) =

∫Wtρ(x, t) u(x, t) dx , (14)

and the balance of momentum reads, thanks to (13):

d

dtML(Wt) =

∫Wtf (x, t) dx +

∫∂Wt

σ(x, t).n(x, t)) ds(x) ,

where n = n(x, t) is the unit outward normal to ∂Wt .


Balance of momentum (2)

The left-hand side of previous equation rewrites (cf. corollary 1):

d

dtML(Wt) =

∫Wtρ(x, t)

Du

Dt(x, t) dx =

∫Wtρ(x, t)u(x, t) dx .

Now, using the divergence theorem 4, one has:∫∂Wt

σ(x, t) n(x, t)ds(x) =

∫Wt

div σ(x, t) dx .

Since Wt is an arbitrary subset, we can gather all previous results to write the linear momentumformula:

ρu = div σ + f . (15)

Interestingly, the linear momentum law can be used in the context of energy considerations.


Balance of energy

Let Wt ⊂ Ωt any evolving portion of fluid, u the velocity field in the frame of reference.

Definition 10. The kinetic energy of Wt at time t is given by:

Ec =1

2

∫Ωt|u(x, t)|2 dµt(x) =

1

2

∫Ωtρ(x, t)|u(x, t)|2 dx ,

The rate of change of kinetic energy of a moving subset Wt of fluid is calculated as follows

d

dtEc =

d

dt

(1

2

∫Wtρ(x, t)|u(x, t)|2 dx

)=

1

2

∫Wtρ(x, t)

D|u(x, t)|2

Dtdx (corollary 1)

=1

2

∫Wtρ(x, t)

(u ·(∂u

∂t+ (u · ∇)u

))(x, t) dx

in other terms,d

dtEc =

∫Wtρu · u dx . (16)


Balance of energy (2)

Theorem 7. The instantaneous rate of change in kinetic energyd

dtEc in any portion of fluid

Wt ⊂ Ωt equals the sum of:1. the power of internal forces within Wt :

−∫Wtσ : D(u) dx ,

2. the mechanical work done by surface loads σ.n (owing to the continuity of the stress tensorin Ωt) applied on ∂Wt : ∫

∂Wt(σ · n) · u dx ,

3. the mechanical work done by external forces on Wt :∫Wtf .u dx .

Hence, we write:∫Wtρu · u dx . = −

∫Wtσ : D(u) dx +

∫∂Wt

(σ · n) · u dx +

∫Wtf .u dx . (17)


Section 1.3

Flows models

Summary of the equations

1. Continuity equation conservation of mass

∂ρ

∂t+ div(ρu) = 0 (18)

2. Momentum equation Newton’s second law

∂(ρu)

∂t+ div(ρu × u) = div σ + f

3. Energy equation first law of thermodynamics

∂(ρEc)

∂t+ div(ρEcu) = −D(u) : σ + f · u


The Navier-Stokes equations

• Physical considerations:we consider the motion of a Newtonian incompressible viscous fluid, with the following as-sumptions on the Cauchy stress tensor σ:

1. σ depends linearly on the velocity gradients ∇u Newtonian fluid;

normal stress : stretching

shear stress : deformation

2. σ is invariant under translations and rigid rotations (Galilean invariance);

3. σ is symmetric (consequence of the balance of angular momentum).

We introduced σ as (viscous fluid µ > 0):

σ = −p Id + λ div u Id + 2µD(u) ,

which for an incompressible fluid (div u = 0) can be simplified:

σ = −p Id + 2µD(u) .


The Navier-Stokes equations (2)

Substituting σ in the expression ρu = div σ + f yields the Navier-Stokes equations for a viscousincompressible fluid filling a domain Ωt , for any t ∈ [0, T ]: ρ

(∂u

∂t+ (u · ∇)u

)= −∇p + 2 div(µD(u)) + f

div u = 0(19)

where ρ = ρ(x, t) denotes the density, u = u(x, t) the velocity of the fluid particle at position xat time t, p = p(x, t) the pressure, and f = f (x, t) the volume forces.

Given µ > 0 and div u = 0, then div∇ut = ∇ div u and thus div(µD(u)) = µ∆u, the "classical"form of Navier-Stokes equations is then:

Given f ; find (u, p) such that

ρ

(∂u

∂t+ (u · ∇)u

)= −∇p + µ∆u + f

div u = 0

. (20)



ρ

(∂u

∂t︸︷︷︸unsteady

acceleration

+ (u · ∇)u︸︷︷︸convectiveacceleration

)

︸︷︷︸iner tia term

= −∇p︸︷︷︸pressuregradient

+ µ∆u︸︷︷︸v iscosity︸︷︷︸

divergence of stress

+ f︸︷︷︸bodyf orces

1. The inertial or non-linear term (u · ∇)u is related to convective acceleration:it contributes to the transfert of kinetic energy.

2. The term −∇p is the pressure gradient:A fluid reacts to attempt to change its volume according to a pressure. This term arisesfrom the isotropic part of the Cauchy-stress tensor (6) p = 1/3 tr(σ). The pressure is onlyinvolved through its gradient in Navier-Stokes equations: accelerating the fluid from high tolow pressure.Pressure can be interpreted as a Lagrange multiplier corresponding to div u = 0.



ρ

(∂u

∂t︸︷︷︸unsteady

acceleration

+ (u · ∇)u︸︷︷︸convectiveacceleration

)

︸︷︷︸iner tia term

= −∇p︸︷︷︸pressuregradient

+ µ∆u︸︷︷︸v iscosity︸︷︷︸

divergence of stress

+ f︸︷︷︸bodyf orces

3. The term µ∆u is the dissipative viscous term;µ is the dynamic viscosity of the fluid.Viscosity operates in a diffusion of a momentum, much alike the diffusion of heat in the heattransfer equation (a Laplacian in space). Diffusion of fluid momentum is the result of frictionbetween fluid particles moving at uneven speed that tend gradually to become the same.

4. The vector field f represents the body forces (forces per unit volume) such as gravity.


Initial and boundary conditions

1. Initial condition:

• initial velocity field must be divergence-free in Ωt and at ∂Ωt :

u(x, 0) = u0(x) , div u0 = 0 .

2. Boundary conditions for viscous fluids

• Need to consider molecular interactions between fluid and surface of a solid body;

• No-slip condition: normal and tangential components of the velocity vanish:

u(x, t) = 0, , ∀x ∈ ∂Ωt , ∀t ∈ I ;

• General situation: ∂Ωt moving at speed v(x, t), then solid and fluid share the samevelocity at the boundary, i.e.

u(x, t) = v(x, t) , ∀x ∈ ∂Ωt , ∀t ∈ I .


The Reynolds number

• Write the equations in a non-dimensional form, independent of the system of units.

• Define the dimensionless variables as follows:

u′ =u

U, x ′ =

x

L, t ′ =

t U

L, p′ =

p

ρU2 .

• Substitute the new variables in the Navier-Stokes equations:∂u′

∂t ′+ (u′ · ∇)u′ = −∇p′ +

1

Re∆u′ + f ′

div u′ = 0

, (21)

• New dimensionless parameter: the Reynolds number of the flow:

Re =UL

ν=ρU2

µU

L

=acceleration forcesviscosity forces

.

provides a measure of the viscosity of the flow.


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TheoreticalandNumericalIssuesofIncompressibleFluidFlows .... Frey, chap1.pdf · Context Objective: describethemotionofafluidfillingadomain tˆRd,t>0,given 0. Representations: motionoffluidflowcanbemodeledintwoways: