Theoretical and Numerical Issues of Incompressible Fluid .... Frey, chap1.pdfTheoretical and...

Theoretical and Numerical Issues of Incompressible Fluid Flows

Chapter 1: Fluid Mechanics

Instructor: Pascal Frey

Sorbonne Université, CNRS4, place Jussieu, Paris

University of Tehran, 2018

The laws of nature are drawn from experience, but to expressthem one needs a special language . . .

H. Poincaré, in Analysis and Physics.

Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 2/ 45

Section 1.1Notations, 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 orderm of u with respect

to xi,m ∈ N∂iju partial derivative (in the distributional sense) of u with respect to xi and

xj

α 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 whereu is a function with values in R3

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


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 = tij ∈ R3×3, u = (u1, u2, u3) ∈ R3 and α ∈ R. We recall :

1. αT = α tij,

2. T1 + T2 = t1ij + t2ij

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. T1 · T2 =

t111 t112 t113t121 t122 t123t131 t132 t133

t

211 t212 t213t221 t222 t223t231 t232 t233

=

3∑

k=1

t1ikt2kj

6. T1 : T2 = tr(T1 · (T2)t) =

3∑i=1

3∑k=1

t1ikt2ik


Gauss-Green theorem

• Let Ω be a bounded, open subset of Rd 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 =

∫∂Ω

uni ds (i = 1, . . . , n) .

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

∂u

∂xiv dx = −

∫Ωu∂v

∂xidx+

∫∂Ω

u v ni dx (i = 1, . . . , n) .

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

1.∫

Ω∆udx =

∫∂Ω

∂u

∂nds;

2.∫

Ω∇v · ∇udx = −

∫Ωu∆v dx+

∫∂Ω

∂v

∂nuds;

3.∫

Ωu∆v − v∆udx =

∫∂Ω

u∂v

∂n− v

∂u

∂nds.


Section 1.2Conservation laws

Mathematical models

Partial Differential Equations (PDE) form the kernel of very many mathematical models andare used 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/rto the data.

• Numerical Simulation: target: imitating / reproducing the behavior of the complex system to analyze andpossibly improve it;

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


Numerical simulation

• has become a consequent part of the mathematical modelling process and is possiblythe sole that 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 na-ture and the behavior of the phenomenon represented as well as to the desired prop-erties of the solutions (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 incompatiblewith one another.

• numerical analysts shall then try to obtain a good compromise between these criteria,preserving 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 mul-tidisciplinary topic to obtain the numerical solution;

• process requires advanced knowledge of theoretical and applied mathematics and ofcomputer science.

• 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 noparticular fluid particle is followed. Each point in the domain Ωt ⊂ Rd -whoseCartesian coordinates are denoted by x = (xi)i=1,...,d - is considered over thetime period, independently of which 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.At each 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 functionφ(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 con-stant in 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 principal 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), andthe speed 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(fu) = u · ∇f + f div u; div(curlu) = div(∇× u) = 0 and curl(∇f) = 0; curl(curlu) = ∇(div u)−∆u.

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

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

Ω∇ · v dx =

∫∂Ω

v nds , (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 wedenote by 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, independentof the 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 aquantity along 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

called the 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 subsetWt ⊂ Ωt, the mass of fluid contained inWt at time t is defined by:

m(Wt) =∫Wt

dµt(x) =∫Wt

ρ(x, t) dx .

Notice that the assumption of existence for ρ is a continuum assumption that no longerholds at 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 subsystemWt ⊂ Ωt. Their descrip-tion relies on Cauchy’s theory of stress tensor.

Definition 5. In a d-dimensional evolving continuum Ωt, Cauchy’s stress tensor σ(x, t)

is a d-dimensional tensor, defined by the property that, for any subsystemWt ⊂ Ωt withsmooth enough boundary ∂Wt, the surface density of forces applied on ∂Wt by the restof the 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 equilib-rium has the following fundamental mathematical translation.

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


Forces (2)

The total force exerted on the fluid insideWt 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),called the 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 linearmapping.

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

σ = −p Id + λdiv u Id + 2µD(u) . (6)

where λ, the volume viscosity, and µ > 0, the dynamic viscosity, are the viscosity coef-ficients of 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 vectorpointing upward.


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 exter-nal volume force f(x, t); the instantaneous mechanical work at time t done by f on aportionWt of Ωt is defined as:∫

Wt

f(x, t) · u(x, t) dx.

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

Ωt

g(x, t) · u(x, t) ds.

When f and u have the same orientation, then f ·u ≤ 0, and the mechanical work doneby f 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 inWt ⊂ Ωt at timet reads:

−∫Wt

σ : 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 bya portionWt of fluid is:

−∫Wt

σ : D(u) dx = −∫Wt

p div(u) + 2µ|D(u)|2 dx = −2∫Wt

µ|D(u)|2 dx.

Hence viscous forces only entail energy dissipation.


Conservation principles

Establishing the partial differential equations that describe the evolution of Ωt accordingto Newtonian 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 thelinear momentum 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 commongeneral sketch: a small portion of fluid, evolving in time, is chosen, and the quantitiesattached to it (mass, momentum, and energy) should satisfy the aforementionedconservation principles.

Giving a precise mathematical meaning to this idea involves differentiation of integrals onmoving domains.


Differentiation of a volume integral

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

g(t) =∫

Ωt

f(x, t) dx ,

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

Theorem 6 (Transport theorem (Liouville’s theorem)). Assume f = f(x, t) is a functionof class 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 andds is the surface measure on ∂Ωt.


Conservation of mass

The rate of change of mass in a subsetWt is

d

dtm(Wt) =

d

dt

∫Wt

ρ(x, t) dx︸︷︷︸=0 (conservation)

.

Using the Transport theorem 6 with f = ρ yields, for allWt 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 todeal with mass quantities:

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

g(t) =∫

Ωt

ρ(x, t)f(x, t) dx.

Then,dg

dt(t) =

∫Ωt

ρ(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 onWt ⊂ Ωt filled by the fluid reads:

F =∫Wt

f(x, t) dx︸︷︷︸external

+∫∂Wt

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

. (13)

Balance of momentum:

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

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

GivenWt = χ(W, t, t0), volume of fluid moving, the linear momentum inWt writes:

ML(Wt) =∫Wt

ρ(x, t)u(x, t) dx , (14)

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

d

dtML(Wt) =

∫Wt

f(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) =

∫Wt

ρ(x, t)Du

Dt(x, t) dx =

∫Wt

ρ(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 .

SinceWt is an arbitrary subset, we can gather all previous results to write the linearmomentum formula:

ρu = div σ + f . (15)

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


Balance of energy

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

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

Ec =1

2

∫Ωt

|u(x, t)|2 dµt(x) =1

2

∫Ωt

ρ(x, t)|u(x, t)|2 dx ,

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

d

dtEc =

d

dt

(1

2

∫Wt

ρ(x, t)|u(x, t)|2 dx

)

=1

2

∫Wt

ρ(x, t)D|u(x, t)|2

Dtdx (corollary 1)

=1

2

∫Wt

ρ(x, t)(u ·

(∂u

∂t+ (u · ∇)u

))(x, t) dx

in other terms,d

dtEc =

∫Wt

ρu · 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 withinWt:

−∫Wt

σ : D(u) dx ,

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

∂Wt

(σ · n) · udx ,

3. the mechanical work done by external forces onWt:∫Wt

f.udx .

Hence, we write:∫Wt

ρu · u dx . = −∫Wt

σ : D(u) dx+∫∂Wt

(σ · n) · udx+∫Wt

f.udx . (17)


Section 1.3Flows 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 followingassumptions 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 + λdiv 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 aviscous incompressible 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 atposition x at 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

)

︸︷︷︸inertial term

= −∇p︸︷︷︸pressuregradient

+ µ∆u︸︷︷︸viscosity︸︷︷︸

divergence of stress

+ f︸︷︷︸bodyforces

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 pressureis only involved through its gradient in Navier-Stokes equations: accelerating the fluidfrom high to low pressure.Pressure can be interpreted as a Lagrange multiplier corresponding to div u = 0.



ρ

(∂u

∂t︸︷︷︸unsteady

acceleration

+ (u · ∇)u︸︷︷︸convectiveacceleration

)

︸︷︷︸inertial term

= −∇p︸︷︷︸pressuregradient

+ µ∆u︸︷︷︸viscosity︸︷︷︸

divergence of stress

+ f︸︷︷︸bodyforces

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 inthe heat transfer equation (a Laplacian in space). Diffusion of fluid momentum is theresult of friction between fluid particles moving at uneven speed that tend gradually tobecome the same.

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

Remark 1. The NS equations are among the very few equations of mathematical physics wherethe nonlinearity does not occur from physical assumption but just from mathematical (kine-matics) arguments.Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 39/ 45

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.


Energy inequality

• consider the flow of a fluid in a region Ω ⊂ Rd bounded by walls and driven by a bodyforce f(x, t). Let pose ν = µ/ρ, we have the set of equations:

• the fluid velocity and pressure are functions which satisfy:

∂u

∂t+ (u · ∇)u+∇p = f

div(u) = 0 and∫

Ωp dx = 0 for t ∈ (0, T )

u(x,0) = u0(x) and u = 0 on ∂Ω

(22)

• the key idea of the analysis is due to Leray and is based on the notion of weak solution.The calculation behind Leray’s theory is the energy inequality.

• If u, p is a smooth solution to (22), then multiplying the momentum equation by u, inte-grating over Ω, integrating by parts, and integrating in time gives:

1

2‖u(t)‖2 +

∫ t0ν‖∇u(τ)‖ dτ =

1

2‖u0‖2 +

∫ t0

(f ′(τ), u(τ))dτ ;

• the energy equality has the physical interpretation:kinetic energy(t) + total energy dissipated over [0, t]

= initial kinetic energy + total power input.Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 42/ 45

Historical landmarks

• Archimedes of Syracuse (287-212 BC)discovered the principle of buoyancy: an immersed body is acted upon by a force equalto the weight of water it displaces (Eureka).

• Leonardo da Vinci (1452-1519)stated the idea of continuity of the fluid continuum, who also performed detailed studiesof waves, jets, and interacting eddies.the concept of momentum in physics was introduced by Galileo (1564–1642).

• Isaac Newton (1642-1727)introduced conservation of linear momentum (force = mass × acceleration). He alsostudied and introduced the first concept of viscosity for laminar flow as a special caseof a linear stress–strain relation.

• Daniel Bernoulli (1700-1782)introduced the first equations of fluid motion coupling the velocity and pressure andthe kinetic theory of gasses, jet propulsion, and manometers.

• Leonhard Euler (1707-1783)the father of fluid mechanics as a mathematical discipline. He derived the correct math-ematical equations of inviscid flow (the Euler equations).Lagrange wrote, “Euler did not contribute to fluid mechanics but created it.”



• C. L. M. H. Navier (1785-1836) and G. Stokes (1819–1903)Navier-Stokes equations first written in their modern form. Navier is credited for devel-oping the equations of viscous flows. Stokes gave the first clear and correct derivationof the viscous terms in the Navier-Stokes equations based on the A. Cauchy (1789–1857)stress principle.

• J. C. Maxwell (1831-1879)derived the fundamental equations of electricity and magnetism and the continuumfluid flow equations by a limiting process beginning with Daniel Bernoulli ’s kinetic theoryof gases, and he studied correct conditions at a boundary.

• O. Reynolds (1842–1912)studied turbulence in real fluids experimentally and showed the correct path to be pre-diction of turbulent flow averages rather than pointwise values.

• H. Poincaré (1854–1912)introduced the method of sweeping - an early numerical method for solving partial dif-ferential equations - and forecasted the future arithmetization of analysis.



• J. Leray (1906-1998)founder of the mathematical theory of the Navier–Stokes equations; introduced the ideaof weak solution of PDEs (turbulent solutions), fixed point and degree theory, connectedtopology to PDEs, and developed one of the major conjectures concerning turbulence.

• A. N. Kolmogorov (1903-1987)introduced the theory of homogeneous, isotropic turbulence, which explains many uni-versal features observed in turbulent flows.

• J. von Neumann (1903–1957)he and R. D. Richtmeyer developed numerical methods for shock problems.

• O. A. Ladyzhenskaya (1922–2004)developed complete, beautiful, and fully rigorous mathematical theory for the Navier-Stokes system, with a correction term to the equations to account for possible nonlineareffects in a stress–strain relation.


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