Mathematical analysis in thermodynamics of incompressible fluids - - PowerPoint PPT Presentation

Mathematical analysis in thermodynamics of incompressible fluids

Josef M´ alek

Mathematical institute of Charles University in Prague, Faculty of Mathematics and Physics Sokolovsk´ a 83, 186 75 Prague 8

June 16, 2008

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 1 / 36

Contents

1

Mathematically self-consistent models of classical mechanics - models for the system Spring - Weight

2

Thermodynamics of incompressible fluids

3

Constitutive equations

4

References

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 2 / 36

Part #1 Mathematically self-consistent models of classical mechanics - models for the system Spring - Weight

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 3 / 36

System Spring - Weight/Description and assumptions

Bodies (weights) modeled as mass-points Three Newton’s postulates:

F = 0 = ⇒ straight-line motion F = d

dt (mv) = m dv dt = m d2x dt2

Any F exerts reaction −F

Motion allowed only in the vertical direction Mass of the spring is neglected

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 4 / 36

System Spring - Weight/Assumptions characterizing material properties

Linear Spring: F2 = (0, −k(y + a), 0) (k > 0) Resistance due to environment is neglected

d2y dt2 + k my = 0

y(0) = y0

dy dt (0) = y1

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 5 / 36

System Spring - Weight/Assumptions characterizing material properties

Linear Spring: F2 = (0, −k(y + a), 0) (k > 0) Resistance proportional to the velocity: F3 = (0, −b dy

dt , 0)

(b > 0)

d2y dt2 + b m dy dt + k my = 0

y(0) = y0

dy dt (0) = y1

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 6 / 36

System Spring - Weight/Assumptions characterizing material properties

Linear Spring: F2 = (0, −k(y + a), 0) (k > 0) Resistance force due to environment depends on the velocity non-linearly: F3 = (0, h

• dy

dt

• , 0)

m d2y

dt2 + h

• dy

dt

• + ky = 0

y(0) = y0

dy dt (0) = y1

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 7 / 36

System Spring - Weight/Assumptions characterizing material properties

Non-linear Spring: F2 = (0, g(y + a), 0) Environment resistance neglected, linear, or non-linear

d2y dt2 + h(dy dt ) + g(y) = 0 d2y dt2 = f (y, dy dt )

Free fall due to gravity: F2 = (0, 0, 0)

d2y dt2 + h(dy dt ) = 0

⇐ ⇒

dv dt + h(v) = 0 dv dt = f (v)

v(0) = v0

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 8 / 36

System Spring - Weight/Mathematically self-consistent models

Simplifying assumptions = ⇒ very crude approximation of the reality Independently how accurate are models we are interested in mathematical self-consistency of the models: notion of solution

existence for arbitrary set of data (T, v0 (or y0 and y1), m, ....) uniqueness continuous dependence of solution on data boundedness of the velocity long time behavior of solutions.

Mathematical self-consistency of models of incompressible fluid thermodynamics Derivation of fluid thermodynamics models stems from the principles

• f classical mechanics
• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 9 / 36

System Spring - Weight/Simple observations

Free fall due to gravity: first order equation for the velocity Mathematical self-consistency of the equation of a ”slightly” generalized form dv

dt = f (v), v(0) = v0. Counterexamples:

existence/boundedness for any time interval - f (v) = v 2 uniqueness - f (v) = v 2/3

m dv

dt + bv = f =

⇒

m 2 d dt |v|2 + b m|v|2 = fv =

⇒ |v(t)|2 ≤ |v0|2e− b

m t + f 2

b2 (1 − e− b

m t)

pro t > 0 Derived models have a limited region where they can be useful

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 10 / 36

Part #2 Thermodynamics of incompressible fluids

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 11 / 36

Fluid

Definition

Fluid is a body that, in time scale of observation of interest, undergoes discernible deformation due to the application of a sufficiently small shear stress v = ∂χ ∂t Fχ = ∂χ ∂X

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 12 / 36

Long-lasting physical experiment

In 1927 at University of Queensland: liquid asphalt put inside the closed vessel, after three years the vessel was open and the asphalt has started to drop slowly.

Year Event 1930 Plug trimmed off 1938 (Dec) 1st drop 1947 (Feb) 2nd drop 1954 (Apr) 3rd drop 1962 (May) 4th drop 1970 (Aug) 5th drop 1979 (Apr) 6th drop 1988 (Jul) 7th drop 2000 (28 Nov) 8th drop

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 13 / 36

Balance equations of continuum physics

Balance of mass, linear and angular momentum, balance of energy and the second law of thermodynamics ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 TT = T

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)
• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 14 / 36

Balance equations of continuum physics

Balance of mass, linear and angular momentum, balance of energy and the second law of thermodynamics ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 TT = T

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)

̺ . . . density v . . . velocity e . . . internal energy T . . . the Cauchy stress q . . . heat flux

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 14 / 36

Balance equations of continuum physics

Balance of mass, linear and angular momentum, balance of energy and the second law of thermodynamics ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 TT = T

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)

̺ . . . density v . . . velocity e . . . internal energy T . . . the Cauchy stress q . . . heat flux Eulerian description - flows of fluid-like bodies No external sources - for simplicity

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 14 / 36

Balance equations of continuum physics/2

B ⊂ Ω fix for all t ≥ 0: d dt

• B

̺ dx = −

• ∂B

̺v · n dS = ⇒ FVM = −

• B

div(̺v) dx = ⇒ ̺t + div ̺v = 0

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 15 / 36

Balance equations of continuum physics/2

B ⊂ Ω fix for all t ≥ 0: d dt

• B

̺ dx = −

• ∂B

̺v · n dS = ⇒ FVM = −

• B

div(̺v) dx = ⇒ ̺t + div ̺v = 0 Choice B = {x ∈ Ω; η(x) > r}, where r ∈ (0, ∞) and η ∈ D(Ω) d dt

• B

̺η dx −

• B

̺v · ∇η dx = 0 = ⇒ weak solution, FEM Oseen, Leray, . . . , Chen, Torres, Ziemer, . . . Feireisl: weak formulation of balance equations - the primary setting classical formulation of balance equations - the secondary setting

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 15 / 36

”Equivalent” formulation of the balance of energy

̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 (BLM) TT = T

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)

is equivalent, provided that v is admissible test function in (BLM), to ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 TT = T (̺e),t + div(̺ev) + div q = T · ∇v

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 16 / 36

”Equivalent” formulation of the balance of energy

̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 (BLM) TT = T

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)

is equivalent, provided that v is admissible test function in (BLM), to ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 TT = T (̺e),t + div(̺ev) + div q = T · ∇v Note that T · ∇v = T · D where D := D(v) is the symmetric part of the velocity gradient

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 16 / 36

Entropy

(̺e),t + div(̺ev) + div q = T · ∇v (1)

Continuum thermodynamics (Callen 1985): there is η (specific entropy density) being a function of state variables, here η = ˜ η(e), fulfilling:

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 17 / 36

Entropy

(̺e),t + div(̺ev) + div q = T · ∇v (1)

Continuum thermodynamics (Callen 1985): there is η (specific entropy density) being a function of state variables, here η = ˜ η(e), fulfilling: ˜ η is increasing function of e = ⇒

1 θ =: ∂ ˜ η ∂e

• r e = ˜

e(η) = ⇒ θ = ∂˜

e ∂η

η → 0+ as θ → 0+ S(t) :=

• Ω ̺∗η(t, ·)dx goes to its maximum as t → ∞ provided that the

body is thermally and mechanically isolated

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 17 / 36

Entropy

(̺e),t + div(̺ev) + div q = T · ∇v (1)

Continuum thermodynamics (Callen 1985): there is η (specific entropy density) being a function of state variables, here η = ˜ η(e), fulfilling: ˜ η is increasing function of e = ⇒

1 θ =: ∂ ˜ η ∂e

• r e = ˜

e(η) = ⇒ θ = ∂˜

e ∂η

η → 0+ as θ → 0+ S(t) :=

• Ω ̺∗η(t, ·)dx goes to its maximum as t → ∞ provided that the

body is thermally and mechanically isolated (1) is equivalent to ∂˜ η ∂e

• ̺
• e,t + v · ∇e
• + div q

θ = T · D(v) θ ̺

• η,t + η · ∇v
• + div

q θ

• = 1

θ

• T · D(v)
• − q · ∇θ

θ2

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 17 / 36

Second law of thermodynamics/1

• ̺η
• ,t + div(̺ηv) + div

q θ

• = ξ

with θξ := T · D(v) − q · ∇θ θ (2)

Second law of thermodynamics: ξ ≥ 0 Stronger requirement: T · D(v) ≥ 0 (entropy production due to work being converted into heat) and − q·∇θ

θ

≥ 0 (entropy production due to heat conduction) We shall use the constitutive equations that automatically meet these requirements Minimum principle for e if e0 ≥ C ∗ in Ω then e(t, ·) ≥ C ∗ in Ω for all t

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 18 / 36

Second law of thermodynamics/2

• ̺η
• ,t + div(̺ηv) + div

q θ

• = ξ

with θξ ≥ T · D(v) − q · ∇θ θ In terms of the internal energy η = ˜ η(e) e,t + div(ev) + div q ≥ T · D(v)

• r, using the balance of energy,
• |v|2

,t − 2 div(Tv) + div

• v|v|2

≤ 0 Suitable weak solution (in the sense of Caffarelli, Kohn, Nirenberg): In addition to equations representing balance of mass, linear momentum and energy we require that solution satisfies one of the formulations of the second law of thermodynamics

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 19 / 36

Incompresibility

Definition

Volume of any chosen subset (at initial time t = 0) remains constant during the motion. for all t: |Vt| = |V0| ⇐ ⇒ det Fχ = 1 Taking the derivative w.r.t. time and using the identity d dt det Fχ = div v det Fχ we conclude that div v = 0

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 20 / 36

Balance equations for Inhomogeneous incompressible fluids

Balance equations ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 (BLM)

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)

Consequences of incompressibility div v = 0 and T = −pI + S div v = 0 ̺t + v · ∇̺ = 0 (̺v)t + div(̺v ⊗ v) − div S = −∇p

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2 + p)v) + div q = div (Sv)
• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 21 / 36

Balance equations for Inhomogeneous incompressible fluids

Balance equations ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 (BLM)

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)

Consequences of incompressibility div v = 0 and T = −pI + S div v = 0 ̺t + v · ∇̺ = 0 (̺v)t + div(̺v ⊗ v) − div S = −∇p

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2 + p)v) + div q = div (Sv)

S and q: additional (the so-called) constitutive equations

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 21 / 36

Balance equations for Inhomogeneous incompressible fluids

Balance equations ̺,t + div(̺v) = 0 (̺v),t + div(̺v ⊗ v) − div T = 0 (BLM)

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2)v) + div q = div (Tv)

Consequences of incompressibility div v = 0 and T = −pI + S div v = 0 ̺t + v · ∇̺ = 0 (̺v)t + div(̺v ⊗ v) − div S = −∇p

• ̺(e + |v|2/2)
• ,t + div(̺(e + |v|2/2 + p)v) + div q = div (Sv)

S and q: additional (the so-called) constitutive equations Homogeneous fluids: the density is constant

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 21 / 36

Balance equations for homogeneous incompressible fluids

div v = 0 (3) v,t + div(v ⊗ v) − div S = −∇p (4) (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv) (5) e,t + div(ev) + div q ≥ S · D(v) (6) Constitutive equations for S and q (next section) Boundary conditions (internal flows) Initial data

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 22 / 36

IBVP

div v = 0 v,t + div(v ⊗ v) − div S = −∇p (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv)

Data

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 23 / 36

IBVP

div v = 0 v,t + div(v ⊗ v) − div S = −∇p (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv)

Data Ω ⊂ R3 bounded open connected container, T ∈ (0, ∞) length of time interval v(0, ·) = v0, e(0, ·) = e0 in Ω α that appears in boundary conditions (thermally and mechanically or energetically isolated body)

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 23 / 36

IBVP

div v = 0 v,t + div(v ⊗ v) − div S = −∇p (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv)

Data Ω ⊂ R3 bounded open connected container, T ∈ (0, ∞) length of time interval v(0, ·) = v0, e(0, ·) = e0 in Ω α that appears in boundary conditions (thermally and mechanically or energetically isolated body) Task Mathematical Consistency of a Model - for any set of data to find uniquely defined, smooth, solution (notion of solution, its existence, uniqueness, regularity)

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 23 / 36

IBVP

div v = 0 v,t + div(v ⊗ v) − div S = −∇p (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv)

Data Ω ⊂ R3 bounded open connected container, T ∈ (0, ∞) length of time interval v(0, ·) = v0, e(0, ·) = e0 in Ω α that appears in boundary conditions (thermally and mechanically or energetically isolated body) Task Mathematical Consistency of a Model - for any set of data to find uniquely defined, smooth, solution (notion of solution, its existence, uniqueness, regularity) Weak solution - solution dealing with averages

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 23 / 36

Boundary conditions

(e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q − div (Sv) = 0 d dt

• Ω

E(t, x) dx

• +
• ∂Ω

[(E + p)v · n + q · n − Sv · n] dS = 0

Mechanically and thermally isolated body, Navier’s slip on [0, T] × Ω:

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 24 / 36

Boundary conditions

(e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q − div (Sv) = 0 d dt

• Ω

E(t, x) dx

• +
• ∂Ω

[(E + p)v · n + q · n − Sv · n] dS = 0

Mechanically and thermally isolated body, Navier’s slip on [0, T] × Ω: v · n = 0 q · n = 0 λ(Sn)τ + (1 − λ)vτ = 0 for λ ∈ (0, 1) uτ := u − (u · n)n λ = 0 = ⇒ no-slip λ = 1 = ⇒ slip

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 24 / 36

Boundary conditions

(e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q − div (Sv) = 0 d dt

• Ω

E(t, x) dx

• +
• ∂Ω

[(E + p)v · n + q · n − Sv · n] dS = 0

Mechanically and thermally isolated body, Navier’s slip on [0, T] × Ω: v · n = 0 q · n = 0 λ(Sn)τ + (1 − λ)vτ = 0 for λ ∈ (0, 1) uτ := u − (u · n)n λ = 0 = ⇒ no-slip λ = 1 = ⇒ slip Energetically isolated body, Navier’s slip on [0, T] × Ω:

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 24 / 36

Boundary conditions

(e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q − div (Sv) = 0 d dt

• Ω

E(t, x) dx

• +
• ∂Ω

[(E + p)v · n + q · n − Sv · n] dS = 0

Mechanically and thermally isolated body, Navier’s slip on [0, T] × Ω: v · n = 0 q · n = 0 λ(Sn)τ + (1 − λ)vτ = 0 for λ ∈ (0, 1) uτ := u − (u · n)n λ = 0 = ⇒ no-slip λ = 1 = ⇒ slip Energetically isolated body, Navier’s slip on [0, T] × Ω: v · n = 0 q · n = −α|vτ|2 (Sn)τ + αvτ = 0 α := (1 − λ)/λ

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 24 / 36

”Equivalent” formulation of the balance of energy/1

div v = 0 v,t + div(v ⊗ v) − div S = −∇p (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv) is equivalent (if v is admissible test function in BM) to div v = 0 v,t + div(v ⊗ v) − div S = −∇p e,t + div(ev) + div q = S · D(v) Helmholtz decomposition u = udiv + ∇gv Leray’s projector P : u → udiv

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 25 / 36

”Equivalent” formulation of the balance of energy/2

div v = 0 v,t + div(v ⊗ v) − div S = −∇p (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv) is equivalent (if v is admissible test function in BM) to div v = 0 v,t + P div(v ⊗ v) − P div S = 0 e,t + div(ev) + div q = S · D(v)

Advantages/Disadvantages

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 26 / 36

”Equivalent” formulation of the balance of energy/2

div v = 0 v,t + div(v ⊗ v) − div S = −∇p (e + |v|2/2),t + div((e + |v|2/2 + p)v) + div q = div (Sv) is equivalent (if v is admissible test function in BM) to div v = 0 v,t + P div(v ⊗ v) − P div S = 0 e,t + div(ev) + div q = S · D(v)

Advantages/Disadvantages + pressure is not included into the 2nd formulation + minimum principle for e if S · D(v) ≥ 0 − S · D(v) ∈ L1 while Sv ∈ Lq with q > 1

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 26 / 36

Part #3 Constitutive equations

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 27 / 36

Newtonian fluids

Definition

The viscosity: the coefficient of the proportionality between the shear rate and the shear stress. Simple shear flow: v(x, y, z) = (v(y), 0, 0) Newton: The resistance arising from the want of lubricity in parts of the fluid, other things being equal, is proportional to the velocity with which the parts are separated from one another. Txy = νv′(y) g(Txy, v′(y)) = 0

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 28 / 36

Generalized Newtonian fluids

Experimental data show that the viscosity may depend on the pressure, shear rate, temperature, concentration, ..., density (if fluid is inhomogeneous) Txy = νv′(y) ν = ν(p, θ, |v′(y)|) S = ν(p, θ, |D(v)|2)D Examples: T = −pI + 2µ0D, tr D = 0 T = −pI + 2µ0|D|r−2D r ∈ [1, ∞) T = −pI + 2µ0

• 1 + |D|2 r−2

2 D

T = −pI + 2µ0 exp(αp)D

• r

T = −pI +

• 1 + αµ(p, θ) + |D|2 r−2

2 D

T = −pI + 2ν(p, ̺, θ)D = −pI + A√̺ exp

• B(p+D̺2)

θ

• D

T = −pI + 2µ0 exp(1/θ − 1/θ0)

• 1 + αµ(p, θ) + |D|2 r−2

2 D

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 29 / 36

Implicitely constituted fluids

More general implicit relations G(Txy, v′(y)) = 0

• r G(p, θ, Txy, v′(y)) = 0

G(p, θ, S, D) = 0 have the ability to capture complicated responses of materials without any need to introduce (non-physical) internal variable constitutive theories, etc. Implicit relations algebraic rate type integral

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 30 / 36

Newtonian versus non-Newtonian fluids

Incompressible Newtonian fluid T = −pI + 2µD, tr D = 0 Departures from Newtonian behavior (at a simple shear flow) Dependence of the viscosity on the shear rate Dependence of the viscosity on the pressure The presence of the yield stress (or other activation or deactivation criteria) The presence of the normal stress differences Stress relaxation Nonlinear creep

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 31 / 36

Fourier fluids - heat conducting fluids

Definition

The heat conductivity: the coefficient of the proportionality between the heat flux q and the temperature gradient ∇θ. Landau, Lifschitz: The heat flux is related to the variation of temperature through the fluid. . . . We can then expand q as a series of powers of temperature gradient, taking only the first terms of the expansion. The constant term is evidently zero since q must vanish when ∇θ does so. Thus we have q = −κ∇θ The coefficient κ is in general a function of temperature and pressure. Examples: q = −κ∇θ q = −κ(θ, p)∇θ q(∇θ) = q(0) + ∂z(0)∇θ + 1/2∂(2)

z (0)∇θ ⊗ ∇θ

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 32 / 36

Implicitely constituted heat conducting fluids

More general implicit relations r(q, ∇θ) = 0 r(q, p, θ, ∇θ, D) = 0 Implicit relations algebraic rate type integral

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alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 33 / 36

Part #4 References

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 34 / 36

References/1

1

• E. Feireisl, J. M´

alek: On the Navier-Stokes Equations with temperature-dependent transport coefficients, Differ. Equ. Nonlinear Mech., pp. Art. ID 90916, 14p., 2006

2

• M. Bul´

ıˇ cek, E. Feireisl, J. M´ alek: Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, to appear in Nonlinear Analysis and Real World Applications, 2008

3

• M. Bul´

ıˇ cek, J. M´ alek, K. R. Rajagopal: Navier’s slip and Evolutionary Navier-Stokes-Like systems with Pressure and Shear-Rate Dependent Viscosity, Indiana University Math. J. 56, 51–85, 2007

4

• M. Bul´

ıˇ cek, J. M´ alek, K. R. Rajagopal: Mathematical analysis of unsteady flows of fluids with pressure, shear-rate and temperature dependent material moduli, that slip at solid boundaries, revised version considered in SIAM J. Math. Anal., 2008

5

• M. Bul´

ıˇ cek, L. Consiglieri, J. M´ alek: Slip boundary effects on unsteady flows of incompressible viscous heat conducting fluids with a nonlinear internal energy-temperature relationship, to appear as the preprint at http://ncmm.karlin.mff.cuni.cz, 2008

6

• M. Bul´

ıˇ cek, L. Consiglieri, J. M´ alek: On Solvability of a non-linear heat equation with a non-integrable convective term and the right-hand side involving measures, to appear as the preprint at http://ncmm.karlin.mff.cuni.cz, 2008

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 35 / 36

References/2

1

• J. M´

alek and K.R. Rajagopal: Mathematical Issues Concerning the Navier-Stokes Equations and Some of Its Generalizations, in: Handbook of Differential Equations, Evolutionary Equations, volume 2, 371-459, 2005

2

• J. Frehse, J. M´

alek and M. Steinhauer: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM J. Math.

• Anal. 34, 1064-1083, 2003

3

• L. Diening, J. M´

alek and M. Steinhauer: On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications, accepted to ESAIM: Control, Optimization and Calculus of Variations, published online, 2007

4

• J. Hron, J. M´

alek and K.R. Rajagopal: Simple Flows of Fluids with Pressure Dependent Viscosities, Proc. London Royal Soc.: Math. Phys. Engnr. Sci. 457, 1603–1622, 2001

5

• M. Franta, J. M´

alek and K.R. Rajagopal: Existence of Weak Solutions for the Dirichlet Problem for the Steady Flows of Fluids with Shear Dependent Viscosities, Proc. London Royal Soc. A: Math. Phys. Engnr. Sci. 461, 651–670, 2005

6

• J. M´

alek, M. R˚ uˇ ziˇ cka and V.V. Shelukhin: Herschel-Bulkley Fluids: Existence and regularity of steady flows, Mathematical Models and Methods in Applied Sciences, 15, 1845–1861, 2005

7

• P. Gwiazda, J. M´

alek and A. ´ Swierczewska: On flows of an incompressible fluid with a discontinuous power-law-like rheology, Computers & Mathematics with Applications, 53, 531–546, 2007

8

• M. Bul´

ıˇ cek, P. Gwiazda, J. M´ alek and A. ´ Swierczewska-Gwiazda: On steady flows of an incompressible fluids with implicit power-law-like rheology, accepted to Adv. Calculus of Variations 2008

• J. M´

alek (MFF UK) Analysis for incompressible fluid flows June 16, 2008 36 / 36