Fluid Dynamics Equations:
From Phenomenological Description towards Kinetic
Theory Formulation
Lecture 3: Fluid models. Closure of fluid dynamic equations
November 14, 2014
Set of fluid dynamic equations
Continuity
dρ
+ ρ∇ · v = 0
dt
Momentum
ρ
dv
= ρ F + ∇T
dt
Energy
dU
= T : ∇v + ∇ · q
dt
Unknowns: ρ, v, U, T, q. Body force F is to be introduced
depending on the problem.
ρ
The set of equations is not closed unless T and q are not
specified explicitly!
Inviscid fluid
Fluid is called inviscid if in the fluid there are no shear stresses;
only normal stresses are observed.
Cauchy formula:
~τn = ~τx cos(n, x) + ~τy cos(n, y ) + ~τz cos(n, z)
From definition of inviscid fluid:
~τx = px ~i,
~τn = pn n~,
~τy = py ~j,
~τz = pz ~k
On the other hand,
n~ = cos(n, x)~i + cos(n, y )~j + cos(n, z)~k
Therefore
pn = px = py = pz = −p
Inviscid fluid
Consequently
~τn = −p~
n,
~τx = −p~i,
~τy = −p~j,
~τz = −p ~k
and the normal stress in inviscid flow does not depend on the
orientation of the elementary area.
p is the pressure.
The stress tensor takes the diagonal form




−p 0
0
1 0 0
T =  0 −p 0  = −p 0 1 0 = −pU
0
0 −p
0 0 1
Viscous fluid
Fluid is called viscous if along with normal stresses, shear
stresses are observed during its motion.
Newton experiment
Consider two plates, the lower one is
fixed, the upper is moving with the velocity v. The force to be applied to the
upper plate to make it moving is
v
f=µ S
h
Then the force per unit area (shear stress) is
τyx = µ
v
h
µ is called shear viscosity coefficient. It depends on the fluid
properties.
Viscous fluid
From the same experiment it appears that the fluid velocity at
the lower plate is zero while at the upper plate it is equal to
the plate velocity, v. The velocity distribution is proportional
to the distance
y
vx = v .
h
Therefore the component of the stress tensor becomes
τyx = µ
dvx
dy
In the considered case, vx = vx (y ), therefore
∂vy
1 ∂vx
1 dvx
εxy =
+
=
;
τxy = 2µεxy
2 ∂y
∂x
2 dy
Viscous Newtonian fluid
Fluid is called viscous newtonian if the following conditions are
satisfied
(1) If the fluid is in the rest or it moves as a rigid bode, then
only normal stresses occur in the fluid
(2) Components of stress tensor are linear functions of
components of strain rate tensor
(3) Fluid is isotropic
Property (1) means that τik = 0 if i 6= k and all εmn = 0
Property (2) allows writing
τ11 = a10 + a11 ε11 + a12 ε22 + a13 ε33 + a14 ε12 + a15 ε23 + a16 ε13
and so on...
Property (3) assures that aik are not varied with co-ordinates
transformations
Viscous Newtonian fluid
After some transformations we obtain (in principle axis of the
rate of strain tensor)
a10 = a20 = a30 = −p;
a12 = a13 = a23 = η;
a11 = a22 = a33 = η + 2µ;
all the remaining aik = 0.
Then the stress tensor is given by
τik = (−p + η∇ · v)δik + 2µεik
µ is the shear viscosity coefficient
η is the second viscosity coefficient related to that of bulk
viscosity, ζ
Coefficients µ and η are uknown!
Viscous Newtonian fluid
More specifically
τxx = −p + η∇ · v + 2µ
τyy
∂vx
,
∂x
∂vy
= −p + η∇ · v + 2µ
,
∂y
τyz = τzy = µ
∂vz
,
∂z
τxz = τzx = µ
τzz = −p + η∇ · v + 2µ
∂vy
∂vx
+
∂y
∂x
∂vy
∂vz
+
∂y
∂z
∂vz
∂vx
+
∂z
∂x
τxy = τyx = µ
Thus the components of the stress tensor are expressed in
terms of velocities and transport coefficients. Pressure can be
found from the equation of state.
Viscous Newtonian fluid. Viscosity coefficients
The best way is to calculate transport coefficients using kinetic
theory methods
Experimental data for the shear viscosity coefficient
Power temperature dependence
n
T
µ = µ0
,
T0
n < 1 = const
µ0 is measured at T = T0 .
Sutherland formula
3/2
C + 273 T
µ = µ0
,
C +T
273
C = const
µ0 is measured at T = 273 K, C is different for various gases.
Viscous Newtonian fluid. Viscosity coefficients
Second viscosity coefficient
Stokes formula is valid for monoatomic gases
3η + 2µ = 0
For polyatomic gases, the bulk viscosity coefficient ζ related to
internal energy relaxation is introduced
2
η=ζ− µ
3
Bulk viscosity coefficient can be expressed in terms of the
relaxation time.
Nonheat-conducting and conducting fluids
Fluid is called nonheat-conducting if the heat flux vector is
equal to zero
qx = qy = qz = 0
Usually, the concept of non-conducting fluid is applied
together with inviscid fluid concept
Fourier law of heat conduction
q = λ∇T
λ is the heat conductivity coefficient
Heat conductivity coefficient can be measured experimentally
It can be also found from the Prandtl number
Pr =
cp µ
,
λ
Pr ≈ const
Alternative way is in using kinetic theory.
Incompressible and compressible fluids
Fluid is called incompressible if it has the constant density
ρ = const
In the general case, the equation of state is
ρ = f (p, T ) or
Φ(ρ, p, T ) = 0.
Perfect gas law (Clapeyron–Mendeleev)
p = ρRT
Van der Waals law (for more dense gases)
a p + 2 (V − b) = RT
V
V is the specific volume, a, b specify molecular interaction and
molecular volume
More complex equations of state for hydrocarbons
Governing equations for inviscid fluid flows
Fluid is inviscid:
~τx = −~ip,
~τy = −~jp,
~τz = −~kp
Fluid is nonheat-conducting
qx = qy = qz = 0
Continuity equation does not change.
Momentum equation
ρ
∂p ~ ∂p ~ ∂p
dv
= ρ F − ~i
−j
−k
dt
∂x
∂y
∂z
Energy
ρ
∂vy
dU
∂vx
∂vz
= −p
−p
−p
dt
∂x
∂y
∂z
Governing equations for inviscid fluid flows
Set of Euler equations
dρ
+ ρ∇ · v = 0
dt
dv
1
= F − ∇p
dt
ρ
dU
= −p∇ · v
dt
This set is to be supplemented with equations of state
ρ
f (p, ρ, T ) = 0
U = U(ρ, T )
Good news: 5 unknowns (ρ, v, U, p, T ) and 5 equations.
Inviscid stationary flows. Boundary conditions
Conditions on the surface. Let S be the surface, n~ normal to
the surface vector
Impermeable surface (impermeability condition)
vn |S = 0
Permeable surface, for instance, porous media. Then for each
point M
vn |S = f (M)
Conditions at the infinite point
v|∞ = v∞ ,
p|∞ = p∞ ,
T |∞ = T∞
Inviscid non-stationary flows. Boundary conditions
Conditions on the surface
Impermeable surface
vn |S = un (M, t)
u(M, t) is the velocity of the point M at the surface
Permeable surface. For each point M
vn |S = f (M, t)
Conditions at the infinite point
v|∞ = v∞ (t),
p|∞ = p∞ (t),
T |∞ = T∞ (t)
Initial conditions
v|t=t0 = v0 (x, y , z),
p|t=t0 = p0 (x, y , z),
T |t=t0 = T0 (x, y , z)
Governing equations for viscous fluid flows
Continuity
dρ
+ ρ∇ · v = 0
dt
Momentum
ρ
∂~τx
∂~τy
∂~τz
dv
= ρF +
+
+
dt
∂x
∂y
∂z
Energy
ρ
dU
∂v
∂v
∂v
∂qx
∂qy
∂qz
= ~τx ·
+ ~τy ·
+ ~τz ·
+
+
+
dt
∂x
∂y
∂z
∂x
∂y
∂z
Closure
qi = λ
∂T
,
∂xi
∂vi
,
∂xi
τik = τki = µ
∂vi
∂vk
+
∂xk
∂xi
µ = µ(p, T ),
η = η(p, T ),
λ = λ(p, T )
τii = −p + η∇ · v + 2µ
f (p, ρ, T ) = 0,
U = U(p, T )
Incompressible flow
For incompressible flows,
ρ = const
Continuity equation
∇·v=0
Therefore, the diagonal components of stress tensor do not
contain ∇ · v
∂vi
∂vi
∂vk
τii = −p + 2µ
, τik = τki = µ
+
∂xi
∂xk
∂xi
Further assumptions
µ = const,
λ = const
Incompressible flow
Navier-Stokes equations
dv
1
= F − ∇p + ν∆v
dt
ρ
ν = µ/ρ is the kinematic viscosity coefficient, Laplace
operator is given by
∆=
∂2
∂2
∂2
+
+
∂x 2 ∂y 2 ∂z 2
Navier-Stokes equation contains second derivatives and is of
elliptic type.
If µ = 0 it is reduced to the Euler equation.
Incompressible flow
Energy equation
dU
= Φ + λ∆T
dt
Φ is the dissipative function specifying the part of kinetic
energy transferred to heat due to viscosity
ρ
" 2
2
2
∂vy
∂vz
∂vx
+2
+2
+
Φ=µ 2
∂x
∂y
∂z
2 2 2 #
∂vx
∂vy
∂vx
∂vz
∂vz
∂vy
+
+
+
+
+
+
∂y
∂x
∂z
∂x
∂y
∂z
For incompressible fluid, U = cv T + const and
dT
= Φ + λ∆T
dt
For a non-moving fluid, we obtain the equation of heat conduction
cv ρ
cv ρ
dT
= λ∆T
dt
Incompressible flow
Final form of governing equations for incompressible viscous
flow
∇·v = 0
cv ρ
dv
dt
1
= F − ∇p + ν∆v
ρ
dT
dt
= Φ + λ∆T
5 unknowns: v, p, T and 5 equations
Energy and Navier-Stokes equations are uncoupled. One can
solve first two equations and then find the temperature field
independently.
Boundary conditions
Conditions on the surface. Let S be the surface, u(M, t) is the
velocity of the point M at time t
Impermeable surface (no-slip conditions)
v|S = u(M, t)
(u(M) = 0 for stationary flows)
For permeable surface v|S = V(M, t)
Condition for the temperature or heat flux
∂T T |S = Tw (M, t) or λ
= q(M, t)
∂n S
Conditions at the infinite point
v|∞ = v∞ (t),
p|∞ = p∞ (t),
T |∞ = T∞ (t)
Initial conditions
v|t=t0 = v0 (x, y , z),
p|t=t0 = p0 (x, y , z),
T |t=t0 = T0 (x, y , z)
General properties of viscous flows
Irreversibility. Flows of inviscid fluids are reversible. For viscous
fluids, the flow is irreversible due to the term ∆v. Viscous flow
is reversible only for a particular case ∆v = 0.
Vorticity. If we consider a viscous fluid flow around a body,
then due to the boundary condition v|S = 0, the class of
potential flow solutions is not applicable. In this case, the
viscous flow is always vortex flow.
Dissipation of kinetic energy. For viscous flows, a part of
kinetic energy always dissipates transforming to the heat
energy as a result of shear stress. For inviscid flows, there is no
dissipation since µ = 0.
Some limit cases
Reynolds number
Re =
VL
ρVL
=
ν
µ
V is the characteristic velocity, L is the characteristic size.
Reynolds number (introduced by
Osborne Reynolds in 1882) is
the main similarity parameter with
respect to viscosity.
It determines various flow regims, transition from laminar to turbulent
flows, and is widely used in fluid
dynamics.
Some limit cases
Boundary layer flows, Re >> 1
In this case, the main part of the flow (area II) can be
described by the Euler equations, except the thin layer near the
surface (I), where viscous effects are of importance. The set of
equations is simplified; the solutions from (I) and (II) are to be
stick together using the boundary conditions at the external
edge of the boundary layer.
Properties of the flow
vx >> √
vy
δ ∼ 1/ Re
∂p/∂y = 0, p = p(x, t)
Some limit cases
Slow flows of viscous fluids, Re << 1
In this case, the Navier-Stokes equations can be reduced to
the Stokes equations
∇·v=0
1
∂v
= ∇p + ν∆v
∂t
ρ
For stationary flows, the last equation reads
µ∆v = ∇p
These equations are linear and have been solved for many
particular cases.
General case
For the general case of arbitrary Reynolds number, the
Navier-Stokes equations are to be solved.
Only a few analytical solutions for the Navier-Stokes equations
have been found (like 1D flows in a round pipe, stationary flow
between parallel plates).
Nowadays, the Navier-Stokes equations in the most cases are
simulated numerically.