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AER1301: KINETIC THEORY OF GASES
Assignment #1
1. A hypersonic wind tunnel is contructed so as to obtain a high-speed flow with M = 7 in
the test section. At that Mach number, the test section gas velocity, density, temperature,
pressure, and viscosity are V = 1743 m/s, ρ = 0.0182 kg/m3 , T = 154 K, p = 8.33 kPa, and
µ = 12.2(10)−6 kg/m-s, respectively. A model 2.54 cm high and 15.24 cm long is to be tested
in the wind tunnel. A pitot tube having an outside diameter of 0.0762 mm will be used to
measure the free-stream stagnation pressure in front of the model. Determine the Reynolds
and Knudsen numbers for the flows about the model and probe and determine whether or
not non-equilibrium rarefied flow effects should be important for the model and the probe,
respectively. Assume the gas particles behave as hard spheres during collisions such that the
mean free path, λ, is given by the expression
λ=
16µ
1
√
,
5ρ 2πRT
where R is the ideal gas constant and p = ρRT .
2. (Problem 1.1 of Textbook) Using index notation show that ∇ · ∇ × A = 0, where the vector,
A, is a function of location, r.
3. (Problem 1.2 of Textbook) In a vacuum, the vector form of Maxwell’s equations is the following:
∂B
,
∇×E=−
∂t
∂E
∇ × B = ◦ µ◦
+ µ◦ j ,
∂t
ρ
∇·E=
,
◦
∇ · B = 0,
where E and B are the electric and magnetic field vectors, ρ is the electric charge density,
j is the electric current density, and µ◦ and ◦ are the magnetic permeability and electric
permittivity of free space, respectively. Write these equations using index notation.
4. For a gas currently not in thermodynamic equilibrium, all of the gas molecules have their
velocity vectors uniformly, but still randomly distributed inside a cubical volume in velocity
space, such that there are no molecules having velocities outside of the cube. The cubical
volume is centred at the origin of velocity space and the length of each side of the cube is 4v◦ .
(a) Obtain an expression for the normalized velocity distribution function, f (v).
(b) Calculate the average velocity, v̄ =< v > and the mean energy per molecule Ē =<
mv 2 /2 >.
(c) If now the particle velocity distribution is such that all molecules are still contained
within a cube of size 4v◦ but instead the cube is centred on the point (v◦ , 0, 0) in velocity
space, provide an expression for this new normalized velocity distribution function and
calculate the mean velocity and energy of the molecules.
Assignment #1 — Page 1 of 3
5. (Problems 2.1 and 2.2 of Textbook) Again consider a gas currently not in thermodynamic equilibrium. All of the molecules of this nonequilibrium gas have their velocity vectors uniformly
distributed inside a sphere of radius v◦ in velocity space, such that there are no molecules
having velocities outside of this sphere. The sphere is centred at the origin of velocity space.
(a) Again obtain an expression for the normalized velocity distribution function, f (v).
(b) Calculate the average velocity, v̄ =< v > and the mean energy per molecule Ē =<
mv 2 /2 >.
(c) If now all of the gas molecules have their velocity vectors uniformly distributed on the
surface of a negligibly thin spherical shell of radius v◦ centred about the origin in velocity
space, what is the normalized velocity distribution function? Calculate new values for
the mean velocity and energy of the molecules in this case.
6. (Problem 2.5 of Textbook) For a gas in thermal equilibrium, find the ratios between the
characteristic speeds of molecules: < v >:< v 2 >1/2 : vm , where v is the molecule speed and
vm is the most probable speed of the molecules.
7. Consider a stationary gas consisting of molecules of mass m in thermal equilibrium with
temperature T .
(a) Derive an expression for the distribution function of translational kinetic energy, fE (E),
where E = mv 2 /2 and fE (E)dE is the fraction of molecules whose kinetic energy lies in
the range between E and E + dE.
(b) Show that Ē =< E >= 3kT /2 and Em = kT /2 where Em is the most probable translational kinetic energy.
(c) Show that the fraction of molecules with translational kinetic energy equal to or greater
than the value E∗ is given by the following expression:
2
1+ √
π
s
E∗
E∗
exp −
kT
kT
s
− erf 

E∗ 
kT
(d) What percentage of the total translational kinetic energy is carried by molecules that
have kinetic energies less than one tenth of the average kinetic energy? In determining
the answer, you should make use of the approximation that
e−x ≈ 1 − x ,
for x 1.
8. (Problems 2.3 and 2.9 of Textbook) The molecular velocity distribution function, f , for a gas
in equilibrium moving with a bulk velocity u = (ux , uy , uz ) is given by
f (v) =
3/2
β
π
h
exp −β (vx − ux )2 + (vy − uy )2 + (vz − uz )2
i
,
where β = m/2kT , T is the gas temperature, and vx , vy , vz are the total velocity components
of the molecules in a stationary coordinate frame. For this drifting Maxwellian velocity
distribution find expressions for:
(a) < mvx2 >
Assignment #1 — Page 2 of 3
(b) < mv 2 /2 >
(c) < mvx vy >
(d) < mvx3 /2 >
and determine the values of these velocity moments when ux = uy = uz = 0.
9. (Problem 2.11 of Textbook) A gas has the following normalized velocity distribution function,
f:
3/2
β
5
2
f (v) =
e−βv 1 + Avx vy + Bvz v 2 −
,
π
2β
where A and B are constants independent of the particle velocity, v, β = m/2kT , T is the
gas temperature, and v 2 = ||v||2 = vx2 + vy2 + vz2 . Calculate < v >, < mvx2 /2 >, < mvy2 /2 >,
< mvz2 /2 >, and < mvv 2 /2 >.
10. (Problem 2.8 of Textbook) The molecules of a polyatomic gas are capable of independently
rotating about the x, y, and z axes, and vibrating about the x and y axes. What is the
specific heat ratio of the gas?
Assignment #1 — Page 3 of 3