# Interdisciplinary Applied Mathematics

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k’j

Pr

. &L

dx J

/

0

1

d

,, (pu , dv 4

P!dy ^ dx>

+ Re

dy

2    /o dv du

3    P’^dy dx)

vi

P (2

dv

dy

du 5 dx )

■V + Pi^ + Ш)-

u +

k’j

Pr

dT dy )

where k is the thermal conductivity and 7 is the ratio of specific heats. The unknowns are the conservative variables, i.e.,

(p, pu, pv, E).

The energy is defined as

E = p[T +1/2(u2 + v2)],

and the pressure p is obtained from the equation of state

P = (Y — 1)PT.

The nondimensionalization is done with respect to reference velocity, density, and    length    scales (i.e.,    U0,    p0,    l0),    and    the    reference temperature    is

chosen as T0 = U2/Cv; here, Cv is the const ant-volume specific heat. The dynamic viscosity p is related to temperature by Sutherland’s law

h_ = fT3/2T0 + S ho To) T + S

where p0    is    the    viscosity    at    the    reference    temperature    T0,    and Si is a

constant, which for air is

Si = 110K.

It is convenient to simplify this equation to a simpler power law of the form

— = ( Jj- ) , with 0.5 < ш < 1.

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