3.2. GOVERNING EQUATIONS FOR MOMENTUM, MASS AND ENERGY 27
which will become useful later. Substituting d by Eq. (3.4b), d" is expressed only in terms
of dT and dp,
d" =
��
cp �� pp
dT +
��
T p �� pT
dp : (3.5b)
Using Eqs. (3.4) and (3.5b), small changes in the dependent thermodynamic variables s, ,
and " can be expressed in terms of changes in the independent thermodynamic variables T
and p. The default unperturbed equilibrium state used in this thesis is water at ambient
temperature T0 = 25:0 C and ambient pressure p0 = 0:1013 MPa.
For future reference we also dene the isochoric heat capacity cV per unit mass and
the ratio of specic heat capacities ,
cV = T
@s
@T
V
; (3.6a)
=
cp
cV
; (3.6b)
from which the following two identities can be derived
T = s; (3.7a)
�� 1 =
2
pT0
0cps
: (3.7b)
3.2 Governing equations for momentum, mass and energy
Besides the thermodynamic variables, the governing equations of thermoviscous acoustics
requires the introduction of the velocity eld v of the uid as well as the stress tensor ,
which for a Newtonian uid is given as 17
= ��p 1 + ; (3.8a)
=
rv + (rv)T
+
b ��
2
3
(r v) 1; (3.8b)
where is denoted the viscous stress tensor, 1 is the unit tensor and the superscript "T"
indicates tensor transposition.
Mass conservation implies that the rate of change @t of the density in a test volume
with surface normal vector n is given by the inux (direction ��n) of the mass current
density v. In dierential form by Gauss's theorem it is
@t = r
�� v
; (3.9a)
which is sometimes referred to as the continuity equation. In order to express changes in
the thermodynamical state by the chosen independent variables p and T, we substitute @t
and r using Eq. (3.4b) and divide by to obtain
T @tp �� p @tT = ��r v �� v (T rp �� p rT): (3.9b)