23
𝜕(𝜌ℎ
𝑜
)
𝜕𝑡
+ 𝑑𝑖𝑣(𝜌ℎ
𝑜
𝒖)
= 𝑑𝑖𝑣(𝒌 𝑔𝑟𝑎𝑑 𝑻) +
𝜕𝜌
𝜕𝑡
+
𝜕(𝑢𝜁
𝑥𝑥
)
𝜕𝑥
+
𝜕(𝑢𝜁
𝑦𝑥
)
𝜕𝑦
+
𝜕(𝑢𝜁
𝑧𝑥
)
𝜕𝑧
+
𝜕(𝑣𝜁
𝑥𝑦
)
𝜕𝑥
+
𝜕(𝑣𝜁
𝑦𝑦
)
𝜕𝑦
+
𝜕(𝑣𝜁
𝑧𝑦
)
𝜕𝑧
+
𝜕(𝑤𝜁
𝑥𝑧
)
𝜕𝑥
+
𝜕(𝑤𝜁
𝑦𝑧
)
𝜕𝑦
𝜕(𝑤𝜁
𝑧𝑧
)
𝜕𝑧
+ 𝑆
ℎ
Eq (3.8)
where S
h
is a source of enthalpy energy, and h
0
is the specific total enthalpy.
3.2.1.4 Navier-Stokes Equations
In the previous equations, the viscous stress components(Ç
ij
) are some unknown variables. For
most fluid flows, these values could be achieved by supplying
the appropriate model, that is
expressed as functions of the local rate of deformation. In three-dimensional flows, the local rate
of deformation is composed of the linear and volumetric deformation rates. [59].
In case
of compressible flows, Newton's law of viscosity is made up of two constant viscosities:
dynamic viscosity, which is connected to linear deformations, and volumetric viscosity, which is
associated with volumetric deformations. As a result, three of the six viscous stress components
are constant and six are changeable. These elements are described as follows:
𝜁
𝑥𝑥
= 2𝜇
𝜕𝑢
𝜕𝑥
+ 𝜆 𝑑𝑖𝑣 𝒖
Eq (3.9)
𝜁
𝑦𝑦
= 2𝜇
𝜕𝑣
𝜕𝑦
+ 𝜆 𝑑𝑖𝑣 𝒖
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