2.12 Newtonian ﬂuid
In Sec. 2.6 we introduced forces in a ﬂuid. At the macroscopic scale of a continuum, we characterise a ﬂuid’s response to applied forces through constitutive models.
The Newtonian (or linear viscous) ﬂuid is the most common constitutive model that represents the behaviour of many liquids and gases. It states that a ﬂuid at rest (or uniform velocity) does not sustain shear stress ; it can be expressed by

(2.41) 
The model is a continuum representation^{9} of Newton’s law of viscosity^{10} which states that shear stress is proportional to velocity gradient by the dynamic viscosity .
It is a speciﬁc case of the more general Stokesian ﬂuid, deﬁned as , where is deformation rate, Eq. (2.33 ). The Newtonian model assumes: (1) the ﬂuid is isotropic, i.e. the value of is independent of the direction in which it is measured; (2) zero bulk viscosity associated with a change in volume o the ﬂuid.
Using in Eq. (2.41 ) ensures that the shear stress is induced by deformation only. Taking the deviatoric part, , ensures viscous stresses are not generated by volume changes, which are represented by .
This is due to the “” operator subtracting from each diagonal component of , giving a total of () from all 3 diagonal components. From Eq. (2.35 ),
Pressure gradient
When we substitute Eq. (2.41) in in Eq. (2.19 ), the pressure part is . The term is equivalent to gradient .
Like divergence of stress in Eq. (2.22 ), the gradient of pressure represents the pressure ﬂux across the surface per unit volume as , according to

(2.42) 

(2.43) 