Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 165

5.1 Rheology of Suspensions

Fluids can be divided according to their rheological behaviour into Newtonian and
non-Newtonian fl uids. Newtonian fl uids follow Newton ' s law of viscosity, which is

expressed in one-dimensional form as

dv
dy

s ¼ g L ð 5 : 1 Þ

It is clear from Eq. ( 5.1 ) that the relation between the velocity gradient (shear
rate) and the shear stress in a Newtonian

fl uid is linear. In order to account for the

solids content in a carrier or base fl uid, Einstein [ 1 , 2 ] derived an equation for the

“ ”

viscosity of a homogeneous Newtonian suspension, comprising spher-
ical particles with a small volume concentration:

effective

g eff ¼ g L 1 ð þ 2 : 5 / V Þ ð 5 : 2 Þ

ϕ V is the volume fraction of the solid phase in a solid-liquid
suspension. Note that Eq. ( 5.2 ) does not take the sizes or positions of the particles
into account, and the theory neglects the effects of particle interaction. The shear
stress in this case is expressed by replacing the liquid-phase viscosity

In this equation,

ʷ L with the

effective viscosity ʷ eff .

dv
dy

s ¼ g eff ð 5 : 3 Þ

After Einstein [ 1 , 2 ], several models for effective viscosity were proposed. They
comprise additional terms to describe the interaction between particles, account for
high concentration of solid particles, their size and their shape. Figure 5.1 shows the
difference in dynamic viscosity between the carrier uid and effective viscosity of a

Newtonian suspension. Because of the complexity of a theoretical description, these
models are based on empirical evaluation.
The rheological behaviour of a homogeneous, non-Newtonian

fl

fl uid in the case

of highly turbulent fl ow or in the case of small concentrations of a solid phase

Fig. 5.1 An example of the
variation of the effective
viscosity with the
concentration of solids