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       Figure 2.7: The dashpot model to illustrate ideal-

      viscous behavior

       Figure 2.8: A shock absorber which can be loaded from both sides [2.24]

      b) The dashpot model

      The dashpot model is used to illustrate the behavior of ideal-viscous fluids or Newtonian liquids, respectively (see Figure 2.7). Mechanically similar examples are gas or liquid shock absorbers (see Figure 2.8).

      Ideal-viscous flow behavior,

      explained by the behavior of a dashpot

      1) When under load

      Under a constant force, the piston is moving continuously as long as the force is applied, pressing the dashpot fluid (e. g. an oil) through the narrow annular gap between the piston and the cylinder wall of the dashpot. When applying forces of differing strength to the dashpot, it can be observed in all cases: The resulting velocity of the piston is proportional the driving force. The proportionality factor corresponds to the internal friction of the fluid within the dashpot, and therefore, to the fluid’s flow resistance or viscosity, respectively.

      2) When removing the load

      As soon as the force is removed, the piston immediately stops to move and remains in the position reached.

      Summary: Behavior of the dashpot model

      Under a constant load, the dashpot fluid is flowing at a constant velocity or deformation rate. After removing the load, the deformation applied to the fluid remains to the full extent. In other words: After a load-and-removal cycle, an ideal-viscous fluid completely remains in the deformed state. This kind of fluids shows absolutely no sign of elasticity.

      Comparison: Dashpot fluid and viscosity law

      For a dashpot the force/velocity law or flow resistance law holds according to Newton:

      F = CN ⋅ v (= CN ⋅ s ̇ )

      with the force F [N], the dashpot constant CN [Ns/m = kg/s], the index N is due to Newton; the piston velocity v [m/s], and the time derivative of the piston’s deflection path s ̇ [m/s].

      Here: F corresponds to the shear stress τ, CN corresponds to the viscosity η, v or s ̇ correspond to the shear rate γ ̇ , and s corresponds to the deformation γ.

      Note: Viscous behavior,

      viscous shear-heating ,

      and lost deformation energy

      Deformation energy acting on a fluid leads to relative motion between the molecules. As a consequence, in flowing fluids frictional forces are occurring between the molecules, causing frictional heating, also called viscous heating. For fluids showing ideal-

      viscous flow behavior, the applied deformation energy is completely used up and can be imagined as deformation work. A part of this thermal energy may heat up the fluid itself and another part may be released as heat to the surrounding environment. During a flow process, the applied deformation energy is used up completely by the fluid, and therefore, it is no longer available for the fluid afterwards, i. e., it is lost. Scientists explain this process as energy dissipation : Here, all the applied deformation energy is lost (dissipated), as it is completely transformed into heat energy.

      When the load is removed, the state of deformation which was reached finally by the fluid is remaining to the full extent. Not even a partial elastic re-formation effect can be observed. Therefore here, an irreversible process has taken place since the shape of the sample remains permanently changed finally, after the load is released from the fluid.

      If fluids are showing ideal-viscous flow behavior, there are absolutely no or at least no significant interactions between their mostly small molecules or particles. Examples are pure solvents, oils and water; and there might be also some diluted polymer solutions and dispersions, however, only if they show a really very low concentration. Since this kind of fluids does not show any visco-elastic gel-like structure, they may tend to separation, and therefore, effects like sedimentation or flotation may occur in mixtures of fluids and in dispersions.

      End of the Cleverly section

2.4Types of flow illustrated by the

      Two-Plates model

      Figure 2.9 illustrates seven different types of laminar flow which may occur in a shear gap: (1) state-at-rest; (2) homogeneous laminar flow, showing a constant shear rate (see also Chapter 2.2); (3) wall-slip , the sample displays very pronounced cohesion while slipping along the walls without adhesion; (4) “plastic behavior” , only a part of the sample is sheared homogeneously (see also Chapter 3.3.4.2c); (5) transient behavior, showing a start-up effect as time-dependent transition until a steady-state viscosity value is reached (occurring above all at low shear rates; see also Chapter 3.3.1b); (6) shear-banding, exhibiting here pronounced cohesion of the medium band (see also Chapter 9.2.2); (7) shear-banding, showing here three different flow velocities or viscosities, respectively (see also here Chapter 9.2.2)