Ideal viscous fluid

Figure H3.2.1 Deformation pattern of a substance in response to shear. (A) An ideal elastic solid subjected to shear. (B) An ideal viscous fluid subjected to shear, h, height AL, displacement in length.

The ideal viscous element can be represented by a dashpot filled with a Newtonian fluid, whose deformation is linear with time while the stress is applied, and is completely irrecoverable (Newton element). In a dynamic mechanical experiment the stress is exactly 90° out of phase with the strain [Pg.412]

A fluid in which the shear stress is proportional to the shear velocity, corresponding to this law, is called an ideal viscous or Newtonian fluid. Many gases and liquids follow this law so exactly that they can be called Newtonian fluids. They correspond to ideal Hookeian bodies in elastomechanics, in which the shear strain is proportional to the shear. A series of materials cannot be described accurately by either Newtonian or Hookeian behaviour. The relationship between shear stress and strain can no longer be described by the simple linear rule given above. The study of these types of material is a subject of rheology. 

When the slab is subject to a single-step shear history y(t) = yoH(t), where H(t) is the Heaviside unit step function, zero for negative t and unit for t zero or positive, the stress response can be used to characterize the rheological properties. When the materials are subjected to a step strain as shown in Fig. 11a, the different stress responses are obtained as shown in Fig. lib. If the material were perfectly elastic, the corresponding stress history would be of the form t(i) = to//(i), constant for t positive (curve a in Fig. lib). If the material were an ideal viscous fluid, the stress would be instantaneously infinite during the step and then zero for all positive t, like a Dirac delta, 5(t) = H t) (curve b in Fig. 11b). For most real materials, like semisolid foods, the stress response shows that neither of these idealizations is quite accurate. The stress usually decreases from its initial value... 

When the shear stress of a liquid is directly proportional to the strain rate, as in Fig. 2.4a, the liquid is said to exhibit ideal viscous flow or Newtonian behavior. Most unfilled and capillary underfill adhesives are Newtonian fluids. Materials whose viscosity decreases with increasing shear rate are said to display non-Newtonian behavior or shear thinning (Fig. 2.4b). Non-Newtonian fluids are also referred to as pseudoplastic or thixotropic. For these materials, the shear rate increases faster than the shear stress. Most fllled adhesives that can be screen printed or automatically dispensed for surface-mounting components are thixotropic and non-Newtonian. A second deviation from Newtonian behavior is shear thickening in which viscosity increases with increasing shear rate. This type of non-Newtonian behavior, however, rarely occurs with polymers. ... 

Liquid-like - Ideal fluid - Purely viscous... 

In a non-ideal (viscous) fluid, momentum may also be transferred across a surface through the molecular motions and interactions within the fluid. These are expressed through the total stress tensor, n, where Ttij is the flux of positive j momentum in the ith direction. 2 is a second-order symmetric tensor, and the rate of flow of momentum as a result of molecular motions is given by (see Figure 3.2) ... 

In fact, it is often possible with stirred-tank reactors to come close to the idealized well-stirred model in practice, providing the fluid phase is not too viscous. Such reactors should be avoided for some types of parallel reaction systems (see Fig. 2.2) and for all systems in which byproduct formation is via series reactions. 

Flow in tubular reactors can be laminar, as with viscous fluids in small-diameter tubes, and greatly deviate from ideal plug-flow behavior, or turbulent, as with gases, and consequently closer to the ideal (Fig. 2). Turbulent flow generally is preferred to laminar flow, because mixing and heat transfer... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Viscoelastic fluids are thus capable of exerting normal stresses. Because most materials, under appropriate circumstances, show simultaneously solid-like and fluid-like behaviours in varying proportions, the notion of an ideal elastic solid or of a purely viscous fluid represents the commonly encountered limiting condition. For instance, the viscosity of ice and the elasticity of water may both pass unnoticed The response of a material may also depend upon the type of deformation to which it is subjected. A material may behave like a highly elastic solid in one flow situation, and like a viscous fluid in another. 

Simple pressure/drag flow. Here we treat an idealization of the down-channel flow in a melt extruder, in which an incompressible viscous fluid constrained between two boundaries of infinite lateral extent (2). A positive pressure gradient is applied in the X-direction, and the upper boundary surface at y - H is displaced to the right at a velocity of u(H) - U this velocity is that of the barrel relative to the screw. This simple problem was solved by a 10x3 mesh of 4-node quadrilateral elements, as shown in Figure 1. 

A knowledge of v can give an indication of the transit time of a plug of chemical or an ensemble of cells through a microfluidic channel network and thus to assess whether there is enough time for complete mixing or chemical reaction. Both Eq. (11) and Eq. (12) are strictly only valid under idealized conditions (i.e. incompressible and non-viscous fluids and steady flow), but can still be helpful for overall estimation and assessment. 

Plug flow is an idealization. Deviations arise with viscous or non-Newtonian fluids. A mathematically simple deviation from the plug flow pattern is that of power law fluids whose velocity in a tube depends on the radial position, /3 = r/R, according to the equation,... 

A flowing fluid is required to do work to overcome viscous frictional forces so that in practice the quantity W0 is always positive. It is zero only for the theoretical case of an inviscid fluid or ideal fluid having zero viscosity. The work W, may be done on the fluid by a pump situated between points 1 and 2. 

For the flow of a viscous fluid past the cylinder, the pressure decreases from A to B and from A to C so that the boundary layer is thin and the flow is similar to that obtained with a non-viscous fluid. From B to D and from C to D the pressure is rising and therefore the boundary layer rapidly thickens with the result that it tends to separate from the surface. If separation occurs, eddies are formed in the wake of the cylinder and energy is thereby dissipated and an additional force, known as form drag, is set up. In this way, on the forward surface of the cylinder, the pressure distribution is similar to that obtained with the ideal fluid of zero viscosity, although on the rear surface, the boundary layer is thickening rapidly and pressure variations are very different in the two cases. 

Torobin, L. B. and Gauvin, W. H. Can. J. Chem. Eng. 38 (1959) 129, 167, 224. Fundamental aspects of solids-gas flow. Part I Introductory concepts and idealized sphere-motion in viscous regime. Part II The sphere wake in steady laminar fluids. Part III Accelerated motion of a particle in a fluid. 

Figure 1.11 represents the cross-section through a spherical particle over which an ideal non-viscous fluid flows. The fluid is at rest at points 1 and 3 but the fluid velocity is a maximum at points 2 and 4. There is a corresponding decrease in pressure from point 1 to point 2 and from 1 fo 4. However, fhe pressure rises to a maximum again at point 3. If fhe ideal fluid is replaced wifh a real viscous fluid then, as the pressure increases towards point 3, the boundary layer next to the particle surface becomes fhicker and then separates from the surface as in Figure 1.12. This separation of the boundary layer gives rise to...

Voigt-Kelvin model or element Model consisting of an ideal spring and dashpot in parallel in which the elastic response is retarded by viscous resistance of the fluid in the dashpot. 

---