Shear stress ideal fluids

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... 

In a fluid under stress, the ratio of the shear stress, r. to the rate of strain, y, is called the shear viscosity, rj, and is analogous to the modulus of a solid. In an ideal (Newtonian) fluid the viscosity is a material constant. However, for plastics the viscosity varies depending on the stress, strain rate, temperature etc. A typical relationship between shear stress and shear rate for a plastic is shown in Fig. 5.1. 

To better understand the way that shear stresses affect a system, we can look at an idealized system. In the example shown in Fig. 6.4 we will sandwich a layer of liquid between two metal plates. We hold the bottom plate stationary, while the top one can slide parallel to the bottom plate at some velocity, v, while maintaining continual contact with the liquid. Between the plates, the top layer of the fluid that is in direct contact with the top plate moves with it. The... 

Such an ideal material is called an inviscid (Pascalian) fluid. However, if the molecules do exhibit a significant mutual attraction such that the force (e.g., the shear stress) is proportional to the relative rate of movement (i.e., the velocity gradient), the material is known as a Newtonian fluid. The equation that describes this behavior is... 

An ideal fluid for which the shear rate is proportional to the shear stress and the viscosity is approximately constant. 

In solutions, the most important physical factors that influence the solubility of ingredients are type of fluid, mixing equipment, and mixing operations. Generalized Newtonian fluids are ideal fluids for which the ratio of the shear rate to the shear stress is constant at a particular time. Unfortunately, in practice, usually liquid dosage forms and their ingredients are non-Newtonian fluids in which the ratio of the shear rate to the shear stress varies. As a result, non-Newtonian fluids may not have a well-defined viscosity [32],... 

Fig. 8.1 Idealized plots of shear rate (y) against shear stress (x) for fluids of various types. (A) Newtonian fluid. (B) Bingham fluid. (C) Shear thinning, (D) Shear thickening. (E) Positive hysteresis 1, 2, 3A thixotropy 1,2, 3B. rhcodestruction, (F) Negative hysteresis with antithixotropy.

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

Friction drag is a strong function of viscosity, and an idealized" fluid with zero viscosity would produce z.ero friction drag since the wall shear stress would be zero. The pressure drag would also be zero in this case during steady flow regardless of the shape of the body since there are no pressure losses. For flow in the horizontal direction, for example, the pressure along a horizontal line is constant (just like stationary fluids) since the upstream velocity is... 

These rheological parameters have been successfully correlated to textural attributes of hardness and spreadabUity and provide information pertaining to the fat crystal network (69). The value of G is useful in assessing the solid-like stmcture of the fat crystal network. Increases in the value of G typically correspond to a stronger network and a harder fat (66). Alternatively, G" represents the fluid-like behavior of the fat system. This value can be related to the spreadability of a fat system, because increases in G" indicate more fluid-like behavior under an applied shear stress. The tan 8 is the ratio of these two values. As the value of 5 approaches 0° (stress wave in phase with stress wave), the G" value approaches zero, and therefore, the sample behaves like an ideal solid and is referred to as perfectly elastic (68). As 8 approaches 90° (stress is completely out of phase relative to the strain). 

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. 

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

In the rheological structure of most food systems there is a viscous element present, and the deformation curves are often highly influenced by the rate of the imposed strain. This is due to the fact that the material relaxes (or flows) while tested under compression and the resultant deformation of this flow is dependent on the nature of the viscous element (Szczesniak, 1963 Peleg and Bagley, 1983). In the viscoelastic food systems, where during processing it is caused to oscillate sinusoidally, the strain curve may or may not be a sine wave. In cases when a periodic oscillatory strain is applied on a food system like fluid material, oscillating stress can be observed. The ideal elastic solid produces a shear stress wave in phase with... 

In physical terms, a simple compressible fluid is one which cannot exert shear stresses without viscous dissipation. In theoretical terms, it is defined as a fluid wherein, during ideal processes, all energy and availability transports with mass transports are collinear with V i.e. o = 0 and to = toj so that V = aV and io V = toV (and tt-V = pV and H/V = [p - pg]V). Then, for ideal relaxation processes Eqs. 12 and 16 become, respectively... 

The behavior of a flowing fluid depends strongly on whether or not the fluid is under the influence of solid boundaries. In the region where the influence of the wall is small, the shear stress may be negligible and the fluid behavior may approach that of an ideal fluid, one that is incompressible and has zero viscosity. The flow of such an ideal fluid is called potential flow and is completely described by the principles of newtonian mechanics and conservation of mass. The mathematical theory of potential flow is highly developed but is outside the scope of this book. Potential flow has two important characteristics (1) neither circulations nor eddies can form within the stream, so that potential flow is also called irrotational flow, and (2) friction cannot develop, so that there is no dissipation of mechanical energy into heat. 

Newtonian Fluid - A term to describe an ideal fluid in which shear stress and shear rate is proportional (e g., water). The proportionality coefficient is called viscosity, which is independent of shear rate, contrary to non-ideal fluids where viscosity is a function of shear rate. Paints and polymer melts are examples of non-Newtonian liquids. 

Fluids treated in the classical theory of fluid mechanics and heat transfer are the ideal fluid and the newtonian fluid. The former is completely frictionless, so that shear stress is absent, while the latter has a linear relationship between shear stress and shear rate. Unfortunately, the behavior of many real fluids used in the mechanical and chemical industries is not adequately described by these models. Most real fluids exhibit nonnewtonian behavior, which means that the shear stress is no longer linearly proportional to the velocity gradient. Metzner [3] classified fluids into three broad groups ... 

The shear stress for ideal flowing gases is given by the following equation and gauges the resistance to establishing a rate of strain (dislocation of the fluid)... 

Generalized vector analysis is presented in this section for fluid flow adjacent to zero-shear interfaces in the laminar regime. The following adjectives have been used to characterize potential flow inviscid, irrotational, ideal, and isentropic. Ideal fluids experience no viscous stress because their viscosities are exceedingly small (i.e., ii 0). Hence, the V r term in the equation of motion is negligible... 

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. ... 

In the case of non-Newtonian liquids according to 77 = /(y) the shear stress is usually expressed by the relationship r = k- y) with k = rj and w = 1 valid for the special case of a Newtonian liquid. The symbol k is the eonsistency and 77 the fluidity (77 < 1 pseudoplastic and n> dilatant). Such fluids are not discussed here. The fluid property viscosity quantifies the inner friction within a fluid or the friction between molecules and is zero for ideal fluids. Increasing temperatures lead to an increase of viscosity in gases but to a reduction of this property in liquids. 

When a constant stress is imposed (its time derivative cr = 0), this equation describes the ideal Newtonian fluid under steady shear flow. When 77 —> 00, this equation describes the ideal elastic solid. The instantaneous response of the solid to an imposed stress is elastic, and the shear modulus E corresponds to the modulus 00 at high frequency. Consequently, the shear stress will relax down to zero exponentially. Under the condition of y = 0, the exponential function (6.18) can be solved from (6.24), which defines the characteristic relaxation time as... 

The transverse shear stress through the melt will be equal through the thickness and will be proportional to the applied rotation rate for a constant viscosity. Polymer melts are not ideal fluids (i.e., Newtonian and constant temperature), so in actual conditions, the transverse shear rate will not be constant through the thickness of the extrudate between the cylinder walls, but this is a good approximation. 

An ideal fluid characterized by a cmistant ration of the shear stress to the rate of shearing in a simple shear deformation and with zero normal stress difference (nonelastic). An analysis to determine whether the material is acceptable for its function. 

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