Purely viscous fluids

Fluids without any sohdlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of derormation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. 

Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation. 

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. 

Following usual conventions, repeated indices indicate summation and fy denotes df/dXj. The permutation S5mibol is used to present the vector cross product in indicial notation. Due to the anisotropic nature, traction and body couples can exist, and thus the angular momentum equation must be considered. For purely viscous fluids this equation says simply that the deviatoric stresses are symmetric. 

The function iKt-t ) may be interpreted as a memory function having a form as shown in Figure 3.14. For an elastic solid, iff has the value unity at all times, while for a purely viscous liquid iff has the value unity at thfe current time but zero at all other times. Thus, a solid behaves as if it remembers the whole of its deformation history, while a purely viscous liquid responds only to its instantaneous deformation rate and is uninfluenced by its history. The viscoelastic fluid is intermediate, behaving as if it had a memory that fades exponentially with time. The purely elastic solid and the purely viscous fluid are just extreme cases of viscoelastic behaviour. 

Here the time derivative of the strain is represented by Newton s dot. This is the response of a purely viscous fluid. Now suppose we consider a combination of these models. The two simplest arrangements that we can visualise is the models in series or parallel. When they are placed in series we have a Maxwell model and in parallel we have a Kelvin (or sometimes a Kelvin-Voigt) model. 

The authors noted that when their friction parameter M= (pG/,) 8/G is real, it is equivalent to the real slip parameter s = fe used by McHale et al. [14]. From this analysis, a real interfacial energy G /8 is related to the slip length b, for a purely viscous fluid, by... 

To determine what stresses are generated in the torsional disk flow of a CEF fluid, we assume that its flow field is that of a pure viscous fluid then we calculate the tensor quantities Vv, y, co, y y, co y, and v Vy that appear in the CEF equation. Obtaining these quantities, we substitute them in the constitutive equation to find out which are the nonzero stress components. 

Thus far we have given exclusive attention to the flow of purely viscous fluids. In practice the chemical engineer often encounters non-Newtonian fluids exhibiting elastic as well as viscous behavior. Such viscoelastic fluids can be extremely complex in their rheological response. The le vel of mathematical complexity associated with these types of fluids is much more sophisticated than that presented here. Within the limits of space allocated for this article, it is not feasible to attempt a summary of this very extensive field. The reader must seek information elsewhere. Here we shall content ourselves with fluids that do not exhibit elastic behavior. 

For ideal elastic materials 8 = 0, whereas for purely viscous fluids 8 = %/2 rad. 

Polymeric (and other) solids and liquids are intermediate in behavior between Hookean, elastic solids, and Newtonian, purely viscous fluids. They often exhibit elements of both types of response, depending on the time scale of the experiment. Application of stresses for relatively long times may cause some flow and permanent deformation in solid polymers while rapid shearing will induce elastic behavior in some macromolecular liquids. It is also frequently observed that the value of a measured modulus or viscosity is time dependent and reflects the manner in which the measuring experiment was performed. Tliese phenomena are examples of viscoelastic behavior. 

In formal rheology, relations between these three tensors are formulated and analyzed. Only for the two extremes of viscoelastic behaviour are such relations simple. For purely elastic materials there is a relation between the stress tensor and the strain tensor it contains the elasticity modulus and the Poisson ratio, accounting for the extent to which extension in one direction is accompamied by concomitant compression in the other two. For purely viscous fluids there is a relation between the stress tensor and the strain rate tensor. As extension in one direction is concomitant with (viscous) compression in the other two, in this case only one viscosity is required. For incompressible Newton fluids eventually an expression with only one viscosity results, see (1.6.1.131. 

Most common fluids of simple structure are Newtonian (i.e., water, air, glycerine, oils, etc.). However, fluids with complex structures (i.e., high polymer melts or solutions, suspensions, emulsions, foams, etc.) are generally non-Newtonian. Examples of non-Newtonian behavior include mud, paint, ink, mayonnaise, shaving cream, polymer melts and solutions, toothpaste, etc. Many two-phase systems (e.g., suspensions, emulsions, foams, etc.) are purely viscous fluids and do not exhibit significant elastic or memory properties. However, many high polymer fluids (e.g., melts and solutions) are viscoelastic and exhibit both elastic (memory) as well as nonlinear viscous (flow) properties. A classification of material behavior is summarized in Table 5.1 (in which the subscripts have been omitted for simplicity). Only purely viscous Newtonian and non-Newtonian fluids are considered here. The properties and flow behavior of viscoelastic fluids are the subject of numerous books and papers (e.g., Darby, 1976 Bird et al., 1987). 

Newtonian fluids are a subclass of purely viscous fluids. Purely viscous nonnewtonian fluids can be divided into two categories (1) shear-thinning fluids, and (2) shear-thickening fluids. Such fluids can be described by a constitutive equation of the general form... 

While the stress tensor component tfor purely viscous fluids can be determined from the instantaneous values of the rate of deformation tensor 4, the past history of deformation together with the current value of 4, may become an important factor in determining t, for viscoelastic fluids. Constitutive equations to describe stress relaxation and normal stress phenomena are also needed. Unusual effects exhibited by viscoelastic fluids include rod climbing (Weis-senberg effect), die swell, recoil, tubeless siphon, drag, and heat transfer reduction in turbulent flow. 

Fully Developed Heat Transfer—Purely Viscous Fluids... 

TURBULENT FLOW OF PURELY VISCOUS FLUIDS IN CIRCULAR TUBES... 

The hydrodynamic entrance length for purely viscous fluids in turbulent pipe flow is approximately the same as for newtonian fluids, being of the order of 10 to 15 pipe diameters... 

Metzner and Friend [73] measured turbulent heat transfer rates with aqueous solutions of Carbopol, corn syrup, and slurries of Attagel in circular-tube flow. They developed a semi-theoretical correlation to predict the Stanton number for purely viscous fluids as a function of the friction factor and Prandtl number, applying Reichardt s general formulation for the analogy between heat and momentum transfer in turbulent flow ... 

A simple correlation has been given by Yoo [13], who compared his results for Carbopol and Attagel solutions with those of previous investigators. Yoo s empirical equation for predicting turbulent heat transfer for purely viscous fluids is given by... 

This equation describes the available data with a mean deviation of less than 5 percent. It is recommended that Eq. 10.68 be used to predict the heat transfer for purely viscous fluids in turbulent pipe flow for values of the power-law exponent n between 0.2 and 0.9 and over the Reynolds number range from 3,000 to 90,000. The recommended procedure is as follows ... 

---