Time-independent fluids, general equations

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

In turbulent flow of time-independent fluids the Reynolds number at which turbulent flow occurs varies with the flow properties of the non-Newtonian fluid. Dodge and Metzner (D2) in a comprehensive study derived a theoretical equation for turbulent flow of non-Newtonian fluids through smooth round tubes. The final equation is plotted in Fig. 3.5-3, where the Fanning friction factor is plotted versus the generalized Reynolds... 

All the non-Newtonian constitutive equations just given are simplifications of the most general time-independent constitutive equation for isotropic, incompressible non-Newtonian fluids that do not exhibit elasticity [4,5],... 

Given the importance of non-Newtonian fluids in chemical engineering generally, it is surprising how relatively little attention the filtration of such liquids has attracted in the scientific literature. Even today, many filtration textbooks do not even mention the problem. It was not until the late 1960s that the first workable filtration equations were developed for the basic, incompressible, time-independent power-law fluids. The constant pressure and constant rate filtration relationships developed by Kozicki et alP were verified experimentally using slurries of calcium carbonate in water and dilute carboxylmethyl cellulose solutions. 

Govier and Aziz (1972) indicated that once the initial period of stabilization is reached, the general form of pressure loss equations are the same as for time-independent non-Newtonian fluids. At the entry to a pipe, the flow may be laminar, but at a certain distance, once the entrance effects are overcome, the flow can transit to turbulence. 

Rivlin and Ericksen (1955) proposed a nonlinear viscoelasticity theory,in which the components of stress at time t in an element of material depend on the gradient of displacement, velocity, acceleration, second acceleration,..., and (n- l)th acceleration in that element at time t. The fluid described by the constitutive equation based on this theory is referred to as a memoryless fluid, the reason being that the components of stress at time t are independent of those experienced up to the time t. A general representation of this constitutive equation is given by (Rivlin and Ericksen 1955)... 

We have discussed stresses and strain rates. A critical objective is to relate the two, leading to equations of motion governing how fluid packets are accelerated by the forces acting on them. Generally, we are working toward a differential-equation description of a momentum balance, F = ma. The approach is to represent both the forces and the accelerations as functions of the velocity field. The result will be a system of differential equations in which velocities are the dependent variables and the spatial coordinates and time are the independent variables (i.e., the Navier-Stokes equations). 

To be more precise, the general tensor equation of Newton s law of viscosity should be obeyed by a Newtonian fluid (2) however, for onedimensional flow, the applicability of eq 1 is sufficient. For a Newtonian fluid, a linear plot of t versus 7 gives a straight line whose slope gives the fluid viscosity. Also, a log-log plot of t versus 7 is linear with a slope of unity. Both types of plots are useful in characterizing a Newtonian fluid. For a Newtonian fluid, the viscosity is independent of both t and 7, and it may be a function of temperature, pressure, and composition. Moreover, the viscosity of a Newtonian fluid is not a function of the duration of shear nor of the time lapse between consecutive applications of shear stress (3). 

As an aside, a possible alternative to the classical reactor modeling aj> proach which consist in solving the temperature equation, is to use the enthalpy equation (1.129) in combination with the enthalpy-temperature relation (1.141). It is generally assumed that the enthalpy for a flowing fluid is the same function of temperature, pressure and composition as that for a fluid at equilibrium. Hence it follows that the two model formulations (1.141) and (1.142) are formally equivalent. As mentioned earlier, the transformation of the thermodynamic relation can be achieved using the total or complete differential for each independent operator at the time (i.e., illustrated using Cartesian coordinates) ... 

A partial differential equation ( PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space or distributed in space and time. (Definition taken from Wikipedia http //en.wikipedia.org/wiki/Partial differenti al equation)... 

In general D q,t) has a nontrivial q -dependence, so it is equally generally incorrect to replace D q,t)t with a -independent F t), hence the closing inequality in the above equation. In a viscoelastic fluid, such as most polymer solutions, the elastic moduli are frequency-dependent. The fluctuation-dissipation theorem then substantially guarantees that the random thermal forces on probe particles have nonzero correlation times, so probe motions in polymer solutions are not described by Markoff processes. The mathematically correct discussion in Berne and Pecora on Brownian particles, including Eq. 9.5, therefore does not apply to probes in polymer solutions. 

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