Models/modeling viscous fluid flows

The diffusion process in general may be viewed as the model for specific well-defined transport problems. In particle diffusion, one is concerned with the transport of particles through systems of particles in a direction perpendicular to surfaces of constant concentration in a viscous fluid flow, with the transport of momentum by particles in a direction perpendicular to the flow and in electrical conductivity, with the transport of charges by particles in a direction perpendicular to equal-potential surfaces. 

In saturated porous media viscous fluid flow is slow. This can be observed in reality as well as in standard experiments. Therefore, dynamic effects will be neglected in the model (x" = o). Furthermore, it will be postulated that the local temperatures of all constituents are equal and that the motions of solid Xs> ice Xb and gel water Xp are the same, i.e., 0 = 0 and xs = Xi = Xp- The distance and response time for movement from gel to ice are negligible. Experiments have shown that the motion occurs in situ, compare Stockhausen Setzer [3],... 

Ramana, M.J.V, Srinivasa, N.C., Ojjela, O., 2007. Viscous fluid flow between two parallel plates with periodic suction and injection. ASME Adv. Model. B Signal Process. Pattern Recogn. 50, 29-37. 

Geidarov, N. A., Zakharov, Y.N., Shokin, Y.I. Solution of the problem of viscous fluid flow with a given pressure differential. Russian Journal of Numerical Analysis and Mathematical Modeling 26(1), 39-48 (2011)... 

Milosevic, H., Gaydarov, N.A., Zakharov, Y.N. Model ofincompressible viscous fluid flow driven by pressme difference in a given channel. International Journal of Heat and Mass Transfer 62, 242-246 (2013). July 2013. ISSN 0017 9310... 

In the second model the viscous fluid flow inside the fracture is taken into account. In this case the propagation model is unsteady. The process unsteadiness is taken into account by the fluid-flow continuity equation. Meanwhile all other equations describing momentum balance, elastic equilibrium, and material rapture are stationary. The dynamics of the propagation process is represented by the static conditions of flow momentum, stress field, and elastic media displacements in various moments of time. 

Infiltration process occurs at a temperature higher than the melting point of the infiltrated metal. It is quite difficult to observe this process directly. The alternative is to simulate the infiltration process by some theoretical models. Considering the molten melt infiltration as a viscous fluid flow, the infiltration speed rate was described by Washburn formula as (Washburn, 1921) ... 

In contrast with the one-dimensional model, the two-dimensional model allows to determine the actual parameter distribution in flow fields of the working fluid and its vapor. It also allows one to calculate the drag and heat transfer coefficients by the solution of a fundamental system of equations, which describes the flow of viscous fluid in a heated capillary. 

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

The derivations of Hadamard and of Boussinesq are based on a model involving laminar flow of both drop and field fluids. Inertial forces are deemed negligible, and viscous forces dominant. The upper limit for the application of such equations is generally thought of as Re 1. We are here considering only the gross effect on the terminal velocity of a drop in a medium of infinite extent. The internal circulation will be discussed in a subsequent section. 

With the above information, it becomes possible to combine viscous characteristics with elastic characteristics to describe the viscoelasticity of polymeric materials.86-90 The two simplest ways of combining these features are shown in Figure 2.49, where a spring having a modulus G models the elastic response. The viscous response is modelled by what is called a dashpot. It consists of a piston moving in a cylinder containing a viscous fluid of viscosity r. If a downward force is applied to the cylinder, more fluid flows into it, whereas an upward force causes some of the fluid to flow out. The flow is retarded because of the high viscosity and this element thus models the retarded movement and flow of polymer chains. 

These equations are called the Navier-Stokes equations, and when supplemented by the state equation for fluid pressure and species transport equations, they form the basis for any computational model describing the flows in fires. For simplicity, several approximations are inherent (see Equation 20.3) (no Soret/Dufour effects, no viscous dissipation, Fickian diffusion, equal diffusion coefficients of all species, unit Lewis number). 

In order to illustrate the specific material properties of polymers, we compare a viscous fluid (silicone oil) with a viscoelastic shear thinning fluid (aqueous polyethylene oxide solution). These fluids are used as model fluids in order to show the flow behavior limits for polymer melts, which corresponds to the behavior of a viscous fluid at very low shear rates and to the behavior of a shear thinning fluid at very high shear rates. 

Let us write the model of nonstationary flow distribution as applied to the problem of search for the maximum pressure rise at a given node of the hydraulic circuit at a fast cut off of the flow in one of its branches (or the largest drop at pipe break) provided that there is an isothermal motion of viscous incompressible fluid subjected to the action of the pressure, friction, and inertia forces (Gorban et al., 2006). find... 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

If we have a model for linear elastic behavior, we must surely have one for Newtonian viscous flow and we do, the dashpot shown also in Figure 13-87. This is simply a piston in a cylinder that can be filled with various Newtonian fluids, each with a different value of the viscosity. Pulling (or pushing) on the piston causes it to move, as the fluid flows past the small gap between the piston and the cylinder walls, but the rate of deformation will depend on the viscosity of the fluid. (Some students who are a bit slow on the uptake or, more probably, trying to give us a hard time, ask what happens when the piston clunks to a stop at the bottom of the cylinder or pops out of the end don t be too literal minded here, this is just a picture representing a type of behavior )... 

Visco-elastic models have been developed for the nonlinear mechanical properties of fluids and solids. For a viscous fluid in simple shear flow, the shear stress, r y y), is a function of the effective viscosity, rj(-y) and the shear rate, y, as follows ... 

In this sub-section the embedded interface method (frequently referred to as a front tracking method) developed for direct numerical simulations of viscous multi-fluid flows is outlined and discussed. The unsteady model is based on the whole held formulation in which a sharp interface separates immiscible fluids, thus the different phases are treated as one fluid with variable material properties. Therefore, equations (3.14) and (3.15) account for both the differences in the material properties of the different phases as well as surface tension effects at the phase boundary. The bulk fluids are incompressible. The numerical surface tension force approximation used is consistent with the VOF and LS techniques [222] [32], hence the major novelty of the embedded interface method is in the way the density and viscosity fields are updated when the fluids and the interface evolve in time and space. 

---