Creeping flows

In many cases, we can safely make another approximation. We neglect the explicit time dependence of i compared to the viscous term and the influence of pressure differences. This corresponds to a steady-state flow, in which the velocity is constant (0i /dt = 0). It is referred to as creeping flow. For creeping flow and in the absence of external forces, the Navier-Stokes equation reduces to 

Written in full and in Cartesian coordinates, the three components of this vector equation read 

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In the absence of body force, the dimensionless form of the governing model equations for two-dimensional steady-state incompressible creeping flow of a viscoelastic fluid are written as... 

Thompson, E., 1986. Use of pseudo-concentration to follow creeping flows during transient analysis. Ini. J. Numer. Methods Fluids 6, 749 -761. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sections if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow... 

After the substitution for Ai and A2 into Equation (5.74) the pressure potential equation corresponding to creeping flow of a power law fluid in a thin curved layer is derived as... 

Viscous Drag. The velocity, v, with which a particle can move through a Hquid in response to an external force is limited by the viscosity, Tj, of the Hquid. At low velocity or creeping flow (77 < 1), the viscous drag force is /drag — SirTf- Dv. The Reynolds number (R ) is deterrnined from... 

Creeping flow (Re <- 1) through porous media is often described in terms or the permeability k and Darcy s Law ... 

For creeping flow ofpower law non-Newtonian fluids, the method of Christopher and Miodleton (Jnd. Eng. Chem. Fundam., 4,422-426 [1965]) may be used ... 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

Darcy s law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and conecting the velocity with porosity yields a Reynolds number in the form ... 

This means that in the range of stresses which are lower than the apparent yield stress (x < Y), a mechanism of plastic flow ( creeping flow) occurs differing completely from that in the common range. 

In practice, the phenomenon of creeping flow at x < Y can usually be neglected. Thus, certainly, it is insignificant in the treatment of filled polymers, though it may be important, for example, in the discussion of the cold flow of filled elastomers. However, we cannot forget the existence of this effect, to say nothing of the particular interest of the physist in this phenomenon, which is probably similar to the mechanism of flow of plastic crystals. 

Since non-Newtonian flow is typical for polymer melts, the discussion of a filler s role must explicitly take into account this fundamental fact. Here, spoken above, the total flow curve includes the field of yield stress (the field of creeping flow at x < Y may not be taken into account in the majority of applications). Therefore the total equation for the dependence of efficient viscosity on concentration must take into account the indicated effects. 

In creeping flow with the inertia term neglected, the velocity distribution rapidly reaches a steady value after a distance of r0 inside a capillary tube. At this stage the velocity distribution showed the typical parabolic shape characteristic of a Poiseuille flow. In the case of inviscid flow where inertia is the predominant term, it takes typically (depending on the Reynolds number) a distance of 20 to 50 diameters for the flow to be fully developed (Fig. 34). With the short capillary section ( 4r0) in the present design, the velocity front remains essentially unperturbed and the velocity along the symmetry axis, i.e. vx (y = 0), is identical to v0. 

BTU/hr. sq.ft. over a wide range of viscosities and rotational speeds. This is equivalent to the thermal resistance of a fluid film equal to about 1/2 the clearance between the helical agitator and the vessel wall. This represents Reynolds numbers in the range of 10 to 10. This is the region of creeping flow where, with no inertial effects, there is little displacement of the fluid adjacent to the wall. 

Figure 10. Pure drag flow of polymer syrup in the wall-blade clearance C of an anchor agitator in creeping flow. All velocities relative to the blade (12),...

The first of these assumptions drops the momentum terms from the equations of motion, giving a situation known as creeping flow. This leaves Vr and coupled through a pair of simultaneous, partial differential equations. The pair can be solved when circumstances warrant, but the second assumption allows much greater simplification. It allows to be given by a single, ordinary differential equation ... 

The Reynolds number is very small (i.e. creeping flow) inertial terms in the equation of motion are neglected. 

In some microfluidic applications liquid is transported with a comparatively low velocity. In such cases, a liquid volume co-moving with the flow experiences inertial forces which are small compared with the viscous forces acting on it. The terms appearing on the left-hand side of Eq. (16) can then be neglected and the creeping flow approximation is valid... 

An indicator of the validity of the creeping flow approximation is the dimensionless Reynolds number ... 

As before, the problem is governed by the creeping flow equations and boundary conditions given earlier [Eqs. (11)—(14)]. The far field boundary condition in this case is... 

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