Creeping viscous flow

In polymer processing, we frequently encounter creeping viscous flow in slowly tapering, relatively narrow, gaps as did the ancient Egyptians so depicted in Fig. 2.5. These flows are usually solved by the well-known lubrication approximation, which originates with the famous work by Osborne Reynolds, in which he laid the foundations of hydrodynamic lubrication.14 The theoretical analysis of lubrication deals with the hydrodynamic behavior of thin films from a fraction of a mil (10 in) to a few mils thick. High pressures of the... 

E. Thompson, Use of Pseudo-concentrations to Follow Creeping Viscous Flows during Transient Analysis, Int. J. Numer. Meth. Fluids, 6, 749-761 (1986). 

Now that we have learned how to solve for the detailed velocity fields for at least one class of flow problems (creeping/viscous flows), we turn to a general introduction to convection effects for heat transfer (primarily) for this class of flows. 

At first glance, three coupled linear third-order PDEs must be solved, as illustrated above. However, each term in the x and y components of the vorticity equation is identically zero because =0 and Vj and Vy are not functions of z. Hence, detailed summation representation of the vorticity equation for creeping viscous flow of an incompressible Newtonian fluid reveals that there is a class of two-dimensional flow problems for which it is only necessary to solve one nontrivial component of this vector equation. If flow occurs in two coordinate directions and there is no dependence of these velocity components on the spatial coordinate in the third direction, then one must solve the nontrivial component of the vorticity equation in the third coordinate direction. 

The angular dependence of the stream function represents one of the Legendre polynomials that is unaffected by the operator for creeping viscous flow in spherical coordinates. In other words,... 

Shortcut Methods for Axisymmetric Creeping Flow in Spherical Coordinates. All the previous results can be obtained rather quickly with assistance from information in Happel and Brenner (1965, pp. 133-138). For example, the general solution for the stream function for creeping viscous flow is... 

Summary of Results for Creeping Viscous Flow Around a Gas Bubble. The shortcut method described above and boundary conditions at a gas-liquid interface are useful to analyze creeping flow of an incompressible Newtonian fluid... 

Creeping Viscous Flow Solutions for Gas Bubbles Which Rise Through Incompressible Newtonian Fluids That Are Stagnant Far from the Submerged Objects. A nondeformable bubble of radius R rises through an incompressible Newtonian fluid such that... 

This motion of the bubble induces axisymmetric two-dimensional flow in the liquid phase such that creeping viscous flow is appropriate. The Reynolds number for this problem is based on the rise velocity of the bubble, its diameter (i.e., 2R), and the momentum diffusivity of the liquid. Since the left sides of both the... 

Hence, two-dimensional axisymmetric potential flow in spherical coordinates is described by = 0 for the scalar velocity potential and = 0 for the stream function. Recall that two-dimensional axisymmetric creeping viscous flow in spherical coordinates is described by E E ir) = 0. This implies that potential flow solutions represent a subset of creeping viscous flow solutions for two-dimensional axisymmetric problems in spherical coordinates. Also, recall from the boundary condition far from submerged objects that sin 0 is the appropriate Legendre polynomial for the E operator in spherical coordinates. The methodology presented on pages 186 through 188 is employed to postulate the functional form for xlr. 

The two roots are n = — 1, 2, which represent a subset of the four roots for the radial function for two-dimensional axisymmetric creeping viscous flow in spherical coordinates (i.e., n = —1, 1, 2, 4). One of the roots for the potential flow problem (i.e., n = 2) is consistent with the functional form of far from submerged objects. The potential flow solution is... 

The solution for n = 1 must be discarded because the fluid is stagnant at large r. Hence, A = 0. The boundary condition at the fluid-sohd interface yields B = QR. The creeping viscous flow solution is... 

Calculate the stream function for axisymmetric fully developed creeping viscous flow of an incompressible Newtonian fluid in the annular region between two concentric tubes. This problem is analogous to axial flow on the shell side of a double-pipe heat exchanger. It is not necessary to solve algebraically for all the integration constants. However, you must include all the boundary conditions that allow one to determine a unique solution for i/f. Express your answer for the stream function in terms of ... 

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

Creeping viscous flow in a semi-infinite channel... 

Viscous flow is a sort of creep. Like diffusion creep, its rate increases linearly with stress and exponentially with temperature, with... 

The extension of an amorphous material under a tensile force can be resolved into three parts first, an immediate elastic extension. Which is immediately recoverable on removing the tensile force Mcondly, a delayed elastic extension which is recoverable slowly and thirdly, a plastic extension, viscous flow, or creep, which cannot be glteovered. With glass at ordinary temperatures, this plastic exten- ion is practically absent. A very slow delayed elastic extension OOCUrs. This effect can be troublesome in work with torsion fibres. The delayed elastic effect in vitreous silica fibres is 100 times less than in other glass fibres, and viscous flow of silica is negligible below OO C (N. J. Tighe, 1956). For exact work vitreous sihea torsion flbres are therefore used. 

Figures 5 and h show how the shape of the creep curve is modified by changes in the constants of the model. The values of the constants are given in Table I. Curve I is the same as shown in Figure 4, curve II shows onlv a small amount of viscous creep, and in curve 111, viscous flow is a prominent part of the total creep. The same data were used in Figures 5 and 6, but notice the dramatic, change in the shapes of the curves when a linear time scale is replaced by a logarithmic time scale. In the model, most of the recoverable creep occurs "Within about one decade of the retardation time.

If there is any viscous flow component to the creep, it should be removed before making the calculation, so... 

Poly(vinyl chloridq) and its copolymers are probably the most important pplymers that are often used in the plasticized state. Even though enough plasticizer is used to shift Tt well below room temperature, the material does not show excessive creep (and has no contribution of viscous flow to the compliance) even after long times under load. This behavior is very similar to that of a cross-linked rubber. However, in this case there are no chemical cross-links the material is held together by a small amount of... 

Creep behavior is similar to viscous flow. The behavior in Equation 14.17 shows that compliance and strain are linearly related and inversely related to stress. This linear behavior is typical for most amorphous polymers for small strains over short periods of time. Further, the overall effect of a number of such imposed stresses is additive. Non-creep-related recovery... 

At high temperatures the glassy phase may become less viscous and even liquid and as a consequence may account for the plastic deformation. However, viscous flow creep is not regarded as a viable creep mechanism for superplasticity due to its limited deformation, which corresponds to the redistribution of the glassy phase and therefore to the squeeze of these secondary phases from grain boundaries subjected to compression.8... 

Jin, Q., Wilkinson, D.S. and Weatherly, G.C., (1999), High-resolution electron microscopy investigation of viscous flow creep in a high-purity silicon nitride , J. Am. Ceram. Soc., 82 (6), 1492-1496. 

Polymer processing flows are always laminar and generally creeping type flows. A creeping flow is one in which viscous forces predominate over forces of inertia and acceleration. Classic examples of such flows include those treated by the hydrodynamic theory of lubrication. For these types of flows, the second term on the left-hand side of Eq. 2.5-18 vanishes, and the Equation of motion reduces to ... 

Interfacial cavitation, microcracking, diffusion and crack branching due to the viscous flow of the glass phase during creep-fatigue. 

---