The rate at which fluid drains from a collapsing channel or crack depends on the interaction between the elastic properties of the solid and the fluid flow. The same interaction controls the rate at which a pressurized fluid can flow into a crack. In this paper, we present an analysis for the interaction between the viscous flow and the elastic field associated with an expanding or collapsing fluid-filled channel. We first examine an axisymmetric problem for which a completely analytical solution can be developed. A thick-walled elastic cylinder is opened by external surface tractions, and its core is filled by a fluid. When the applied tractions are relaxed, a hydrostatic pressure gradient drives the fluid to the mouth of the cylinder. The relationship between the change in dimensions, time, and position along the cylinder is given by the diffusion equation, with the diffusion coefficient being dependent on the modulus of the substrate, the viscosity of the fluid, and the ratio of the core radius to the exterior radius of the cylinder. The second part of the paper examines the collapse of elliptical channels with arbitrary aspect ratios, so as to model the behavior of fluid-filled cracks. The channels are opened by a uniaxial tension parallel to their minor axes, filled with a fluid, and then allowed to collapse. The form of the analysis follows that of the axisymmetric calculations, but is complicated by the fact that the aspect ratio of the ellipse changes in response to the local pressure. Approximate analytical solutions in the form of the diffusion equation can be found for small aspect ratios. Numerical solutions are given for more extreme aspect ratios, such as those appropriate for cracks. Of particular note is that, for a given cross-sectional area, the rate of collapse is slower for larger aspect ratios. With minor modifications to the initial conditions and the boundary conditions, the analysis is also valid for cracks being opened by a pressurized fluid.

Pure substances can often be cooled below their melting points and still remain in the liquid state. For some supercooled liquids, a further cooling slows down viscous flow greatly, but does not slow down self-diffusion as much. We formulate a continuum theory that regards viscous flow and self-diffusion as concurrent, but distinct, processes. We generalize Newton's law of viscosity to relate stress, rate of deformation, and chemical potential. The self-diffusion flux is taken to be proportional to the gradient of chemical potential. The relative rate of viscous flow and self-diffusion defines a length, which, for some supercooled liquids, is much larger than the molecular dimension. A thermodynamic consideration leads to boundary conditions for a surface of liquid under the influence of applied traction and surface energy. We apply the theory to a cavity in a supercooled liquid and identify a transition. A large cavity shrinks by viscous flow, and a small cavity shrinks by self-diffusion.

The parallel-disk viscometer is a widely used instrument for measuring the rheological properties of Newtonian and non-Newtonian fluids. The torque-rotational speed data from the viscometer are converted into viscosity and other rheological properties of the fluid under test. The classical no-slip boundary condition is usually assumed at the disk-fluid interface. This leads to a simple azimuthal flow in the disk gap with the azimuthal velocity linearly varying in the radial and normal directions of the disk surfaces. For some complex fluids, the no-slip boundary condition may not be valid. The present investigation considers the flow field when the fluid under test exhibits wall slip. The equation for slow steady azimuthal flow of Newtonian fluids in parallel-disk viscometer in the presence of wall slip is solved by the method of separation of variables. Both linear and nonlinear slip functions are considered. The solution takes the form of a Bessel series. It shows that, in general, as a result of wall slip the azimuthal velocity no longer linearly varies in the radial direction. However, under conditions pertinent to parallel-disk viscometry, it approximately remains linear in the normal direction. The implications of these observations on the processing of parallel-disk viscometry data are discussed. They indicate that the method of Yoshimura and Prud’homme (1988, “Wall Slip Corrections for Couette and Parallel-Disk Viscometers,” J. Rheol., 32(1), pp. 53–67) for the determination of the wall slip function remains valid but the simple and popular procedure for converting the measured torque into rim shear stress is likely to incur significant error as a result of the nonlinearity in the radial direction.

A micromechanically based composite model is proposed to study the viscoelastic behavior of solid-filled rubber composites. A nonlinear So-Chen’s (1991) mechanical model which describes the viscoelastic behavior of the rubber matrix is proposed to relate volume-average deformation and stress within the two-phase composite inclusion to the remote (macroscopic) fields. The influence of the volume fractions of inclusions on the overall creep strain of a rubber-matrix composite is investigated at the level of dilute concentration. The creep rate of the rubber matrix, which depends nonlinearly on the creep strain and the primary creep and secondary creep resulting from the viscous flow of creep deformation, is also considered in addition to the usual steady-state, or secondary, creep. The method developed for the calculation of the incremental process is based upon Eshelby’s (1957) equivalence principle of an inhomogeneity-transformation problem and Mori-Tanaka’s (1973) idea of mean-field stress. In order to examine the applicability of the model as well as the nonlinear stretch parameter, a series of experiments on solid-filled silicone rubbers has been carried out, which included constant rate of tensile tests and creep tests. It is demonstrated that this simple, albeit approximate micromechanical modeling is capable of predicting the volume fraction dependence of the time dependent creep, with characteristic consistency with the known elastic behavior.

Expressions are derived for the gradient, divergence, Laplacian, curl, and material time derivative in terms of general curvilinear coordinates using physical components of all vector quantities. The results are conveniently expressed in terms of two new coefficients, involving physical and reciprocal base vectors. An application to the problem of viscous flow and heat transfer in arbitrarily smoothly curved ducts is presented. In particular, helical ducts are considered.

We calculate the free surface shapes upstream and downstream of an obstacle obstructing a thin liquid layer on a moving surface, taking into account gravity and surface tension. We assume low Reynolds number viscous flow, a two-dimensional layer, and small ratio of vertical to horizontal length scale. The upstream and downstream shapes are very different. The upstream liquid piles up against the obstacle to provide an overpressure sufficient to drive the Poiseuille component of the lubrication flow under the obstacle. The downstream liquid is disturbed only by surface tension.

An asymptotic approximation technique is shown to substantially decrease the number of interations necessary for the accurate computation of steady solutions for viscous incompressible flows. A numerical study of flows exterior to a circular cylinder indicates that the technique reduces the required number of iterations by a factor of 10. The flux formulation of the vorticity stream function solution for the flow of a viscous incompressible fluid external to a circular cylinder requires the solution of the Poisson equation for the stream function. The use of the FACR (1), fast direct Poisson equation solver leads to serious errors in the computation of the stream function near the cylinder. However it is shown that these errors are eliminated through the introduction of a perturbation stream function.

The velocity field for creeping viscous flow around a solid sphere due to a spherically symmetric thermal field is determined and a simple thermal generalization of Stokes’ formula is obtained. The velocity field due to an instantaneous heat source at the center of the sphere is obtained in closed form and an application to the storage of heat-generating nuclear waste is discussed.

A model is suggested to study the viscid-inviscid interaction associated with steady incompressible flow past wedges of arbitrary angles. It is shown from this analysis that the determination of the nearly constant pressure (base pressure) prevailing within the near wake is really the heart of the problem and this pressure can only be determined from these interactive considerations. The basic free streamline flow field is established through two discrete parameters which should adequately describe the inviscid flow around the body and the wake. The viscous flow processes such as boundary-layer buildup along the wedge surface, jet mixing, recompression, and reattachment which occurs along the region attached to the inviscid flow in the sense of the boundary-layer concept, serve to determine the aforementioned parameters needed for the establishment of the inviscid flow. It is found that the point of reattachment behaves as a saddle point singularity for the system of equations describing the viscous recompression process. Detailed results such as the base pressure, pressure distributions on the wedge surface, and the wake geometry as well as the influence of the characteristic Reynolds number are obtained. Discussion of these results and their comparison with the experimental data are reported.

The steady laminar axisymmetric flow of a Newtonian viscous homogeneous incompressible fluid through constricted or widened circular tubes is studied. Assuming that the tube wall is slightly curved with the axial distance as described by an exponential function, an analytical method is presented based on the perturbation method in connection with a transformation that maps the curved tube domain into a rectangular region. The results for the velocity, the pressure, and the wall shear stress for various values of the Reynolds number and of the parameters specifying the exponentially curved wall, are represented and discussed in detail. The necessity of the assumption that the flow is two dimensional is justified.

A flow model has been devised to deal with the viscid-inviscid interaction of a class of two-dimensional incompressible separated flow problems. It is suggested that the corresponding inviscid flow of these problems is described by the free streamline theory with few unspecified parameters and their values are, in turn, determined by the viscous flow considerations. The problem of a flow past a backward facing step is selected for study in detail. The viscous flow components of turbulent jet mixing, recompression, and reattachment are delineated and studied individually. When they are later combined, it is found that the point of reattachment behaves as a saddle-point-type singularity in the system of differential equations describing the viscous flow process. This feature is employed to the determination of the aforementioned free parameters and thus the establishment of the overall corresponding inviscid flow field. The resulting base pressure coefficient for the specific case agrees reasonably well with the available experimental data. Additional calculations are performed to demonstrate the influence of higher Reynolds numbers and the values of the similarity (or spread rate) parameter σ of the “constant pressure” turbulent jet mixing process. Further studies of redevelopment of the viscous flow after reattachment, the turbulent exchange within the recompression and redevelopment regions, and the effect of wind tunnel-wall interference on the overall flow patterns have been suggested and discussed.

The steady incompressible viscous flow induced in a cylindrical tank by the rotation of its bottom was studied both theoretically and experimentally. The complete Navier-Stokes equations are expressed in terms of the tangential velocity, vorticity, and meridional stream function. The transformed equations are solved numerically using an alternating-direction implicit scheme and a nonuniform grid. The general validity of the numerical solution was demonstrated by the agreement between the computed and experimental results.

The results of an experimental investigation of the free turbulent mixing of wakes and jets in axial pressure gradients are presented. The data include static pressure and velocity profiles and the turbulent intensity which is presented in terms of the parameter u ′ cL 2 ¯ ( Δ u ) max 2 . It is hypothesized that the representation of the Reynolds stress by a generalized Clauser eddy viscosity model is scaled by this parameter. The experimentally observed dependence of this turbulence quantity on flow field dimensionality and the imposed pressure gradient places more stringent demands on the form of the eddy viscosity than has been shown before. However, the experimental data reveal some fortuitous behavior which aids in the specification of the spatial dependence of the turbulence parameter, leaving the scaling to be determined primarily by the initial conditions, i.e., the state of the turbulence in the near field. Substantial lateral static pressure gradients were observed in all two-dimensional cases studied. It is shown that the boundary-layer form of the viscous flow equations are inadequate in such cases, and a numerical solution of a system of equations that includes an approximate form of the lateral momentum equation provides predictions in good agreement with the data for the mean flow field.

Viscous drops, confined by the slow axisymmetric straining motion of a viscous fluid, are considered when the surface tension is weak. The shape of the drops is determined using slender-body theory, and it is found that steady solutions only exist for sufficiently small drop viscosities. Nonuniqueness exists, with bifurcation from a simple quadratic solution. At high drop viscosities, when there are no steady solutions, a description of the unsteady elongation of shape-preserving drops is obtained. This is the bursting phenomenon described experimentally by Taylor [1].

Creeping viscous flow in a nonuniform pipe is analyzed by taking the local velocity profile and pressure gradients to be those of flow in a cone-shaped region. With this approach it is not necessary to assume that the pipe radius is a slowly varying function of distance along the axis.

A formulation of the finite-element method for both two-dimensional and axisymmetric slow viscous flow is presented. Its application to flow of large particles in a channel or a circular cylindrical tube is discussed. The particles are assumed to be disk-shaped and are disposed axisymmetrically in the tube. The velocity profiles, the pressure gradient, and the shear stress on the wall are determined as functions of kinematic parameters. The conditions are selected to represent an idealization of the motion of red blood cells and plasma in capillary blood vessels.

An analytical method is presented for the solution of two-dimensional solidification of fluid in motion over a semi-infinite plate. The method is applicable to other two-dimensional free boundary problems involving convection of the liquid phase. The solution is based upon an instantaneous source method which transforms the moving free boundary problem to a stationary domain. The transformed problem is solved by a Laplace transformation which results in a two-dimensional elliptic problem in an irregularly shaped region. An approximate point matching method is employed to solve the elliptic problem. Interface motion is obtained from the solution of a nonlinear integrodifferential equation of the Volterra type. Numerical solutions are presented to verify the validity of the model used in the analytical method and to examine the accuracy of the analytical solution.