Slow Viscous Flow Research Articles

Integral equation solution for the flow due to the motion of a body of arbitrary shape near a plane interface at small Reynolds number

The problem of determining the slow viscous flow due to an arbitrary motion of a particle of arbitrary shape near a plane interface is formulated exactly as a system of three linear Fredholm integral equations of the first kind, which is shown to have a unique solution. A numerical method based on these integral equations is proposed. In order to test this method valid for arbitrary particle shape, the problem of arbitrary motion of a sphere is worked out and compared with the available analytical solution. This technique can be also extended to low Reynolds number flow due to the motion of a finite number of bodies of arbitrary shape near a plane interface. As an example the case of two equal sized spheres moving parallel and perpendicular to the interface is solved in the limiting case of infinite viscosity ratio.

Slow Viscous Flow Due to the Rotation of Two Finite Hinged Plates

Two-dimensional slow viscous flow due to the rotation of two finite hinged plates in an unbounded medium is investigated on the basis of the Stokes approximation under the condition that the fluid velocity remains finite everywhere. The symmetric flow caused when the plates rotate oppositely about the vertex and antisymmetric flow due to rotation of the plates as a whole are considered separately. A formal expression for the flow is obtained by solving a pair of simulatneous Wiener-Hopf equations. The velocity induced at infinity is given as a function of the angle between the plates and the asymptotic behaviors of the flow near the vertex are discussed. Normal and shear forces and moment exerted on each plate are also calculated.

Thermal effects on slow viscous flow between rotating eccentric spheres with axial gravitational field

The steady flow of a viscous stratified fluid between two eccentric spheres is considered. The velocity and temperature fields are expanded in powers of the Reynolds number Re and the eccentricity δ. Calculations are made up to O( Re s δ 1), where s + t = 2. The thermal and eccentricity influences on the flow patterns are discussed and illustrated. The heat transfer characteristics are also presented for various cases.

Numerical modeling of intense heat release in a slow viscous gas flow

The pulsed heat release in an essentially subsonic flow of viscous heat-conducting gas in a plane channel is numerically modeled. The presence of two time scales makes it possible to calculate the initial stage of the gas dynamic process within the framework of the inviscid gas model. In this phase a monotonie difference scheme of second order of accuracy is employed. The description of the later stages of the process is based on the investigation of a system of equations obtained from the complete Navier-Stokes equations for a compressible gas on the assumption that the Mach number and the hydrostatic compressibility parameter are small. The results of the calculations are presented for the case of a rectangular channel.

On two-dimensional slow viscous flows past obstacles in a half-plane

SynopsisWe consider a cylinder with arbitrary cross section moving in a viscous incompressible fluid parallel to a plane wall. Formal asymptotic expansions of the solution for small Reynolds numbers are constructed by using boundary integral equations of the first kind. In contrast to the problem without a wall, we show that there exists a unique solution to the zeroth order problem. However, the problem considered here is still singular in the sense that we find the Stokes paradox in the next higher order problem. A justification of the formal asymptotic expansion for the first two terms is established rigorously.

Singular perturbations for the exterior three-dimensional slow viscous flow problem

In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A 0 + A 1 Re + O( Re 2 ln Re −1) as the Reynolds number Re → 0 +, where the vectors A 0 and A 1 can be obtained from the method of matched inner and outer expansions.

Axisymmetric Slow Viscous Flow Due to the Motion of Two Equal Coaxial Disks

Axisymmetric slow viscous flow due to the on-axis translation of two equal coaxial disks is investigated on the basis of the Stokes approximation. A formal expression for the flow field is obtained by sloving a mixed boundary value problem. The drag on the disk is expressed in power series of the ratio of the radius to the distance between the disks. Analysis and improvement of series for the drag are carried out. The asymptotic behavior or the drag is estimated, when two disks are nearly piled up.

Slow Viscous Flow Due to Sliding of a Semi-Infinite Plate over a Plane

Two-dimensional slow viscous flow in a region bounded by a plane wall and an inclined semi-infinite flat plate at a distance is investigated on the basis of Stokes' approximation. The motion is caused by the translation of the plane wall parallel to itself. A formal expression for the flow is obtained by solving a pair of simultaneous Wiener-Hopf equations. Streamlines and stress distributions on the plate are determined by evaluating the formal expression. The case in which the flow is caused by a pressure difference between up- and down-stream infinity with the plane wall at rest is also considered. When the plate is not perpendicular to the plane, it is found that separation occurs at the leading edge of the plate for both cases and that for the flow due to pressure difference a viscous eddy of which size diminishes as the inclination angle approaches 90° appears adjacent to the broader side of the plate.

Bounds on fluid permeability for viscous flow through porous media

General properties of variational bounds on Darcy’s constant for slow viscous flow through porous media are studied. The bounds are also evaluated numerically for the penetrable sphere model. The bound of Doi depending on two-point correlations and the analytical bound of Weissberg and Prager give comparable results in the low density limit but the analytical bound is superior for higher densities. Prager’s bound depending on three-point correlation functions is worse than the analytical bound at low densities but better (although comparable to it) at high densities. A procedure for methodically improving Prager’s three point bound is presented. By introducing a Gaussian trial function, the three-point bound is improved by an order of magnitude for moderate values of porosity. The new bounds are comparable in magnitude to the Kozeny–Carman empirical relation for porous materials.

Slow viscous flow inside a torus—the resistance of small tortuous blood vessels

The hydrodynamic resistance of a buckled microvessel in the form of a tightly would helix is approximated by studing the Stokes flow inside a torus. The unidirectional flow is driven by a constant tangential pressure gradient. The solution is obtained by an eigenfunction expansion in toroidal coordinates. The ratio of volume flow carried by the torus to that carried by a straight tube is computed as a function of the vessel radius: coil radius ratio. An asymptotic expansion for this flux ratio is also obtained. The results show that the resistance of a moderately curved vessel is slightly less than the resistance of a straight one, whereas the resistance of a greatly curved vessel is at most 3 3% greater than the straight one.

Asymptotic analysis of Stokes flow in a tortuous vessel

Steady slow viscous flow is considered inside a vessel with circular cross section. The centerline curvature is specified as a function of arc length. The Stokes equations are written in orthogonal curvilinear coordinates. The primary small parameter is the slenderness ratio ϵ \epsilon , which is the ratio of vessel radius to vessel length or wavelength. The product of centerline curvature and vessel length is assumed to be of order unity. A transverse drift appears at O ( ϵ 2 ) O\left ( {{\epsilon ^2}} \right ) that is proportional to the rate of change of curvature. Contours of axial velocity show a primary peak shifted toward the inside wall and a secondary peak grows toward the outside wall as curvature is increased. The flux ratio or relative hydrodynamic conductance is calculated to O ( ϵ 4 ) O\left ( {{\epsilon ^4}} \right ) and includes the effect of variable curvature. The present calculations tend to indicate that the sinusoidal mode of buckled micro-vessel could offer substantially more resistance to flow than the helical buckled mode.

On the Slow Broadside Motion of a Flat Plate along the Center-Line of a Fluid-Filled Two-Dimensional Channel

Two-dimensional slow viscous flow due to the broadside translation of a flat plate along the center-line between two parallel walls is investigated on the basis of the Stokes approximation. An iterative solution is obtained by reducing the problem to a Fredholm integral equation of the second kind. Series expressions in power of ratio of widths of plate to channel are obtained for the drag on the plate and additional pressure difference between the ends of the channel due to the presence of the plate. Limiting values of the drag and the additional pressure drop are estimated from the series expression, when the plate nearly blocks the channel.

Two-Dimensional Slow Viscous Flow Past a Plate Midway between an Infinite Channel

Steady two-dimensional slow viscous flow past a finite plate midway between an infinite channel composed of two parallel planes is investigated. The plate is held parallel to plane walls and the fluid velocity is assumed to have a parabolic distribution at upstream infinity. The problem is reduced to a mixed boundary value problem and two methods of solution for large and small plates are presented. Local and total skin frictions exerted on the plate and the pressure distirbution on the channel wall are calculated.

Some applications of a galerkin‐collocation method for boundary integral equations of the first kind

AbstractHere we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order ‐ 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B‐splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L2. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.

Computational models of liquid metal ion sources

This work defines a computational model of the motion of ions from a liquid metal ion source in the interelectrode region. An Algol 68 program has been written which calculates the Laplace potential in an axially symmetric region by a variable mesh finite difference method, for arbitrary shaped boundaries, and the results have been compared with those of the finite element method. The effect of space charge can be important for these sources due to the large current densities near the emitter and a method for determining these effects by an iterative solution of Poisson's equation is described. Finally, the liquid motion on the needle is considered, which has been shown to be described, for a wide variety of different liquid metals and needle sizes, by the linear equations of slow viscous flow.

Axisymmertic Stokes flow images in spherical free surfaces with applications to rising bubbles

AbstractA theorem is derived for the hydrodynanuc image of an axially symmetric slow viscous (Stokes) flow in a sphere which is impermeable and free of shear stress. A second theorem establishes a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with its centre on the axis of symmetry, a flow past the rigid or shear-free inverse of that surface or sphere.The theorems are used to simplify the proofs of a number of known results for images of point singularities in plane and spherical rigid and free boundaries, and for a pair of bubbles rising steadily in line in a viscous fluid. They also give for the first time accurate numerical solutions for the velocities of each of a larger number of spherical bubbles rising quasi-steadily in line. These enable one to assess the accuracy of simple approximations to those velocities.

Slow Viscous Flow around an Inclined Fence on a Plane

Two-dimensional slow viscous flow around an inclined fence on an infinite plane is investigated on the basis of the Stokes approximation. A formal expression for the flow is obtained by solving a pair of simultaneous Wiener-Hopf equations. Force and moment exerted on the fence are calculated and features of the flow are discussed.

Two-dimensional oxidation

This paper introduces the first general two-dimensional oxidation model based on steady-state oxygen diffusion and the slow incompressible viscous flow of oxide. The moving-boundary problem is solved through a novel numerical technique based on pressure/velocity iteration and a boundary-value approach. The simulation results obtained for oxide shape and bird's beak size versus pad-oxide thickness are in excellent agreement with experiments. This model is also able to calculate stress on the silicon interface during oxidation; this calculated value (6 × 109dyne/cm2) also agrees with the measurement and reveals the tendency of stress to decrease with increasing pad-oxide thickness.

Axisymmetric Stokes Flow Due to a Point Force near a Circular Disk

Axisymmetric slow viscous flow due to a stokeslet situated along the axis of a circular disk is examined. The problem is formulated as a mixed boundary value problem, and a formal expression for the stream function is obtained. By evaluating the formal expression, it is found that a finite toroidal eddy always exists on the opposite side from the stokeslet and that an additional eddy adjacent to the stokeslet appears for values of the radius of the disk larger than a critical value.

An integral equation procedure for the exterior three-dimensional slow viscous flow

An integral equation procedure for the exterior three-dimensional slow viscous flow