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A screen composed of in-plane thin strips is embedded in a porous medium. The screen is either normal or parallel to the applied pressure gradient which forces a flow through the anisotropic porous medium. The principal axes of anisotropy are assumed to be aligned with that of the screen. The governing equation is fourth order and cannot be factored as in the isotropic case. The solutions are found by eigenfunction superposition (with complex eigenvalues) and point match. Anisotropy has first-order effects on the flow and the drag on the screen. Extrapolation yields fundamental results for the drag of a single slat in an anisotropic porous medium.

The Falkner–Skan flow over a wedge is classic in boundary layer theory. We consider the heat or mass transfer from a source at the vertex of the wedge. The interactions of the thermal boundary layer and momentum boundary layer lead to nonlinear similarity equations which are integrated numerically. There exists a mixing index that depends on the Prandtl number and the wedge opening angle. Attention is paid to special cases such as forced convection in Blasius flow past a semi-infinite plate and the Hiemenz stagnation flow normal to a plate.

Fluid flow in saturated porous media imbedded with a solid inclusion may be described by the Darcy–Brinkman equation. When the Darcy number is small, a boundary-layer theory, similar to Prandtl's theory for viscous flow, is established. The pressure and shear forces are predicted for Darcy–Brinkman flows over a variety of solid inclusions.

The filling of a membrane tube with self-weight by a fluid is studied theoretically for the first time. The fundamental problem depends on two non-dimensional parameters, β and γ which represent the importance of internal pressure and membrane self-weight respectively. The nonlinear equations are solved by an efficient numerical integration method. It is found that membrane self-weight has considerable effect on the tube geometry and the tension of the membrane at low filling pressures.

Analytic approximate formulas for flow and heat transfer through a porous medium in narrow crevices are derived. The Poiseuille number and the Nusselt number depend on the crevice geometry and the product of the aspect ratio and the porous medium factor, the latter being inversely proportional to the square root of the Darcy number. Exact numerical solutions show the approximate formulas are valid up to an aspect ratio of 0.3. The results are applicable to flow through porous rock fissures and biological clefts.

This article is aimed to present an analytic study of electro-osmotic (EO) pumping of a micro-duct with inserted fin vanes. Finned structures are known to be an efficient and very important tool in conducting heat generated in fluid transport system, such as Joule’s heating in micro-fluidic devices. The present semi-analytical analysis is performed under the Debye–Huckel approximation (DHA), enabling us to explicitly investigate the combined effects of various parameters for optimizing the EO pumping rate while retaining substantial fin vanes for heat removal. A mathematical model based on the solutions of two fundamental EO flows is introduced to explain how the choice of the fin vane width may optimize the EO pumping rate in the general case. Moreover, we present the optimized EO flow rates in diagrams plotted on the plane of zeta potentials, which may serve as an easily used reference for engineering design and applications.

The Ritz method for slip flow and constant flux heat transfer in ducts of general shape is established. The efficient Ritz method is applied to the isosceles triangular duct, yielding useful tables for the Poiseuille number and the Nusselt numbers. The results are generic, such that they can be applied to both rarefied gas flows and superhydrophobic flows.

Using intrinsic coordinates, the slip flow in a minute meandering channel is studied by perturbation about the small ratio of curvature to inverse half gap width. The exact solution for an annulus shows this ratio can be as large as 0.5 with less than 1% error. Velocity slip on the walls and the pressure drop depend on the slip factor. Formula for the pressure drop in a channel with a single bend is derived.

The forced convection problem for a developing thermal boundary layer in a parallel shear flow is studied. If the shear flow has a power-law velocity profile, exact similarity thermal boundary layer solutions in terms of Gamma functions can be found. Specifically, three types of thermal boundary conditions are considered: a step temperature change, a step flux change, and a concentrated heat source. The latter is also analogous to mass diffusion form an isolated source. The mixing index for mass diffusion is found exactly.

The vertical axially moving string with non-negligible weight is solved for the first time. Using an efficient initial value method, the natural frequencies are determined for various axial speeds and string densities. Nonexistence of standing waves is explained.

This paper presents the governing equation of motion, boundary conditions and exact vibration frequencies of a segmented rod where the segments are connected by hinges with elastic rotational springs of constant stiffness. The mass of each segment is assumed to be evenly distributed along the length of the rod. Another discrete model called Hencky bar-chain model (short for HBM; which is equivalent to the finite difference model for discretizing continuous rod) assumes the rod mass to be lumped at the ends instead and a different set of boundary conditions are adopted clamped end. The vibration results of a clamped–clamped segment rod are compared with those of the HBM. It is shown that the HBM underestimates the vibration frequencies when compared to the segmented rod model for a finite number of segments while both models furnish vibration solutions that converge to the solutions of Euler beam for infinitely large number of segments.

Starting flow due to a suddenly applied pressure gradient in a circular tube containing two immiscible fluids is solved using eigenfunction expansions. The orthogonality of the eigenfunctions is developed for the first time for circular composite regions. The problem, which is pertinent to flow lubricated by a less viscous near-wall fluid, depends on the ratio of the radius of the core region to that of the tube, and the ratios of dynamic and kinematic viscosities of the two fluids. In general, a higher lubricating effect will lead to a longer time for the starting transient to die out. The time development of velocity profile and slip length are examined for the starting flows of whole blood enveloped by plasma and water enveloped by air in a circular duct. Owing to a sharp contrast in viscosity, the starting transient duration for water/air flow can be ten times longer than that of blood/plasma flow. Also, the slip length exhibits a singularity in the course of the start-up. For blood with a thin plasma skimming layer, the singularity occurs very early, and hence for the most part of the start-up, the slip length is nearly a constant. For water lubricated by air of finite thickness, the singularity may occur at a time that is comparable to the transient duration of the start-up, and hence, an unsteady slip length has to be considered in this case.

A modified Ritz method for solving nonuniform slip flow in a duct is applied to the semicircular duct and the isosceles triangular duct. These ducts are important in microfluidics. Detailed flow fields and Poiseuille numbers show the large effects of nonuniform slip. A rare exact solution for the semicircular duct with nonzero slip is also found.

A heated triangular cylinder is suddenly cooled in a constant temperature bath. The transient heat conduction problem is transformed to the Helmholtz equation related to the vibration of membranes. Using the membrane analogy, exact analytic solutions for the transient heat conduction problem for three triangular cross sections are found.

The classic problem of a circular tube or ring buckled by external pressure is investigated by including geometric nonlinearity, material nonlinearity and initial residual stress. Perturbation theory for buckling and immediate postbuckling agree well with numerical integration. It is found that for softening materials the postbuckling may be unstable and catastrophic snap through may occur.

Free convection with velocity slip and temperature jump in a vertical polygonal duct with a heated circular core is semianalytically solved by the methods of eigenfunction expansion and point match. The conditions of the core wall being at uniform wall temperature or uniform heat flux are considered. It is shown that the temperature and flowfields can be very different from those without the velocity slip and temperature jump. The optimum core radius and the corresponding maximum flow rate are numerically determined for some values of the slip length, temperature jump coefficient, and the number of sides of the polygon.

The constant flux forced convection in a rectangular duct with two highly conductive (H1) walls and two poorly conductive (H2) walls is studied for the first time. This mixed problem is solved analytically using a modified single series method. The Nusselt number is determined for various duct aspect ratios. Depending on the aspect ratio, hot spots and cold spots may occur either on the H1 walls or on the H2 walls.

Analytical solutions are derived for steady, fully developed, buoyancy-driven flow in a vertical annular microchannel, of which either the inner or outer wall exhibits superhydrophobic velocity slip and temperature jump and the inner wall is maintained either at constant wall temperature or constant heat flux. For the four possible cases of hydrodynamic and thermal boundary conditions, the flow rate is determined as a function of the core size, slip length, and temperature jump coefficient. Asymptotic limits are obtained for very large slips and temperature jumps. The effects of slips and temperature jumps on the two issues, namely, the optimum core radius for the maximum flow rate and the singular increase of the flow rate for a very small core radius, are investigated in particular.

The slip flow and temperature jump boundary conditions are derived for H2 constant flux forced convection in ducts. An exact solution is found for the fully developed equilateral triangular duct. Closed-form formulas for the temperature distribution and the Nusselt number are presented. The exact H2 Nusselt number for the no-slip case is 308/163.

The vibration of a two-section column under gravity effects is studied using an efficient initial value method. The problem is governed by the length and area ratios of the two sections in addition to gravity. For the standing column, the stability boundaries show looped properties. The first three frequencies for both standing and hanging two-section columns are determined. For given total volume or mass, there exists an optimum length and area ratios of the two sections for the highest fundamental frequency.

Slip flow occurs in a wide variety of practical chemical engineering processes. This work models, for the first time, the interaction of surface slip and system rotation normal to the flow direction. It is found that the flow fields are affected considerably. For Poiseuille flow, slip increases the longitudinal flow rate at low rotation, but decreases it at high rotation. For Couette flow, slip decreases the longitudinal drag. The solutions are also exact solutions of the Navier-Stokes equations.

The forced flow through a channel with bumpy walls which sandwich a porous medium is studied. The problem models micro-fluidics where, due to the small size of the channel width, the surface roughness of the walls is amplified. The Darcy-Brinkman equation is solved analytically through small perturbations on the ratio of bump amplitude to the half width of the channel. The first- order perturbation solutions give the three-dimensional velocity effects of the bumpiness and the second-order perturbation solutions give the increased resistance due to roughness. The problem depends heavily on the non-dimensional porous medium parameter $$k$$ which represents the importance of length scale to the square root of permeability. Our solutions reduce to the clear fluid limit when $$k$$ is zero and to the Darcy limit when $$k$$ approaches infinity.

Viscous flow in a tube is basic in fluid mechanics. However, there are cases where the tube is filled with a porous medium, such as those in filters, catalytic reactors or matrix-filled biological pores. In these cases the appropriate governing equation is the DarcyBrinkman equation, where both the resistance of the matrix and the bounding walls are taken into account [1, 2]. There are many articles on the flow in a straight tube filled with a porous medium, notably Refs. [3– 10]. This paper considers the flow in a curved tube filled with a porous medium. Previous literature include the works of Avramenko and Kuznetsov [11, 12] who solved analytically for the flow in a curved channel and a curved rectangular tube. Full numerical solution was also done for a helically coiled tube [13]. We shall introduce a Ritz method to treat the flow through a porous medium in a general curved tube, then apply the method to a curved tube of elliptic cross section. The Ritz method and its convergence have been known in elasticity (e.g. [14]), but its applications

The stability and natural vibration of a standing tapered vertical column under its own weight are studied. Exact stability criteria are found for the pointy column and numerical stability boundaries are determined for the blunt tipped column. For vibrations we use an accurate, efficient initial value numerical method for the first three frequencies. Four kinds of columns with linear taper are considered. Both the taper and the cross section shape of the column have large influences on the vibration frequencies. It is found that gravity decreases the frequency while the degree of taper may increase or decrease frequency. Vibrations may occur in two different planes.

Oscillatory channel flow in a rotating system is considered. The Navier–Stokes equations reduce to the Ekman equations that are solved exactly. The results show the interaction between oscillation frequency and rotation rate. Resonance occurs when the oscillation frequency is twice the rotation rate.

A vibrating hanging tapered string (chord, chain, cable) is studied both analytically and numerically. The proper boundary condition for a tip-mass-less string is derived. It is found that the frequencies depend heavily on the taper, tip mass and the shape of the cross section.

The fully developed flow and H1 heat transfer in a polygonal duct filled with a Darcy–Brinkman medium is studied. The efficient method of boundary collocation is used. The problem is governed by the duct shape and a non-dimensional parameter s which characterizes the inverse square root of permeability. Asymptotic formulas for small and large s are derived.

An inclined, gravity driven, open membrane trough is used as a low-cost fluid transport conduit. The membrane shape and the fluid velocity are determined numerically. The optimum opening width for maximum flow is found to be 0.651 of the membrane perimeter.

The Stokes flow through a wavy or corrugated channel with surface slip is studied. The correct Navier's partial slip condition is applied and perturbation solutions about the small amplitude to channel width ratio are obtained. As in Stokes slip flow over a sphere, the resistance is not zero even when slip is infinite. The resistance (due to the interaction of waviness and slip) is larger when the corrugations of the two plates are out of phase than that when they are in phase.

We show that a unique, nontrivial, natural convection state exists under the Boussinesq approximation and completely passive boundary conditions.

A simple mass–spring system with an attached hanging column is investigated. The problem is formulated and the frequencies obtained with an efficient initial value method. Under forced vibration, the amplitude of the mass may be greatly reduced by adding a hanging column. The possibility of using such a hanging column as a dynamic vibration absorber is shown for the first time.

We present solutions for the boundary layer between two uniform shear flows flowing in the same direction. In the upper layer, the flow has shear strength a, fluid density ρ1, and kinematic viscosity ν1, while the lower layer has shear strength b, fluid density ρ2, and kinematic viscosity ν2. Similarity transformations reduce the boundary-layer equations to a pair of ordinary differential equations governed by three dimensionless parameters: the shear strength ratio γ = b/a, the density ratio ρ = ρ2/ρ1, and the viscosity ratio ν = ν2/ν1. Further analysis shows that an affine transformation reduces this multi-parameter problem to a single ordinary differential equation which may be efficiently integrated as an initial-value problem. Solutions of the original boundary-value problem are shown to agree with the initial-value integrations, but additional dual and quadruple solutions are found using this method. We argue on physical grounds and through bifurcation analysis that these additional solutions are not tenable. The present problem is applicable to the trailing edge flow over a thin airfoil with camber.

The Stokes flow through a barrier with multiple pores is solved using a cylinder approximation and eigenfunction expansions. The effect of the proximity of neighboring pores on the resistance is determined.

This paper presents a continued study to our previous work on electroosmotic (EO) flow in a channel with vertical baffle plates by further investigating EO flow through an array of baffle plates arranged in parallel to the channel walls. The flow may be driven either in the direction along or in the direction transverse to the plates, thus distinguishing the longitudinal EO pumping (LEOP) and the transverse EO pumping (TEOP). In both types of EO pumping, it is more interesting to examine the cases when the baffle plates develop a higher zeta potential (denoted by α) than that on the channel walls (β). This semi-analytical study enables us to compare between LEOP and TEOP in the pumping efficiency under similar conditions. The TEOP case is more difficult to solve due to the higher order governing partial differential equations caused by the induced non-uniform pressure gradient distribution. In particular, we examine how the EO pumping rates deviate from those predicted by the Helmholtz-Smoluchowski veloci...

An infinite beam attached to an elastic foundation is buckled by an axial force. The beam is weakened by one or more joints or partial cracks. The governing equations are solved analytically and an exact nonlinear characteristic equation gives the buckling criterion. It is found that the buckling force depends on the foundation stiffness and the rotational resistance of the joints. The buckling modes are complex, and may be either antisymmetric or symmetric.

The present study is aimed to investigate the effects of wavy roughness on electro-osmotic (EO) flow over a wavy plate. The waviness of the plate is modeled by the product of two cosinoidal functions, and the roughness (e) is defined to be the ratio of the wavy amplitude to the Debye length. The effects are examined with respect to the roughness e and different wave numbers (α and β) of the plate waviness. The analysis of the EO flow over the wavy plate is carried out for the applied electric potential, the potential for the electric double layer, as well as the EO flow velocity and pressure field under the Debye-Huckel approximation by using a boundary perturbation method. It is found that the velocity component along the direction of the applied electric field is modified by a second-order term of the roughness, though the same velocity component near the wavy wall exhibits periodic behaviors in phase with the plate waviness. The mean flow rate deficit (e2μ2) due to the surface roughness presents a soph...

The fully developed flow and constant flux heat transfer in super-elliptic ducts filled with a porous (Darcy–Brinkman) medium are studied. Super-elliptic ducts resemble rectangular ducts with rounded corners. An efficient Ritz method is used to determine the velocity and temperature fields. Extensive tables for friction factor–Reynolds number product and Nusselt number are given.

Analytical solutions are found for the transient starting flow due to a sudden pressure gradient in cylindrical, rectangular, and parallel plate ducts fill with a Darcy– Brinkman porous medium. It is found that, for all geometries, the initial velocity front is flat. It eventually becomes more parabolic for small porous media parameter s but remains flat for large s. The boundary layer thickness is of order (1/s). The transient is also shorter (proportional to exp(−s2t)) for large s.

This paper describes a time-domain boundary element method developed to analyze the interactions of acoustic and elastic waves near the interfaces between water and an anisotropic elastic solid. Two models are analyzed with one being the interface between two half spaces of fluid and solid and the other being a fluid region sandwiched by half space domains of anisotropic elastic solids. Both monopole and dipole point sources are used to generate an initial pressure wave in the fluid. Some snapshots of the transient wave behavior near the fluid-solid interfaces are given. The effect of the anisotropy in the solid on the pressure waveforms in the fluid is discussed. The numerical results allow detailed arrival identification and interpretation of acoustic and elastic waves propagating along the fluid-solid interfaces.

The exact series solutions for the laminar flow in a semi-elliptic duct are presented. The present work studies the semi-elliptic duct with the minor axis as the straight wall, which complements that of Alassar and Abushoshah who used the major axis. Properties of the two types of semi-elliptic ducts are given, including the asymptotic Poiseuille numbers.

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.