Steady surface wave pattern in a shear flow

The subject under investigation concerns the steady surface wave patterns created by small concentrated disturbances acting on a non-uniform flow of a heavy fluid. The initial value problem of a point disturbance in a primary flow having an arbitrary velocity distribution (U(y), 0, 0) in a direction parallel to the undisturbed free surface is formulated. A geometric optics method and the classical integral transformation method are employed as two different methods of solution for this problem. Whenever necessary, the special case of linear shear (i.e. U(y) = 1+ϵy)) is chosen for the purpose of facilitating the final integration of the solution.

The asymptotic form of the solution obtained by the method of integral transforms agrees with the leading terms of the solution obtained by geometric optics when the latter is expanded in powers of small ϵ r.

The overall effect of the shear is to confine the wave field on the downstream side of the disturbance to a region which is smaller than the wave region in the case of uniform flows. If U(y) vanishes, and changes sign at a critical plane y = y_cr (e.g. ϵy_cr = -1 for the case of linear shear), then the boundary of this asymmetric wave field approaches this critical vertical plane. On this boundary the wave crests are all perpendicular to the x-axis, indicating that waves are reflected at this boundary.

Inside the wave field, as in the case of a point disturbance in a uniform primary flow, there exist two wave systems. The loci of constant phases (such as the crests or troughs) of these wave systems are not symmetric with respect to the x-axis. The geometric optics method and the integral transform method yield the same result of these loci for the special case of U(y) = U_o(1 + ϵy) and for large Kr (ϵr ˂˂ 1 ˂˂ Kr).

An expression for the variation of the amplitude of the waves in the wave field is obtained by the integral transform method. This is in the form of an expansion in small ϵr. The zeroth order is identical to the expression for the uniform stream case and is thus not applicable near the boundary of the wave region because it becomes infinite in that neighborhood. Throughout this investigation the viscous terms in the equations of motion are neglected, a reasonable assumption which can be justified when the wavelengths of the resulting waves are sufficiently large.

I. The effects of variations in certain parameters upon the efficiency of in vivo hemodialysis with a Kiil dialyzer. II. Interfacial structure of concurrent air-water flow in a two-inch diameter horizontal tube.

Part I

Regression analyses are performed on in vivo hemodialysis data for the transfer of creatinine, urea, uric acid and inorganic phosphate to determine the effects of variations in certain parameters on the efficiency of dialysis with a Kiil dialyzer. In calculating the mass transfer rates across the membrane, the effects of cell-plasma mass transfer kinetics are considered. The concept of the effective permeability coefficient for the red cell membrane is introduced to account for these effects. A discussion of the consequences of neglecting cell-plasma kinetics, as has been done to date in the literature, is presented.

A physical model for the Kiil dialyzer is presented in order to calculate the available membrane area for mass transfer, the linear blood and dialysate velocities, and other variables. The equations used to determine the independent variables of the regression analyses are presented. The potential dependent variables in the analyses are discussed.

Regression analyses were carried out considering overall mass-transfer coefficients, dialysances, relative dialysances, and relative permeabilities for each substance as the dependent variables. The independent variables were linear blood velocity, linear dialysate velocity, the pressure difference across the membrane, the elapsed time of dialysis, the blood hematocrit, and the arterial plasma concentrations of each substance transferred. The resulting correlations are tabulated, presented graphically, and discussed. The implications of these correlations are discussed from the viewpoint of a research investigator and from the viewpoint of patient treatment.

Recommendations for further experimental work are presented.

Part II

The interfacial structure of concurrent air-water flow in a two-inch diameter horizontal tube in the wavy flow regime has been measured using resistance wave gages. The median water depth, r.m.s. wave height, wave frequency, extrema frequency, and wave velocity have been measured as functions of air and water flow rates. Reynolds numbers, Froude numbers, Weber numbers, and bulk velocities for each phase may be calculated from these measurements. No theory for wave formation and propagation available in the literature was sufficient to describe these results.

The water surface level distribution generally is not adequately represented as a stationary Gaussian process. Five types of deviation from the Gaussian process function were noted in this work. The presence of the tube walls and the relatively large interfacial shear stresses precludes the use of simple statistical analyses to describe the interfacial structure. A detailed study of the behavior of individual fluid elements near the interface may be necessary to describe adequately wavy two-phase flow in systems similar to the one used in this work.

I. Studies on bacteriophage DNA's and minicircular DNA's in M. lysodeikticus and E. coli 15. II. Flow dichroism of DNA solutions

Part I

Chapter 1.....A physicochemical study of the DNA molecules from the three bacteriophages, N1, N5, and N6, which infect the bacterium, M. lysodeikticus, has been made. The molecular weights, as measured by both electron microscopy and sedimentation velocity, are 23 x 10⁶ for N5 DNA and 31 x 10⁶ for N1 and N6 DNA's. All three DNA's are capable of thermally reversible cyclization. N1 and N6 DNA's have identical or very similar base sequences as judged by membrane filter hybridization and by electron microscope heteroduplex studies. They have identical or similar cohesive ends. These results are in accord with the close biological relation between N1 and N6 phages. N5 DNA is not closely related to N1 or N6 DNA. The denaturation T_m of all three DNA's is the same and corresponds to a (GC) content of 70%. However, the buoyant densities in CsCl of Nl and N6 DNA's are lower than expected, corresponding to predicted GC contents of 64 and 67%. The buoyant densities in Cs₂SO₄ are also somewhat anomalous. The buoyant density anomalies are probably due to the presence of odd bases. However, direct base composition analysis of N1 DNA by anion exchange chromatography confirms a GC content of 70%, and, in the elution system used, no peaks due to odd bases are present.

Chapter 2.....A covalently closed circular DNA form has been observed as an intracellular form during both productive and abortive infection processes in M. lysodeikticus. This species has been isolated by the method of CsC1-ethidium bromide centrifugation and examined with an electron microscope.

Chapter 3.....A minute circular DNA has been discovered as a homogeneous population in M. lysodeikticus. Its length and molecular weight as determined by electron microscopy are 0.445 μ and 0.88 x 10⁶ daltons respectively. There is about one minicircle per bacterium.

Chapter 4.....Several strains of E. coli 15 harbor a prophage. Viral growth can be induced by exposing the host to mitomycin C or to uv irradiation. The coliphage 15 particles from E. coli 15 and E, coli 15 T^- appear as normal phage with head and tail structure; the particles from E. coli 15 TAU are tailless. The complete particles exert a colicinogenic activity on E.coli 15 and 15 T^-, the tailless particles do not. No host for a productive viral infection has been found and the phage may be defective. The properties of the DNA of the virus have been studied, mainly by electron microscopy. After induction but before lysis, a closed circular DNA with a contour length of about 11.9 μ is found in the bacterium; the mature phage DNA is a linear duplex and 7.5% longer than the intracellular circular form. This suggests the hypothesis that the mature phage DNA is terminally repetitious and circularly permuted. The hypothesis was confirmed by observing that denaturation and renaturation of the mature phage DNA produce circular duplexes with two single-stranded branches corresponding to the terminal repetition. The contour length of the mature phage DNA was measured relative to φX RFII DNA and λ DNA; the calculated molecular weight is 27 x 10⁶. The length of the single-stranded terminal repetition was compared to the length of φX 174 DNA under conditions where single-stranded DNA is seen in an extended form in electron micrographs. The length of the terminal repetition is found to be 7.4% of the length of the nonrepetitious part of the coliphage 15 DNA. The number of base pairs in the terminal repetition is variable in different molecules, with a fractional standard deviation of 0.18 of the average number in the terminal repetition. A new phenomenon termed "branch migration" has been discovered in renatured circular molecules; it results in forked branches, with two emerging single strands, at the position of the terminal repetition. The distribution of branch separations between the two terminal repetitions in the population of renatured circular molecules was studied. The observed distribution suggests that there is an excluded volume effect in the renaturation of a population of circularly permuted molecules such that strands with close beginning points preferentially renature with each other. This selective renaturation and the phenomenon of branch migration both affect the distribution of branch separations; the observed distribution does not contradict the hypothesis of a random distribution of beginning points around the chromosome.

Chapter 5....Some physicochemical studies on the minicircular DNA species in E. coli 15 (0.670 μ, 1.47 x 10⁶ daltons) have been made. Electron microscopic observations showed multimeric forms of the minicircle which amount to 5% of total DNA species and also showed presumably replicating forms of the minicircle. A renaturation kinetic study showed that the minicircle is a unique DNA species in its size and base sequence. A study on the minicircle replication has been made under condition in which host DNA synthesis is synchronized. Despite experimental uncertainties involved, it seems that the minicircle replication is random and the number of the minicircles increases continuously throughout a generation of the host, regardless of host DNA synchronization.

Part II

The flow dichroism of dilute DNA solutions (A₂₆₀≈0.1) has been studied in a Couette-type apparatus with the outer cylinder rotating and with the light path parallel to the cylinder axis. Shear gradients in the range of 5-160 sec.^-1 were studied. The DNA samples were whole, "half," and "quarter" molecules of T4 bacteriophage DNA, and linear and circular λb₂b₅c DNA. For the linear molecules, the fractional flow dichroism is a linear function of molecular weight. The dichroism for linear A DNA is about 1.8 that of the circular molecule. For a given DNA, the dichroism is an approximately linear function of shear gradient, but with a slight upward curvature at low values of G, and some trend toward saturation at larger values of G. The fractional dichroism increases as the supporting electrolyte concentration decreases.

On the transport properties of fluid-particle flow

The hydrodynamic forces acting on a solid particle in a viscous, incompressible fluid medium at low Reynolds number flow is investigated mathematically as a prerequisite to the understanding of transport processes in two-phase flow involving solid particles and fluid. Viscous interaction between a small number of spherical particles and continuous solid boundaries as well as fluid interface are analyzed under a “point-force” approximation. Non-spherical and elastic spherical particles in a simple shear flow area are then considered. Non-steady motion of a spherical particle is briefly touched upon to illustrate the transient effect of particle motion.

A macroscopic continuum description of particle-fluid flow is formulated in terms of spatial averages yielding a set of particle continuum and bulk fluid equations. Phenomenological formulas describing the transport processes in a fluid medium are extended to cases where the volume concentration of solid particles is sufficiently high to exert an important influence. Hydrodynamic forces acting on a spherical solid particle in such a system, e.g. drag, torque, rotational coupling force, and viscous collision force between streams of different sized particles moving relative to each other are obtained. Phenomenological constants, such as the shear viscosity coefficient, and the diffusion coefficient of the bulk fluid, are found as a function of the material properties of the constituents of the two-phase system and the volume concentration of solid. For transient heat conduction phenomena, it is found that the introduction of a complex conductivity for the bulk fluid permits a simple mathematical description of this otherwise complicated process. The rate of heat transfer between particle continuum and bulk fluid is also investigated by means of an Oseen-type approximation to the energy equation.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

Three different categories of flow problems of a fluid containing small particles are being considered here. They are: (i) a fluid containing small, non-reacting particles (Parts I and II); (ii) a fluid containing reacting particles (Parts III and IV); and (iii) a fluid containing particles of two distinct sizes with collisions between two groups of particles (Part V).

Part I

A numerical solution is obtained for a fluid containing small particles flowing over an infinite disc rotating at a constant angular velocity. It is a boundary layer type flow, and the boundary layer thickness for the mixture is estimated. For large Reynolds number, the solution suggests the boundary layer approximation of a fluid-particle mixture by assuming W = W_p. The error introduced is consistent with the Prandtl’s boundary layer approximation. Outside the boundary layer, the flow field has to satisfy the “inviscid equation” in which the viscous stress terms are absent while the drag force between the particle cloud and the fluid is still important. Increase of particle concentration reduces the boundary layer thickness and the amount of mixture being transported outwardly is reduced. A new parameter, β = 1/Ω τ_v, is introduced which is also proportional to μ. The secondary flow of the particle cloud depends very much on β. For small values of β, the particle cloud velocity attains its maximum value on the surface of the disc, and for infinitely large values of β, both the radial and axial particle velocity components vanish on the surface of the disc.

Part II

The “inviscid” equation for a gas-particle mixture is linearized to describe the flow over a wavy wall. Corresponding to the Prandtl-Glauert equation for pure gas, a fourth order partial differential equation in terms of the velocity potential ϕ is obtained for the mixture. The solution is obtained for the flow over a periodic wavy wall. For equilibrium flows where λ_v and λ_T approach zero and frozen flows in which λ_v and λ_T become infinitely large, the flow problem is basically similar to that obtained by Ackeret for a pure gas. For finite values of λ_v and λ_T, all quantities except v are not in phase with the wavy wall. Thus the drag coefficient C_D is present even in the subsonic case, and similarly, all quantities decay exponentially for supersonic flows. The phase shift and the attenuation factor increase for increasing particle concentration.

Part III

Using the boundary layer approximation, the initial development of the combustion zone between the laminar mixing of two parallel streams of oxidizing agent and small, solid, combustible particles suspended in an inert gas is investigated. For the special case when the two streams are moving at the same speed, a Green’s function exists for the differential equations describing first order gas temperature and oxidizer concentration. Solutions in terms of error functions and exponential integrals are obtained. Reactions occur within a relatively thin region of the order of λ_D. Thus, it seems advantageous in the general study of two-dimensional laminar flame problems to introduce a chemical boundary layer of thickness λ_D within which reactions take place. Outside this chemical boundary layer, the flow field corresponds to the ordinary fluid dynamics without chemical reaction.

Part IV

The shock wave structure in a condensing medium of small liquid droplets suspended in a homogeneous gas-vapor mixture consists of the conventional compressive wave followed by a relaxation region in which the particle cloud and gas mixture attain momentum and thermal equilibrium. Immediately following the compressive wave, the partial pressure corresponding to the vapor concentration in the gas mixture is higher than the vapor pressure of the liquid droplets and condensation sets in. Farther downstream of the shock, evaporation appears when the particle temperature is raised by the hot surrounding gas mixture. The thickness of the condensation region depends very much on the latent heat. For relatively high latent heat, the condensation zone is small compared with Ʌ_D.

For solid particles suspended initially in an inert gas, the relaxation zone immediately following the compression wave consists of a region where the particle temperature is first being raised to its melting point. When the particles are totally melted as the particle temperature is further increased, evaporation of the particles also plays a role.

The equilibrium condition downstream of the shock can be calculated and is independent of the model of the particle-gas mixture interaction.

Part V

For a gas containing particles of two distinct sizes and satisfying certain conditions, momentum transfer due to collisions between the two groups of particles can be taken into consideration using the classical elastic spherical ball model. Both in the relatively simple problem of normal shock wave and the perturbation solutions for the nozzle flow, the transfer of momentum due to collisions which decreases the velocity difference between the two groups of particles is clearly demonstrated. The difference in temperature as compared with the collisionless case is quite negligible.

I. Stokes flow past a thin screen. II. Viscous flows past porous bodies of finite size

Part I

The slow, viscous flow past a thin screen is analyzed based on Stokes equations. The problem is reduced to an associated electric potential problem as introduced by Roscoe. Alternatively, the problem is formulated in terms of a Stokeslet distribution, which turns out to be equivalent to the first approach.

Special interest is directed towards the solution of the Stokes flow past a circular annulus. A "Stokeslet" formulation is used in this analysis. The problem is finally reduced to solving a Fredholm integral equation of the second kind. Numerical data for the drag coefficient and the mean velocity through the hole of the annulus are obtained.

Stokes flow past a circular screen with numerous holes is also attempted by assuming a set of approximate boundary conditions. An "electric potential" formulation is used, and the problem is also reduced to solving a Fredholm integral equation of the second kind. Drag coefficient and mean velocity through the screen are computed.

Part II

The purpose of this investigation is to formulate correctly a set of boundary conditions to be prescribed at the interface between a viscous flow region and a porous medium so that the problem of a viscous flow past a porous body can be solved.

General macroscopic equations of motion for flow through porous media are first derived by averaging Stokes equations over a volume element of the medium. These equations, including viscous stresses for the description, are more general than Darcy's law. They reduce to Darcy's law when the Darcy number becomes extremely small.

The interface boundary conditions of the first kind are then formulated with respect to the general macroscopic equations applied within the porous region. An application of such equations and boundary conditions to a Poiseuille shear flow problem demonstrates that there usually exists a thin interface layer immediately inside the porous medium in which the tangential velocity varies exponentially and Darcy's law does not apply.

With Darcy's law assumed within the porous region, interface boundary conditions of the second kind are established which relate the flow variables across the interface layer. The primary feature is a jump condition on the tangential velocity, which is found to be directly proportional to the normal gradient of the tangential velocity immediately outside the porous medium. This is in agreement with the experimental results of Beavers, et al.

The derived boundary conditions are applied in the solutions of two other problems: (1) Viscous flow between a rotating solid cylinder and a stationary porous cylinder, and (2) Stokes flow past a porous sphere.