Plane surface suddenly set in motion in a non-Newtonian fluid

Summary The flow field of a fluid being called the third order fluid or the fluid of grade three is considered for a non-Newtonian flow in the vicinity of a plane wall, suddenly set in motion. The velocity field of the flow is described by a fourth order, non-linear partial differential equation. The solution of this differential equation shows that for short time a strong non-Newtonian effect is present in the velocity field. However, for long time the velocity field becomes a Newtonian one.     

Calculation of He II flow in tubes

The flow of superfluid helium through a tube with different temperatures at the ends differs considerably from that of a Newtonian fluid. The strong dependence of the thermodynamic properties on temperature, the internal convection mechanism and the structure of superfluid turbulence causes unusual flows. The equations for the flow of He II are integrated using a new one-dimensional, steady state model to study the flow in a tube. A wide range of driving conditions is studied. The temperature and pressure profiles along the tube fall into four classes. A dimensionless parameter called is defined which determines the progression through the four classes of behavior. The deviation of the flow from Newtonian is measured by . Significant maxima of the temperature and pressure can occur between the ends of the tube for large values of . The shapes of the profiles and the mass flux depend primarily on , the geometry and the boundary conditions. Formulas are presented which relate the variables of interest to the boundary conditions. These formulas result from averaging the equations of motion along the tube. A general and unified approach, based on , is presented for analyzing experimental data and designing new experiments. It is shown that the common practice of neglecting the pressure term in the energy equation results in poor prediction for many situations. The occurrence of large maxima of pressure and temperature imply that the interpretation of some of the experimental data of the literature should be reconsidered.     

Unsteady MHD flow of a non-newtonian fluid due to eccentric rotations of a porous disk and a fluid at infinity

Summary An exact solution of the unsteady flow of a second-order fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is investigated. It is once again shown that for uniform suction or uniform injection at the disk an asymptotic profile exists for the velocity distribution. The effects of the magnetic field, the material parameters of the second-order fluid, suction and injection on the velocity distribution are studied. Further, from the solution of a rigid disk, it is found that for parameter >.01, a non-Newtonian effect is present in the velocity field. However, for <.01 the velocity field becomes a Newtonian one.     

Thermal boundary layer flow of a micropolar fluid past a wedge with constant wall temperature

Summary The steady laminar flow of micropolar fluids past a wedge has been examined with constant surface temperature. The similarity variables found by Falkner and Skan are employed to reduce the streamwise-dependence in the coupled nonlinear boundary layer equation. Numerical solutions are presented for the heat transfer characteristics with Pr=1 using the fourth-order Runge-Kutta method, and their dependence on the material parameters is discussed. The distributions of dimensionless temperature and Nusselt number across the boundary layer are compared with the corresponding flow problems for a Newtonian fluid over wedges. Numerical results show that for a constant wedge angle with a given Prandtl number Pr=1, the effect of increasing values ofK results in an increasing thermal boundary thickness for a micropolar fluid, as compared with a Newtonian fluid. For the case of the constant material parameterK, however, the heat transfer rate for a micropolar fluid is lower than that of a Newtonian fluid.Nomenclature h Dimensionless microrotation - j Micro-inertia density - K Dimensionless parameter of vortex viscosity - m Falkner-Skan power-law parameter - Re Reynolds number - T Temperature - u, v Fluid velocities in thex andy directions, respectively - U Free stream velocity - x Streamwise coordinate along the body surface - y Coordinate normal to the body surface Greek symbols Thermal diffusivity - Wedge angle parameter - Spin gradient viscosity - Pseudo-similarity variable - Vortex viscosity - Absolute viscosity of the fluid - v Kinematic viscosity - Dimensionless temperature - Density of the micropolar fluid - Angular velocity of micropolar fluid - Stream function     

Effect of Liquid Inertia and Wall Porosity on the Process of Momentum Transfer in a Biobearing With Injection of a Lubricant

We consider the flow of a compressed Newtonian fluid in a biological bearing that has a porous wall, i.e., in a hip joint. The system is modeled by two curvilinear surfaces and a porous wall bounded by a curved impermeable surface. The flow in the gap is considered with account for inertia forces. The Navier–Stokes and Poisson equations used are separated with the aid of the MorganCameron approximation. A solution which relies on the averaging of inertia forces is obtained in a closed form. A bearing of spherical shape is considered as an example.     

Mixed convection of micropolar fluids along a vertical wavy surface

Summary This paper presents numerical results for the steady-state mixed convection in micropolar fluids along a vertical wavy surface. The problem has been formulated by a simple trnasposition theorem, and the spline alternating-direction implicit method has been applied to solve the governing momentum, angular momentum and energy equations. The influence of the micropolar parameters (R and ), the amplitude-wave length ratio and the Gr/Re² number on the skin-friction coefficient and Nusselt number have been studied. Results demonstrate that the skin friction coefficient and local Nusselt number consist of a mixture of two harmonics in micropolar fluids and in Newtonian fluids. As the vortex viscosity parameter (R) increases, the heat transfer rate decreases but the skin friction increases. In addition, when the spin gradient viscosity parameter () increases, the heat transfer rate and the skin friction decreases. However, the heat transfer rate of a micropolar fluid is smaller than a Newtonian fluid, but the skin friction of a micropolar fluid is larger than a Newtonian fluid under all circumstances.     

Exact solutions for the flow of a dipolar fluid on a suddenly accelerated flat plate

Summary The Laplace transformation is used to determine the exact general solution for the unsteady and irrotational flow of an incompressible dipolar fluid set to motion by the acceleration of a flat plate from rest. The general solution is found for arbitrary values of the dipolar constantsd andl. In particular, attention is focused on the case of sudden plate motion for which the velocity field, the displacement thickness, the boundary layer thickness, and the dipolar stress component _yyx have been determined. Special cases of the dipolar constants are also considered. It is shown that for a special boundary value of _yyx , the velocity distribution becomes independent of time. In addition, some significant new results concerning steady flows are presented. Finally, results obtained are compared to the corresponding case for a viscous Newtonian fluid.     

Flow of Newtonian fluid in non-uniform tubes with variable wall permeability with application to flow in renal tubules

Summary The hydrodynamical aspects of flow through proximal renal tubule have been investigated. Assuming renal fluid as Newtonian fluid, flow through diverging/converging tubes with variable wall permeability has been considered. Solutions have been obtained for approximate Navier-Stokes equations with boundary conditions which include a dynamic condition, i.e., leakage flux at the wall depends on variable wall permeability and transboundary pressure drop. Numerical solutions, using fourth order Runge-Kutta method, and approximate analytic solutions, using perturbation method, have been obtained. A comparison of the numerical solution with approximate analytic solution, shows a good agreement (difference less than 4%) between the two solutions for small values of ||, a tube non-uniformity parameter. The velocity profiles at different positions along the axis, the axial distribution of wall shear stress, flow rate and leakage flux have been obtained. For a given value of wall permeability, in diverging (converging) tubes the fractional reabsorption FR is more (less) than its corresponding value in uniform tubes. Further, FR increases (decreases) as the wall permeability increases (decreases) as a linear function of axial distance. The results for flow with constant/variable permeability through uniform tubes and for flow through diverging/converging tubes with constant permeability can be obtained as special cases of this analysis. It is shown that by considering the divergent tube model with linear increase of wall permeability along the axis, an improvement of about 20% in total reabsorption can be achieved over the uniform tube model with constant wall permeability. It is concluded that the approach of using a dynamic boundary condition for leakage flux at the wall has an advantage over the method of prescribing the leakage flux at the wall for this physical problem. Using a set of data, relevant to a physiological situation, implications of the results on glomerular tubular balance have been briefly discussed.     

Rheological behavior of a confined bead-spring cube consisting of equal Fraenkel springs

A general bead-spring model is used to predict linear viscoelastic properties of a non-Hookean bead-spring cube immersed in a Newtonian fluid. This K×K×K cube consist of K ³ beads with equal friction coefficients and 3K ²(K–1) equal Fraenkel springs with length q. The cube has a topology based upon a simple cubic lattice and it is confined to a container of volume V _s=(Kq)³. The confined cube is subjected to a small-amplitude oscillatory shear flow with frequency , where the directions of the flow velocity and its gradient coincide with two principal directions of the simple cubic bead-spring structure. For this flow field an explicit constitutive equation is obtained with analytical expressions for the relaxation times and their strengths. It is found that the resulting relaxation spectrum belonging to a K×K×K Fraenkel cube has the same shape as the one belonging to a `two-dimensional' K×K cubic network consisting of equal Hookean springs. On the other hand, the dynamic moduli G() and G() belonging to a K×KK Fraenkel cube appear to have the same frequency-dependency as the ones belonging to a `three-dimensional' K×KK cube consisting of equal Hookean springs.     

Nonlinear and curvature effects on peristaltic flow of a viscous fluid in an asymmetric channel

Summary. The flow of an incompressible viscous fluid driven by the travelling waves along the boundaries of an asymmetric channel is studied when inertia and streamline curvature effects are not negligible. The channel asymmetry is produced by choosing the wave train on the walls to have different amplitudes and phases. An asymptotic solution is obtained to second order in , a ratio of channel width to the wavelength, giving the curvature effects. A domain transformation is used to transform the channel of variable cross section to a uniform cross section, and this facilitates in easy way of finding closed form solutions at higher orders. The relation connecting the pressure gradient and time rate of flux is a cubic leading to non-uniqueness of flux. A uniqueness criterion is derived which restricts the parameters to get a unique flux for a prescribed pressure gradient. The effects of inertia and curvature on pumping, trapping and shear stress are discussed for symmetric and asymmetric channels and compared with the existing results in the literature. Even under a favorable pressure gradient the possibility of fluid flow in a direction opposite to the direction of the waves propagating on the walls is detected as in the case of some non–Newtonian fluids. It is noticed that the effects of Reynolds number and asymmetry may play an important role in producing mixing. Another interesting observation is that the shear stress distribution on the walls vanishes at some points and this will not indicate any flow separation.     

Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative

Summary In this paper the unsteady Couette flow of a generalized Maxwell fluid with fractional derivative (GMF) is studied. The exact solution is obtained with the help of integral transforms (Laplace transform and Weber transform) and generalized Mittag-Leffler function. It was shown that the distribution and establishment of the velocity is governed by two non-dimensional parameters η, b and fractional derivative α of the model. The result of classical (Newtonian fluid and standard Maxwell fluid) Couette flow can be obtained as a special case of the result given by this paper, and the decaying of the unsteady part of GMF displays power law behavior, which has scale invariance.     

The Effect of a Heat Source on the Adiabatic Heating of a Fluid in the Vicinity of a Critical Point

For a fluid with near-critical parameters, the adiabatic heating referred to as the piston effect arising as a result of compression is investigated. The heating from a source of constant power is treated. One-dimensional analysis and numerical simulation of the effect are performed, and a comparison is made with a similar process developing from a source of constant temperature. It is found that, as the critical point is approached, the piston effect of the type under consideration ceases to depend on the parameters of the fluid, this qualitatively differing from the effect triggered by an isothermal source.     

Simulation and SANS Studies of gelation under shear

Computer simulations of a two-dimensional Lennard-Jones fluid undergoing spinodal decomposition are reported for the system subjected to planar Couetic flow. Key results are the images of the atomic structure and plots of the corresponding pair correlation functions. A companion small-angle neutron scattering (SANS) study of shear-innuenced gelation in colloidal silica suspensions at volume fractions =0. l, (1.13, 0.18, and 0.24 is discussed. It is found that the scattered intensity wave vector curves from the unsheared gels obey a power law lor < 0.34. At higher volume fractions, the power law does not seem to be followed. Shear. however. induces an apparent fractal structure in the gel at = 0.24 Results from the computer and the SANS experiments indicate that the spinodal decomposition process and the getalion mechanism have features in common.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder, Colorado, U.S.A.     

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

The Poiseuille flow of a KBKZ-fluid, being a nonlinear viscoelastic model for a polymeric fluid, is studied. The flow starts from rest and especially the transient phase of the flow is considered. It is shown that under certain conditions the steady flow equation has three different equilibrium points. The stability of these points is investigated. It is proved that two points are stable, whereas the remaining one is unstable, leading to several peculiar phenomena such as discontinuities in the velocity gradient near the wall of the pipe (spurt) and hysteresis. Our theoretical results are confirmed by numerical calculationsof the velocity gradient.     

Low Reynolds number non-Newtonian flow in slowly varying axisymmetrical tubes

Summary Low Reynolds number flow of a non-Newtonian fluid through an axisymmetric tube whose radius varies slowly in the axial direction is analysed by asymptotic approximations. Expressions for the pressure drop along the tube and shear stress at the wall are obtained and compared with the Newtonian results. Some of the highlights of the investigations are (i) for a tube of exponentially increasing radius flow separation for R 3 is only possible at sufficiently long way down the tube while the low Reynolds number solutions do not break down when R 1 and (ii) for a locally constricted tube the Reynolds number for flow separation is less than its Newtonian value when 4K ₂>17K ₁. Here is a small parameter characterizing the radius variation,R the Reynolds number andK ₁ andK ₂ the visco-elastic and cross viscosity parameters respectively.With 1 Figure     

Harmonic waves in a prestressed thin elastic tube filled with a viscous fluid

In this work a theoretical analysis is presented for wave propagation ina thin-walled prestressed elastic tube filled with a viscous fluid. Thefluid is assumed to be incompressible and Newtonian, whereas the tubematerial is considered to be incompressible, isotropic and elastic.Considering the physiological conditions that the arteries experience, sucha tube is initially subjected to a mean pressure P_i and anaxial stretch _z. If it is assumed that in the course ofblood flow small incremental disturbances are superimposed on this initialfield, then the governing equations of this incremental motion are obtainedfor the fluid and the elastic tube. A harmonic-wave type of solution issought for these field equations and the dispersion relation is obtained.Some special cases, as well as the general case, are discussed and thepresent formulation is compared with some previous works on the samesubject.     

The critical conditions of the thermal regime in a generalized couette flow

We examine the nonisothermal steady-state flow of a Newtonian fluid between two parallel plates with consideration of the energy dissipation and in the assumption of a hyperbolic relationship between viscosity and temperature under various temperature boundary conditions. It is assumed that the upper plate is moving at a constant speed and that there is a pressure difference across the space between the plates in the direction of plate motion.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 1, pp. 86–94, July, 1969.     

Simulation of front evolving liquid film flowing down an inclined plate using level set method

In this paper, the projection/level set method is used to simulate the liquid film flow down on an inclined plate for both Newtonian and non-Newtonian fluids. Special numerical treatments are designed to cope with the viscous terms. The simulation is carried out for different inclined plate angles ranging from 20 to 90. The effects of the inclined angle, surface tension, and shear thinning on the flow are discussed. It is found that surface tension reduces the contact angle, increases the film thickness, and promotes a fuller flow front. The effect of shear thinning is to increase the velocity near the wall, make the contact angle larger, and reduce the film thickness. It is also found that the variation of rheological properties and surface tension may produce a complex flow behaviour due to the swollen flow front.The authors are grateful to Dr. M. Sussman (Florida State University) and X-J Fan (Institute of High Performance Computing) for their helpful discussions.     

Numerical simulation of the motion and deformation of a non-Newtonian shear-thinning drop suspended in a Newtonian circular Couette flow using DR-BEM

The motion and deformation of a non-Newtonian shear-thinning drop suspended in a Newtonian circular Couette flow is studied using a boundary element numerical simulation. Non-linear effects from the dependency of the viscosity on the velocity field are treated in an implicit manner and the resultant domain integral is transformed into an equivalent series of boundary integrals using the Dual Reciprocity Method. The non-homogeneous (non-linear) system of algebraic equations resulting from the discretization of the boundary element formulation is solved using a modified Newton–Raphson method for drops with values of the power law index of $n=0.8$ and $0.6$ and compared to the corresponding Newtonian cases ($n=1$). The viscosity of the fluid inside the drop follows the truncated power law model. By using this model, the shear-thinning behaviour of the viscosity is correctly represented while avoiding the shear thickening which can be observed using the standard power law in small gradient flows. The simulations showed that the non-Newtonian drops had larger deformations than the corresponding Newtonian drops due to a general decrease in the viscosity. The value of the local viscosities was found to be dependant not only on the velocity field created by the motion of the internal cylinder, but strongly dependant on the surface tension forces. The rate of deformation of the drops was greater in the beginning of the simulation and decreased toward the end showing the drops found a more or less stable shape.