Showing papers on "Herschel–Bulkley fluid published in 1979"

A mathematical introduction to fluid mechanics

Abstract: Contents: The Equations of Motion.- Potential Flow and Slightly Viscous Flow.- Gas Flow in One Dimension.- Vector Identities.- Index.

The motion of small particles in non-Newtonian fluids

Abstract: There are many problems in which the motion of a small particle, bubble or drop in a non-Newtonian fluid differs in an important qualitative way from its corresponding motion in a Newtonian fluid. From a theoretical point of view such problems are conveniently separated into two groups. In the first, some aspect of the particle's motion only exists, for small Reynolds number, because the suspending fluid is non-Newtonian. Examples of this class include the cross-stream (or lateral) motion of spherical particles in a unidirectional shear flow, rotational motion of an orthotropic particle in sedimentation (leading to a deterministic equilibrium orientation), and cross-orbital drift in the rotation of an axisymmetric particle in shear flow. In these cases, a major change in the orientation or position of the particle can result from small instantaneous contributions of non-Newtonian rheology to the particle's motion, provided that these act “cumulatively” over a sufficiently long period of time. An analytical description of the fluid mechanics relevant to this process may thus be based on the asymptotic limit of a nearly-Newtonian fluid using the so-called “retarded-motion” expansion, and a relevant constitutive model for viscoelastic materials is the Rivlin—Ericksen nth-order fluid. Comparison between theory and experiment shows excellent qualitative (and frequently quantitative) agreement for such problems even when the flow is too rapid, in a rheological sense, for strict adherence to the requirements of a retarded-motion expansion. The second major class of problems is that in which the observed difference between Newtonian and non-Newtonian behavior is due to an important, O(1) change in the fluid motion at all times. In this case, the only possible theoretical description which is valid in more than an asymptotic sense is one based on a full non-linear constitutive model, including “memory”, and thus a solution of the equations of motion is generally possible only via numerical methods. Unlike the first class of problems, an important determining factor in successful match between experiment and theory is therefore a judicious (fortunate?) choice of the constitutive model. In the second part of this paper, I shall discuss some examples of numerical and experimental studies which pertain to particle motions in the regime of strongly viscoelastic flows.

Viscoelastic fluid theories based on the left Cauchy-Green tensor history

Abstract: After discussing the way the left Cauchy-Green tensor history can be used to formulate theories of viscoelastic fluids and showing that this class includes the simple fluids, some comments are made on three second-order fluid models regarding their stability in initial value problems. One second-order fluid model is due toColeman andNoll, the second was proposed by the author and based on the left Cauchy-Green tensor, and the third is due toHarnoy. It is shown that all such second-order fluid models are inadequate because they cannot be stable in initial value problems and possess positive storage moduli as well.

On the unsteady flow of viscous fluid between two plates

Abstract: A solution is found for Navier-Stokes equations for the unsteady laminar flow of a viscous incompressible fluid between two horizontal parallel plates, the upper one is moving with an arbitrary velocityU(t) and the other at rest, under the action of an arbitrary pressure gradient.

Stability of a Two-Layer Film Flow of Viscous Fluids down an Inclined Plane

Abstract: The stability of a two-layer film flow of viscous fluids down an inclined plane is studied by the use of the derivative expansion method. Multiple scales are introduced, and two nonlinear asymptotic equations are derived for the elevation of the free and internal surfaces. The stability of the two-layer flow is governed by four nondimensional parameters of ratios of density, kinematic viscosity, surface tension and layer thickness. For specific case of water and benzene layers, it is shown that, when the heavier fluid (water) is in the lower layer, the two-layer flow is similar to a single fluid flow. On the other hand, when the heavier fluid is in the upper, layer reversion is expected to occur under a certain condition of the film thickness ratio.

Vortex on the surface of a viscous fluid

Abstract: The system of Navier-Stokes equations is solved for boundary conditions corresponding to the case when an axisymmetric tangential transversal load acts at the surface of a gravity viscous incompressible fluid of infinite depth. An integral representation is obtained for the shape of the free surface under the prolonged effect of a stationary vortex load. The example of a tangential load, similar to a concentrated vortex, is examined. In this case a column is squeezed out of the fluid, the height of the column being directly proportional to the square of the moment of the transverse tangential forces and inversely proportional to the square of the product of the dynamic fluid viscosity and the area of the tangential stress distribution. The depth of the annular funnel being formed in front of the column is determined.

Effect of fluid microstructure on stability of plane two-layer flows

Abstract: The stability of plane two-layer Couette and Poiseuille flows, where the lower layer consists of a Grad-model fluid and the upper layer is a viscous Newtonian fluid, is investigated. The disturbances are assumed to be of the long-wave type, and the analysis involves expansion in wave numbers and is limited by two approximations. Numerical calculations are made for some values of the parameters. The calculations indicate that the rotational energy of the fluid in the lower layer has a destabilizing effect on the flow.