Orthogonal Rheometer Research Articles

A numerical study of the flow of a K—BKZ fluid between plates rotating about non-coincident axes

The flow of a Currie fluid in an orthogonal rheometer has been studied. The numerical strategy includes projection of the solution onto a polynomial subspace with a B-spline basis, and parametric continuation. In contrast to earlier work, we have uncovered no evidence for bifurcation and multiplicity of solution. For values of γ ⩽ 0.9750, where γ is the non-dimensional distance between the non-coincident axes, the results are in good agreement with our earlier work while for γ ⩾ 0.9750 we obtain solutions by using a parameter continuation method, which also gave good agreement without difficulties with convergence as encountered in other studies. The numerical work is in conplete agreement with the Lemma established earlier. The solutions exhibit strong boundary layer development with increasing Reynolds number and significant interaction between fluid inertia and fluid elasticity. For the K—BKZ fluid under investigation, the surface traction, which has identical values on the two plates in all cases, is strongly dependent on the magnitude of Re, Ws, and γ.

Swirling flows of viscoelastic fluids

Few problems in Newtonian fluid mechanics can match the attention which has been given to the study of the flow of a Newtonian fluid between two parallel plates rotating about a common axis. Most of these analyses are based on the similarity transformation of Karman to study axi-symmetric solutions. Recently, Parter and Rajagopal have proved the existence of solutions which do not possess axi-symmetry. Here, we shall discuss the analogous problems which arise within the context of viscoelastic fluids. We shall also discuss in detail the flow of viscoelastic fluids between infinite parallel plates rotating about non-coincident axes. When the angular speeds of both the plates are the same, the flow reduces to that in an orthogonal rheometer. We shall discuss such flows in some detail. Finally, we shall discuss the case when the two angular speeds are not the same.

On the calculation of the exponential function of a matrix

A method due to Putzer is presented to calculate the exponential function of a matrix and this is used to determine the deformation gradient history of the motion in the Maxwell orthogonal rheometer.

Flow of K-BKZ fluids between parallel plates rotating about distinct axes: shear thinning and inertial effects

It has been shown that for any simple fluid, a flow field of the form u = –Ω[ y - g( z)], v = Ω[ x – f( z], w = 0, which is appropriate for modeling the flow in a orthogonal rheometer, is dynamically possible. The functions f( z) and g( z) depend on the choice of constitutive equation. In the present paper, these are calculated for a class of K-BKZ fluids which exhibit shear thinning. The results are then used to study the interaction of shear thinning and inertial effects on the flow field in an orthogonal rheometer.

A class of exact solutions for the flow of a micropolar fluid

The flow of a micropolar fluid in an orthogonal rheometer is considered. It is shown that an infinite number of exact solutions characterizing asymmetric motions are possible. The expressions for pressure in the fluid, the components of the forces and couples acting on the plates are obtained. The effect of microrotation on the flow is brought out by considering numerical results for the case of coaxially rotating disks.

On exact solutions of the unsteady navierstokes equations—the vortex with instantaneous curvilinear axis

The unsteady pseudo plane motions have been investigated in which each point of the parallel planes is subjected to non-torsional oscillations in their own plane and at any given instant the streamlines are concentric circles. Exact solutions are obtained and the form of the curve Γ, the locus of the centers of these concentric circles, is discussed. The existence of three infinite sets of exact solutions, for the flow in the geometry of an orthogonal rheometer in which the above non-torsional oscillations are superposed on the disks, is established. Three cases arise according to whether ω is greater than, equal to or less than σ, where ω is angular velocity of the basic rotation and σ is the frequency of the superposed oscillations. For a symmetric solution of the flow these solutions reduce to a single unique solution. The nature of the curve Γ is illustrated graphically by considering an example of the flow between coaxial rotating disks.

Flow of viscoelastic fluids between plates rotating about distinct axes

We discuss the flow of BKZ fluids in an orthogonal rheometer. Some analytical results are proved, and numerical solutions are obtained for the Currie model. These solutions show a boundary layer behavior at high Reynolds numbers and the possibility of discontinuous solutions or nonexistence at high Weissenberg numbers.

MHD flow of a second grade fluid in an orthogonal rheometer

The magnetohydrodynamics (MHD) flow of a conducting, homogeneous incompressible Rivlin-Ericksen fluid of second grade contained between two infinite, parallel, insulated disks rotating with the same angular velocity about two noncoincident axes, under the application of a uniform transverse magnetic field, is investigated. This model represents the MHD flow of the fluid in the instrument called an orthogonal rheometer, except for the fact that in the rheometer the rotating plates are necessarily finite. An exact solution of the governing equations of motion is presented. The force components in the x and y directions on the disks are calculated. The effects of magnetic field and the viscoelastic parameter on the forces are discussed in detail.

EFFECTS OF MIXING SOYBEAN FLOUR DOUGHS ON THEIR DYNAMIC MODULI MEASURED IN AN ORTHOGONAL RHEOMETER

ABSTRACTAn orthogonal rheometer was used to determine the dynamic loss moduli and storage moduli of defatted soybean flour doughs. Doughs of 30 to 50% moisture were prepared by mixing for 1 to 10 min in a Brabender laboratory mixing head. Both moduli decreased as dough moisture increased. Moduli increased as mixing time increased. Moduli were initially constant as percent strain increased. After 1 1/2% strain, moduli decreased with increasing strain. No effect of resting time was found.

Flow of a BKZ Fluid in an Orthogonal Rheometer

First Page

On the flow of a simple fluid in an orthogonal rheometer

On the flow of a simple fluid in an orthogonal rheometer

An exact solution of the navier-stokes equation the vortex with curvilinear axis

This paper studies the steady motion of a viscous incompressible fluid the streamlines of which are, in each plane parallel to a fixed plane Π 0, concentric circles. The locus Γ of the centers of the circles is a skew curve. All motions which are exact solutions of the Navier-Stokes equation are studied and the possible forms of the curve Γ are determined. It is shown that the motions so obtained give an infinite set E of solutions for the motion of the fluid in the instrument called orthogonal rheometer. A subset of E constituted by solutions which are stable with regard to arbitrary disturbances is determined.

DETERMINATION OF DYNAMIC MODULI OF SOY DOUGHS USING AN ORTHOGONAL RHEOMETER1

AbstractThe orthogonal rheometer in principle provides a relatively simple means of applying a dynamic strain to polymers to determine the dynamic storage and loss moduli (G′, G″) as functions of frequency and strain. In applying this technique to soy doughs, surface evaporation of water from the samples was found to be a serious problem, but when evaporation was controlled, good reproducibility for G′ and G″, was obtained. The moduli were very sensitive to dough moisture level, with G’ changing by a factor of 10 over the moisture range 40 to 50%. Plate diameter and percent strain both affected the absolute values of G′ and G″, though the effects were relatively small. Log G′ and log G″ increased linearly with decreasing nitrogen solubility index. The two major advantages of the orthogonal rheometer for such food systems are the ease of operation and the fact that sample thickness can be chosen to obviate problems associated with sample heterogeneity. Nevertheless, the effects of plate diameter and percent strain need more extensive study.

A numerical study of the deformation and burst of a viscous drop in general shear flows

The time-dependent deformation and burst of a viscous drop in an arbitrary shear flow at zero Reynolds number is studied. The viscosities of the drop and the suspending fluid are assumed to be equal. A numerical scheme to track the (non-axisymmetric) drop shape in time is presented, and used to investigate the deformation induced by two-dimensional shear and orthogonal rheometer flows. Steady deformations, critical flow rates and burst modes are determined, and compared with asymptotic (small and large) deformation theories, and with experiment.

A comment on the material functions for steady circular shear in the orthogonal rheometer

A comment on the material functions for steady circular shear in the orthogonal rheometer

Viscous Dissipation in the Flow Between Eccentric Rotating Disks (Orthogonal Rheometer)

The orthogonal rheometer has become increasingly popular for determination of the viscoelastic properties of fluids. Here we present an analytical solution for the steady-state temperature distribution arising from the dissipation due to flow between infinite, isothermal eccentric rotating disks. This information is necessary to experimentalists concerned with obtaining accurate rheological characterizations for materials whose properties are temperature dependent, e.g., polymer melts. The results, temperature profiles and the maximum temperature rise, are presented using two dimensionless parameters which indicate the effects of inertia and fluid elasticity. Thus, the results can be easily used by practitioners to estimate the importance of dissipation in their experiments. For conditions of practical interest, temperature rises large enough to be of concern may be encountered.

Inertia effects in rheometrical flow systems

The flow field of a linear viscoelastic material in the orthogonal rheometer, taking fluid inertia into account, has been studied theoretically and an exact solution is given. The flow field of a Newtonian liquid is included in this solution as a special case. The forces on the plates are readily deduced from this solution. The paper concludes with an energy consideration.

Further Studies of Fluid Nonlinearity: The Orthogonal Rheometer and the Oscillating Sphere

Nonlinear effects have been considered in computations of orthogonal rheometer flow and in flow caused by the oscillation of a sphere in an unbounded fluid. Predictions of fluid response, based on a Bird-Carreau constitutive model, are qualitatively different from those which follow from linear theory. Of particular interest are the corrections to force components in the orthogonal rheometer analysis and the generation of a steady secondary flow in the oscillating sphere problem. Magnitudes of these nonlinear effects are presented for a variety of fluid and flow parameters.

The Limit of Linear Viscoelastic Response in Polymer Melts as Measured in the Maxwell Orthogonal Rheometer

The Maxwell orthogonal rheometer was used to measure the limit of linear viscoelastic response (LLVER) as a function of frequency for three linear polyethylenes at 170°C, for six atactic polystyrenes at 170°C, and for Phenoxy A at 212°C. For the elastic component of the stress, linear viscoelastic response was found to exist in all cases up to finite strains of approximately 50%. The viscous component of the stress exhibited linear viscoelastic response up to approximately 100% strain and sometimes as high as 120%, the maximum strain tested. Two constitutive equations (Bird and Carreau's theory with Gordon and Schowalter's generalization and Tanner's network rupture theory) are noted for the dependence they predict of the strain LLVER as a function of frequency. Bird's theory, with Gordon and Schowalter's generalization, predicts that the elastic strain limit times the frequency will be a constant as the frequency is varied; Tanner's theory predicts that the elastic strain limit will be a constant with varying frequency. Data are shown to support Tanner's theory. Although the theories predict similar relations for the viscous strain limit, since no viscous limit is found up to 120% strain, this portion of the theories could not be checked. Plots of the reduced in-phase modulus, G′(γ)/GL′, and the reduced in-phase viscosity, η′(γ)/ηL′, versus strain, γ, (GL′ and ηL′ are the modulus and viscosity below the limit) yield very similar curves for all the polymers. This is in agreement with the limited data from MacDonald, Marsh, and Ashare. The reduced in-phase modulus is constant as the strain is increased up to the elastic LLVER. A gradual decrease then occurs for all of the polymers; at 120% strain, the reduced modulus has decreased between 20 and 30%.

Dynamic melt properties of polymer blends

AbstractThe dynamic mechanical properties of blends of polymer melts were measured using the orthogonal rheometer. Two‐phase blends, polyethylene‐polystyrene, polyethylene‐poly‐(methylmethacrylate), and polystyrene‐polymethylmethacrylate, were studied. The in‐phase and out‐of‐phase moduli were measured over the range of composition and at frequencies between 10−4 and 10 revolutions/sec.The out‐of‐phase modulus increases in a monotonic manner with composition. The in‐phase modulus, however, shows a maximum with composition in two cases. Examination of the relaxation spectra of these blends shows that when no maximum occurs it can be written as an additive function of the spectra of the components. In the case where a maximum is observed in the modulus the measured spectrum of the blend is shifted in frequency relative to the calculated one. This is tentatively attributed to slight interpretation and solubility of one phase in the other in these cases.