Channel Flows Of Viscoelastic Fluids Research Articles

Lateral migration of a deformable fluid particle in a square channel flow of viscoelastic fluid

Cross-stream migration of a deformable fluid particle is investigated computationally in a pressure-driven channel flow of a viscoelastic fluid via interface-resolved simulations. Flow equations are solved fully coupled with the Giesekus model equations using an Eulerian–Lagrangian method and extensive simulations are performed for a wide range of flow parameters to reveal the effects of particle deformability, fluid elasticity, shear thinning and fluid inertia on the particle migration dynamics. Migration rate of a deformable particle is found to be much higher than that of a solid particle under similar flow conditions mainly due to the free-slip condition on its surface. It is observed that the direction of particle migration can be altered by varying shear thinning of the ambient fluid. With a strong shear thinning, the particle migrates towards the wall while it migrates towards the channel centre in a purely elastic fluid without shear thinning. An onset of elastic flow instability is observed beyond a critical Weissenberg number, which in turn causes a path instability even for a nearly spherical particle. An inertial path instability is also observed once particle deformation exceeds a critical value. Shear thinning is found to be suppressing the path instability in a viscoelastic fluid with a high polymer concentration whereas it reverses its role and promotes path instability in a dilute polymer solution. It is found that migration of a deformable particle towards the wall induces a secondary flow with a velocity that is approximately an order of magnitude higher than the one induced by a solid particle under similar flow conditions.

A novel LES method for predicting drag reduction in viscoelastic fluid based on the time period of turbulence structures

Direct numerical simulations (DNS) of turbulent channel flow of viscoelastic fluid were performed with the Oldroyd-B model as a constitutive equation to identify the temporal characteristics of turbulence structures. First, using the results of the DNS in a minimal flow unit (MFU), temporal variations in the energy input and dissipation rates in fully developed wall turbulence of viscoelastic fluid were observed and compared with those of Newtonian fluid. Then, a modified time scale reflecting the local turbulence state was proposed by considering the temporal variation in the viscous stretching stress intensity. It was also found that the idea of the modified time scale could be extended and applied locally to viscoelastic fluid turbulence in a more significant computational domain than MFU to investigate the temporal characteristics of the turbulence structures. Finally, using the local modified time scale, a prediction method employing large eddy simulation for Newtonian fluid turbulence was extended for viscoelastic fluid turbulence. We proved that the drag reduction of viscoelastic fluid turbulence could be accurately predicted with a drag reduction ratio of approximately 40% by using the local modified time scale and the model in the constitutive equation. • The characteristics of viscoelastic fluid turbulence were observed from DNS results. • Active and hibernating turbulence states were compared with those in Newtonian fluid. • A local modified time scale which reflected the local turbulence state was proposed. • LES was extended using the modified time scale for viscoelastic fluid turbulence. • The proposed LES could predict the drag reduction of viscoelastic fluid turbulence.

Well-conditioned ultraspherical and spectral integration methods for resolvent analysis of channel flows of Newtonian and viscoelastic fluids

Modal and nonmodal analyses of fluid flows provide fundamental insight into the early stages of transition to turbulence. Eigenvalues of the dynamical generator govern temporal growth or decay of individual modes, while singular values of the frequency response operator quantify the amplification of disturbances for linearly stable flows. In this paper, we develop well-conditioned ultraspherical and spectral integration methods for frequency response analysis of channel flows of Newtonian and viscoelastic fluids. Even if a discretization method is well-conditioned, we demonstrate that calculations can be erroneous if singular values are computed as the eigenvalues of a cascade connection of the frequency response operator and its adjoint. To address this issue, we utilize a feedback interconnection of the frequency response operator with its adjoint to avoid computation of inverses and facilitate robust singular value decomposition. Specifically, in contrast to conventional spectral collocation methods, the proposed method (i) produces reliable results in channel flows of viscoelastic fluids at high Weissenberg numbers (∼500); and (ii) does not require a staggered grid for the equations in primitive variables.

Localized stress amplification in inertialess channel flows of viscoelastic fluids

Nonmodal analysis typically uses square-integrated quantities to characterize amplification of disturbances. However, such measures may be misleading in viscoelastic fluids, where polymer stresses can be strongly amplified over a small region. Here, we show that when using a localized measure of disturbance amplification, spanwise-constant polymer-stress fluctuations can be more amplified than streamwise-constant polymer-stress fluctuations, which is the opposite of what is observed when a square-integrated measure of disturbance amplification is used. To demonstrate this, we consider a model problem involving two-dimensional pressure-driven inertialess channel flow of an Oldroyd-B fluid subject to a localized time-periodic body force. Nonmodal analysis of the linearized governing equations is performed using recently developed well-conditioned spectral methods that are suitable for resolving sharp stress gradients. It is found that polymer-stress fluctuations can be amplified by an order of magnitude while there is only negligible amplification of velocity fluctuations. The large stress amplification arises from the continuous spectrum of the linearized problem, and may put the flow into a regime where nonlinear terms are no longer negligible, thereby triggering a transition to elastic turbulence. The results suggest an alternate mechanism that may be useful for understanding recent experimental observations of elastic turbulence in microchannel flows of viscoelastic fluids. • Nonmodal amplification of localized time-periodic body forces is examined. • Poiseuille flow of Oldroyd-B fluids is considered. • Stress fluctuations can be amplified much more than velocity fluctuations. • Stress amplification is highly localized in space. • Results suggest a mechanism for transition to elastic turbulence in microchannels.

Comparative study on numerical performances of log-conformation representation and standard conformation representation in the simulation of viscoelastic fluid turbulent drag-reducing channel flow

In this paper, a new derivation process of the log-conformation governing equation for viscoelastic fluid flows is presented by using the Taylor series definition of the matrix logarithm. Based on the log-conformation representation (LCR) and standard conformation representation (CR) methods, the turbulent drag-reducing channel flow of viscoelastic fluid described by the Oldroyd-B constitutive model is simulated by the finite difference method. The comparison illustrates that the turbulent drag reduction (DR) effect under the condition of a low Weissenberg number (Wi = 1) or moderate Weissenberg number (Wi = 5) can be successfully reproduced by the CR method but is very difficult to be obtained by the LCR method at the same grid resolution if the commonly used interpolation approaches in the computing domain (i.e., log domain) are employed. Further research reveals that the interpolation method of log-conformation tensor involved is one of the dominant reasons responsible for the disability to obtain a turbulent DR effect by using the LCR method. If the interpolation is performed in a physical domain, the turbulent DR effect can be reproduced by using the LCR method. If the interpolation involved in the CR method is carried out in a log domain, the turbulent DR phenomenon can still be simulated but with a weakened DR effect. In sum, this study demonstrates that the commonly used interpolation approaches in the log domain should be responsible for the poor performance of the LCR method.

Turbulent transport dissimilarity with modulated turbulence structure in channel flow of viscoelastic fluid

This experimental study investigated the turbulent transport dissimilarity with a modulated turbulence structure in a channel flow of a viscoelastic fluid using simultaneous particle image velocimetry and planar laser-induced fluorescence measurements. An instantaneous dye concentration field with fluctuating velocity vectors showed that mass was transferred by hierarchically large-scale wavy motions with inclination. A co-spectral analysis showed that the spatial phase modulation of the streamwise velocity and dye concentration fluctuations for the wall-normal velocity fluctuation corresponded to the relaxation time. The occurrence of intense dye concentration fluctuation and small streamwise velocity fluctuation in a thin boundary layer caused dissimilar turbulent transport because of the non-zero negative correlation of the streamwise velocity and dye concentration fluctuations for the wall-normal velocity fluctuation only on large scales. This explains the turbulent transport dissimilarity which leads to the zero averaged Reynolds shear stress and non-zero wall-normal turbulent mass flux.

Amplification of localized body forces in channel flows of viscoelastic fluids

The study of nonmodal amplification of distributed body forces in channel flows of viscoelastic fluids has provided useful insights into the mechanisms that may govern the initial stages of transition to elastic turbulence. However, distributed body forces are not easy to implement in experiments and so there is a need to examine amplification of localized body forces. In this work, we use the linearized governing equations to examine such amplification in Poiseuille flow of FENE-CR fluids. We first identify the wall-normal location at which the impulsive excitations experience the largest amplification and then analyze the kinetic energy of the fluctuations and the resulting flow structures. For both a Newtonian fluid at high Reynolds numbers and a viscoelastic fluid at low Reynolds numbers, the largest amplification occurs for disturbances that are located near the channel wall. Our analysis of the energy evolution shows that a localized body force in the spanwise direction has the largest impact and that the streamwise velocity component is most affected. For viscoelastic fluids we observe the development of vortical structures away from the source of impulsive excitation. This feature is less prominent in Newtonian fluids and it may provide a mechanism for triggering the initial stages of transition to elastic turbulence.

Dynamics of particle migration in channel flow of viscoelastic fluids

The migration of a sphere in the pressure-driven channel flow of a viscoelastic fluid is studied numerically. The effects of inertia, elasticity, shear-thinning viscosity, secondary flows and the blockage ratio are considered by conducting fully resolved direct numerical simulations over a wide range of parameters. In a Newtonian fluid in the presence of inertial effects, the particle moves away from the channel centreline. The elastic effects, however, drive the particle towards the channel centreline. The equilibrium position depends on the interplay between the elastic and inertial effects. Particle focusing at the centreline occurs in flows with strong elasticity and weak inertia. Both shear-thinning effects and secondary flows tend to move the particle away from the channel centreline. The effect is more pronounced as inertia and elasticity effects increase. A scaling analysis is used to explain these different effects. Besides the particle migration, particle-induced fluid transport and particle migration during flow start-up are also considered. Inertial effects, shear-thinning behaviour, and secondary flows are all found to enhance the effective fluid transport normal to the flow direction. Due to the oscillation in fluid velocity and strong normal stress differences that develop during flow start-up, the particle has a larger transient migration velocity, which may be potentially used to accelerate the particle focusing.

Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids

AbstractWe study the modal and non-modal linear instability of inertia-dominated channel flow of viscoelastic fluids modelled by the Oldroyd-B and FENE-P closures. The effects of polymer viscosity and relaxation time are considered for both fluids, with the additional parameter of the maximum possible extension for the FENE-P. We find that the parameter explaining the effect of the polymer on the instability is the ratio between the polymer relaxation time and the characteristic instability time scale (the frequency of a modal wave and the time over which the disturbance grows in the non-modal case). Destabilization of both modal and non-modal instability is observed when the polymer relaxation time is shorter than the instability time scale, whereas the flow is more stable in the opposite case. Analysis of the kinetic energy budget reveals that in both regimes the production of perturbation kinetic energy due to the work of the Reynolds stress against the mean shear is responsible for the observed effects where polymers act to alter the correlation between the streamwise and wall-normal velocity fluctuations. In the subcritical regime, the non-modal amplification of streamwise elongated structures is still the most dangerous disturbance-growth mechanism in the flow and this is slightly enhanced by the presence of polymers. However, viscoelastic effects are found to have a stabilizing effect on the amplification of oblique modes.

Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids

AbstractAmplification of deterministic disturbances in inertialess shear-driven channel flows of viscoelastic fluids is examined by analysing the frequency responses from spatio-temporal body forces to the velocity and polymer stress fluctuations. In strongly elastic flows, we show that disturbances with large streamwise length scales may be significantly amplified even in the absence of inertia. For fluctuations without streamwise variations, we derive explicit analytical expressions for the dependence of the worst-case amplification (from different forcing to different velocity and polymer stress components) on the Weissenberg number ($\mathit{We}$), the maximum extensibility of the polymer chains ($L$), the viscosity ratio and the spanwise wavenumber. For the Oldroyd-B model, the amplification of the most energetic components of velocity and polymer stress fields scales as${\mathit{We}}^{2} $and${\mathit{We}}^{4} $. On the other hand, the finite extensibility of polymer molecules limits the largest achievable amplification even in flows with infinitely large Weissenberg numbers: in the presence of wall-normal and spanwise forces, the amplification of the streamwise velocity and polymer stress fluctuations is bounded by quadratic and quartic functions of$L$. This high amplification signals low robustness to modelling imperfections of inertialess channel flows of viscoelastic fluids. The underlying physical mechanism involves interactions of polymer stress fluctuations with a base shear, and it represents a close analogue of the lift-up mechanism that initiates a bypass transition in inertial flows of Newtonian fluids.

Nonmodal amplification of stochastic disturbances in strongly elastic channel flows

Nonmodal amplification of stochastic disturbances in elasticity-dominated channel flows of Oldroyd-B fluids is analyzed in this work. For streamwise-constant flows with high elasticity numbers μ and finite Weissenberg numbers We, we show that the linearized dynamics can be decomposed into slow and fast subsystems, and establish analytically that the steady-state variances of velocity and polymer stress fluctuations scale as O ( We 2 ) and O ( We 4 ) , respectively. This demonstrates that large velocity variance can be sustained even in weakly inertial stochastically driven channel flows of viscoelastic fluids. We further show that the wall-normal and spanwise forces have the strongest impact on the flow fluctuations, and that the influence of these forces is largest on fluctuations in the streamwise velocity and the streamwise component of the polymer stress tensor. The underlying physical mechanism involves polymer stretching that introduces a lift-up of flow fluctuations similar to vortex tilting in inertia-dominated flows. The validity of our analytical results is confirmed in stochastic simulations. The phenomenon examined here provides a possible route for the early stages of a bypass transition to elastic turbulence and might be exploited to enhance mixing in microfluidic devices.

Worst-case amplification of disturbances in inertialess flows of viscoelastic fluids

AbstractAmplification of deterministic disturbances in inertialess channel flows of viscoelastic fluids is studied by analyzing the frequency responses from spatio-temporal body forces to the velocity fluctuations. In strongly elastic flows, we show that velocity fluctuations can exhibit significant amplification even in the absence of inertia. Our analysis demonstrates that streamwise-constant disturbances are the most amplified. Explicit expressions are established for the worst-case amplification of velocity fluctuations arising from different components of the body forces. These show that amplification from the wall-normal and spanwise forces to the streamwise velocity component scales quadratically with the Weissenberg number. The underlying physical mechanism involves stretching of polymer stress fluctuations by a background shear, which is a close analog of the vortex titling mechanism that is responsible for amplification in inertial flows of Newtonian fluids.

Transient growth without inertia

We study transient growth in inertialess plane Couette and Poiseuille flows of viscoelastic fluids. For streamwise-constant three-dimensional fluctuations, we demonstrate analytically the existence of initial conditions that lead to quadratic scaling of both the kinetic energy density and the elastic energy with the Weissenberg number, We. This shows that in strongly elastic channel flows of viscoelastic fluids, both velocity and polymer stress fluctuations can exhibit significant transient growth even in the absence of inertia. Our analysis identifies the spatial structure of the initial conditions (i.e., components of the polymer stress tensor at t=0) responsible for this large transient growth. Furthermore, we show that the fluctuations in streamwise velocity and the streamwise component of the polymer stress tensor achieve O(We) and O(We2) growth, respectively, over a time scale O(We) before eventual asymptotic decay. We also demonstrate that the large transient responses originate from the stretching of polymer stress fluctuations by a background shear and draw parallels between streamwise-constant inertial flows of Newtonian fluids and streamwise-constant creeping flows of viscoelastic fluids. One of the main messages of this paper is that at the level of velocity fluctuation dynamics, polymer stretching and the Weissenberg number in elasticity-dominated flows of viscoelastic fluids effectively assume the role of vortex tilting and the Reynolds number in inertia-dominated flows of Newtonian fluids.

Energy amplification in channel flows of viscoelastic fluids

Energy amplification in channel flows of Oldroyd-B fluids is studied from an input–output point of view by analysing the ensemble-average energy density associated with the velocity field of the linearized governing equations. The inputs consist of spatially distributed and temporally varying body forces that are harmonic in the streamwise and spanwise directions and stochastic in the wall-normal direction and in time. Such inputs enable the use of powerful tools from linear systems theory that have recently been applied to analyse Newtonian fluid flows. It is found that the energy density increases with a decrease in viscosity ratio (ratio of solvent viscosity to total viscosity) and an increase in Reynolds number and elasticity number. In most of the cases, streamwise-constant perturbations are most amplified and the location of maximum energy density shifts to higher spanwise wavenumbers with an increase in Reynolds number and elasticity number and a decrease in viscosity ratio. For similar parameter values, the maximum in the energy density occurs at a higher spanwise wavenumber for Poiseuille flow, whereas the maximum energy density achieves larger maxima for Couette flow. At low Reynolds numbers, the energy density decreases monotonically when the elasticity number is sufficiently small, but shows a maximum when the elasticity number becomes sufficiently large, suggesting that elasticity can amplify disturbances even when inertial effects are weak.

Effect of non-normal interactions on the interfacial instability of multilayer viscoelastic channel flows

The effect of non-normal interactions on the interfacial instability of multilayer pressure driven channel flow of viscoelastic fluids in the limit of vanishingly small Reynolds number has been investigated. Specifically, it is shown that for the upper convected Maxwell (UCM), the Oldroyd-B, and the modified Phan-Thien–Tanner (MPTT) constitutive models the short time dynamics are significantly influenced by non-normal effects both in two- and three-layer flows. Moreover, the departure from the normal exponential behavior is most pronounced in the velocity perturbations. Furthermore, in two- and three-layer symmetric flows, the non-normal effects that arise from the interaction of the interfacial mode with bulk mode dissipate very quickly and therefore do not significantly influence the growth of interfacial disturbances. However, in asymmetric three-layer flows the non-normal behavior that results from the interaction of the two interfacial modes can be very significant and can persist over very long time periods. Hence, non-normal behavior can play a very significant role in determining the growth/decay rate of interfacial disturbances. Our analysis shows that the discrepancy observed in the measured and predicted growth rates of interfacial disturbances in earlier investigations [J. Non-Newtonian Fluid Mech. 79 (1998) 315] can be attributed to the presence of non-normal interactions.

The effect of transient viscoelastic properties on interfacial instabilities in superposed pressure driven channel flows

In a recent study, Ganpule and Khomami (submitted to J. Non-Newtonian Fluid Mech.) have shown that in order to accurately describe the experimentally observed interfacial instability phenomenon in superposed channel flow of viscoelastic fluids, a constitutive equation that can accurately depict not only the steady viscometric properties of the experimental test fluids, but also their transient viscoelastic properties must be used in the analysis. In the present study, the effect of differences in transient viscoelastic properties which can arise either due to the differences in the predictive capabilities of various constitutive models or from the presence of multiple modes of relaxation on the interfacial instabilities of the superposed pressure driven channel flows has been investigated. Specifically, a linear stability analysis is performed using nonlinear constitutive equations which predict identical steady viscometric properties but different transient viscoelastic properties. It is shown that different nonlinear constitutive equations give rise to the same mechanism of interfacial instability, but the boundaries of the neutral stability contours and the magnitudes of the growth/decay rates, especially at intermediate and shortwaves, are shifted due to the overshoots in the transient viscoelastic responses predicted by the constitutive equations. In addition, the effect of the presence of multiple modes of relaxation on interfacial stability is studied using single and multiple mode upper convected Maxwell (UCM) fluids and it is shown that pronounced differences in the intermediate and shortwave linear stability predictions arise due to the fact that the increase in the number of modes gives rise to additional fast as well as slow relaxation modes and the presence of these additional relaxation modes gives rise to differences in the transient viscoelastic response of the fluids in the absence of any overshoots. The effect of fluid inertia on the interfacial stability of viscoelastic liquids is examined and it is shown that at longwaves, inertia has a pronounced effect on the stability of the interface, whereas at shortwaves, elastic and viscous effects dominate. Furthermore, the mechanism of viscoelastic interfacial instabilities is studied by a careful examination of disturbance eigenfunctions as well as performing a disturbance energy analysis. The results indicate that the mechanism of viscoelastic interfacial instabilities can be described in terms of interaction of mechanisms of purely viscous and purely elastic instabilities. However, since more than one mechanism for the instability is at work, the disturbance energy analysis can not clearly distinguish between them due to the fact that the eigenfunctions used in the energy analysis contain the information regarding both viscous and elastic effects. Hence, the mechanism of the instability must be determined by a careful examination of disturbance eigenfunctions.

An investigation of interfacial instabilities in the superposed channel flow of viscoelastic fluids

Interfacial instabilities of superposed pressure driven channel flow of viscoelastic fluids are investigated theoretically using linear stability analysis. Nonlinear constitutive equations which accurately depict the steady as well as transient viscoelastic properties of typical polymeric melts and solutions with various degrees of flexibility and accuracy are used to assess the constitutive complexity required to accurately describe the stability characteristics of this class of flows by comparing the results of the stability analysis with the experimental results of Wilson and Khomami (J. Non. Newtonian Fluid Mech. 41 (1992) 255; J. Rheol. 37 (1993) 315) and Khomami and Ranjbaran (Rheol. Acta 36 (1997) 1–22). It is shown that the multimode Giesekus model, which can accurately describe the steady as well transient behavior of the polymeric test fluids used in the experiments, can quantitatively describe the interfacial instability phenomenon in terms of the neutral stability contour as well as the growth/decay rate behavior. A rigorous energy analysis based on a newly developed disturbance-energy equation for viscoelastic flows is performed to investigate the mechanism of the purely viscous and purely elastic interfacial instabilities in pressure and drag driven channel flows. The mechanism of shortwave purely viscous instability is found to be due to the viscosity mismatch and the subsequent perturbation vorticity mismatch at the interface (i.e. interfacial friction), whereas the mechanism of the longwave purely viscous instability is found to be due to the bulk Reynolds stresses. The mechanism of purely elastic instability is found to be due to the coupling between the perturbation velocity and the jump in normal stresses across the interface at longwaves as well as shortwaves. An examination of perturbation velocity field reveals that for purely elastic longwave instability the jump in the normal stresses across the interface leads to a perturbation back flow in the bulk resulting in either accumulation (destabilizing) or depletion (stabilizing) of the fluid below the crest of the perturbed interface. In the case of purely elastic shortwave instability, coupling of the jump in the normal stresses across the interface and the perturbation velocity leads to perturbation vorticities adjacent to the interface which drive the instability.

A theoretical investigation of interfacial instabilities in the three layer superposed channel flow of viscoelastic fluids

The effects of viscosity and elasticity stratification on interfacial instabilities of symmetric and asymmetric three-layer superposed pressure driven channel flows have been investigated theoretically using linear stability analysis. The results indicate that the presence of an additional interface has a significant effect on the stability of the flow. Specifically, it has been shown that resonant instabilities as well as finite wavenumber interfacial instabilities could occur in this class of flows. Moreover, it has been demonstrated that a thin layer of less viscous fluid adjacent to the solid wall can stabilize longwave disturbances in both symmetric and asymmetric superposed Newtonian flows. In addition, in asymmetric superposed Newtonian flows it has been shown that when the jump in the shear rate across both the interface is positive then both the interfacial modes are unstable. In case of purely elastic instabilities, longwave disturbances can be stabilized if the more elastic fluid occupies the majority of the channel, irrespective of its position. Moreover, in both purely elastic and viscous instabilities the dominant mode of the instability has been shown to have wavenumbers of O(1). It has also been demonstrated that stability of viscoelastic flows can not be deduced based on the guidelines developed for purely elastic and viscous instabilities (i.e., the effects are non-additive). Furthermore, nonlinear constitutive equations which accurately depict the steady as well as transient viscoelastic properties of typical polymeric melts and solutions with various degrees of flexibility and accuracy have been used to assess the constitutive complexity required to accurately describe the stability characteristics of this class of flows by comparing the results of the stability analysis with the experimental results of Khomami and Ranjabaran [9]. It is shown that the multimode Giesekus model, which can accurately describe the steady as well transient behavior of the polymeric test fluids used in the experiments, can quantitatively describe the interfacial instability phenomenon in terms of the neutral stability contour as well as the growth/decay rate behavior when the effect of interfacial tension is taken into account. A rigorous energy analysis based on a disturbance–energy equation for viscoelastic flows has also been performed to investigate the mechanism of the purely viscous and purely elastic interfacial instabilities in pressure driven channel flows. The mechanisms of purely viscous and purely elastic instabilities of the three-layer flows are found to be identical to those of the two-layer flows. Namely, the mechanism of shortwave purely viscous instability is found to be due to the viscosity mismatch and the subsequent perturbation vorticity mismatch at the interface (i.e., interfacial friction), whereas the mechanism of the longwave purely viscous instability is found to be due to the bulk Reynolds stresses. The mechanism of purely elastic instability is found to be due to the coupling between the perturbation velocity and the jump in normal stresses across the interface at longwaves as well as shortwaves. Finally, the possibility of non-normal interactions between the two interfacial modes or interfacial and bulk modes has been demonstrated and it has been shown that these interactions should be carefully considered when the disturbance growth/decay rates are experimentally determined.