Viscosity Of External Fluid Research Articles

Lipid bilayer hydrodynamic drag

The hydrodynamic drag at a lipid bilayer surface determines in part the flow properties of suspensions of cells and liposomes. Given the fluidity of lipid bilayers, it is not obvious a priori whether solid-like no-slip, liquid-like no-stress, or intermediate boundary conditions apply at the water-bilayer interface. Though no-slip conditions have been widely assumed for many decades, this fundamental aspect of membrane rheology has, to our knowledge, never been directly measured for free bilayers. We applied light sheet fluorescence microscopy to image freely diffusing phospholipid vesicles and determined the hydrodynamic drag coefficient $C \pi \eta R$, where $\eta$ is the external fluid viscosity, $R$ is the vesicle radius, and the dimensionless $C$ characterizes the flow boundary condition. We find that $C = 5.92 \pm 0.13$ (stat.) $\pm 0.16$ (syst.), matching the theoretical value of $C=6$ for a no-slip boundary and far from the $C=4$ value for a zero shear stress boundary

Gelled Crude Oil Particle Deformation under Hydraulic Suspension Transport Investigated by Numerical Simulation

In hydraulic suspension transport, the gelled crude oil particles are known to change shape in response to local flow conditions. The phase field method is used to simulate the deformation for the gelled crude oil particle with the same internal and external fluid viscosities. For this purpose, an Image Pro Plus software records the major and minor axises and thus, analyzes digitally its deformation degree. Initially spheroidal oil droplet is simulated and it is shown that the gelled crude oil particle exhibits deformation behavior as the movement and its final stable shape close to oval. The deformation behavior is response to shear stress with a time of 0.08s delayed. In the range of delayed time, the gelled crude oil particle deforms with a little degree and after the delayed time, the deformation degree increases seriously.

Orbital drift of capsules and red blood cells in shear flow

Many numerical studies have considered the dynamics of capsules and red blood cells in shear flow under the condition that the axis of revolution of such bodies remained aligned in the shear plane. In contrast, several experimental studies have shown that the axis of revolution of red blood cells could drift away from the shear plane in a certain range of controlling parameters. In this article, we present three-dimensional numerical simulations on the orientation dynamics of capsules in simple shear flow with different initial undeformed shapes, namely, prolate, oblate, and biconcave disk. It is observed that unlike rigid ellipsoids in Stokes flow, capsules reorient their axis of revolution either towards the vorticity axis while undergoing a precessing motion or towards the shear plane while undergoing a kayaking-type motion. The specific dynamics are observed to depend on initial shape, capillary number, and the ratio of the internal to external fluid viscosity. Near the physiological values of the viscosity ratio, the biconcave shape performs a rolling motion like a wheel. If the viscosity ratio is reduced below the physiological range, a transition to the kayaking dynamics is observed with increasing capillary number. The critical shear stress at which the rolling-to-kayaking transition occurs is found to be dependent on the viscosity ratio.

On the problem of slipper shapes of red blood cells in the microvasculature

Red blood cells (RBC) are known to exhibit non symmetric (slipper) shapes in the microvasculature. Vesicles have been recently used as a model for RBC and numerical simulations proved the existence of slipper shapes under Poiseuille flow (both in unconfined and confined geometry). However, in our recent numerical simulations the transition from symmetric (parachute) shape to the slipper one was found to take place upon decreasing the flow strength, while experiments on RBCs showed the contrary. In this work we show that if the viscosity contrast (ratio between the internal over external fluid viscosities) is different from unity, as is the case with RBCs, the transition from parachute to slipper shape occurs upon increasing the flow strength, in agreement with experiments. We provide the phase diagram of shapes in the microcirculation. The slipper is found to have a higher speed than the parachute (for the same parameters), that we believe to be the basic reason for its prevailing in the microvasculature. We provide a simple geometrical picture that explains the slipper flow efficiency over the parachute one. Finally, we show that there exists in parameter space regions of co-existence of slipper/parachute shapes and suggest simple experimental protocols to test these findings. The coexistence of shapes seems to be supported by experiments, though a systematic experimental study is lacking. A potential application of this work is to guide designing flow-based experiments in order to link the shape of RBCs to pathologies affecting cell deformability, such as sickle cell diseases, malaria, and those affecting blood hematocrit, as in polycythemia vera disease.

How does confinement affect the dynamics of viscous vesicles and red blood cells?

Despite its significance in microfluidics, the effect of confinement on the transition from the tank-treading (steady motion) to the tumbling (unsteady motion) dynamical state of deformable micro-particles has not been studied in detail. In this paper, we investigate the dynamics of a single viscous vesicle under confining shear as a general model system for red blood cells, capsules, or viscous droplets. The transition from tank-treading to tumbling motion can be triggered by the ratio between internal and external fluid viscosities. Here, we show that the transition can be induced solely by reducing the confinement, keeping the viscosity contrast constant. The observed dynamics results from the variation of the relative importance of viscous-, pressure-, and lubrication-induced torques exerted upon the vesicle. Our findings are of interest for designing future experiments or microfluidic devices: the possibility to trigger the tumbling-to-tank-treading transition either by geometry or viscosity contrast alone opens attractive possibilities for microrheological measurements as well as the detection and diagnosis of diseased red blood cells in confined flow.

Dynamic rheology of a dilute suspension of elastic capsules: effect of capsule tank-treading, swinging and tumbling

Three-dimensional numerical simulations are used to study the effect of unsteady swinging and tumbling motion on the rheology of a dilute suspension of oblate-shaped elastic capsules. Unlike a suspension of initially spherical capsules undergoing the steady tank-treading motion for which the rheology is constant in time, the suspension of non-spherical capsules is time-dependent due to the unsteady capsule motion. In a simple shear flow, the non-spherical capsules undergo a transition from the tank-treading/swinging to the tumbling motion with a reduction in the shear rate or an increase in the ratio of the internal to external fluid viscosities. We find that the time-averaged rheology obtained for the non-spherical capsules undergoing the unsteady motion is qualitatively similar to that obtained for the spherical capsules undergoing the steady tank-treading motion, and that the tank-treading-to-tumbling transition has only a marginal effect. The time-averaged rheology exhibits a shear viscosity minimum when the capsules are in a swinging motion at high shear rates but not at low shear rates. This is a remarkable departure from the behaviour of a vesicle suspension which exhibits a shear viscosity minimum at the point of transition. We find that the shear viscosity in a capsule suspension can decrease as well as increase with increasing viscosity ratio during both tank-treading and tumbling motions, while that of a vesicle suspension always decreases in tank-treading motion and increases in tumbling motion. We then seek to connect the time-dependent rheology with the time-dependent membrane tension, capsule orientation, deformation and tank-treading velocity. At low shear rates, the numerical results exhibit a similar trend to that predicted by analytical theory for rigid ellipsoids undergoing tumbling motion. The trend differs during swinging motion due to the periodic deformation and time-dependent variation of the membrane stress. The elastic component of the shear stress is minimum when the capsules are maximally compressed, and is maximum when the capsules are maximally elongated. In contrast, the viscous component is related to the periodic variation of the tank-treading velocity synchronized with the swinging motion, and the rate of capsule elongation or compression. The swinging or tumbling velocity makes no contribution to the time-dependent rheology.

Mass Transfer from Ensembles of Newtonian Fluid Spheres at Moderate Reynolds and Peclet Numbers

In this work, the rate of mass transfer from an ensemble of mono-size spherical Newtonian droplets (free from surfactants) to a Newtonian continuous phase has been numerically studied at moderate Reynolds and Peclet numbers. A simple spherical cell model (so-called free surface cell model) has been used to account for inter-drop hydrodynamic interactions. Extensive numerical results have been obtained to elucidate the effects of the Reynolds number ( Re o), the ratio of internal to external fluid viscosity ( k), the volume fraction of the dispersed phase ( ɛ) and the Schmidt number ( Sc) on the local and average Sherwood number ( Sh) over the ranges of conditions: 1 ≤ Re o ≤ 200, 0.2 ≤ ɛ ≤ 0.6, 0.1 ≤ k≤ 50 and 1 ≤ Sc ≤ 10 000. It has been observed that the effects of viscosity ratio on the local and average Sherwood number is less significant for small values of the Peclet number ( Pe) for all values of dispersed phase concentration. As the value of the viscosity ratio increases, the average Sherwood number decreases for all values of the droplet concentration and the Reynolds number. Based on the present numerical results, a simple predictive correlation is proposed which can be used to estimate the rate of inter-phase mass transfer in a liquid–liquid system in a new application. However, it is also appropriate to add here that at higher concentrations, fluid spheres interact significantly, deform and coalesce. All these effects are neglected in this study. Therefore, the present results are valid only for dilute to moderate concentration of the dispersed phase.

Sedimentation in Emulsions of Mono-size Droplets at Moderate Reynolds Numbers

The steady incompressible flow of a Newtonian fluid through an ensemble of Newtonian fluid spheres has been numerically investigated using a finite difference method based SMAC algorithm. For both, dispersed and continuous phases, the convective terms have been discretized using the QUICK scheme, and diffussive terms with the second-order central differencing scheme. A simple concentric sphere cell model has been employed to account for inter-particle hydrodynamic interactions. Extensive numerical results on pressure drag ( C dp ), friction drag ( C df ) and total drag ( C D ) coefficients have been presented as functions of the Reynolds number ( Re o ) for the continuous phase, the volume fraction of the dispersed phase (∈), and the ratio of internal to external fluid viscosity ( k ) over the ranges of conditions: 1 ≤ Re o ≤ 500, 0.2 ≤ ∈ ≤ 0.6 and 0.1 ≤ k ≤ 50. Based on the numerical values of the total drag coefficients obtained in this work, a simple correlation has been proposed which can be used to predict the rate of sedimentation in liquid–liquid systems.

Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces

A drop in an axisymmetric extensional ow is studied using boundary integral methods to understand the effects of a monolayer-forming surfactant on a strongly deforming interface. Surfactants occupy area, so there is an upper bound to the surface concentration that can be adsorbed in a monolayer, Γ∞. The surface tension is a highly nonlinear function of the surface concentration Γ because of this upper bound. As a result, the mechanical response of the system varies strongly with Γ for realistic material parameters. In this work, an insoluble surfactant is considered in the limit where the drop and external fluid viscosities are equal.For Γ<Γ∞, surface convection sweeps surfactant toward the drop poles. When surface diffusion is negligible, once the stable drop shapes are attained, the interface can be divided into stagnant caps near the drop poles, where Γ is non-zero, and tangentially mobile regions near the drop equator, where the surface concentration is zero. This result is general for any axisymmetric fluid particle. For Γ near Γ∞, the stresses resisting accumulation are large in order to prevent the local concentration from reaching the upper bound. As a result, the surface is highly stressed tangentially while Γ departs only slightly from a uniform distribution. For this case, Γ is never zero, so the tangential surface velocity is zero for the steady drop shape.This observation that Γ dilutes nearly uniformly for high surface concentrations is used to derive a simplified form for the surface mass balance that applies in the limit of high surface concentration. The balance requires that the tangential flux should balance the local dilatation in order that the surface concentration profile will remain spatially uniform. Throughout the drop evolution, this equation yields results in agreement with the full solution for moderate deformations, and underscores the dominant mechanism at high deformation. The simplified balance reduces to the stagnant interface condition at steady state.Drop deformations vary non-monotonically with concentration; for Γ<Γ∞, the reduction of the surface tension near the poles leads to higher deformations than the clean interface case. For Γ near Γ∞, however, Γ dilutes nearly uniformly, resulting in higher mean surface tensions and smaller deformations. The drop contribution to the volume averaged stress tensor is also calculated and shown to vary non-monotonically with surface concentration.

Large deformation of red blood cell ghosts in a simple shear flow.

Red blood cells are known to change shape in response to local flow conditions. Deformability affects red blood cell physiological function and the hydrodynamic properties of blood. The immersed boundary method is used to simulate three-dimensional membrane-fluid flow interactions for cells with the same internal and external fluid viscosities. The method has been validated for small deformations of an initially spherical capsule in simple shear flow for both neo-Hookean and the Evans-Skalak membrane models. Initially oblate spheroidal capsules are simulated and it is shown that the red blood cell membrane exhibits asymptotic behavior as the ratio of the dilation modulus to the extensional modulus is increased and a good approximation of local area conservation is obtained. Tank treading behavior is observed and its period calculated.

The effect of viscosity on free vibrations of submerged fluid-filled spherical shells

In order to clarify the effect of fluid viscosity on the vibration of submerged elastic shells, the axisymmetric free oscillations of a fluid-filled spherical shell immersed in a sound field are studied. The dynamic response of the shell is determined by the classical normal mode method, while a boundary layer approximation is employed for the fluid medium. In the absence of viscosity, the shell motion is always damped due to the compressibility of the fluid outside the shell. It is shown that, except for the appearance of natural frequencies with a large damping component, the presence of surrounding fluid outside a fluid-filled shell produces only small changes in the real part of the frequency spectra. The analysis of the influence of viscosity reveals that the viscosity has essentially no effect on the frequencies of shells of moderate thickness. However, the viscous damping is predominant for the non-radiating modes of a fluid-filled submerged shell and the damping is due solely to viscosity for all modes if the outer fluid is assumed incompressible. Moreover, for very thin shells, viscous effects are noticeable on both frequencies and total damping, as the frequency is close to the upper branch of the in vacuo frequency. It also is found that in the presence of compressible flow outside the shell, the damping effect of viscosity can be either positive or negative. Thus, the damping is reduced by fluid viscosity for certain cases. This implies that both the damping of the shell motion and the acoustic radiation damping are reduced by fluid viscosity, since the viscous dissipation damping must remain positive. It also is noted that in the case of negative viscous damping, the inviscid damping is always predominant and the viscous contribution is rather insignificant. A detailed examination indicates that the viscosity of the contained fluid will, in general, produce a damping effect, except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. On the other hand, the viscosity of the surrounding fluid will, in general, increase the damping when frequency of vibration is less than the critical frequency, which is somewhat above the upper branch of the in vacuo frequency. The damping is reduced by the viscosity of external fluid for frequencies associated with large inviscid damping, or when the frequency of vibration is greater than the critical frequency.

The slow motion of two touching fluid spheres along their line of centers

The creeping motion along their line of centers of two fluid spheres in contact is analyzed. An exact solution is presented. Corrections to the Hadamard—Rybezynski equation are tabulated for various particle radii ratios and particle fluid to external fluid viscosity ratios. In the limit of infinite particle viscosity, these corrections are shown to agree with previous calculations for rigid spheres.

The transverse force on a drop in an unbounded parabolic flow

The steady flow in and around a deformable liquid sphere moving in an unbounded viscous parabolic flow and subject to an external body force is calculated for small values of the ratio of the Weber number to the Reynolds number in the creeping-flow regime. It is found that, in addition to the drag force, the drop experiences a force orthogonal to the undisturbed flow direction. When the body force is absent (neutrally buoyant drop), this lift force tends to drive the drop inwards to the axis, where the undisturbed flow velocity is maximum, i.e., towards a position of lower velocity gradient. In the case for which the parabolic flow profile is a Poiseuille flow profile, the lift force is given by the expression. \[ {\bf F}_1 =-6\pi\mu\epsilon U_0\frac{\alpha +\frac{2}{3}}{\alpha + 1}\bigg(\frac{a}{R_0}\bigg)^4{\bf b}F[1+o(\epsilon)]. \] Here a is the radius of the undeformed sphere, R0 is the radial distance from the position of maximum undisturbed flow U0 at the profile axis to the position of zero flow, ε is the ratio of the Weber number to the Reynolds number, given by ε=μU0T−1, where μ is the external fluid viscosity and T is the surface tension of the drop, α is the ratio of the drop and external fluid viscosities, b is the radial vector from the flow axis to the centre of mass of the drop, and F is a function of α and a dimensionless parameter dependent on the body force that is determined in the analysis. Reasonable agreement is found between the observations by Goldsmith & Mason (1962) of the axial drift of liquid drops in Poiseuille flow and the predictions of the theory herein.