A. J. Mestel

Electrohydrodynamic stability of a slightly viscous jet

Many electro-spraying devices raise to a high electric potential a pendant drop of weakly conducting fluid, which may adopt a conical shape from whose apex a thin, charged jet is emitted. Such a jet eventually breaks up into fine droplets, but often displays surprising longevity. This paper examines the stability of an incompressible cylindrical jet carrying surface charge in a tangential electric field, allowing for the finite rate of charge relaxation. The viscosity is assumed to be small so that the shear resulting from the tangential surface stress can be large, even for relatively small fields. This shear can suppress surface tension instabilities, but if too large, it excites electrical ones. For imperfect conductors, surface charge is redistributed by the rapid fluid reaction to variations in tangential stress as well as by conduction. Phase differences between the effects due to the tangential field and the surface charge lead to charge ‘over-relaxation’ instabilities, but the maximum growth rate can still be lower than in the absence of electric effects.

Steady flow in a helically symmetric pipe

Fully developed flow in an infinite helically coiled pipe is studied, motivated by physiological applications. Most of the bends in the mammalian arterial system curve in a genuinely three-dimensional way, so that the arterial centreline has not only curvature but torsion and can be modelled by a helix. Flow in a helically symmetric pipe generalizes related problems in axisymmetry (Dean flow) and two-dimensionality, but the geometry ensures that even irrotational flow has a cross-pipe component. Fully developed helical flows driven by a steady pressure gradient are studied analytically and numerically. Varying the radius and pitch of the helical pipe, the effects of curvature and torsion on the flow are investigated

Unsteady blood flow in a helically symmetric pipe

Fully developed flow in a helical pipe is investigated with a view to modelling blood flow around the commonly non-planar bends in the arterial system. Medical research suggests that the formation of atherosclerotic lesions is strongly correlated with regions of low wall shear and it has been suggested that the observed non-planar geometry may result in a more uniform shear distribution. Helical flows driven by an oscillating pressure gradient are studied analytically and numerically. In the highfrequency limit an expression is derived for the second-order steady flow driven by streaming from the Stokes layers. Finite difference methods are used to calculate flows driven by sinusoidal or physiological pressure gradients in various geometries. Possible advantages of the observed helical rather than planar arterial bends are discussed in terms of wall shear distribution and the inhibition of boundary layer separation

Behaviour of a conducting drop in a highly viscous fluid subject to an electric field

We consider the slow deformation of a relatively inviscid conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field. The general behaviour is to deform and elongate in the direction of the field. Detailed numerical computations, based on a boundary integral formulation, are presented. For fields below a critical value, we obtain the evolution of the drop to an equilibrium shape; above the critical value, we calculate the drop evolution up to breakup. At breakup it appears that smaller droplets are emitted from the ends of the drop with a charge greater than the Rayleigh limit. As the electric field strength is increased the ejected droplet size decreases. A further increase in field strength results in the mode of breakup changing to a thin jet-like structure being ejected from the end. The shape of all drops is very close to spheroidal up to aspect ratios of about 5. Also, for fields just above the critical value there is a period of slow deformation which increases in duration as the critical field strength is approached from above. Slender-body theory is also used to model the drop behaviour. A similarity solution for the slender drop is obtained and a finite-time singularity is observed. In addition, the general solution for the slender-body equations is presented and the solution behaviour is examined. The slender-body results agree only qualitatively with the full numerical computations. Finally, a spheroidal model is briefly presented and compared with the other models.

Flow through a charged biopolymer layer

The polyelectrolyte layer coating mammalian cells, known as the glycocalyx, is important in communicating flow information to the cell. In this paper, the layer is modelled as a semi-infinite, doubly periodic array of parallel charged cylinders. The electric potential and ion distributions surrounding such an array are found using the Poisson-Boltzmann equation and an iterative domain decomposition technique. Similar methods are used to calculate Stokes flows, driven either by a shear at infinity or by an electric field, parallel or transverse to the cylinders. The resulting electric streaming currents due to flow over endothelial cells, and the electrophoretic mobilities of red blood cells are deduced as functions of polymer concentration and electrolyte molarity. It is shown that only the top portion of the layer is important in these effects.

Steady flow in a dividing pipe

The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the bifurcation, but wall shear on the curved wall is found to increase from its undisturbed value downstream

Helical Flow Around Arterial Bends for Varying Body Mass

The three dimensionally curved aortic arch is modeled as a portion of a helical pipe. Pulsatile blood flow therein is calculated assuming helical symmetry and an experimentally measured pressure pulse. Appropriate values for the Womersley and Reynolds numbers are taken from allometric scaling relations for a variety of body masses. The flow structure is discussed with particular reference to the wall shear, which is believed to be important in the inhibition of atheroma. It is found that nonplanar curvature limits the severity of flow separation at the inner bend, and reduces spatial variation of wall shear.

Maximal accelerations for charged drops in an electric field

The equilibrium shapes of highly conducting, charged drops accelerating in an electric field are found. The maximum possible charge for a given field stength and surface tension is calculated. A spheroidal approximation often used for uncharged drops is generalized to include charge and is found to agree very well with the numerical solution when account is taken of the asymmetric surface stress. The charge and field values giving rise to the maximum possible acceleration of the drop are found with a view to optimizing the efficiency of certain electrospraying devices. The effects of air resistance are considered, at both low and high Reynolds numbers. In the former case, internal motion of the drop contributes to the shape, but the results are broadly similar to those for the accelerating drop. At high Reynolds number potential air flow with a constant pressure wake is assumed. It is found that the drop shape can be oblate in this case and may be stable for higher charge and field values.

Kinematic dynamo action in a helical pipe

Steady incompressible laminar flow of an electrically conducting fluid down a helically symmetric pipe is investigated with regard to possible dynamo action. Both the fluid motion and the magnetic field are assumed to be helically symmetric, with the same pitch. Such a velocity field can be represented by its down-pipe component,

Nonlinear dynamos in laminar, helical pipe flow

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