be described with a weak formulation as follows: given 0 , find,for
each k 0, uk+1 i Vi such that ai(uk+1 i , v)=(f,v)i v V 0 i with
uk+1 i | = k. 108 Domain decomposition methods Then find k+1 i
Vi such that as i(k+1 i , v)=(f,E# 1 v|)1 a1(uk+1 1 , E# 1

increasing vortex spacing due to diminished inter-ring interactions. For a
fixed volume of ejected fluid, the thrust increases with jet ratio A/D, at
the expense of efficiency. Trends are based on the assumption of
negligible vortex formation during the r

formulation of the Dirichlet-Neumann method. Given 0,solve for each
k 0 : uk+1 1 = f in D1, uk+1 1 = 0 on D1 D,
uk+1 1 = k on , and uk+1 2 = f in D2, uk+1 2 = 0
on D2 D, uk+1 2 n = uk+1 1 n on , with k+1 := uk+1 2| +
(1 )k (4.22) , where is a positive acc

vortex rings. J. Fluid Mech. 80, 465- 495. Mohseni, K. 2001 Statistical
equilibrium theory for axisymmetric flows: Kelvins variational
principle and an explanation for the vortex ring pinch-off process. Phys.
Fluids 13, 1924-1931. Mohseni, K. and Gharib,

dL dp 2 Plotting u against y gives figure 1.2. BOUNDARY LAYER. The
velocity grows from zero at the surface to a maximum at height . In
theory, the value of is infinity but in practice it is taken as the height
needed to obtain 99% of the mainstream veloci

formation of a starting vortex ring during bell contraction but,
unexpectedly, we observed the formation of a stopping vortex ring
during bell relaxation and an interaction of these two vortices to form a
lateral vortex superstructure in the wake of the m

measurement inputs. (a) acceleration, (b) velocity, and (c) position. Gray
line, fineness ratio result; black line, mVmA baseline data set. Values are
normalized by maximum in the plot. 131 Using the direct fineness ratio
measurement and hemiellipsoid app

Re = 0.16, it appears that left-right symmetry holds good. But through
examination of the figure indicates that the left-right symmetry is
approximately correct. The left-right symmetry is broken slightly
because of the fact that interactions among eddies

the vortex ring celerity. The difficulty with these definitions in a general
starting flow is that the former is valid strictly for a constant-diameter
vortex generator, and the latter requires knowledge of the motion of the
leading vortex ring, which may

scale fluid transport for their function have repeatedly and often
independently converged on the use of jet flows. During flow initiation
these jets form fluid vortex rings, which facilitate mass transfer by
stationary pumps (e.g., cardiac chambers) and

vorticity located further from the axis of symmetry relative to a static
nozzle case. In addition, the normalized energy supplied by the vortex
generator is increased in this process. We do not observe a delay in the
onset of vortex ring disconnection fro

diameter data as independent trends. It is that 141 decoupling that has
led to spurious conclusions regarding the correlation between vortex
formation and animal pump kinematics. 8.2.2 Laboratory apparatus To
test this result experimentally, we studied je

in the previous section, the vortex ring velocity in the laboratory frame
of reference was measured in these experiments based on the location of
peak vorticity in the cores. By subtracting the counter-flow velocity
from this measured value, the ring velo

symmetric elliptic equations and the numerical difficulties arising in
their study. 24 Navier-Stokes equations Chapter 2 Regularity results In
this chapter we recall some basic fact regarding uniqueness,and
regularity for the solutions of the Navier-Stoke

6.3.4 Leading vortex ring energy The dimensionless vortex ring energy
E* is an especially useful parameter for investigating the dynamics of
these starting flows, as it invokes the dynamical invariants being
delivered from the vortex generator. Measuring

identically equal and opposite, the net contribution to the animal
swimming and flying dynamics can be non-zero. An important example
is the wake capture mechanism observed prominently in insect flight
(e.g., Dickinson et al., 1999) and fish swimming (e.g

Discrepancies shortly after flow initiation at time T = 0 are due to
difficulties in measuring the flow field close to the nozzle exit, and due
to vortex ring rollup and overpressure effects previously documented by
Didden (1979) and Krueger (2001), which

model (e.g., figures 7.2a, 7.4). By contrast, the area approximation was
rough, both qualitatively and quantitatively (figures 7.2b). Interestingly,
the crude area approximation was found to have negligible effect on the
accuracy of dynamical model predic

jellyfish locomotor to the complex trans-mitral and trans-aortic blood
flows in the animal heart. Throughout the diverse catalogue of biological
systems utilizing pulsatile flow, the generated starting flow typically
emerges from a nozzle or orifice with

variation studied in these experiments. The diameter profiles were
measured from frontal view video recordings of the nozzle undergoing
each actuation program. The measured diameter at each instant is an
effective value computed as De = 2(Ae/) 1/2, where

by the support spars, but did not measurably affect the observed vortex
dynamics. Dynamic nozzle programs are repeatable to within 3%. Figure
6.2 Programs of nozzle exit diameter temporal variation. Figure 6.3 Rate
of diameter change for each program of n

Rosenfeld, M. and Mohseni, K. 2002 On the effect of pipe boundary
layer growth on the formation of a laminar vortex ring generated by a
piston/cylinder arrangement. Theor. Comput. Fluid Dyn. 15, 303-316.
Widnall, S. E. and Sullivan, J. P. 1973 On the stab

(strictly regarding symmetric problems,see Remark 4.2.8) were known
till 1986. In the sequel the operators Si, for i = 1, 2, act on the Hilbert
space (X, . X ), and S = S1 + S2. Theorem 4.2.6. Suppose that a) S2 is
continuous: 2 > 0 : !S2, " 2 X X , X; b)

2 X 2 Q2 s 2 2 X . We calculate the Q2-norm of T and we
obtain T 2 Q2 = 2 Q2 2 < (S1 + S2), > +2 Q1 2 (S1 + S2)
2 Q2 . By recalling hypothesis b) we get that !(S1 + S2), " (1 + 2)
2 X (1 + 2) s 2 2 Q2 = C1 2 Q2 . Since Q1 2 is continuous
with continuity

practice. 6.2 Apparatus and Experimental Methods 6.2.1 Apparatus
design Several challenges were faced in designing an apparatus to
achieve temporal variation of a nozzle exit diameter, without introducing
undue complexity or unwanted artifacts into the ex

decreasing classes of temporal nozzle exit diameter variation. Many of
the trends observed here cannot be predicted by quasi-steady analysis
unsteady mechanisms must be considered. Optimal nozzle actuation will
be dictated by the particular application of

Methods more advanced than the current heuristic will be necessary to
determine the additional non-zero elements of the added-mass tensor
that arise for the large class of animal vortex wakes that do not
propagate rectilinearly or unidirectionally. Noneth

measurements.164 9.4
Discussion.
.166 9.4.1 Implications for medusan swimming
behavior.166 9.4.2 Implications for medusan
feeding behavior.170 9.4.3 Implications
for prolate medusae.171 9.4.4
A note on fluid dynamic and geometric
scaling.174 9.5 Chapter
R

ECH 141
Problem Set #2 (Due 1/25/17)
1. A rectangular tank that is 7 m wide is layered with 8 m of oil ( = 880 kg/m3), 6 m of water, and
4 m of mercury ( = 13,600 kg/m3). The tank is surrounded by air at atmospheric pressure and the
top is open. Calculate

(1822),375394. [Nec67] J. Necas, Les methodes directes en theorie
des equations elliptiques,Masson et Cie, Editeurs,Paris,1967. [Ned80]
J.-C. Nedelec, Mixed finite elements in R3,Numer. Math. 35
(1980),no. 3,315341. [Ned86] J.-C. Nedelec, Anew family of m

The counter-flow technique described here is the first empirical
demonstration of effective pinch-off delay. The authors intend to shift
focus to methods of pinch-off delay that do not rely on external devices,
such as might be accomplished by manipulatio

section we briefly explain how is it possible to prove more regularity for
the solutions of the Navier-Stokes equations. We recall that if f = 0, u0
V and the boundary of D R3 is smooth, then there exists a fully
classical solution (u, p) (C(D (0, T)4, o

contrast, the counter-flow implemented in protocol LD4-CF05-6 was
initiated after the formation number and therefore could not affect the
shear layer dynamics or the pinch-off process. 2.4 Comparison with
Maxworthy (1972) The work of Maxworthy (1972) prov

axisymmetric counter-flow to maintain the vortex ring within the
measurement window for longer periods of time, with the goal of
observing ring growth due to fluid entrainment. 9 Instantaneous
streamlines of the flow in the reference frame of the vortex r