An analytical model of the glass flow du

Dumitru-Ion ARSENIE and Ichinur OMER, Determination of the pressure ina pipeline with uniformly distribution of the discharge 5 Florin B ˘ ALT ˘ ARET ¸ U, Numerical prediction of air flo

Dumitru-Ion ARSENIE and Ichinur OMER, Determination of the pressure in

a pipeline with uniformly distribution of the discharge 5 Florin B ˘ ALT ˘ ARET ¸ U, Numerical prediction of air flow pattern in a ventilated

Florin B ˘ ALT ˘ ARET ¸ U, Cornel MIH ˘ AIL ˘ A, Numerical simulation of a horizontal

Galina CAMENSCHI, The effects of the temperature - dependent viscousity on

Claude CARASSO and Ruxandra STAVRE, Numerical simulation of a jet of

Adrian CARABINEANU, Numerical and qualitative study of the problem of

Mircea Dimitrie CAZACU, On partial differential equations of the viscous liquid relative flow through the turbomachine blade channel 45 Mircea Dimitrie CAZACU and Loredana NISTOR , Numerical solving of the bidimensional unsteady flow of a viscous liquid, generated by displacement of a flat

Eduard - Marius CR ˘ ACIUN, Behaviour of the −42m piezoelectric crystal

Liviu Florin DINU , An example of interaction between two gasdynamic objects:

a piecewise constant solution and a model of turbulence 67 Alexandru DUMITRACHE , An Analytical Model of the Glass Flow during the

C FALUP-Precurariu, D MINEA, Oana FALUP-PRECURARIU, Laura DRACEA, The dynamic of mechanical forces on lung properties 87 Constantin FETECAU , Nonsteady shearing flow of a fluid of Maxwellian type 93

2 Sumar Sommaire Contents

Florin FRUNZULIC ˘ A and Lucian IORGA, An adaptive method for structured

Anca Marina MARINOV, A Two Dimensional Mathematical Model for lating Water and Chemical Transport in an Unsaturated Soil 149 Alexandru M MOREGA, A Numerical Analisys of Laminar Transport Processes

Sebastian MUNTEAN, Romeo F SUSAN-RESIGA, Ioan ANTON and tor ANCUS ¸A, Domain Decomposition Approach for 3D Flow Computation in

Elena PELICAN, Constantin POPA, Approximate orthogonalization of early independent functions with applications to Galerkin-like discretization tech-

Dumitru POPESCU, Stelian ION and Maria Luiza FLONTA,

Appearance of pores through black lipid membranes due to collective thermic

Mihai POPESCU, Optimality and Non-Optimality Criteria for Singular Control193 Lucica ROSU, Liliana SERBAN, Dan PASCALE, Cornel CIUREA and Carmen MAFTEI, The study of the water stability in canals with rectangular

Valeriu Al SAVA, Translation flows of non-local memory-dependent micropolar

Romeo F SUSAN-RESIGA and Hafiz M ATASSI, Nonreflecting Far-Field Conditions for Unsteady Aerodynamics and Aeroacoustics 211 Victor TIGOIU, On the Uniqueness of the Solution of the Initial and Boundary

Annual National Conference ”Caius Iacob” of Fluid Mechanics and Technical Applications

anniversary sessionInstitute of Applied Mathematics ”Caius Iacob” –ten years

In the Romanian scientific life it is acknowledged as a tradition the factthat the researchers in the Fluid Mechanics field participate at a scientificmeeting every year in October

This scientific event, also known as National Colloquium on Fluid

Me-chanics and its Technical Applications for a long time, initiated since 1959 by

the Society of Mathematical Science has become, thanks to all fluid ics researchers’ effort, an annual meeting It take place in various economic

mechan-or universitary centers all over the country

In addition to the usual significance of this event, the meeting that willghater us this year has a special meaning: it celebrate ten years since theInstitute of Applied Mathematics ”Caius Iacob” of Romanian Academy wasfounded The process of setting up such an institute has begun after thebreaking up of the scientific institutes of Romanian Academy and has beencarried on by the Colloquies

As for this year’s Conference, we do consider it necessary that this

tra-dition should be preserved and that an old goal should be achieved: ”that a

large amount of productive units, as well as the youth in Universities should

be involved in such scientific events: [ ] that the Fluid Mechanics course should be introduced in every University” (Caius Iacob and C Ciobanu,

preface of the Proceedings of National Colloquium on Fluid Mechanics, Iasi,1978)

3

4

Analele Universitˇat¸ii Bucure¸sti, Matematicˇa

Anul L(2001), pp 5–10

Determination of the pressure in a pipeline with

uniformly distribution of the discharge

Dumitru Ion ARSENIE and Ichinur OMER

November 2, 2001

Abstract - In the literature, the most studied is the samples pipelines in the steady and unsteady conditions We propose to study the waterhammer phenomenon in the pipeline with uniformly distribution of the discharge By using the impulse variation theorem, we obtain the variation of the medium velocity, the variation of the velocity of elastic wave propagation and the variation of the pressure along the pipeline We have determined these when in the final section occurs a quick totally close manoevre of the valve We applie the results into a numerical example, using an original program.

Key words and phrases : pipeline with uniformly distribution of the discharge, pressure, discharge, velocity of the elastic wave propagation, waterhammer.

Mathematics Subject Classification (2000) : 76A05

We shall obtain the variation of the medium velocity, the variation of thevelocity of elastic wave propagation and the variation of the pressure alongthe pipeline In the end of the paper we present a numerical example

Useful notations : p - the pressure, β - ununiform coefficient of the rate (Boussinesq), ρ - water density, v - flow velocity, A - area of the transversal section, Q - discharge, t - the time, V - the volume, ∆p-the medium pressure variation on the segment, v x,t - the velocity in the x section and at t time,

c j=∆x

∆t - the medium velocity of elastic wave propagation on the segment

5

6 D.-I Arsenie , I Omer

j, j = 1, n − 1,the segment j being delimited by the sections i and i+1,

A (L − x) - the velocity in the x

section at initial time before the stop manoeuvre of the valve For the section

x of the segment j and after the moment when the waterhammer wave has

arrived in this section, the velocity is:

In the numerical example, the pipeline with the uniformly distribution of the

discharge was divided into equals lengths segments ∆x In the x - coordinate

sections which delimit these segments, we applied the precedents formulas,

considering: x = i∆x; i = 1, n.

The first section is the final section (going out of the pipeline with theuniformly distribution of the discharge) and the last section is the initialsection (entrance of the water in the pipeline with the uniformly distribution

of the discharge), the sections order being in the same direction like thewaterhammer propagation direction

One of the used hypothesis refers to the way of the discharge variationwhich leaves the pipe and which has an uniformly distribution in the initial

steady flow We considered that the specifically discharge q0, corresponding

to the pressure p0, is changed when the steady condition, respective when the

pressure in the pipeline become p0+ ∆p, by the orifice-nozzle type relation:

the-In the projection on the water flow direction (the horizontally axis), thescalar relation is

Determination of the preassure 7

Figure 1: The calculus scheme

Because the pipeline’s axis is horizontal, the weight of the liquid (− → G )

and the reaction of the pipeline’s wall (− → R ) don’t appear in this equation.

Replacing the expression of the velocity’s derivative by a finit difference:

8 D.-I Arsenie , I Omer

In the relation (6) result that have appeared two unknown variables

y i+1 and ∆p i+1 Between these two variables exists the Jukovski relation:

∆p i+1 = −ρc i+1 ∆v i+1 = −ρc j y i+1 ∆v i+1 (8)For solve this problem we used an iterative procedure, considering the

initial approach ∆p j ∼ = ∆p i , which permits to calculate y i+1 , respective c i+1

Then with the relation (8) we determine ∆p i+1 and thus we may calculate

again ∆p j in the next approach In the numerical example treated, the

iterative process stopped when the relative error of the value ∆p j lowersunder 1%

The velocity v i+1is:

and the variation of velocity:

For the final segment 1, the previous relations become

Determination of the preassure 9

10 D.-I Arsenie , I Omer

we determined the variation of the medium velocity, the variation of thevelocity of elastic wave propagation and the variation of the pressure alongthe pipeline (for four segments)

4 Conclusions

We observe in the Figure (2) that the medium velocity, the velocity of elasticwave propagation and the pressure diminish along the pipeline, strating fromthe final section of the pipeline where is the valve It is possible that insome situations the pressure’s variations becomes so small, that they may

be neglected The quality aspect (the attenuation of the waterhammer alongthe pipeline) doesn’t meet in the simple pipeline

References

[1] Trofin, P., Water Supply E.D.P., Bucure¸sti, 1983.

[2] Jeager, Ch., Fluid Transients in Hydro-Electric Engineering Practice

Blackie, London, 1977.

[3] Cioc, D Hydraulic E.D.P., Bucure¸sti, 1975.

Dumitru Ion Arsenie

Ovidius University Constantza, B-dul Mamaia nr.124,

Analele Universitˇat¸ii Bucure¸sti, Matematicˇa

Anul L(2001), pp 11–16

Numerical prediction of air flow pattern in a

ventilated roomFlorin B ˘ALT ˘ARET¸ UNovember 2, 2001

Abstract - This paper is concerned with the numerical simulation of turbulent air flow in a ventilated room with ceiling slot air supply and return The room is operated isothermally with the walls and supply air at the same temperature In order to avoid the shortcut of the flow, the distance between the supply slot and the vertical wall is chosen as to produce the attachment of the inlet jet on the vertical wall by the Coand˘a

effect Turbulence k −ε model with finite volume method and the SIMPLE computational

algorithm are used to study the air flow in the room Numerical results and comparison with experimental data of time averaged velocity and turbulence characteristics in the room are presented.

Key words and phrases : finite volume method, k − ε model, indoor air flow

Mathematics Subject Classification (2000) : 76A05

1 Introduction

Indoor air quality and thermal comfort are a major concern of modern times,because most of the time are spent in interior activities It is therefore nec-essary to study the air flow patterns in the room, which has a great influence

on the spreading of contaminants and on the temperature distribution withinthe room This paper describes a numerical investigation, using turbulence

k − ε model with finite volume method and the SIMPLE computational

al-gorithm, in order to validate the capability of a personal code to predict theair flow pattern inside a ventilated room

2 Experimental measurements

Xu et al [7] conducted experimental measurements at the University of

Minnesota Interior measurements were made using a computer controlledthree-axis positioning system, with a position accuracy within 0.4 mm in

11

12 F Bˇaltˇaret¸u

all three directions The chamber used has inside dimensions of 1.95 mwide, 1.45 m high and 1.95 m in depth, in order to simulate a single roomwith a scalin factor of 2.0 The air was supplied downward through a slot1.97 m long and 5 cm wide, designed to provide fully developed turbulentchannel flow at the exit The return was of the same design, and both supplyand return slots were located 0.36 m from the nearest wall The inlet airvelocity was 200 ft/min (1.02 m/s) All the other experimental conditionsare specified in [7]

Air flow pattern 13

Because of the known limitations of the standard k − ε model, which uses a

single time scale, poor predictions are obtained in a certain number of casesand especially in the case of free jets and in the case of reattachment flows.For this reason, we considered the Chen & Kim (1987) [2] turbulence modelmodification, which improves the results for non-equilibrium turbulence Byconsidering an additional time scale, a new term is added into (6):

Ã

U j ∗ φ − Γ ∗ ∂φ ∗

∂x ∗ j

14 F Bˇaltˇaret¸u

and the constants take the values κ = 0.41, B = 5.0.

For inlet section, the values for k and ε are taken:

−a E φ E + a P φ P − a W φ W = [a N φ N + a S φ S + b] (21)For all the variables, under-relaxation is used:

φ P = αφ n + (1 − α) φ o (22)with the under-relaxation factors listed below:

α 0.3 0.7 0.3 0.7 0.2 0.3 0.5 0.7 0.5 0.7

In order to avoid physically incorrect negative values of k and ε, we used

a limitation procedure, which consist in imposing positive minimum values

Air flow pattern 15

6 Numerical results

The numerical results are presented in figure 1 and figure 2

(a) Data of Xu et al [7] (b) Present simulation

Figure 1: Comparision of velocity distribution

Figure 2: Turbulence kinetic energy distribution

7 Conclusions

The general pattern of air flow obtained using a personal code is very

sim-ilar with the pattern presented by Xu et al [7] One can observe that the

inlet jet attaches the right side vertical wall at a distance of approximatively0.83 m from the floor, in very good agreement with experimental data Theturbulence kinetic energy distribution is also very similar with the exper-imental one, excepting the outlet region This fact occurs because of the

lack of k − ε model to produce an accurate evaluation of the kinetic energy

production term for the regions with large velocity gradients

16 F Bˇaltˇaret¸u

References

[1] B˘alt˘aret¸u, F., Mathematical and numerical modelling of transport

phe-nomena in buoyant jets Ph.D Thesis, U.T.C.B., 2001.

[2] Chen, Y.S., Kim, S W., Computation of turbulent flows using an

ex-tended k − ε model, NASA CR-179204, 1987.

[3] Jones, W.P., Launder, B E., The prediction of laminarization with a

two-equation model of turbulence, Int J Heat Mass Transfer, 15(1972),

301-314

[4] Patankar, S V., Numerical Heat Transfer and Fluid Flow Hemisphere,

New York, 1980

[5] Spalding, D B., A novel finite difference formulation for differential

ex-pressions involving both first and second derivatives, Int J Num Meth.

Engng., 4(1972), 551-559.

[6] Wilcox, D C., Turbulence modeling for CFD DCW Industries, La

Ca˜nada, CA, 1994

[7] Xu, J., Liang, H., Kuehn, T H., Comparison of numerical predictions

and experimental measurements of ventilation in a room, Proceedings

ROOMVENT’94 (1994), 2, 213-227.

Florin B˘alt˘aret¸u

U.T.C.B (Civil Engineering Technical University of Bucharest)

Thermodynamics and Heat Transfer Department

Bd Pache Protopopescu 66, sect.2, RO-73232, Bucharest-39, ROM ˆANIAE-mail: fiorin@rnc.ro

Analele Universitˇat¸ii Bucure¸sti, Matematicˇa

Anul L(2001), pp 17–22

Numerical simulation of a horizontal buoyant jet

deflected by the Coand˘ a effectFlorin B ˘ALT ˘ARET¸ U and Cornel MIH ˘AIL ˘A

November 2, 2001

Abstract - In the present paper, the reattachment of a horizontal buoyant jet on a plane wall - by the Coand˘a effect - is investigated numerically using a finite volume method for the solution of time averaged continuity, Navier-Stokes, and energy equations The critical condition of the Coand˘a effect breaking the buoyancy effect is reported The numerical method is based on the SIMPLE approach for velocity and pressure coupling, and the

k − ε turbulence model including buoyancy correction is used The results are expected to

be of importance for indoor air distribution and heating and air-conditioning of industrial spaces.

Key words and phrases : finite volume method, k −ε model, buoyancy, Coand˘a effect

Mathematics Subject Classification (2000) : 76A05

1 Introduction

An important problem relating the heating or cooling of a large interiorspace using air jets is how to control the buoyancy effect Considering theheating case, it is essential to limit the action of the warm air in the occupiedzone, otherwise the air rises, and the local heating effect is lost A veryeffective method for ”capturing” the jet is the Coand˘a effect (attachment ofthe jet in the near presence of a solid surface), and such a flow is often called

”an offset jet” [5], [6] However, critical conditions occur in the case of anon-isothermal jet, because the buoyancy effect and the Coand˘a effect act inopposite directions This paper deals with numerical simulation of describedphenomena, having as an objective to find out the critical situation

2 Experimental measurements

A series of experiments was conducted by Yamada et al [9], by using a

model chamber, whose upper side and opposite end were opened The jet

17

18 F B ˘Alt˘aret¸u, C Mih˘ail˘a

considered is plane and horizontal, with initial velocity of 3,4 and 5 m/s

The distance between the inlet slot and the floor was considered 0.6 m to0.8 m

3 Key factors on flow development

The key factors on flow development are:

- the distance between the floor and the inlet slot, H s;

- the initial temperature difference, ∆T0, between the warm jet and the biant air;

am the initial jet velocity, U0;

- the width of the inlet slot, h;

- the ambiant temperature, T a;

- the pressure values in the regions above and below the jet

Most of the factors mentioned above are present in the Archimedes number:

Ar = gh

U2 0

which is often used to characterize the evolution of a free non-isothermal

jet As shown by Yamada et al [9], an more complete number for a

non-isothermal offset jet is:

K = Ar

µ

H s h

We chosed as buoyancy measure parameter the term B = G B /ε, and the

Chikamoto et al correction function [4]:.

Some numerical results are presented in figures 1 and 2 Numerical results

show that the attachment occur for K ≤ 0.2 In order to have a stable attachment, the K ≤ 0.15 condition is proposed.

20 F B ˘Alt˘aret¸u, C Mih˘ail˘a

6 Conclusions

The reattachment of a horizontal buoyant jet on a plane wall - by the Coand˘aeffect - is investigated numerically The critical condition of the Coand˘aeffect breaking the buoyancy effect is reported, as a measure of the non-

dimensional factor K.

(a) Velocity distribution (b) Temperature distribution

(c) Velocity distribution (d) Temperature distribution

Figure 1: U = 3.5 m/s: ∆T = 17.5 ◦ C and ∆T = 15 ◦ C

Numerical simulation 21

(a) Velocity distribution (b) Temperature distribution

(c) Velocity distribution (d) Temperature distribution

(e) Velocity distribution (f) Temperature distribution

Figure 2: U = 5 m/s: ∆T = 35 ◦ C, ∆T = 33.5 ◦ C, ∆T = 30 ◦ C

22 F B ˘Alt˘aret¸u, C Mih˘ail˘a

References

[1] B˘alt˘aret¸u, F., Mathematical and numerical modelling of transport

phe-nomena in buoyant jets Ph.D Thesis, U.T.C.B., 2001.

[2] B˘alt˘aret¸u, F., Numerical prediction of air flow pattern in a ventilated

room, An Univ Bucure¸sti, Mat., 50 (2001), 11–16.

[3] Chen, Y.S., Kim, S W., Computation of turbulent flows using an

ex-tended k − ε model, NASA CR-179204, 1987.

[4] Chikamoto, T., Murakami, S., Kato, S., Numerical simulation of velocity

and temperature fields within atrium based on modified k − ε model incorporating damping effect due to thermal stratification, Proceedings

ISRACVE (1992), 501-510.

[5] Hoch, J., Jiji, L M., Two-dimensional turbulent offset jet-boundary

in-teraction, ASME J Fluid Eng., 103(1981), 1, 154-161.

[6] Holland, J.T., Liburdy, J A., Measurements of the thermal

characteris-tics of heated offset jets, Int J Heat Mass Transfer, 33(1990), 1, 69-78 [7] Patankar, S V., Numerical Heat Transfer and Fluid Flow Hemisphere,

New York, 1980

[8] Wilcox, D C., Turbulence modeling for CFD DCW Industries, La

Ca˜nada, CA, 1994

[9] Yamada, N., Kubota, H., Kurosawa, K., Yoshida, Y., Hanaoka, Y.,

Local space heating by covering with a warm plane jet, Proceedings

ROOMVENT’94 (1994), 2, 299-308.

Florin B˘alt˘aret¸u, Cornel Mih˘ail˘a

U.T.C.B (Civil Engineering Technical University of Bucharest)

Thermodynamics and Heat Transfer Department

Bd Pache Protopopescu 66, sect.2, RO-73232, Bucharest-39, ROM ˆANIAE-mail: fiorin@rnc.ro

Analele Universitˇat¸ii Bucure¸sti, Matematicˇa

Anul L(2001), pp 23–30

The effects of the temperature - dependent viscousity on flow in cooled channel

Galina CAMENSCHINovember 2, 2001

Abstract - The mathematical model of an incompressible linear viscous fluid motion with

the dynamical viscousity coefficient µ depending only on the temperature in a thin cooled

channel with small curved walls is presented assuming a relation between the Reynolds number and the ratio of middle breadth and the length of the channel The governing system of equations leads, following the mentioned assumptions, to differential equations for the pressure and the temperature The velocity field equations depends on the pressure,

the function µ(T ) and the geometry of the channel The boundary conditions refers to

the adherence of the fluid to the channel walls, the prescription of the temperature on them and then pressure difference at the exit and imput in the channel The geometry

of the channel walls and the function µ(T ) being given the pressure and the temperature

can be determined Particulary, the problem can be used as a model for basaltic fissure eruptions.

Key words and phrases : cooled channel, viscous fluid, temperature-dependent cosity

vis-Mathematics Subject Classification (2000) : 76D99

1 Introduction

Let S0 and S1 be two fixed rigid vertical walls with small curvature In the

Ox1x2x3 coordinate system, where Ox1x3 belongs to the tangent plane in

O to S0, the S1 surface equation is x2 = h(x1, x3) (fig.1) In the actualconfiguration

B t = {(x1, x2, x3)/x1∈ (−∞, +∞), x2∈ h(x1, x3), x3∈ (0, l)}

the motion is governed by Cauchy’s equation, the continuity equation, theconstitutive equation and the internal energy equation in spatial description

23

where ~a is the accelaration vector, T and D the stress tensor and the

deformation rate tensor

x

x

x x

3 3

2 1

= l

B t

Figure 1: The geometry of the problem

We neglect the body forces ( ~b = ~0 )and the heat sources (r = 0) as well

as the scalar product T · D for the incompressible fluid motion The internal energy density is considered by e = c T where c is the specific heat which is

a constant The heat flux vector ~q is given by the Fourier law

Temperature - dependent viscousity 25

where

a = κ

The boundary conditions corresponding to the adherence of the fluid to

the walls, to the constant temperatures T1 and T0 at the walls as well as the

imput in B t, are expressed in the form

v1(x1, 0, x3) = v2(x1, 0, x3) = v3(x1, 0, x3) = 0,

v1(x1, h(x1, x3), x3) = v2(x1, h(x1, x3), x3) = v3(x1, h(x1, x3), x3) = 0,

T (x1, 0, x3) = T (x1, h(x1, x3), x3) = T1, T (x1, x2, 0) = T0.

(7)

Let l be the characteristic length for the x i , (i = 1, 3) directions and

d (the mean thicknes of the channel) the characteristic length for the x2

direction Nondimensionalising the equations system (3)-(5) using the values

of the characteristic time, speed and pressure given by

∂x0 1

¶+

µ

µ0∂v02

∂x0 1

¶

+ ²2 ∂

∂x0 3

µ

µ0∂v10

∂x0 3

+ µ0∂v30

∂x0 1

+ ²4 ∂

∂x0 1

µ

µ0∂v

0 2

∂x0 1

¶+

¶

+ ²2 ∂

∂x0 2

µ

2µ0∂v02

∂x0 2

¶+

¶

+ ²2 ∂

∂x0 3

µ

µ0∂v30

∂x0 2

+ µ0∂v10

∂x0 3

¶+

µ

µ0∂v02

∂x0 3

¶

+ ²2 ∂

∂x0 3

µ

2µ0∂v03

∂x0 3

¶

.

(16)

Temperature - dependent viscousity 27The equation for the temperature results from (5), via (9) and (10) as

∂x0

22

2 The velocity field and the presure equation

Using the estimations (14) in (16) and (17) one can write the dimensionalsystem equations for the linear viscous incompressible fluid motion having

the dynamical viscousity coefficient µ(T ) in the cooled channel in the form

.

(18)

The equation (18)1,3, via (18)2 and the boundary conditions (7) lead to the

v1 and v3 velocity components determination in the form

h(xZ1,x3 )

0

x2µ(T ) dx2·

x2

Z0

h(xZ1,x3 )

0

x2µ(T ) dx2·

∂x2 3





η

Z0





η

Z0

Temperature - dependent viscousity 29

The boundary condition on the wall S1 : v2(x1, h(x1, x3), x3) = 0, allows

us the determination of the pressure equation





x2

Z0

ξ µ(T ) dξ



 dx2−

h3

Z0





x2

Z0





x2

Z0





x2

Z0





x2

Z0





x2

Z0

It easy to see that if the h(x1, x3) function for the surface S1 is given and

µ(T ) is also given (as in [1] for example), the pressure can be determined

from the last equation

30 Galina Camenschi

Then introducing the functions h(x1, x3) and µ(T ) in (19) - (21) the

velocity field can be also obtained

From the equation (18)5 the absolute temperature can be determined

with the initial condition T (0, x1, x2, x3) = T ∗ (x1, x2, x3)

For the integration of the pressure equation we assume as usual that

pressure difference ∆p = p(x1, 0) − p(x 1, l) is given and is a constant.

References

[1] J.J Wylie, J R Lister - The effects of temperature dependent viscousity

on flow in a cooled channel with application to basaltic fissure eruptions,

J.Fluid Mech., 305, (1995), 239-261.

[2] K R Hefrich - Thermo-viscous fingering of flow in a thin gap: a model

magma flow in dikes and fissures, J.Fluid Mech., 305, (1995), 219-238.

[3] D Berscovici - A theoretical model of cooling viscous gravity currents

with temperature dependent viscousity, Geophys Res Lett., 21, (1994),

1177-1180

Galina Camenschi

Department of Mechanics, University of Bucharest ,

14 Academiei Str., 70109 Bucharest, Romania

Analele Universitˇat¸ii Bucure¸sti, Matematicˇa

Anul L(2001), pp 31–36

Numerical simulation of a jet of ink

Claude CARASSO and Ruxandra STAVRE

November 2, 2001

Abstract - The hydrodynamics of a jet of ink, exiting from a printer is considered A one-dimensional model is obtained from the Navier-Stokes equations, by using a rational asymptotic expansion A partial differential system for the first order approximation is obtained and some numerical computations are performed.

Key words and phrases : breakup of the jet, similar solutions

Mathematics Subject Classification (2000) : 76B10, 76D05, 76M45

1 Introduction

This paper deals with a one-dimensional model evolution equation, whichdescribes the hydrodynamics of a jet of ink, exiting from a printer Themathematical model is obtained from the Navier-Stokes equations by us-ing a rational asymptotic expansion of the unknown functions This is aproblem with applications in the industrial process of conception of printerswith jet of ink By giving the vibrations of the liquid at the nozzles of thereservoir, we study, both from the theoretical and the numerical points ofview the breakup of the viscous jet This problem was previously studied

in [1] In this paper, the authors used the Cosserat model for obtaining,under some hypotheses, a nonlinear partial differential system Some nu-merical computations were also performed Experimental observations of jetbreakup phenomena have been carried out in [2], [3], [4], [6] The study of aninfinite jet breakup is performed in [5] The model, derived from the Stokesequations, is employed in extensive simulations to compute breakup timesfor different initial conditions; satellite drop formation is also supported bythe model and the dependence of satellite drop volumes on initial conditions

is studied In Section 2, by using a cylindrical coordinate system, we givethe dimensionless equations, the boundary and the initial conditions whichdescribe the unsteady flow of a semi-infinite, incompressible Navier-Stokesjet By assuming that the ratio between the radius of the nozzles to the

31

32 R Stavre

dimensional axial length scale is an asymptotically small parameter whichcan be used in an asymptotic expansion of the unknown functions, we de-rive, in Section 3, the system of partial differential equations for the firstorder approximation Our interest is in the description of the breakup phe-nomenon In Section 4 we show that the system describing the leading orderapproximation has a singularity with the jet radius vanishing and the fluidvelocity becoming infinite after a finite time, at some axial location Thenext section deals with the numerical study of the problem The system forthe first order approximation is solved by using a finite difference method.The numerical experiments show that the breakup of the jet depends on theinitial conditions

2 The mathematical model

We consider the evolution of a semi-infinite jet of incompressible viscousfluid (ink), exiting from the nozzles of a printer We suppose that the flow

is axisymmetric Initially the jet is a semi-infinite cylinder of radius r0

(the radius of a nozzle) and constant velocity By giving the vibrations ofthe fluid at the nozzles, we obtain a free boundary problem, in cylindrical

coordinates, (r, z) If ρ > 0 is the constant density of the fluid, µ > 0 the constant viscosity of the fluid, α > 0 the constant surface tension coefficient,

u, v and p the radial velocity, the axial velocity and the pressure of the fluid,

respectively, R the unknown radius of the jet and v0, v1, f, r0 positive givenconstants,we obtain for the nondimensional variables:

Numerical simulation of a jet of ink 33

v s The unknown flow

region, Ω(t) is defined by:

Ω(t) = {(r, z)/z > 0, 0 < r < R(z, t)}, ∀ t ≥ 0. (5)

3 The asymptotic expansions

Since the lenght scale L does not have a physical meaning, we suppose that

r0 << L, so that k can be considered as an asymptotically small parameter.

We use as asymptotic parameter the coefficient of the highest-order

deriva-tive of (1) We seek the solution (u, v, p, R) of the problem (1)-(4), as an expansion in powers of k2, of the form:

34 R Stavre

4 The breakup of the jet

In the sequel we shall prove that the system (10) has a solution with thefollowing property: the jet radius vanishes and the fluid velocity becomesinfinite after a finite time, at some axial location The next theorem provesthis property

Theorem 1 There exists (z b , t b ) ∈ IR ∗

3τ 2a+b−2δ f2g 00 + 6τ 2a+b−2δ f f 0 g 0 − Aτ a−δ f 0+

B(bτ 2a+b−1 f2g − δτ 2a+b−1 φf2g 0 + τ 2a+2b−δ f2gg 0 ) = 0.

(15)

Choosing a, b and δ so that τ does not appear explicitly in (15) (i e.

a = 1, b = −1/2, δ = 1/2) and introducing the new variable f1 = Af, denoted also by f, the system (15) becomes:

(

f 0 (φ − 2g) − f (2 + g 0 ) = 0, 3(f2g 0)0 − f 0+B

Numerical simulation of a jet of ink 35

with a1, b1 ∈ IR ∗ and p, n ∈ Z Introducing the expansions (17) in (16) it

(p − 2)a1φ p + (p − 3)a2φ p−1 + O(φ p−2 ) − (2p + n)a1b1φ p+n−1 −

((2p + n − 2)a2b1+ (2p + n − 1)a1b2)φ p+n−2 + O(φ p+n−3 ) = 0,

We study in the sequel the following cases:

i) p < p + n − 1 In this case we obtain from (18) a contradiction ii) p = p + n − 1 (18)1 becomes:

a1((p−2)−(2p+1)b1)φ p +((p−3)a2−(2p−1)a2b1−2pa1b2)φ p−1 +O(φ p−2 )=0 (19)

In this case max(2p + 2n − 1, 2p + n, p − 1) =max(2p + 1, p − 1).

ii1) 2p + 1 > p − 1 Then p > −2 and max(2p + 1, p − 1) = 2p + 1 The coefficient of φ 2p+1 in (18)2 is Ba21b21 − Ba21b1, which yields b1 = 1 Introducing this value in (19) we get p = −3, in contradiction with p > −2.

ii2) 2p + 1 < p − 1 Then p < −2 and max(2p + 1, p − 1) = p − 1 From

(18)2 we obtain, since a16= 0, p = 0, again a contradiction with p < −2.

ii3) 2p + 1 = p − 1 In this case p = −2 and the identification of the coefficients (for φ −2 , φ −3 in (4.6)1 and for φ −3 , φ −4 , φ −5 in (18)2 leads us

to a contradiction The last case is

iii) p > p + n − 1 In this case n < 1 and (18)1 yield p = 2; hence max(2p + 2n − 1, 2p + n, p − 1) = max(2n + 3, n + 4, 1) =max(n + 4, 1).

iii1) n + 4 < 1 Then max(n + 4, 1) = 1 and from (18)2 we obtain

−pa1 = 0, which is impossible for p = 2.

iii2) n + 4 ≥ 1 Then max(n + 4, 1) = n + 4 and (18)2 gives −B(n + 1)a2

1b1 = 0 which yields n = −1 Hence the only possible values for p and n are 2 and −1, respectively For the above determined values of a, b, δ, p, n,

the relations (14) become:

char-36 R Stavre

influence of the boundary conditions on the breakup For solving the lem (10)-(12) we use a finite difference method Since for nonlinear problemsthe stability depends on the structure of the finite difference procedure, we

prob-chose an implicite scheme We denote by ∆z and ∆t the mesh size on Oz and Ot, respectively and by f n

j the value f (j∆z, n∆t) for any function f The values for the physical data, used for the computations, are: ρ = 800

kg/m3, µ = 5 · 10 −3 Pa·s, α = 10 −3 N/m, r0 = 3 · 10 −6 m, v0 = 5 m/s,

v1 = 1 m/s For the parameters L and v s , without a physical meaning, we

chose the values 4 · 10 −5 m and 0.05 m/s, respectively; ∆z = 0.4, ∆t = 0.1.

Experimental observations show that sometimes the jet radius does notvanish The numerical computations, performed for different values of the

frequence of the vibrations, f, are in agreement with the experiments For smaller values of f (f < 34 · 103 Hz), we obtain after 150 iterations in time

a stabilization of the oscillations of the jet radius around its initial value If

we increase the number of iterations in time, we obtain a similar behaviour

of the jet radius The breakup phenomenon appears for f > 34000 Hz.

References

[1] Carasso, C.,Largillier, A.,Regal M-C.,Formation des gouttes dans un

flu-ide non newtonien soumis `a un champs ´electrique, 2 eme colloque chilien de math´ematiques appliqu´ees, C Carasso, C Conca, R Coreea,

franco-J P Puel ed., CEPADUES Editions,(1991), 133-144

[2] Chaudhary, K C.,Maxworthy, T., The nonlinear capillary instability of a

liquid jet Part 2-3 Experiments on jet behavior before droplet formation,

J Fluid Mech., 96, (1980), 275-297

[3] Donelly, R J.,Glaberson, W., Experiments on the capillary instability of

a liquid jet, Proc Roy Soc Lond A 290, (1966), 547-556.

[4] Goedde, E F.,Yuen, M C., Experiments on liquid jet instability, J Fluid

Mech., 40, (1970), 495-511

[5] Papageorgiu, D T., On the breakup of viscous liquid threads, Institute for

Computer Applications in Science and Engineering Report 95-1, (1995)

[6] Peregine, D H., Shoker, G.,Symon, A., The bifurcation of liquid bridges,

J Fluid Mech., 212, (1990), 25-39

Ruxandra Stavre

Institute of Mathematics ”Simion Stoilow”, Romanian Academy,

P O Box 1-764 RO-70700 Bucharest, Romania

E-mail: rstavre@stoilow.imar.ro

Analele Universitˇat¸ii Bucure¸sti, Matematicˇa

Anul L(2001), pp 37–44

Numerical and qualitative study of the problem of incompressible jets with curvilinear walls

Adrian CARABINEANUNovember 2, 2001

Abstract - The jet flow problem concerning the discharge of a fluid (from an orifice in

a container) into the atmosphere is studied herein in the framework of the Kirchhoff model The problem is reduced to the study of a system of nonlinear equations Using Leray-Schauder’s fixed point theorem we prove that the system of functional equa- tions has at least one solution Then we present a semi-inverse method which gives us the possibility to calculate numerically the unknown free lines for jets whose walls consist of semi-infinite straight lines and arcs of circle

Helmholtz-Key words and phrases : incompressible flow, free-lines, jet, topological-degree

Mathematics Subject Classification (2000) : 76B10, 35J25

1 Introduction

The free - boundary streamline flow is still an open research field Recentpapers dedicated to this subject are dealing either with numerical [2] or ana-litical [4] methods of investigation The jet flow problem is concerned withthe discharge of a fluid from an orifice (in a fixed vessel or container) into

an atmosphere at constant pressure For purposes of theory the convenient

idealization assumes that the vessel has two semi-infinite walls $1 and $2

(extending to infinity upstream) each of them consisting of a semi-infinitestraight portion and a finite curvilinear portion nearby the orifice (figure

1) In this paper we assume that the walls $1 and $2 are simmetric with

respect to the Ox - axis.

We assume that the wall $1 consists of an arc of circle having the radius

R and the length νπR and a semi-infinite straight line, streching to infinity

upstream (x → −∞) and making with the Ox - axis the angle (1 − µ) π where 0 ≤ ν ≤ µ ≤ 1

2 Let A and B be the edges of the orifice of the jet (i.e.

the endpoints of the walls $1 and $2) and let L = |z A − z B | be the length

of the orifice of the jet

37

38 A Carabineanu

Two free lines λ1 and λ2 detach from the edges of the orifice (named

the detachement points) and extend to infinity downsteam The domain bounded by the walls of the vessel and by the free lines is the flow domain.

We neglect the gravity and we consider that the jet emerges because ofthe difference of the pressures inside and outside the vessel We considerthat the fluid is ideal, incompressible and the fluid flow is plane, steady and

irrotational Denoting by v = (u, v) the velocity of the fluid we have from

the conditions of irrotationality and mass conservation

Figure 1: Flow domain

From the Cauchy - Riemann condition (2) we deduce that the function

f (z) = ϕ (x, y) + iψ (x, y) (named the complex potential) is holomorphic,

and denoting by w (z) = u (x, y) − iv (x, y) the complex velocity, we have

Denoting by l (s) the length of the arc from $1 having the endpoints

z (exp (is)) and z (0) we deduce from (5) and (16) that

(In the sequel we shall use the notations T (s) = T (cos s, sin s) , Θ (s) =

Θ (cos s, sin s) , θ (s) = θ (cos s, sin s).)

The function z (ζ) mapps the unit half - circle onto a curve consisting of

half - lines and arcs of circle According to Schwarz’s principle concerning

the analytic continuation the function z (ζ) can be extended in a vicinity of the half - circle {exp (is) ; s ∈ [0, π]} Taking into account (16) one deduces that the function Ω(ζ) can also be extended in a vicinity of the half - circle

{exp (is) ; s ∈ [0, π]} The conjugate harmonic functions T and Θ satisfy

∂s is the tangential derivative.

Using U Dini’s formula and seeking for Ω(ζ) such that Ω(0) = 0 we get

−iΩ (ζ) = 1

π

Z 2π0

∂T (s)

∂n ln (exp (is) − ζ) ds. (19)