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Chapter 2

MATHEMATICAL DESCRIPTION OF CONSERVATION

EQUATIONS INVOLVING AN INTERFACE

Donnans reference to an interface as a two-dimensional molecularworld, the dynamics of which is analogous to that of the ordinarythree-dimensional world of homogeneous phases in bulk provides atext for this paper, for here we examine the dynamics of substances

that may be called Newtonian fluids in the interfacial state. We as-sume that the two-dimensional molecular world can be representedas a geometric surface and the material therein as an isotropic fluidcontinuum.

L. E. Scriven, Dynamics of a fluid interface (1960)

The equations of conservation of mass and momentum are well known

for bulk phases (Bird et al., 1960). However, the governing equations for flows

involving free surfaces are more difficult to formulate (Scriven, 1960; Aris, 1989).

In this chapter the field equations that are used in later chapters are developed

in detail so that the assumptions involved are made evident. We start with

the tensorial framework developed in the pioneering work by Scriven (1960) and

reported again by Edwards et al. (1991). The purpose of the derivation presented

here is only to help the reader understand the governing equations. Additional

details can be found in the literature references.

Our discussion begins with the description of the classical discontinuous

interface as exact expressions can be obtained for this case. Then, we generalize

to an interface that has a small non-zero thickness, h, within which the values of

the properties change smoothly, from those corresponding to one phase, to those

corresponding to the other. The former description is obtained from the latter in

17


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the limit as h 0.

2.1 Transport at a Discontinuous Interface

This section develops equations that describe the dynamics of Newtonianfluids that contain a discontinuous interface. The familiar three-dimensional bulk

properties, such as volumetric density with units of mass per unit volume, have

counterparts in the two-dimensional region of the interface that we denote with

a superscript s. For example, s is the mass per unit interfacial area. So just

as the more familiar bulk equations of mass and momentum can be presented,

analogous surface equations exist, which are crucial in formulating interfacial

boundary conditions. In this section we assume that the surface is also Newtonian

and possesses certain properties that are direct analogs to the bulk Newtonian

constitutive equations. Thus, for example, the analog of bulk pressure in force per

unit area is surface tension with units of force per unit length.

Consider the material pillbox in Figure 2.1 with volume V straddling a

discontinuous moving interface with area A. The domain is decomposed as:

V = V A (2.1)

where the overbar denotes a bulk-phase quantity and the locus of the entire volume

of the pillbox V is the union of the locus of points of the bulk volume V and the

locus of points of the interface A. The bulk volume V is the union of the bulk

volumes V1 and V2 on either side of the interface:

V = V1 V2 (2.2)

The total closed surface V bounding the volume V can be decomposed as the

union of the surface area bounding the bulk volume V and the closed curve

bounding the interface A:

V = V A (2.3)


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with the area enclosing the bulk volume V the union of the areas A1 and A2

enclosing the bulk volumes on either side of the interface:

V = A1 A2 (2.4)

Here the interfacial normal, n, is defined as pointing from phase 2 to phase 1.

2.1.1 Linear Momentum Balance

We can write a balance for linear momentum for the pillbox:

d

dt

V

v dV +

A

svs dA

=

V

P dS +

A

Ps dL+

V

g dV +

A

sg dA

(2.5)

The first term on the left hand side of this equation represents the rate of change

in the total amount of linear momentum v per unit volume of V. The second

term on the left represents the rate of change in total amount of linear momentum

svs per unit area in the interfacial region A. The first term on the right is the

diffusive flux of linear momentum into V through V, where P is the (tensile)1

stress tensor in units of force per unit area of V, and dS is the outward directed

normal to V. The second term on the right hand side of equation (2.5) is the

diffusive flux of linear momentum into A through A, where Ps is the (tensile)

surface stress tensor in units of force per unit length of A, and dL is the outward

directed normal to A. The third term is the rate of supply of momentum to V by

the action of long range forces (in this case gravity), where g is the gravitational

force per unit volume of V. The fourth term is the rate of supply of momentum

to A by the action of long range forces, where sg is the gravitational force

per unit area of A. At this point we need to use four theorems which are

presented here without proof (Edwards et al., 1991). Before doing so, we define

the following operators and tensors. Let the position vector x =

x1, x2, x3

be the

1 Some texts (Bird et al., 1960) define P as compressive with the appropriate sign change.


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Fluid 1

F = 0

A

Interface

n A1

A2

Fluid 2

F = 1

n

V2

V1

dS1

dL

dS1

dS2

dS2

A

n

n

Figure 2.1: A material volume V which intersects the discontinuous interfacebetween fluids 1 and 2.

3-D Cartesian coordinates of a point in space, let

q1, q2, q3

be the coordinates

in another general curvilinear coordinate system and let the functional relation

between the two coordinate systems be x = x

q1, q2, q3

. Let

q1, q2

be the 2-D

surface coordinate system and let the equation of the surface be xs = xs

q1, q2


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(Edwards et al., 1991). We can construct basis vectors for these 3-D and 2-D

spaces:

gi x

qi, (i = 1, 2, 3) ; a

xsq

, ( = 1, 2) (2.6)

The spatial and surface reciprocal basis vectors can be defined such that:

gigj ji , (i, j = 1, 2, 3) ; aa

, (, = 1, 2) (2.7)

Where ji is the Kronecker delta:

ji

1, if i = j0, if i = j

(2.8)

The spatial and surface gradient operators are then defined as:

3

i=1

gi

qi; s

2=1

a

q(2.9)

We can further define the dyadic spatial and surface unit tensors:

I 3

i=1

gigi; Is 2

=1

aa = I nn (2.10)

The surface unit normal is constructed using the surface basis vectors:

n a1a2

|a1a2|(2.11)

The gradient along a direction normal to the interface is defined as (Brackbill et

al., 1992):

N n (n) (2.12)

and its gradient tangent to the interface is the surface gradient operator:

s = N (2.13)


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Theorem 2.1 The surface divergence theorem for a surface A surrounded by a

closed curve A:

A

Ps dL =A

s (IsPs) dA (2.14)

Theorem 2.2 The surface divergence theorem for a fluid volume V possessing a

surface of discontinuity A:

V

P dV =

V

P dS

A

(P1 P2) n dA (2.15)

Theorem 2.3 The volumetric Reynolds transport theorem for a moving volumeV(t):

d

dt

V

v dV

=

V

tv + (vv)

dV (2.16)

Theorem 2.4 The surface Reynolds transport theorem for a convected material

surface A(t):

d

dt

A

svs dA

= A

tsvs +s (v

svss)

dA (2.17)

Using these four theorems the momentum balance becomes:

V

tv + (vv)

dV +

A

tsvs +s (v

svss)

dA

=V

P dV +A

(P1 P2) n dA +A

s (IsPs) dA

+

V

g dV +

A

sg dA

(2.18)


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or:

V

tv + (vv) Pg

dV

+

A

tsvs +s (v

svss) s (IsPs) sg (P1 P2) n

dA = 0

(2.19)

Since V and A are arbitrarily chosen this yields the bulk and surface linear

momentum equations:

tv + (vv) P g = 0 (2.20)

tsvs +s (v

svss) s (IsPs) sg (P1 P2) n = 0 (2.21)

Also, for the entire domain of V = V A including the interfacial region we have:

V

tv + (vv) P g

dV

+V

t

svs

+

(vsvs

s

) s

(IsP

s

)

sg{

n(

x

xs)} dV

+

V

[ (P1 P2) n ] {n (x xs)} dV = 0

(2.22)

where {n (x xs)} is the Dirac delta function for the scalar normal distance

from the interface, n (x xs) defined such that f(x) (x a) dx = f(a).

Collecting terms, equation (2.22) becomes:

V

tv + (vv) Pg [(P1 P2) n] {n (x xs)} dV = 0 (2.23)

Since the volume V is arbitrary we have what we call the volumetric linear


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momentum equation:

tv + (vv) = P + g + (P1 P2) n {n (x xs)} (2.24)

2.1.2 Mass Balances

Equations (2.20) and (2.21) are in fact quite general in that if we substitute

mass density for momentum density ( v, s svs) into the bulk and surface

momentum equations (2.20) and (2.21), and assume that the diffusive flux of mass

term is zero (P, Ps), and the rate of supply of mass term is zero (g, sg 0),

we obtain the bulk and surface continuity equations:

t +

(v

) = 0 (2.25)

s

t+s (v

ss) = 0 (2.26)

Using the continuity equations, (2.25) and (2.26), the bulk, surface and volumetric

momentum equations then become (using the fact that evidently IsPs = Ps,

Edwards et al., 1991):

v

t+ (v) v = P+g (2.27)

s

vs

t+ (vss) v

s

= sP

s + sg+ (P1 P2) n (2.28)

v

t+ (v) v

= P + g + (P1 P2) n {n (x xs)} (2.29)

If we invoke the condition that =constant (incompressible fluids) in both phases

the bulk continuity equation becomes:

v = 0 (2.30)

However, it is noted by Edwards et al. (1991) that since s is not generally constant

in the interfacial region:

svs = 0 (2.31)


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2.1.3 Constitutive Equations

Let us start with the constitutive equation for the bulk stress tensor of a

Newtonian fluid:

P = pI + (2.32)

=

2

3

(I:D) I + 2D (2.33)

D =1

2

(v) + (v)

(2.34)

where p is the pressure, is the viscous stress tensor, is the dilatational viscosity,

is the shear viscosity, D is the rate of deformation tensor, and the double dot

product follows the nesting conventionmn:pq

= (np

) (mq

). If we invoke thecondition that =constant (incompressible fluids) in both bulk phases, equations

(2.32)(2.34) become (since I:D = v = 0):

= 2D (2.35)

By analogy the (Boussinesq-Scriven) constitutive equation for the surface

stress tensor is (Scriven, 1960):

Ps = Is + s (2.36)

s = (s s) (Is:Ds) Is + 2sDs (2.37)

Ds =1

2

(sv

s) Is + Is (sv)

(2.38)

where is the interfacial tension, s is the surface viscous stress tensor, s is the

surface dilatational viscosity, s is the surface shear viscosity, and Ds is the surface

rate of deformation tensor. Assuming that the surface is clean s s 0, then

equations (2.36) to (2.38) become simply:

Ps = Is (2.39)


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2.1.4 Surface Stress Boundary Condition

Now, if it is assumed that there is no material accumulation at the interface

so that s 0, our surface linear momentum equation (2.28) becomes:

(P1 P2) n = sPs (2.40)

If we insert the constitutive equation (2.39) into (2.40) we obtain the surface stress

boundary condition:

(P1 P2) n = s (Is) = (sIs) + Is (s) = 2Hn +s (2.41)

where the mean curvature H is defined by:

2H sn (2.42)

Here we have used the relation [sIs] = 2Hn (Edwards et al., 1991). Note that

from this definition, (2.42), H is a positivescalar when the unit surface normal n

points in the direction of the concave side of the surface.

2.2 Interfacial Relations

2.2.1 CSF Method Formulation

Equations for mass (2.25), momentum (2.30), constitutive equations (2.32)

to (2.34) and equation (2.41) as the boundary condition at the free surface can be

used to solve multiphase flow problems (e.g., the derivation of the Young-Laplace

equation in appendix B). However, an alternative route, more convenient for finite

volume numerical methods, is to use the surface momentum equation (2.29). This

includes the surface forces as accelerations and constitutes the Continuous Surface

Force Method (CSF) of Brackbill et al. (1992).

If we insert the surface stress boundary equation (2.41) into the volumetric


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momentum balance (2.29) we obtain:

v

t+ (v) v

= P + g (2Hn +s) {n (x xs)} (2.43)

At this point, we need a mathematical description of the interface. The derivation

of the following equations given here (Richards et al., 1993) follows a slightly

different path from that in the CSF reference (Brackbill et al., 1992), but reaches

the same final result. The equation of the interface can be expressed by the

(discontinuous) Volume of Fluid (VOF) function:

F(x) 0, fluid 11/2, at the interface

1, fluid 2

(2.44)

We may also define a mollified VOF function, F(x), such that within a transition

region of finite thickness, h, it is a smoothly varying series of nested contours where

0 F 1. A definition for such a function is (Brackbill et al., 1992):

F(x) 1

h3

V

F(xs)(xs x) d3xs (2.45)

limh0

F(x) = F(x) (2.46)

where (xs x) is an interpolation function (such as a B-spline) with the following

properties (in addition to being differentiable and decreasing monotonically with

increasing |x|): V

(x) dV = h3 (2.47)

(x) = 0 for |x| h

2

(2.48)

The CSF interface normal (which points from fluid 1 into fluid 2) is defined

by:

n F

|F|(2.49)


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Thus, the CSF choice of normal is the opposite of the Edwards et al. (1991)

normal: n = n. The surface boundary condition becomes with the CSF normal

definition:

(P1 P2) n = n +s (2.50)

and the surface momentum equation (2.43) now becomes:

v

t+ (v) v

= P + g + (n +s) {n (x xs)} (2.51)

where the mean curvature, (not to be confused with dilatational viscosity), is

now:

sn = 2H (2.52)

Note that is a positive scalar when the unit surface normal n points in the

direction of the concave side of the surface. Equations (2.50)(2.52) indicate that

the pressure is greater on the concave side of the interface, and that if there is a

surface tension gradient, the fluid will flow from regions of lower to higher surface

tension (Landau and Lifshitz, 1959). For example, the effect of a non-zero surface

tension gradient (known as the Marangoni effect) due to a non-zero temperature

gradient has been recently investigated by Sasmal and Hochstein (1993) in the

context of the VOF method.

The expression for curvature can be simplified (Brackbill et al., 1992):

(sn)

= [Is] n

= [{I nn} ] n

= [ nn] n

= (n) + [nn] n

(2.53)


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The last term in equation (2.53) is:

[nn] n = n [(n) n] = n [nn] = n

1

2 (nn) n [n]

(2.54)

Now (nn) = 0 and the last term in equation (2.54) becomes (Aris, 1989):

[n] = [F] = 0 (2.55)

so that finally we can replace sn with n in equation (2.51):

= (n) (2.56)

We can define the volumetric surface force from the right hand side of the volu-

metric momentum equation (2.51) neglecting the surface tension gradient term,

Fsv(x), for an interface of finite thickness as:

limh0

Fsv(x) (x) n(x) {n(xs) (x xs)} (2.57)

Now the VOF equation of the interface can be written (Richards et al., 1993):

F(x,t) = (F2 F1) H {n(xs) (x xs)} (2.58)

where H(x) is the Heaviside step function:

H(x)

1, for x 00, for x < 0

(2.59)

We can take the spatial gradient of equation (2.58), and by using the chain rule

obtain:F(x) = (F2 F1) H{n(xs) (x xs)}

= (F2 F1) n(x) {n(xs) (x xs)}

= limh0

F(x)

(2.60)


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Inserting equation (2.60) into the volumetric surface force definition (2.57):

Fsv(x) = (x)F(x)

F2 F1(2.61)

so that finally (with F2 F1 1):

limh0

Fsv(x) = (x) F(x) (2.62)

If we restrict ourselves to situations where the surface tension gradient s = 0,

the surface momentum equation (2.51) becomes:

vt

+ (v) v = P + g + (x)F(x) (2.63)2.2.2 Interface Kinematic Relation

Suppose we have a point fluid particle moving through 3-D space. At time

t = 0 the position of the particle is specified by and at a later time the particle

is at position x. The spatial position can be represented parametrically by (Aris,

1989):

x = x (, t) (2.64)

The point trajectory equation may be inverted (assuming a non-singular Jacobian,

i.e., that the fluid particle does not break up during the motion or that two particles

do not occupy the same space at the same time) to give the initial position or

material coordinates of the particle which is at any position x at time t:

= (x, t) (2.65)

Any property of the fluid, say (, t), may be observed along the particle path.

The description of the change of this property (, t) may be changed into a

spatial description by equation (2.65):

(x, t) = [ (x, t) , t] (2.66)


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This says that the value of the property at position x and time t is the same as

the value appropriate to the particle at (x, t). The material description may be

derived from the spatial description (2.64):

(, t) = [x (, t) , t] (2.67)

meaning that the value as seen by the particle at time t is the value of the position

it occupies at that time. Let the change in the property observed at a fixed point

x be:

t

t

x

(2.68)

Let the change in the property observed when moving with the particle be:

D

Dt

t

(2.69)

The velocity of the particle is the material derivative of its position ( = xi) and

is defined by:

v (x, t)

x

t

(2.70)

The two derivatives (2.68), (2.69) may be related by differentiating the material

description (2.67) and using the chain rule:D

Dt

t

=

t (, t) =

t [x (, t) , t] =

t

x

+

x

t

x

t

(2.71)

or:D

Dt=

t+ v (2.72)

Let the interface consist of the same materialparticles moving at velocity vs = vs1

= vs2

and let the material function describing their position be the VOF function

( =F(x, t)) as defined above. Then using the VOF definition (2.44) the kinematic

equation for the interface becomes by differentiation:

DF

Dt=

F

t+ vs F = 0 (2.73)


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This equation assumes that the particles move at the same velocity as the interface,

which may not be the case if mass transfer is occurring between the interface and

the bulk phases (Edwards et al., 1991).

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