Flow of a Bingham fluid in contact with a Newtonian fluid

Flow of a Bingham Fluid in Contact

with a Newtonian Fluid*

Elena Comparini and Paola Mannucci†

Dipartimento di Matematica ‘‘U.Dini,’’ Viale Morgagni 67ra, 50134 Firenze, Italy Submitted by M. C. Nucci

Received November 20, 1997

We study the problem modeling the flow of a Bingham fluid in contact with a newtonian fluid, playing the role of lubricant. This is a free boundary problem coupled by means of diffraction conditions with a boundary value problem of parabolic type. We examine the steady state solutions, the evolutive case with particular regard for the asymptotic behaviour of the solution, and a regularized model related to the appearance of a new rigid zone. 䊚 1998 Academic Press

Key Words: free boundary problem; Bingham fluid; stationary solutions; asymp-totic behaviour of the solution.

INTRODUCTION

The lubrication of one fluid by another is a particularly important branch of two-fluid dynamics: if one fluid has a large viscosity, it may be lubricated by a less viscous fluid. A review of the most interesting models and techniques in chemical engineering and in medicine can be found in w x w x1 , 10 , and 12 .w x

In view of applications in lubricate transport, it can be interesting to study the flow of a Bingham fluid in contact with a newtonian fluid. In fact the Bingham fluid behaves like a viscous fluid if the shear stress exceeds a yield value, and like a rigid body otherwise. During the motion the possible formation of rigid zones inside the medium may cause a deceleration of the medium.

Then, in order to avoid the consequent possible stopping of the flow, a newtonian fluid with small viscosity playing the role of lubricant can be placed between the Bingham fluid and the boundary.

* Work partially supported by MURST National Project ‘‘Problemi non lineari. . . . ’’

†

E-mail: comparin@udini.math.unifi.it. 359

0022-247Xr98 $25.00

Copyright䊚 1998 by Academic Press All rights of reproduction in any form reserved.

We recall that the one-dimensional flow of Bingham fluids has been w x

studied both in planar and in cylindrical symmetry 2᎐4 : there the prob-lem is formulated as a free boundary probprob-lem, where the surfaces dividing fluid and rigid zones are the free boundaries.

Equations of motion are obtained from mass and momentum conserva-tion, coupled with a constitutive law for the shear stress, expressed as a linear function of the shear rate. The equation of fluid motion yields a parabolic equation in the fluid zone, and the equation of motion of the

Ž w x. solid zone reduces to a free boundary condition see 2 .

w x w x

In particular, in 3 and 4 the behaviour of the solution has been investigated, taking into account the possible formation of rigid zones inside the fluid.

Here we propose a model for the flow of two immiscible fluids between

Ž .

two parallel plates at a distance 2 Lq l : the Bingham fluid flows between two symmetric slabs, both occupied by the newtonian fluid.

Let y be the coordinate along the motion direction and let x be the coordinate in the direction perpendicular to the plates.

< <

We suppose that the Bingham fluid occupies the zone x - L, and the < <

newtonian fluid occupies the two symmetric zones, L- x - L q l. Under the hypothesis of laminar flow, the velocity field at each point of

Ž Ž .. the representative xy plane is given by vs 0,¨ x, t .

Let us denote by < < ␩ ,₁ L- x - L q l, ␩ s

½

Ž

0.1

.

< < ␩ ,2 x - L, < < ␳ ,1 L- x - L q l, ␳ s

½

_{< <}

Ž

0.2

.

␳ ,₂ x - L,

respectively, the viscosity and the density of the two fluids. < <

In the zone x - L, occupied by the Bingham fluid, the only nonzero element ␴ of the stress tensor is given by the constitutive law

¨x

¡

_{␩2 x¨ q␶0< <}, ¨x/ 0,

~

¨x

␴ s

Ž

0.3

.

¢

w

y␶ , ␶ ,0 0

x

¨xs 0,

where␶ G 0 represents the yield stress of the Bingham fluid.₀ < <

In the zone L- x - L q l, occupied by the newtonian fluid, the stress is defined by

The equation of motion for the fluids reduces then to < <

␳¨ts␴ q f ,x x F L q l, t ) 0,

Ž

0.5

.

where f represents the termy⭸ pr⭸ x, depending only on t, with p being the pressure.

Ž .

Note that Eq. 0.5 holds in the whole region occupied by both the fluids. However, the stress, which is continuous at each point of the slab, is not differentiable at the interfaces xs "L between the two fluids.

w x Then we cannot provide a global formulation of the problem as in 9 , but we have to split the problem into two coupled ones.

In the region occupied by the newtonian fluid, the governing equation Ž0.5 together with 0.4 gives. Ž .

< <

␳1 t¨ y␩1 x x¨ s f , in L- x - L q l.

Ž

0.6

.

In the region occupied by the Bingham fluid, we define a free boundary

Ž . Ž .

problem. Inserting 0.3 into 0.5 one obtains in the fluid zone the equation

␳2 t¨ y␩2 x x¨ s f , in



yL - x - sy

Ž .

t

4

j s



q

Ž .

t - x - L

4

Ž

0.7

.

"_{Ž .}

where we denote by xs s t the equations of the rigid boundaries, where the stress assumes the yield value"␶ .0

Free boundary conditions are given by the condition of no deformation Ž "Ž . .

of the free boundary, i.e., ¨_x s t , t s 0, corresponding to imposing

␴ s "␶ , and by the equation of motion for the rigid core.0

Ž The problem is completed assuming boundary conditions at xs " L q .

l , e.g., no-slip conditions, and initial conditions.

w x

As done in previous works 3, 4 , we have to point out the existence of a unilateral constraint, related to the presence of rigid zones inside the Bingham fluid. In fact, in the fluid zone occupied by the Bingham fluid, the shear rate ¨x cannot change its sign, because of the constitutive law Ž0.3 . Then we have to impose that when. ¨x vanishes somewhere in the Bingham zone, then at these points it must be ␴ s "␶ .0

A detailed description of the problem will be given in next section. The following two sections contain the study of how the steady state solutions depend on the parameters characterizing the two fluids.

The study of the evolutive case is the object of Section 4, with particular regard for the asymptotic behaviour of the solution. Finally, the possible onset of further rigid zones is considered in the regularized model pre-sented in Section 5.

1. STATEMENT OF THE PROBLEM Let␶ , ␳, ␩ be fixed reference values.

Define the transformation

x ␩

˜

xs , ts t.

˜

_{Lq l} 2 ␳ L q l

Ž

.

Denoting by ␴ ␩ ␶0 ␴ s ,

˜

_␶ ¨

˜

s _{␶ L q l} ¨, ␶ s

˜

0 _␶ ,

Ž

.

f ␩_i ␳_i l

˜

fs

Ž

Lq l ,

.

␩ s

˜

i , ␳ s

˜

i , ␭ s , ␶ ␩ ␳ L

the nondimensional variables and constants involved in the problem, we Ž . Ž .

can rewrite equations 0.3᎐ 0.5 . Dropping the tildes we obtain

1 1 < < < < ␳¨ts␴ q f ,x in

½

x -

5 ½

j - x F 1 , t ) 0, 1.1

5

Ž

.

1q␭ 1q␭ with ␴ defined by 1

¡

_{␩¨ , F x F 1,< <} 1 x 1q␭ ¨x 1

~

␩ ¨ q␶ , < <x - ,¨ / 0, ␴ s 2 x 0_{< <} x

Ž

1.2

.

¨x 1q␭ 1 < <

w

y␶ , ␶ ,

x

x - ,¨ s 0,

¢

0 0 x 1q␭ Ž . Ž . and ␳, ␩ as in 0.1 and 0.2 . < < Ž .

In the region x - 1r 1 q␭ , the fluid zone of the Bingham fluid

Žwhere_¨x/ 0 is separated from the rigid zone where. Ž _¨xs 0 by the free. "_{Ž .}

boundaries xs s t .

Ž . Ž .

Inserting 1.2 into 1.1 , and writing 1 < < D_1Ts

½

- x - 1, 0 - t - T ,

5

1q␭ 1 y D2Ts y

½

- x - s t , 0 - t - T

Ž .

5

1q␭ 1 q j s t - x -

½

Ž .

, 0- t - T ,

5

1q␭ ¨1, in D1T, ¨s

_½

_{¨ , in D ,} 2 2T

we have ␳i i t¨ y␩i i x x¨ s f , in D , iiT s 1, 2,

Ž

1.3

.

¨1

Ž

"1, t s 0,

.

t) 0,

Ž

1.4

.

1 1 ¨1

ž

" , t

/

s¨2

ž

" , t ,

/

t) 0,

Ž

1.5

.

1q␭ 1q␭ 1 ␴ "

ž

, t

/

s 0, t) 0,

Ž

1.6

.

1q␭ ¨2 x

Ž

s"

Ž .

t , t

.

s 0, t) 0,

Ž

1.7

.

␳ ¨

Ž

s"

Ž .

t , t

.

y f s"

Ž .

t s y␶ , t) 0,

Ž

1.8

.

Ž

2 2 t

.

0 ¨i

Ž

x , 0

.

s_¨i0

Ž .

x , is 1, 2, 0- x - 1, s"

Ž .

0 s s0",

Ž

1.9

.

< "_{< Ž . w x Ž .}

with s₀ - 1r 1 q␭ and f being the jump of f. Equation 1.4 is a

Ž . Ž .

no-slip boundary condition; 1.5 and 1.6 express the continuity of

veloc-Ž .

ity and stress at the surfaces xs "1r 1 q␭ separating the two fluids. Ž .

Equation 1.7 is the condition of no deformation of the free boundaries "_{Ž .}

xs s t bounding the rigid core in the Bingham fluid, due to the

assumption that on the free boundary the strain rate is zero. The further Ž .

condition 1.8 on the free boundary can be derived directly from the Ž w x.

equation of motion of the solid zone see 13 .

For the sake of simplicity, we will suppose that f is a positive constant. The methods used here can be extended, with minor changes, to the

Ž . Ž w x. general case fs f t see 2 .

We recall that we look for nonnegative solutions of our problem such that ¨x/ 0 inside the fluid region of the Bingham zone.

We remark that in the theory of flow involving two fluids is proved the

w x w x

nonuniqueness of the solution as discussed in 10 and 12 . However, restricting ourselves to symmetric solution w.r.t. the axis xs 0 guarantees

Ž . Ž .

that the solution of problem 1.3᎐ 1.9 is unique. This will be the argu-ment of Sections 3 and 4.

2. STEADY STATE SOLUTIONS

For the sake of simplicity, we will study the stationary problem only in the half-plane x) 0, remarking that the same method can be applied for

x- 0. Prescribing the value ¨0) 0 of the velocity of the Bingham fluid on xs 0, we will look for all the solutions analyzing how they depend on the

Ž . parameters f,␶ , and ␴ 1 .0

Ž . Integrating Eq. 0.6 ,

␴ x s yf x y 1 q ␴ ,

Ž .

Ž

.

1 ␴ s ␴ 1 , 0 - x - 1.1

Ž .

Ž

2.1

.

Ž . Ž .

Figure 1 shows that in the plane ␴ , x Eq. 2.1 gives, for any f and ␴ 1 , the value of the stress at each point of the domain.

Ž . Ž . Ž .

From 1.2 and 2.1 , we obtain ¨_⬁, the stationary solution of 1.1 integrating

¡

␴ x

Ž .

1 , - x - 1, ␩₁ 1q␭ ␴ x y ␶

Ž .

0 1 0- x - ,␴ ) ␶ ,₀ ␩2 1q␭ X

~

¨_⬁

Ž .

x s

Ž

2.2

.

␴ x q ␶

Ž .

0 1 , 0- x - ,␴ - y␶ ,0 ␩2 1q␭ 1

w

x

0, 0- x - ,␴ g y␶ , ␶ .

¢

0 0 1q␭

Ž . Ž . Ž .

Imposing ¨_⬁0 s¨₀, ¨_⬁1 s 0 no-slip conditions , we have that the fol-lowing identity must hold:

1 _X

¨0q

_H

¨⬁

Ž .

x dxs 0.

Ž

2.3

.

0

As we can observe in Fig. 1, different kinds of behaviour of the stationary solution are allowed, with the possible presence of one or two boundaries dividing the rigid zone from the fluid one in the Bingham fluid.

Let us denote by

1

x1s min

½

, max 0, sup

½



␴ x ) ␶

Ž .

0

4

5

5

,

1q␭ x

1

x2s min

½

, max 0, inf

½



␴ x - y␶

Ž .

0

4

5

5

,

1q␭ x

Ž .

with 0F x F x F 1r 1 q1 2 ␭ , the abscissae of the boundaries separating rigid and fluid zones in the region occupied by the Bingham fluid.

Ž .

Integrating 2.3 in the zone occupied by the newtonian fluid and

Ž . Ž . recalling 2.1 and 2.2 , x1yf x y 1 q

Ž

.

␴ y ␶1 0 1r 1q␭Ž .yf x y 1 q

Ž

.

␴ q ␶1 0 ¨0q

_H

dxq

_H

dx ␩ ␩ 0 2 x2 2 1 ␭ f ␭ s y

ž

␴ q1

/

.

Ž

2.4

.

␩ 1 q ␭1 2 1q␭ Ž Ž ..

Imposing¨_⬁1r 1 q␭ G 0, we obtain the condition

f ␭

␴ F y1 .

Ž

2.5

.

2 1q␭ Ž .

In order to evaluate the left-hand side of 2.4 we have to distinguish many

Ž . Ž Ž ..

cases according to the value of ␴ 0 and ␴ 1r 1 q ␭ , taking into account the possible presence of either one or two boundaries dividing the rigid zone from the fluid one in the region occupied by the Bingham fluid. For each case, we obtain a relationship between ␴ , f, ␶ , supposing the_{1 0} constants␩ , ␩ , ␭ fixed._{1 2}

We refer to the lines represented in Fig. 1 in order to describe all possible stationary solutions, whose profiles are given in Fig. 3, at the end of the present section.

Ž Ž ..

Case 1. ␴ 1r 1 q ␭ ) ␶ . The whole region is fluid, because the0 applied stress is greater than the threshold value everywhere. Writing

Ž . ␩ s ␩ r␩ , Eq. 2.4 becomes2 1 2 2 1q␩␭ ␴ y ␶

Ž

.

1 0 1q␭ 1q␭ 2 ␭ q 1 y 1 y

Ž

␩

.

2 fq 2␩2 0¨ s 0.

Ž

2.6

.

1q␭

Ž

.

Ž .

Moreover␴ , ␶ , f are subject to constraint 2.5 and, from the hypothesis1 0 Ž Ž .. ␴ 1r 1 q ␭ ) ␶ ,0 ␭ ␴ ) y␣ f q ␶ ,1 0 ␣ s .

Ž

2.7

.

1q␭ < Ž Ž ..<

Case 2. ␴ 1r 1 q ␭ - ␶ . Close to the newtonian fluid there is a₀

layer in which the Bingham fluid is rigid. We have to distinguish two possibilities:

Ž .

Case 2.1. ␴ 0 ) ␶ . The zone close to x s 0 is fluid and is divided₀

Ž .

from the rigid zone by a boundary located at x₁s ␴ y ␶ rf._{1 0} The constants ␴ , f, ␶ satisfy_{1 0}

␴2_{q 1 q␩␣}2 _f2_{q 2 1 q␩␣ ␴ f y 2␴ ␶ y 2␶ f q 2␩¨ fq␶}2_{s 0.}

Ž

.

Ž

.

1 1 1 0 0 2 0 0 2.8

Ž

.

Ž . Ž Ž ..

Moreover because of the constraints for ␴ 0 and ␴ 1r 1 q ␭ , besides Ž2.5 the following inequalities hold:.

y␣ f y ␶ - ␴ - y␣ f q ␶ ,0 1 0

Ž

2.9

.

␴ ) yf q ␶ .1 0

Ž

2.10

.

< Ž .<

Case 2.2. ␴ 0 - ␶ . The whole region occupied by the Bingham fluid0 is rigid and the conditions are

2␣␴ q ␣2_{fq 2␩¨ s 0, 2.11}

Ž

.

1 1 0

Ž . Ž . together with 2.5 , 2.10 , and

yf y␶ - ␴ - yf q ␶ .0 1 0

Ž

2.12

.

Ž Ž ..

Case 3. ␴ 1r 1 q ␭ - y␶ . The zone of Bingham fluid close to the₀

We have to impose

␴ - y␣ f y ␶ .1 0

Ž

2.13

.

Ž .

Case 3.1. ␴ 0 - y␶ . Again the two regions are fluid. Because of the₀

Ž . Ž .

condition ␴ 0 - y␶ ,0 ¨ is decreasing everywhere and condition 2.4 gives the equation

2 2 2 ␭ 1q␩␭ ␴ q ␶ q 1 y 1 y ␩ fq 2␩ ¨ s 0

Ž

.

1 0

Ž

.

2 2 0 1q␭ 1q␭

Ž

1q␭

.

2.14

Ž

.

Ž . Ž . together with 2.5 , 2.13 , and

␴ - y␶ y f.1 0

Ž

2.15

.

< Ž .<

Case 3.2. ␴ 0 - ␶ . Close to x s 0 there is a rigid layer divided from₀

Ž .

a fluid one by the boundary x₂s ␴ q ␶ rf q 1._{1 0} Ž . Equation 2.4 becomes 2 2 2 ␴ q ␶ q 2␣ 1 y ␩ ␴ f q 2␣␶ f q 1 y ␩ ␣ f y 2␩¨ fs 0

Ž

1 0

.

Ž

.

1 0

Ž

.

2 0 2.16

Ž

.

Ž . Ž . and the ␴ , f, ␶ have to satisfy 2.5 , 2.13 , and1 0

yf y␶ - ␴ - yf q ␶ .0 1 0

Ž

2.17

.

Ž .

Case 3.3. ␴ 0 ) ␶ . In this case there are two boundaries dividing a0 rigid zone from two fluid external ones. The equations of the boundaries are

␴ y ␶1 0 ␴ q ␶1 0

x1s q 1, x2s q 1.

f f

The equation for␴ , f, ␶ is_{1 0}

2 1 y4␴ ␶ q1 0

Ž

1q␩␭ f␴ q 2

.

1

ž

y 2 f␶

/

0 1q␭ 1q␭ 2 ␭ ₂ y 1 y

Ž

␩

.

2 y 1 f q 2␩2 0¨ fs 0,

Ž

2.18

.

1q␭

Ž

.

Ž . Ž . together with conditions 2.5 , 2.13 , and

Ž .

The study of the equations obtained from 2.4 in different cases can be carried on fixing the values of one of the parameters. The representation

Ž .

in the plane␶ , ␴ , e.g. see Fig. 2 , has been obtained setting f s 1._{0 1}

Remark 2.1. The stationary flow of two newtonian fluids can be

investi-Ž w x.

gated with same methods setting ␶ s 0 see 12, Sect. 1.3 .0

We obtain three possible cases corresponding to Case 1, Case 3.1, and Case 3.3, respectively. The velocity profiles are analogous to the ones in

Ž Fig. 3, where the possible constant intervals are reduced to one point the

. rigid parts disappear .

Ž . Ž .

In Case 1 Eq. 2.6 with constraint 2.7 gives 2 2

␩␭ y 1 f y 2␩ ¨

Ž

1q␭ ) 0.

.

Ž

2.20

.

Ž

.

2 0

Ž . 2

We remark that this case cannot hold if ␩ r␩ ␭ - 1, that is, if the2 1 viscosity of fluid 1 is larger than the viscosity of fluid 2.

Ž . Ž .

In Case 3.1 we have from 2.14 and 2.15

2 2 ␩␭ q 2␩␭ q 1 f y 2␩¨

Ž

1q␭ - 0.

.

Ž

2.21

.

Ž

.

2 0 Ž . FIG. 2. ␴ 1 versus ␶ .₀

Case 3.3 holds if ␩␭2_{y 1 f y 2␩ ¨ 1q␭}2 _{) 0,}

Ž

.

Ž

.

2 0 2.22

Ž

.

2 2 ␩␭ q 2␩␭ q 1 f y 2␩¨

Ž

1q␭ - 0.

.

Ž

.

2 0

3. STEADY SYMMETRIC CASE

Ž .

If we consider axisymmetrical solutions, the assumption ␴ 0 s f q ␴1 s 0 gives a further relationship between the two parameters f and ␴ .1 This guarantees for any f and␶ a unique stationary solution.₀

Ž . Ž .

For x) 0, Eq. 2.1 reduces to ␴ x s yfx. Ž Ž ..

Referring again to Fig. 1, in the case␴ 1r 1 q ␭ G y␶ , we have that₀ all the Bingham fluid is rigid. This case corresponds to previous Case 2.2,

Ž Ž .. with ␴ 1r 1 q ␭ - 0.

Ž .

Hence from 2.11 we obtain the expression of ¨₀ as a function of f, 2

f 1

¨0s 1y

ž

/

,

Ž

3.1

.

2␩1 1q␭ Ž Ž .. with the condition obtained imposing ␴ 1r 1 q ␭ G y␶0

fF␶ 1 q ␭ .0

Ž

.

Ž

3.2

.

The expression for ¨_⬁ is then

2

¡

f 1 1 1y

_ž

_/

, 0F x F , 2␩1 1q␭ 1q␭

~

¨_⬁

Ž .

x s

Ž

3.3

.

f 1 2 1y x , - x F 1.

Ž

.

¢

2␩₁ 1q␭ Ž Ž ..

If ␴ 1r 1 q ␭ - y␶ the Bingham fluid presents a rigid zone of width0 s_⬁s␶ rf, in contact with a fluid one close to the newtonian fluid, as in₀

Case 3.2. Ž . Equation 2.16 becomes 2 2 f 1 ␶0 1 ¨0s

ž

y

/

q␩ 1 y

ž

ž

/

/

,

Ž

3.4

.

2␩2 1q␭ f 1q␭ Ž . which holds, recalling 2.13 , if

The explicit expression for ¨_⬁ in this case is the following: 2 2

¡

f 1 ␶₀ 1 ␶₀ y q␩ 1 y

_ž

_/

, 0F x F ,

ž

/

ž

/

2␩₂ 1q␭ f 1q␭ f 2 f 1 ₂ 2␶0 1 y x y y x

ž

/

ž

/

½

2␩₂ 1q␭ f 1q␭

~

¨_⬁

Ž .

x s 2 1 ␶0 1 q␩ 1 y

ž

_ž

_/

/

5

, - x - , 1q␭ f 1q␭ f 1 2 1y x , - x - 1.

Ž

.

¢

2␩1 1q␭ 3.6

Ž

.

4. TIME DEPENDENT SOLUTIONS

Let us consider the evolution problem defined in Section 1, looking for symmetric solutions. For x) 0 we have ␳i i t¨ y␩i i x x¨ s f , in D , iiT s 1, 2,

Ž

4.1

.

1 1 ¨1

ž

, t

/

s¨2

ž

, t ,

/

t) 0,

Ž

4.2

.

1q␭ 1q␭ 1 1 ␩1 1 x¨

ž

, t

/

s␩2 2 x¨

ž

, t

/

y␶ ,0 t) 0,

Ž

4.3

.

1q␭ 1q␭ ¨2 x

Ž

s t , t

Ž .

.

s 0, t) 0,

Ž

4.4

.

␳ ¨

Ž

s t , t

Ž .

.

y f s t s y␶ ,

Ž .

t) 0,

Ž

4.5

.

Ž

2 2 t

.

0 s 0

Ž .

s s ,0 ¨i

Ž

x , 0

.

s¨i0

Ž .

x , is 1, 2, s0F x F 1, 4.6

Ž

.

Ž . with 0- s - 1r 1 q0 ␭ . Ž . Ž .

Equation 4.3 expresses the continuity of the stress at xs 1r 1 q␭ if

Ž . Ž .

s t - 1r 1 q␭ .

Ž . Ž .

The functions¨i0 x in 4.6 satisfy the following assumptions:

¨ g C3_{, ¨ G 0, ¨}X _{F 0, ¨}Y _{F 0, 4.7}

Ž

.

i0 i0 i0 i0

We recall that we are looking for nonnegative classical solutions such that ¨_{2 x}- 0 in D ._2T

Concerning local existence and uniqueness of the solution of problem Ž4.1. Ž᎐ 4.6 we refer to 6 and 11 .. w x w x

In particular we note that classical methods can be applied to the problem satisfied by the function zs¨t,

␳ z y ␩ z s 0,i i t i i x x in D , iiT s 1, 2,

Ž

4.8

.

1 1 z 1, t1

Ž

.

s 0, z1

ž

, t

/

s z2

ž

, t ,

/

t) 0,

Ž

4.9

.

1q␭ 1q␭ 1 1 ␩ z1 1 x

ž

, t

/

s␩ z2 2 x

ž

, t ,

/

t) 0,

Ž

4.10

.

1q␭ 1q␭ 1 ␶₀ z2

Ž

s t , t

Ž .

.

s

ž

fy

/

, t) 0,

Ž

4.11

.

␳2 s t

Ž .

1 ␶₀ z2 x

Ž

s t , t

Ž .

.

s _{␩ s t}

˙

s t ,

Ž .

t) 0,

Ž

4.12

.

Ž .

2 1 _Y z x , 0i

Ž

.

s

Ž

␩i i¨ q f ,

.

0F x F 1.

Ž

4.13

.

␳i

This is a Stefan problem in D_2T coupled by means of diffraction

condi-Ž .

tions on xs 1r 1 q␭ with a standard parabolic problem in D . To_1T Ž . Ž .

prove regularity for the solution of problem 4.8 ᎐ 4.13 , we need hypothe-Ž Ž ..

ses on the second derivative of the initial data see 4.7 .

In order to study the continuation of the solution, and its behaviour, we

w x Ž . Ž .

can follow the methods of 4 , noting that problem 4.1 ᎐ 4.6 can be transformed into a free boundary problem with Cauchy data prescribed in

Ž .

s t in the zone occupied by the Bingham fluid for the function defined as

follows:

x

␳_{2 ␰} f

C2

Ž

x , t

.

s y

_H

d␰

_H

ž

z2

Ž

␨ , t y

.

/

d␨ .

Ž

4.14

.

␩2 s tŽ . s tŽ . ␳2

The function C satisfies a problem like the reaction diffusion problem for₂ Ž w x w x.

the concentration of oxygen in a living tissue see 5 and 7 . We remark that

w x

Recalling the results of 7 we have that the solution could either exist for any T or become extinct or blow up. Note that it could happen also that the model loses its sense because the constraint ¨_{2 x}- 0 is violated at some time.

Ž . Ž

Moreover we have to exclude that s t vanishes at some time i.e., the .

extinction of the rigid zone , because at that time we would have a maximum for ¨, where the stress would be zero, inside the fluid domain.

We can state that only one of the following cases can happen: Ž .a there exists a global solution;

Ž .b ᭚t such that the whole region occupied by the Bingham fluid is

Ž .

rigid extinction of the solution in D_{2 t} ;

Ž .c ᭚t such that the constraint0 ¨2 x- 0 is violated on x s 1r 1 qŽ ␭. at ts t .₀

Ž . Ž .

In order to characterize cases a᎐ c , we have first to exclude the occurrence of blow-up and the disappearance of the rigid zone. This will be done by means of the following estimates and by means of a compari-son with suitable supersolutions.

Ž .

PROPOSITION4.1. Under assumption 4.7 we ha¨e

¨iG 0, in D ,iT

Ž

4.16

.

1 ¨i x- 0, in DiTj x s

½

, 0- t - T ,

5

Ž

4.17

.

1q␭ f f ¨i t- max

½

,

5

, in D .iT

Ž

4.18

.

␳1 ␳2

Moreo¨er, in the case ␳ - ␳ , if we assume also_{1 2}

␳1 f ¨1 x x

Ž

x , 0

.

-

ž

_␳ y 1

/

_␩ - 0,

Ž

4.19

.

2 1 we ha¨e f ¨i t- _␳ , in D .iT

Ž

4.20

.

2

Proof. The estimates can be easily obtained from the maximum princi-ple. Here we have denoted by T the maximum existence time of the solution, both in the case of global solution and in the case of existence up

Ž Ž . Ž ..

Ž . Ž .

Remark 4.1. Inequalities 4.18 and 4.20 provide estimates of ¨_{i x x} depending on the density of the two fluids. Then

if ␳ G ␳ ,1 2 ¨2 x x- 0 in D2T, 4.21

Ž

.

if ␳ - ␳ ,1 2 ¨1 x x- 0 in D1T. Ž . If ␳ - ␳ and 4.19 holds1 2 ¨i x x- 0, in D .iT

Ž

4.22

.

Ž .

COROLLARY 4.1. Under hypotheses 4.7 , blow-up of the solution cannot happen.

w x

Proof. Recalling 8 , the occurrence of blow-up is related to the onset Ž .

of a negative set for the function C defined in 4.14 ; however, in this case2 Ž .

estimate 4.17 implies C2) 0 in D .2T

Now we will construct supersolutions and subsolutions in order to Ž . Ž .

characterize cases a᎐ c . The existence of supersolutions will enable us also to exclude the extinction of the rigid zone.

Ž . Ž . Ž . Ž .

Let ¨i, s, T and ¨, s, T be the solutions of problem 4.1᎐ 4.6 corre-Ž Ž . Ž . . Ž Ž . Ž . .

sponding to the data ¨_i x, 0 , s 0 , f and _¨i x, 0 , s 0 , f , respectively.

PROPOSITION4.2. Suppose that

s 0

Ž .

) s 0 ,

Ž .

_¨i x

Ž

x , 0

.

)_¨i x

Ž

x , 0 ,

.

Ž

4.23

.

and either Ž .i ␳ G ␳ , f F f1 2 or Ž .ii ␳ - ␳ , f s f._{1 2} Then s t

Ž .

) s t ,

Ž .

0- t - min T , T s T ,

Ž

1 2

.

Ž

4.24

.

˜

¨i

Ž

x , t

.

F_¨i

Ž

x , t ,

.

in D ,iT

Ž

4.25

.

˜

¨i x

Ž

x , t

.

G¨i x

Ž

x , t ,

.

in D ,iT

Ž

4.26

.

where 1

˜

D_1Ts

½

- x - 1, 0 - t - T ,

5

1q␭ 1

˜

D2Ts s t - x -

½

Ž .

, 0- t - T .

5

1q␭

w x

Proof. The result follows using the methods of 2 applying the

maxi-˜

mum principle to the functions⌬w s_i ¨_{i x}y¨_{i x} in D , taking into account_iT

Ž .

the conditions satisfied on xs 1r 1 q␭ :

1 1 ␩ ⌬w_{1 1}

_ž

, t

_/

s␩ ⌬w_{2 2}

_ž

, t ,

_/

1q␭ 1q␭ 1 ␳₁ 1 ␳₁ ␩ ⌬w_{1 1 x}

_ž

, t

_/

s ␩ ⌬w_{2 2 x}

_ž

, t

_/

q f y f

_ž

_/

_ž

y 1 .

_/

1q␭ ␳₂ 1q␭ ␳₂ Ž . Ž . The proposition above enables us to compare the solution of 4.1᎐ 4.6 with suitable stationary solutions.

Ž . Ž . Ž .

Let ¨_⬁, s_⬁ and ¨_⬁, s_⬁ be the stationary solutions defined in 3.6 corresponding to f and f, respectively, with f- f.

Ž . Ž . Ž .

Let ¨, s, T the solution of 4.1 ᎐ 4.6 .

Ž .

PROPOSITION4.3. Suppose that ␳ G ␳ . Suppose that 4.7 holds and1 2

X X s_⬁- s 0 - s ,

Ž .

_{⬁ ¨}i⬁

Ž .

x -_¨i x

Ž

x , 0

.

-_¨i⬁

Ž .

x , fF f F f. 4.27

Ž

.

Then s_⬁- s t - s ,

Ž .

_{⬁ ¨}i⬁

Ž .

x -_¨i

Ž

x , t

.

-_¨i⬁

Ž .

x ,

_Ž

_4.28

_.

X X ¨i⬁

Ž .

x -¨i x

Ž

x , t

.

-¨i⬁

Ž .

x .

Proof. Follows immediately from the previous proposition.

Ž . Ž .

PROPOSITION 4.4. Suppose that ␳ - ␳ . Suppose that 4.7 and 4.27_{1 2} hold and ␳_{1 Y} ␳₁ f ␳_{1 Y} ¨1⬁

Ž .

x -_¨1 x x

Ž

x , 0

.

-

ž

y 1

/

q _¨1⬁

Ž .

x ,

Ž

4.29

.

␳2 ␳2 ␩1 ␳2 Y Y ¨2⬁

Ž .

x -_¨2 x x

Ž

x , 0

.

-_¨2⬁

Ž .

x .

Ž

4.30

.

Then X s_⬁- s t - s ,

Ž .

_⬁ ¨i⬁

Ž .

x -¨i

Ž

x , t

.

-¨i⬁

Ž .

x -¨i x

Ž

x , t

.

-¨i⬁

Ž .

x . 4.31

Ž

.

Ž .

Proof. Let us first consider the comparison with ¨_⬁, s ._⬁

Ž . Ž .

Noting that s 0 ) s , then s t ) s at least up to a time t s t ._{⬁ ⬁ 0} Ž .

Suppose that s t₀ s s . Then the function z s_{⬁ i} ¨_{i t} satisfies problem Ž4.8. Ž᎐ 4.13 in D .. i t₀

Ž .Ž .

Set Ms 1r␳₂ fy f ) 0. Then, using maximum principle, from

as-Ž . Ž Ž . .

sumptions on the data we get z x, t_i ) yM for t - t , and z s t , t_{0 2 0 0} Ž .

would be a minimum; hence, noting that s t

_˙

₀ - 0 we get a contradiction Ž .

with 4.12 . Hence s) s and the lower estimates for_⬁ ¨i x x. The reverse inequalities can be proved in a similar manner.

Ž .

Other estimates in 4.31 follow as consequences. Ž .

PROPOSITION4.5. Suppose that assumptions 4.7 hold. Then ␶0 s t

Ž .

) ) 0

Ž

4.32

.

F1 with ␶0 5 5 F1) max

½

,␩i ¨i x x

Ž

x , 0

.

5

.

Ž

4.33

.

s0 Ž .

Proof. The stationary solution 3.6 corresponding to fs F satisfies₁ the hypotheses of Propositions 4.3 and 4.4.

Looking for a possible upper bound for s we need more hypotheses on the data

Ž . Ž .

PROPOSITION4.6. Suppose 4.7 holds, f)␶ 1 q ␭ , and0

max



␩i i x x¨

Ž

x , 0

.

4

' ya F y␶ 1 q ␭ .i 0

Ž

.

Ž

4.34

.

x Then ␶₀ 1 s t

Ž .

- F ,

Ž

4.35

.

F₂ 1q␭ with ␶0 ␶ 1 q ␭ F F F min0

Ž

.

2

½

, a .i

5

Ž

4.36

.

s₀ Ž .

Proof. If ␳ G ␳ , then for any a G ␶ 1 q ␭ we can choose F as in1 2 i 0 2 Ž4.34 and consider the stationary solution corresponding to F . Then the. 2

Ž . hypotheses of Proposition 4.3 are verified and 4.35 holds.

If ␳ - ␳ , in order to verify the hypotheses of Proposition 4.4, we have_{1 2} to suppose

␳1 ␳1

a1G y

ž

y 1 f q

/

␶ 1 q ␭ ,0

Ž

.

a2G␶ 1 q ␭ .0

Ž

.

The previous results guarantee that the following theorem holds.

Ž . Ž .

THEOREM 4.1. Suppose that 4.7 and 4.34 hold. Suppose that fG

Ž .

␶ 1 q ␭ . Then there exists a unique solution for any t.0

In the case of global existence we can give a further estimate: PROPOSITION4.7. Under the hypotheses of Theorem 4.1 we ha¨e

⬁ ␶0 y f dt - q⬁.

Ž

4.37

.

H

ž

_{s t}

/

Ž .

0 Ž .

Proof. Inequality 4.37 can be obtained from the Green’s identity ␳ u¨dxq␩ u

Ž

¨y u¨

.

dts 0,

H

i i i i x i x

⭸DiT

setting uis z andi ¨s x.

Ž . Ž . Ž .

Noting that, from 4.32 and 4.35 , s_⬁' lim_{␶ ª⬁}s t exists and that

Ž .

z x,t tends to 0 uniformly as tª ⬁, we obtain

⬁ f ␩₂ ␶₀ ␶₀ 2 1y s s y f dt y y␩ ¨

Ž

s , 0

.

q f

Ž

⬁

.

H

ž

/

2 2 0 2 ␳2 0 s 1q␭ 2 f 1 q

Ž

␩ y ␩2 1

.

1y

ž

/

. 2␩1 1q␭ Ž .

Recall that, if f-␶ 1 q ␭ , the corresponding stationary solution is0

Ž . Ž . Ž .

given by 3.3 and s_⬁' 1r 1 q␭ . Then 4.37 cannot be satisfied. Therefore we have

Ž . Ž .

PROPOSITION 4.8. Suppose that 4.7 holds and that, if ␳ - ␳ , 4.191 2

Ž .

holds. Suppose that f-␶ 1 q ␭ . Then there exists a time t such that_{0 0}

Ž . Ž .

s t₀ s 1r 1 q␭ .

Ž . Ž . Ž .

Proof. If f-␶ 1 q ␭ then case a cannot hold. Moreover case c is0

Ž .

excluded because_{¨2 x x} stays negative see Remark 4.2 .

Under the hypotheses of the previous proposition, the solution exists again for any t, with the whole region occupied by the Bingham fluid that stays rigid. In fact if we consider the problem for tG t , it is always₀

Ž .

possible to find a stationary solution defined by 3.3 , such that ¨₁)¨_1⬁ in D_{1 t T}.

0

Assuming further hypotheses on the data, we can give some monotonic-ity results.

PROPOSITION4.9. Suppose that the assumptions of Proposition 4.3 or 4.4 are satisfied. we ha_¨e that

Ž .i if¨_{i x x x}Žx, 0.- 0 then

a t

Ž .

- s t - 0,

˙

Ž .

for any t) 0,

Ž

4.38

.

¨i x x x

Ž

x , t

.

- 0, in D ,iT

Ž

4.39

.

Ž .

where a t is a negati¨e function which is bounded for any t;

Ž .ii if¨i x x xŽx, 0.) 0 and¨i x xŽx, 0.- y␶ r␩ x then0 i

␩2 0- s t -

˙

Ž .

, t) 0,

Ž

4.40

.

␳ s2 ␶0 ¨i x x

Ž

x , t

.

- y_{␩ x}, ¨i x x x

Ž

x , t

.

) 0, in D .iT

Ž

4.41

.

i Ž . Ž . w x

Proof. We obtain estimates 4.38 and 4.39 as done in 2 . Concern-Ž .

ing case ii we apply the maximum principle to the function ¨_is z y_i Ž1r␳ f y ␶ rx .i.Ž 0 .

Ž . Let us now investigate case c .

Ž . If ␳ - ␳ , for suitable initial data not satisfying condition 4.19 , then1 2

Ž Ž . .

¨2 x can vanish at 1r 1 q␭ , t . Then at time t s t , our model loses its0 0 validity and we need regularization in order to study the behaviour of the solution after t .0

5. REGULARIZATION PROBLEM

Ž Ž Suppose that ␳ - ␳ and that there exists a time t such that_{1 2 0} ¨_{2 x} 1r 1

. . Ž .

q␭ , t s 0, then at this time the stress at x s 1r 1 q ␭ reaches the0 threshold value.

We formulate a regularized free boundary problem with a new free

Ž . Ž .

boundary xs ⌺ t appearing at 1r 1 q␭ , at time t . Then an internal0 rigid zone appears: it is separated from the fluid zone by this new free

w x boundary and it is in contact with the newtonian fluid 4 .

For the sake of simplicity we shift the time origin to ts t . Let₀ 1 D2Ts

½

Ž

x , t : 0

.

- s t - x - ⌺ t -

Ž .

Ž .

, 0- t - T ,

5

1q␭ 1 D1Ts

½

Ž

x , t

.

- x - 1, 0 - t - T ,

5

1q␭ ␳i i t¨ s␩i i x x¨ q f , in D ,iT

Ž

5.1

.

¨1

Ž

1, t

.

s 0, t) 0,

Ž

5.2

.

1 ¨2

Ž

⌺ t , t s

Ž .

.

¨1

ž

, t ,

/

t) 0,

Ž

5.3

.

1q␭ 1 f 1 1 y ⌺ t

Ž .

¨2 t

Ž

⌺ t , t y

Ž .

.

s ␶ q ␩0 1 1 x¨ , t ,

ž

1q␭

/

␳2 ␳2

ž

1q␭

/

5.4

Ž

.

¨2 x

Ž

⌺ t , t s 0,

Ž .

.

t) 0,

Ž

5.5

.

¨2 x

Ž

s t , t

Ž .

.

s 0, t) 0,

Ž

5.6

.

1 ␶₀ ¨2 t

Ž

s t , t

Ž .

.

s _␳

ž

fy

/

, t) 0,

Ž

5.7

.

s t

Ž .

2 1 ¨i

Ž

x , 0

.

s¨i0

Ž .

x , 0F x F 1, s 0

Ž .

s s ,1 ⌺ 0 s

Ž .

. 1q␭ 5.8

Ž

.

Ž .

Equation 5.4 is the rigid motion equation for the zone between ⌺ and

Ž .

1r 1 q␭ , where we took into account the continuity of the stress on

Ž .

xs 1r 1 q␭ .

Ž . Ž . Existence and uniqueness of the solution of problem 5.1᎐ 5.8 can be proved again by means of classical methods, studying the problem satisfied by zis¨i t.

According to the results of the previous section, the assumptions on the data are 1 0- s -1 , 1q␭ f ¨i

Ž

x , 0

.

) 0, ¨i x

Ž

x , 0

.

- 0, 0-¨i t

Ž

x , 0

.

- , ␳1

Ž

5.9

.

᭚ x: ¨2 x x

Ž

x , 0

.

- 0, for s1- x - x, 1 ¨2 x x

Ž

x , 0

.

) 0, for x- x - . 1q␭

˙

Ž .

Moreover, in order to obtain ⌺ 0 - 0, we need also

1 1 z2 x

Ž

x , 0

.

) 0, in ␦ y - x - ,

Ž

5.10

.

1q␭ 1q␭ Ž . for some 0-␦ - 1r 1 q ␭ . Ž .

In fact, assuming z_{2 x} continuous at xs 1r 1 q␭ , we get, at least for a

˙

Ž .

small time t,⌺ t - 0, recalling

˙

⌺

z2 x

Ž

⌺ t , t s y

Ž .

.

Ž

␳ z ⌺ t , t y f .2 2

Ž

Ž .

.

.

␩2

We conclude by giving some estimates following from the maximum principle that hold in D :iT

1 ¨1

Ž

x , t

.

) 0, _¨2

Ž

x , t

.

)_¨2

ž

, 0

/

) 0,

Ž

5.11

.

1q␭ ¨i x

Ž

x , t

.

- 0,

Ž

5.12

.

f 0-¨i t

Ž

x , t

.

- _␳ ,

Ž

5.13

.

1 s t

Ž .

) 0.

Ž

5.14

.

A detailed analysis of this case could be performed using the methods w x

of 3 .

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Chem. Engrg. 1 1 1983 .

Ž . Ž . 2. E. Comparini, A one-dimensional Bingham flow, J. Math. Anal. Appl. 169 1 1992 ,

127᎐139.

3. E. Comparini, Regularization procedures of singular free boundaries problems in rota-Ž . rota-Ž .

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4. E. Comparini and E. De Angelis, Flow of a Bingham fluid in a concentric cylinder Ž . Ž .

viscometer, Ad¨. Math. Sci. Appl. 6 1 1996 , 97᎐116.

5. J. Crank and R. S. Gupta, A moving boundary problem arising from the diffusion of Ž .

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Ž . 6. A. Fasano and M. Primicerio, General free boundary, J. Math. Anal. Appl. 57 1977 ,

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7. A. Fasano and M. Primicerio, New results on some classical parabolic free boundary Ž .

problems, Quart. Appl. Math. 38 1980 , 439᎐460.

8. A. Fasano, M. Primicerio, S. D. Howison, and J. R. Ockendon, Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension, Quart. Appl.

Ž . Ž .

9. R. Gianni and R. Ricci, An elliptic᎐parabolic problem in Bingham fluid motion, Rend. Ž .

Mat. Uni¨. Trieste 1997 .

10. D. D. Joseph, K. Nguyen, and G. S. Beavers, Non-uniqueness and stability of the configuration of flow of immiscible fluids with different viscosities, J. Fluid Mech. 141 Ž1984 , 319. ᎐345.

11. L.-S. Jiang, M. Primicerio, and W.-Q. Xie, A class of diffraction problems in diffusive Ž . Ž .

absorption phenomena, Ann. Mat. Pura Appl. 4 1990 .

12. D. D. Joseph and W. W. Renardy, ‘‘Fundamentals of Two-Fluid Dynamics,’’ Vol. 3, Springer-Verlag, BerlinrNew York, 1992.

13. L. I. Rubinstein, ‘‘The Stefan Problem,’’ Amer. Math. Soc. Transl., Vol. 27, Amer. Math. Soc., Providence, 1971.

14. K. Vajravelu, P. V. Arunachalam, and S. Sreenadh, Unsteady flow of two immiscible Ž . Ž . conducting fluids between two permeable beds, J. Math. Anal. Appl. 196 3 1995 , 1105_᎐1117.