Stability char acter istics under inclined r otation

 − + + + + + Ω

− ^{r i} _l

l C

F i l F

F

DC k ω ω

, (4.13a)

=0

wl , (4.13b)

( )(

⁰

) ( )(

⁰ 1 1

)

⁰

42 1

1

41 − − + =

− _{− + l− l+}

c l

l c

l C C

FG i f C FG C

Dw f , (4.13c)

( )(

⁰

) ( )(

⁰ 1 1

)

⁰

32 1

1

31 − − + =

− _{− + l− l+}

c l

l c

l C C

FG i f C FG C

ς f . (4.13d)

The boundary conditions at z→∞ become

→0

Cl , w_l →0, Dw_l →0,ς_l →0. (4.14a~d)

The coefficients f , m= 1, 2, 3, 4 and n= 1, 2 are functions of the basic-state velocity, _mn which are shown in Appendix A. Note that because the maximum value of the index l needs to be taken as a finite value L , (4.12) accounts for a set of simultaneous _n ordinary differential equations of order 16L_n +8 in the computations. Equation (4.12) along with the boundary conditions (4.13) and (4.14) constitute a complex-eigenvalue problem, which can be solved by the shooting technique (Chung and Chen, 2000).

5. Stability char acter istics under inclined r otation

There are five physical mechanisms affecting the stability in the present system.

Firstly, the reduction of buoyancy in the z-direction (i.e. the direction of the basic density gradient) due to inclination is a stabilizing factor (Chung & Chen 2000).

Secondly, the gravity (buoyancy) component in the (x, y)-plane (parallel to the melt/solid interface) due to inclination is destabilizing. Thirdly, the rotation component in the z-direction contributed by spin and/or precession corresponding to the Coriolis force parallel to the (x, y)-plane is stabilizing. The fourth is the induced helical flow and the last is the precession component in the (x, y)-plane. It was shown by Kropp & Busse (1991), Busse & Kropp (1992) and Matthews & Cox (1997) that

an imposed shear flow or a rotation vector on the (x, y)-plane is stabilizing when they act individually. However, when acting simultaneously, they may together play a stabilizing or a destabilizing role depending on their relative orientation and amplitude-ratio. Similarly, the interaction between basic flow and precession can also apply to the present system. In addition, we note that both the buoyancy reduction in the z-direction and the rotation component in the z-direction have the so-called symmetric action, affecting all instability modes equally in different directions. The other three mechanisms, however, may or may not act symmetrically depending on whether spin is imposed. More precisely, when the system rotates with precession only, these three mechanisms have stationary-oriented components on the (x, y)-plane, which consequently destroy the stability symmetry in the (x, y)-plane. But nevertheless, when the system rotates with spin with/without precession, these three mechanisms change their directions periodically with the same frequency of spin and thus all instability modes traveling in different directions can sense their effects periodically. In this instance the stability symmetry in the (x, y)-plane holds.

We examine the stability characteristics of the system under four different kinds of rotation: (I) the system rotating vertically, (II) the system rotating by inclined precession, (III) the system rotating by inclined spin, and (IV) the system rotating by inclined spin and precession. These four cases are influenced by either part or all of these five stability mechanisms. The physical values considered are shown in Table 1, corresponding to lead-tin alloy (Coriell et al. 1980).

5.1 System rotating vertically

Since the system rotates vertically, there is no induced basic flow (Lu and Chen

1997) so that the number of the physical parameter involved is largely reduced.

Specifically, in (4.6) and (4.7), since S_n =0 and u_b =v_b =0, the linear perturbation equations are thus independent of φ so that the symmetry in the (x, y)-plane holds._α In addition, the spin Taylor number T and the precession Taylor number _as T_ap can be combined into a single Taylor number T ._a

The numerical results are shown in figure 4, illustrating three cases of different Taylor numbers: T_a¹² =0, 1 and 2. Figures 4a, 4c and 4e show the neutral curves of the morphological modes M1, M2 and UM (thin curves), the convective modes C1 and C2 (thick curves) and the mixed modes X1 and X2 (dotted curves); figures 4b, 4d and 4f show the corresponding wave speeds c_r =−ω_i α . In this figure and the subsequent ones, for the convenience of discussion, we have adopted the same labels as used by Forth and Wheeler (1992) to denote the instability modes and the gray shadow to denote the unstable region. To help read the physical meaning of these figures, we discuss first the case of α =0.5 and S_k =5 in figure 4a, which is morphologically unstable to M2. When S decreases to a value below the neutral_k curve of M2, the system becomes unstable to C1. For a higher wave number, for example α =10 and S_k =5, the system is unstable to M1. When the value of S_k decreases to be lower than the critical value while remaining positive, the system turns into stable. If the value of S_k continues to decrease to zero and becomes negative, then the system becomes morphologically unstable to UM. Note that the neutral curves of X1, X2 join M1 to form C2, and join C1 to form M2. Moreover, the coalescence of the neutral curves makes the wave speeds of X1 and X2 separate into two distinct branches: the wave speed of the left branch is generally larger than that of the right branch (see the dotted curves of figure 4b).

Figure 4a, similar to figure 7 of Forth and Wheeler (1992), corresponds to a case without rotation, in which the interaction between the convective mode and morphological mode predominates the system. The neutral curves exhibit a so-called folding structure for the X1 and X2 modes. Namely, they have the same stability criteria but travel in opposite directions; i.e. X1 moves forward with wave speed +c_r and X2 moves backward with wave speed −c_r . The neutral curves of the stationary modes C1 and M1 are coalesced by those of X1 and X2, forming two other stationary modes M2 and C2. The UM mode is also stationary but is physically unrealistic because it occurs in the region of non-positive Sekerka number. According to Forth and Wheeler (1992), the C1 mode is characterized by the flow rising from the troughs of the deformed interface and descending towards the peaks, whereas the M1 mode circulates in the opposite sense. Owing to the interaction between these two modes, the buoyancy-driven rising plume is shifted laterally between the trough and the peak and the interface tends to freeze on one side while dissolve on the other side of the trough, leading to the formation of the traveling modes X1, X2.

We consider two cases when vertical rotation is applied: T_a =1 and T_a =4 which are equivalent to the rotation speeds of 0.5 rpm (figures 4c and 4d) and 1 rpm (figures 4e and 4f) respectively. Results show that, due to vertical rotation, the coalescence between the neutral curves of C1 and X1, X2 become disconnected and M2 disappears. This is because C1 and X1, X2 are stabilized by vertical rotation through the action of Coriolis force, as commonly found in the similar system regarding the buoyancy-driven convection (for example Lu and Chen 1997). The M1 mode is virtually unaffected due to its short characteristic wavelength compared to the convective modes (Forth and Wheeler 1992). This implies a situation that M1 will

eventually dominate the system over C1 once the vertical rotation speed becomes large enough, as can be seen in figure 4e. The wave speeds of X1 and X2 also decrease with increasing rotation speed due to the stabilization by Coriolis force.

5.2 System rotating by inclined precession

As mentioned in section 4.1, when the system rotates with inclined precession the stability is orientation-dependent so that it is necessary to consider

°

≤

≤

°

−90 φ_α 90 for a complete analysis. It is impractical to consider many values of φ because of the tremendous computational effort required. We therefore choose α

five representative values: φ_α =90° , 50_° , 0_° , ₋74_° , and ₋80_° for the illustration. Results are shown in figure 5 where the tilt angle and the precession speed are fixed respectively at φ_n =10° and T_ap =1 (equivalent to 0.5 rpm). In this case the buoyancy reduction and the precession component in the z-direction have the action equally to the disturbances traveling in different directions in the (x, y)-plane.

In contrast, the gravity and the precession components act in the y-direction only and the basic flow varies direction with height by changing from 220_° (measured with respective to the x-axis) at the interface to 180° in the far field, both breaking the stability symmetry in the (x, y)-plane.

By comparing figure 5 with figure 4, two stability effects due to inclined precession are noticed. First, all the instability modes are oscillatory. Second, the neutral curve of X1 (originally traveling forward with +c_r ) is now smoothly connected onto M1 to form M(X1) and that of X2 (originally traveling backward with

cr

− ) is smoothly connected onto C1 to form C(X2). Similar phenomena can be found in Forth and Wheeler (1992), who investigated the influence of a shear flow imposed in the x-direction on the coupled convective and morphological instabilities of a binary alloy. In the present system, like the imposed shear flow considered by Forth and Wheeler, the simultaneous presence of the gravity and precession components in the y-direction and the induced spiral flow has two similar effects: First, it induces the

overstability for the modes; secondly, it destroys both the stability symmetry in the (x, y)-plane and the folding structure between the X1 and X2 modes.

By observing the wave speeds in figures 5h and 5j, one may infer that there is likely to exist a marginal angle corresponding to a zero wave speed at which the forward traveling mode M(X1) switches into the backward traveling mode M(X2) and the backward traveling modes M1, C1 and C(X2) switch into the forward traveling modes M1, C1 and C(X1), respectively. The marginal angle in the present case is possibly located between φ_α =−74° and φ_α =−80° , determined by the collaboration among the three asymmetry-driving mechanisms; i.e. the induced flow, the precession and the gravity. A similar result of marginal angle was shown by Forth and Wheeler (1992), in which the marginal direction was perpendicular to the imposed shear flow because the instabilities propagating in this direction could not sense the imposed flow.

As far as the stability criterion is concerned, it is seen that the critical Sekerka number of M1 is virtually independent of the propagating angle φ . In contrast, the _α criteria for C1, C(X1) and C(X2) are quite sensitive to the variation of φ . Namely, _α both the convective modes C1 and C(X2) are most unstable in the direction along

°

=90

φα (parallel with the y-axis) as shown in figure 5a, and are greatly suppressed along φ_α =0° (parallel with the x-axis) as shown in figure 5e. This is because, due to inclined precession, both the gravity and precession act in the y-direction only so that they have no effect on the modes traveling in the x-direction. Whereas the modes traveling in the x-direction are stabilized by the other three factors － the buoyancy reduction in the z-direction, the rotation vector in the z-direction and the induced basic flow. Note that for the case in figure 5e the basic flow plays a stabilizing role to

the modes traveling in the x-direction because the modes cannot sense the precession component, which in the instance is acting in the y-direction. To other modes propagating in the directions not equal to φ_α =0°, it is inferred that the interaction between the basic flow and the precession component in the y-direction are destabilizing. This inference is made based on the work of Matthews and Cox (1997), who examined the buoyancy-driven convection under the interaction of a horizontal rotation vector and an imposed shear flow, finding that the imposed shear and rotation may together play a destabilizing role. We will discuss in more detail about the application of their work in the case involving both precession and spin.

5.3 System rotating by inclined spin

In this case, except the precession component in the (x, y)-plane, all other four mechanisms are active. As in the previous cases, the buoyancy reduction in the z-direction and the rotation component in the z-direction (equal to the spin vector for the present case) will not cause any interference to the stability symmetry in the (x, y)-plane. Moreover, because of spin, both the induced flow and the gravity component in the (x, y)-plane changes the directions in synchrony with spin; namely, the induced flow and gravity component also rotate with the spin frequency Ω. It follows that instability modes traveling in different directions will have equal stability condition and the system will retain the stability symmetry in the (x, y)-plane. Note also that the basic flow velocity increases with increasing inclined angle and decreases with increasing spin frequency Ω. Bearing these features in mind, we examine the system’s stability under the effects of inclined spin.

We show in figure 6 the neutral curves for the case of Ω=80 (equivalent to

2 2

1 ≈

Tas or 1 rpm for the present system) with the inclination angle φ varying from _n

°

10 to 25°. A major outcome due to inclined spin is seen by comparing figure 6a with figure 4a: The mixed modes X1, X2 and the convective modes C1, C2 are largely stabilized while the morphological mode M1 is slightly stabilized. When the inclination angle increases (Figures 6c and 6e), the stabilization due to inclined spin is enhanced. By comparing figure 6a with figure 5a, we examine the difference between inclined spin and inclined precession and find that the mixed modes X1, X2 and convective mode C2 absent in the inclined precession case show up in the spin case.

Both the folding structure of the X1 and X2 modes and the stability symmetry in the (x, y)-plane destroyed in the inclined precession case also recover here. Note that C1, C2, M1 and UM are of c_r =0 (see figures 6b, 6d and 6f), indicating that these modes move in synchrony with the motion of spin. On the other hand, X1 and X2 are of c_r ≠0, indicating that the mixed modes move non-synchronously with spin and their frequencies are modulated by ω_i =−αc_r (see equation (4.11)).

5.4 System rotating by inclined spin and precession

We examine the stability characteristics for the case Ω=80 (equivalent to

2 2

1 ≈

Tas ) and φ_n =20° for various values of (−1)^npT_ap¹² and the results are shown in figure 7. In this case, all the five stability mechanisms are active and because of spin they influence equally on all instability modes traveling in different directions.

Consequently, the stability symmetry in the (x, y)-plane and the folding structure of the mixed modes are retained. Moreover, M1, C1 and C2 move synchronously with spin while X1 and X2 moves non-synchronously. It is shown in figure 7 that the stability criterion of M1 remains virtually the same for different T_ap . The

most-unstable mode is the C1 occurring at (−1)^npT_ap¹² =−0.8 (figure 7c) and the modes C1, C2, X1 and X2 are significantly suppressed at other values of T ; for _ap example at (−1)^npT_ap¹² =−1 (figure 7a), (−1)^npT_ap¹² =0.5 (figure 7e) and

1 )

1

(− ^npT_ap¹² = (figure 7g). The comparison of the neutral curves of the C1 mode in figure 7c, 7e and 7g indicates that C1 becomes unstable at larger negative values of

2

) 1

1

(− ^npT_ap because the stabilizing action due to the rotation component in the z-direction (in terms of

( )

¹ ap¹² as¹²

n n

az C T T

T = − ^p + ) becomes smaller at larger negative values of (−1)^npT_ap¹². However, the same reason fails to explain why the C1 mode in

figure 7a is more stable again than that of figure 7c despite that (−1)^npT_ap¹² =−1 in figure 7a is even more negative. A plausible explanation for this can be obtained from the interaction between the basic flow and the precession component in the (x, y)-plane, which is discussed in more detail below.

We summarize in figure 8 the stability characteristics in terms of the relationship between S_k^c and (−1)^npT_ap¹². The stable region, marked by the gray shadow, is enclosed by the stability critical values of the M1, C1 and UM modes. The stability criterion of the M1 mode is virtually unchanged with varying (−1)^npT_ap¹². However, the C1 mode is significantly stabilized when (−1)^npT_ap¹² >−0.5 but is slightly

destabilized when (−1)^npT_ap¹² <−0.5 except in the small region near (−1)^npT_ap¹² =−1 where the C1 mode is also stabilized. To interpret the stabilization of C1 in this small region, we illustrated in figure 9 the variations of T , _az ψ and T_eT_ap¹² versus

2

) 1

1

(− ^npT_ap , where ψ is the relative direction (in radians) of the basic flow measured with respect to the precession component in the (x, y)-plane.

Although the direction of the basic flow changes with height in the Ekman layer, it is reasonable to choose the representative value of ψ at the melt/solid interface because the convective mode is largely confined in the solute boundary layer near the interface. The parameter T_eT_ap¹² accounts for the amplitude ratio of the precession component in (x, y)-plane to the basic flow, which is obtained by (3.12) in the limits of the large Schmidt number S and large Lewis number _c L . The results of figure 9 _e show that the rotation component in the z-direction accounted for by T decreases _az with decreasing (−1)^npT_ap¹², indicating that the stabilizing effect of T gradually _az diminishes as (−1)^npT_ap¹² decreases. This explains the phenomenon that the convective C1 mode becomes unstable as (−1)^npT_ap¹² decreases from figure 7g through 7e to 7c. Note that in figure 9 there are two zeros of T_eT_ap¹² , one at

0 )

1

(− ^npT_ap¹² = and the other close to (−1)^npT_ap¹² =−1 , implying that the amplitude-ratio of the basic flow to the precession component is quite large near these two points. Matthew and Cox (1997) investigated the system of buoyancy-driven convection under the interaction between an imposed shear flow and a rotation vector lying in the plane parallel to the shear flow. They found that the convective instability tends to be suppressed when the following two conditions both hold: (1) the shear flow is relatively strong compared to the rotation and (2) these two mechanisms have virtually the same directed vorticities (see figure 5 of Matthew & Cox 1997);

Applying their findings to our case, to have the same directed vorticities, the relative

orientation measured from the precession component in the (x, y)-plane to the basic flow needs to be close to 90°. Because the relative orientation in the region

0 )

1 (

1≤ − ¹² <

− ^npT_ap shown in figure 9 is about ψ ≈60° and the basic flow is

relatively strong compared to the precession component near (−1)^npT_ap¹² =−1 and 0

) 1

(− ^npT_ap¹² = where T_eT_ap¹² →0, we infer that the interaction between the basic flow and the precession component in the (x, y)-plane is stabilizing to C1 near these two points. This then gives the reason why the C1 mode becomes more stable in figure 7a where (−1)^npT_ap¹² =−1 than in figure 7c where (−1)^npT_ap¹² =−0.8. For

further larger negative values of (−1)^npT_ap¹² than (−1)^npT_ap¹² =−1, as shown in figure 9 the relative orientation becomes ψ ≈−45° , indicating that the precession component in the (x, y)-plane and the shear flow have virtually opposite directed vorticities and so the interaction between them becomes destabilizing, making C1 become unstable again.

To elucidate this scenario more concretely, we illustrate another example by comparing figure 7e to 6c. Both cases have Ω=80 and φ_n =20°, while figure 6c has (−1)^npT_ap¹² =0 and figure 7e has (−1)^npT_ap¹² =0.5. Namely, both cases have the same intensity of buoyancy reduction in the z-direction and the same gravity component in the (x, y)-plane but the stabilizing rotation vector in the z-direction in the inclined-spin case (figure 6c) is weaker than that in the inclined-spin-with-precession case (figure 7e). The comparison shows that the critical value of the C1 mode in figure 6c is about S_k =−6 and in figure 7e is about

−3

k =

S , indicating surprisingly that C1 is more stabilized in figure 6c although the rotation vector in the z-direction is weaker there. This result also can be explained by

taking into account the interaction between the basic flow and the precession component in the (x, y)-plane. For the inclined-spin case in figure 6c, there is no precession applied, i.e.(−1)^npT_ap¹² =0, so the basic flow is itself a stabilizing factor. In contrast to that, for the inclined-spin-with-precession case of figure 7e, the value

5 . 0 )

1

(− ^npT_ap¹² = corresponds to ψ ≈−130° as shown in figure 9, implying that the precession component and the basic flow have virtually opposite-directed vorticities and they together play a destabilizing role. In words, the destabilizing action due to the collaboration of the precession component in the (x, y)-plane and the basic flow has prevailed over the stabilizing action by the rotation vector in the z-direction for the case in figure 7e, rendering C1 be much stabilized in figure 6c.

6. Conclusions

We have analyzed the stability characteristics of a directionally solidifying binary alloy under inclined rotation. Before the onset of instability occurs, the basic state is mainly a strong helical shear flow induced by inclination and modified by rotation, moving along the melt/solid interface. The corresponding basic-state temperature and concentration remain the same as those of the cases where the system is standing vertically with/without rotation. The induced helical flow, which increases in magnitude with increasing inclined angle and decreases with increasing rotation speed, consists of three components: The solutal-layer flow, the thermal-layer flow and the Ekman-layer flow. For a lead-tin alloy of large Lewis number, the

We have analyzed the stability characteristics of a directionally solidifying binary alloy under inclined rotation. Before the onset of instability occurs, the basic state is mainly a strong helical shear flow induced by inclination and modified by rotation, moving along the melt/solid interface. The corresponding basic-state temperature and concentration remain the same as those of the cases where the system is standing vertically with/without rotation. The induced helical flow, which increases in magnitude with increasing inclined angle and decreases with increasing rotation speed, consists of three components: The solutal-layer flow, the thermal-layer flow and the Ekman-layer flow. For a lead-tin alloy of large Lewis number, the

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