The Stability of a Rotating Cartesian Plume in the Presence of Vertical Boundaries ()
1. Introduction
The study of the dynamics of compositional plumes is important for many real life applications in industry ([1] - [6] ), geophysics ([7] -[25] ) and environment ([26] -[34] ). While the presence of the compositional plumes can be harmful (e.g., in iron bars), it is useful in geophysics (e.g., the hot compositional plumes, that rise from the inner core boundary of the Earth into the outer core interact with the rotation and magnetic field of the Earth and may contribute to the Geodynamo). Such a wide range of applications has motivated many studies on various aspects of the dynamics of compositional plumes. These studies are experimental and theoretical. The experimental works on the dynamics of compositional plumes observed that the plume flow seems to be stable (Sample and Hellawell [35] ; Chen and Chen [36] ; Hellawell et al. [37] ). The laboratory studies by Hellawell et al. [37] find that the plumes are thin and long, and its top part tends to break up and disappears. Classen et al. [38] experimentally studied the dynamics of compositional plumes under the influence of vertical rotation to find that the plumes are unstable and break into blobs. On the other hand, the theoretical works on the stability of the plumes showed that the Cartesian plume is always unstable in the absence of rotation (Eltayeb and Loper [16] ) and in the presence of rotation (Eltayeb and Hamza [18] ). These studies assumed that the plume rises vertically in a fluid of unbounded domains. While the experimental studies were conducted in bounded regions to show that the plume was stable, the theoretical models were conducted in unbounded domains to find that the plume was unstable. Thus it is of interest to examine the influence of the vertical boundaries on the dynamics of the plumes. The mathematical model by Al Mashrafi and Eltayeb [6] investigated the influence of the two fixed vertical boundaries on the dynamics of the plumes. They tested the stability of non-rotating Cartesian plumes in a bounded domain to find that the presence of two vertical boundaries affects the stability, but the plumes remain unstable. Moreover, they found that the plume was stable when it was close to the boundary but had a large thickness and the material diffusion is potent in the thin layer between the plume and the nearest boundary.
Motivated by real life applications and laboratory results, we study here the influence of vertical rotation on the dynamics of bounded Cartesian plume. In general, the purpose of this study is to extend the theoretical model by Al Mashrafi and Eltayeb [6] on the dynamics of a Cartesian compositional plume in bounded regions to include the action of vertical rotation. The model by Al Mashrafi and Eltayeb [6] consisted of a column of buoyant fluid of finite thickness, 
, rising vertically in another less buoyant fluid bounded by two fixed vertical walls located at 
and
. The system was infinite in the 
and 
directions. In the current study, we consider that the whole system rotates about the vertical with a uniform angular speed, 
(see Figure 1).
In Section 2, we formulate the model mathematically and state the boundary conditions of the system. The presence of rotation introduces an additional parameter, 
, which is a measure of the Coriolis force relative to the viscous force, and this parameter referred to hereinafter as the rotation parameter, defined by
(1)
where 
is the Taylor number, 
is the kinematic viscosity and 
is the unit of length (see Equation (6) below). In Section 3, we investigate the influence of the vertical boundaries on the linear stability of a rotating
Figure 1. The geometry of the problem showing the profile of the basic state concentration of light material, 
, representing a plume of width, 
, and concentration, 1 , rising vertically in a finite fluid of width, 
, and concentration, 0. The system rotates uniformly about the vertical with angular speed,
. The plume divided the system into three regions: region 2 represents the plume whereas regions 1 and 3 represent the surrounding fluid.
bounded Cartesian plume. The problem of the rotating plume was studied by Eltayeb and Hamza [18] in the absence of boundaries. In Section 4, we discuss the effect of the boundaries on the stability of a rotating plume. The growth rate is maximised over the wave numbers plane 
in the parameter space. In Section 5, we make some concluding remarks.
2. Formulation of the Problem
We consider a two-component fluid, in which the concentration of the solvent component (light material) is 
and the temperature is
, rotating uniformly about the vertical with angular velocity
. The fluid has kinematic viscosity, 
, and thermal diffusivity, 
, and material diffusion is negligible. The dimensionless equations of the system have been derived by Eltayeb and Hamza [18] . They are
(2)
(3)
(4)
(5)
Here R, 
and 
are the Grashoff number, the Prandtl number and the rotation parameter, respectively, defined by
(6)
where 
is the Taylor number and
, 
and 
are characteristic units of velocity, length and concentration, respectively (Al Mashrafi and Eltayeb [6] ), and
, 
, 
, 
, 
, 
, 
, 
, 
are the velocity vector, the pressure, the constant gravitational acceleration, the upward unit vector, the time, the thermal expansion coefficient, the compositional expansion coefficient, the density, the uniform temperature gradient and the subscript “
” in the Equation (2) refers to reference values.
We consider a basic concentration profile
(7)
which defines a plume of thickness, 
, rising with velocity 
in the presence of mean temperature 
and mean pressure 
such that
(8)
(9)
which are the same equations obtained in the absence of rotation. The presence of rotation does not affect the basic state, and the solution of (8) and (9) is the same as in the absence of rotation. We include it here for easy reference:
(10)
where
, 
and 
are defined by
(11)
3. The Stability Analysis
In this section, we use the perturbation Equations (14)-(17) to investigate the linear stability of the basic state solution given by (10). We assume that the interface at the plane 
is given a small harmonic disturbance of the form
(12)
where 
and 
are the wavenumber components in the 
plane, 
is the growth rate and 
refers to the complex conjugate.
The disturbance (12) will propagate into the fluid, and affect the second interface and the variables of the system to produce the perturbations. Consequently, the interface at 
can be written in the form
(13)
where 
is a measure of the amplitude of the interface and it is evaluated as a part of the solution. The perturbations introduced into the system are governed by the dimensionless equations (see Al Mashrafi and Eltayeb [6] )
(14)
(15)
(16)
(17)
where
, 
, 
and 
are the perturbations in velocity, pressure, temperature and concentration, respectively.
The perturbation variables take the form
(18)
in which the factors
, 
, and 
are introduced in the variables
, 
and
, respectively, for convenience.
Substituting the variables (18) into the Equations (14)-(17) and after some arrangements we get the following equations
(19)
(20)
(21)
(22)
(23)
(24)
(25)
subject to the boundary conditions
(26)
(27)
(28)
(29)
where the variable 
is related to the vertical component of vorticity and is defined as
(30)
and we have introduced the notation
(31)
(cf. Al Mashrafi and Eltayeb [6] ).
We use the same method adopted in Al Mashrafi and Eltayeb [6] and expand the perturbation variables and the growth rate in the small parameter
, thus
(32)
where 
indicates any of the perturbation variables
, 
, 
, 
, 
and
.
It turned out that the leading order terms in the equations determines the stability of the system. The relevant equations and the boundary conditions are then obtained from (19)-(24) and (26)-(29) by neglecting the terms with
. In order to facilitate comparison with the results of the non-rotating case, we shall maintain the subscript 0. The equations are given by
(33)
(34)
(35)
(36)
(37)
(38)
The associated boundary conditions are obtained from (26)-(29) by introducing the subscript 0 to all the variables and the subscript 1 to
.
The system (33)-(38) together with the boundary conditions poses an eigenvalue problem for the growth rate, 
, which determines the stability of the system. The real part 
governs the variations of the amplitude of the disturbance with time, and hence it determines the stability of the disturbance. If it is negative for all possible values of the wavenumbers 
and
, then the plume is stable, while if at least one pair of 
and 
gives a positive value, then the plume is unstable. If 
vanishes for all values of the wavenumbers, the plume is neutrally stable. If the preferred mode occurs for
, 
both non-zero, it is referred to as a 3-dimensional mode (oblique), and if
, it is called 2-dimensional (vertical). The case 
and 
is found not to occur. The imaginary part 
determines the phase speed of the disturbance. The vertical phase speed 
and the horizontal phase speed 
are defined as
(39)
We note that 
is defined only if
.
We operate on Equation (33) with
, and use Equations (34)-(36) to get
(40)
The general solution of the differential Equation (40) can be written in the form
(41)
where the superscript 
refers to the three regions of the system (see Figure 1), 
, 
are constants, and 
are the roots of the cubic equation
(42)
with 
is given by
(43)
We use the Equations (34)-(36) and the solution (41) to obtain
(44)
The Equation (37) can be solved in the form a complementary function and particular solution to find
(45)
where
, 
are constants, and
, 
are given by
(46)
Now we apply the boundary conditions at
, 
, 
and solve the resulting algebraic equations for the constants
, 
, 
, 
for 
The details of the solution are given in the Appendix A.
The growth rate is given by the quadratic equation
(47)
where 
and 
are given in the Appendix A.
Solving Equation (47) yields
(48)
and the displacement 
is given by
(49)
in which 
and 
are defined in the Appendix A.
We note that, as in the absence of rotation, the system has two modes. The upper sign in the expression (48) corresponds to the modified varicose mode (MV) and the lower sign refers to the modified sinuous mode (MS).
4. Discussions of the Results
The growth rate (48) is evaluated numerically in the 
plane as a function of the parameters
, 
, 
and 
for both modes. The contours of 
in the 
plane are plotted for sample values of the parameters
, 
, 
and 
in the Figures 2-4.
In Figure 2, we present a comparison between the contours of the two modes MV and MS for different values of the Taylor number, 
, when
, 
and
. We note that the MV mode always possesses a minimum with negative growth rate and one or two maxima with positive growth rates while the MS mode has
Figure 2. Contours of the growth rate , 
, of the modes MV, as in (a), (c), (e) and MS, as in (b), (d), (f) for the rotating bounded Cartesian plume where
, 
,
. Here 
for (a), (b); 
for (c), (d); and 
for (e), (f). Note that the preferred mode is MS.
Figure 3. Contours of the growth rate, 
, of the modes MV, as in (a), (c), (e) and MS, as in (b), (d), (f) for
, 
,
. Here 
for (a), (b); 
for (c), (d); and 
for (e), (f). Note that the preferred mode of the instability is 2-dimensional when
.
a minimum with negative growth rate only when 
is small and one maximum with positive growth rate which is larger than that of the corresponding MV mode. This indicates that the MS mode is preferred for this set of parameters. The local maxima correspond to 3-dimensional modes and the largest maximum always increases with the increase in the rotation parameter
.
Figure 4. Isolines of the growth rate , 
, as in (a), (c), (e) and 
as in (b), (d), (f) for
, 
,
. Here 
for (a), (b); 
for (c), (d); and 
for (e), (f). Note that at fixed point
, the values of 
and 
are nearly same but there different in the sign.
Figure 3 presents the influence of the distance 
between the plume and the nearest sidewall on the stability of the plume when the thickness of the plume and the rotation parameter are held fixed at 
and
. The figure is plotted for three different values of
: 
for subfigures (a), (b); 
for subfigures (c), (d); and 
for subfigures (e), (f). When the plume is situated half-way between the sidewalls
, the MV mode has a negative growth rate everywhere and hence stable while the MS mode has a growth rate that is positive everywhere and hence is preferred. The local maximum possesses a vanishing horizontal wave number and hence propagates vertically. As the plume moves towards a wall, the growth rates of both modes increase and the MV mode develops local maxima with positive growth rates but they are not preferred because the MS mode local maximum increases as well and becomes 3-dimenaional.
Figure 4 illustrates the influence of the plume thickness on the contours of the growth rate for fixed values of the rotation parameter and the distance between the two sidewalls when the plume is situated half-way between the two sidewalls. As 
increases from
, the growth rate for MS mode, which is positive everywhere, increases and that of the MV mode, which is negative everywhere, decreases until 
reaches a critical value, 
, when the growth rate of the MS mode decreases and that of the MV mode increases.
The contours of Figures 2-4 indicate that the preferred mode of instability is the modified sinuous (MS) mode. This indication is quantified by calculating the maximum growth rate and the associated wave numbers and phase speeds for different values of the parameters. For fixed values of
, 
, 
and
, we maximize over the wave numbers 
and 
by demanding that
(50)
The solution of (50) gives the values 
and 
at maximum growth rate, 
, calculated from (48) for these values of 
and 
to give, together with the corresponding value 
of the phase speed, the parameters 
of the preferred mode. For fixed parameters
, 
, 
and
, and particular mode, all possible local maxima of 
are identified and the largest value taken together with the corresponding wavenumbers and phase speeds as defining the preferred mode for that set of parameters for that mode. This is carried out for both modified varicose (MV) and modified sinuous (MS) modes, and the largest is chosen as the preferred mode. A sample of the results is given in Figures 5-9.
Figure 5 and Figure 6 illustrate the influence of the distance between the boundaries on the preferred mode of instability of the rotating plume. In Figure 5, the preferred mode parameters are plotted as a function of 
for fixed 
and two values of the distance between the sidewalls, 
, for a plume situated half-way between the sidewalls. It is found that the preferred mode is the MS mode and the influence of the boundaries tends to stabilise the plume but only reduces the growth rate slightly. The horizontal and vertical wavenumbers are increased while the phase speeds are reduced as the distance, 
, between the two walls is reduced.
Figure 6 presents the preferred mode of instability as a function of 
and fixed 
where the plume is equidistant from the two walls and 
takes the values 10 and 20. The growth rate increases rapidly as 
increases from zero until it reaches a maximum when the plume occupies the middle half of the region between the sidewalls. As the thickness of the plume increases further, the growth rate decreases if 
is small but stays at the maximum value for increasing 
if the distance 
is large. In both case, the growth rate drops to zero as the wall is approached and the plume nearly fill the whole region between the sidewalls. The wavenumbers decrease as 
increases from zero but they soon jump to larger values and increase with
. In both cases the MS mode is preferred.
Figure 7 illustrates the dependence of the preferred mode on 
for
, 
and 
takes two values: 
and
. As the Taylor number increases from
, the maximum growth rate 
increases rapidly until 
reaches a certain value that increases as the distance between plume and the nearest sidewall increases, after which the growth rate varies much more slowly. The wavenumbers of the preferred mode show sudden changes at a small value of 
indicating a change of local maximum as
, increases through a value,
. The horizontal wavenumber, 
, is zero for small 
corresponding to 2-dimensional motions, but as 
reaches
, 
jumps to a nonzero value and increases thereafter. The vertical wavenumber increases rapidly as 
increase to 
and then decreases as 
increases further. The vertical phase speed of the preferred mode also suffers a change at 
and the jump depends on how far the plume is from the nearest sidewall.
In Figure 8, we present the dependence of the preferred mode on the distance 
between the plume and the nearest sidewall. We note that the growth rate decreases when the plume moves towards the location half-way
between the two sidewalls. Both wavenumber components increase steadily as the distance between plume and the nearest sidewall increases. The vertical phase speed however behaves differently as 
increases; it decreases steadily with increasing 
if 
is small but when 
is large, it decreases slowly reaching a minimum before it increases at a moderate rate. In all cases, it is the MS mode that provides the preferred mode.
Figure 9 shows the dependence of the critical modes 
on
. The growth rate increases gradually until 
reaches a critical thickness, 
, and then it decreases. This indicates that there is a critical thickness representing the most unstable plume. The behaviour of the wavenumber components and phase speed with 
is quite complicated. The horizontal wavenumber vanishes for small 
when 
is small. For large, 
, behaves in a similar way when 
is close to
. The vertical wavenumber, on the other hand decreases as 
increases from zero and jumps to a larger value when the local maximum changes, and then varies very slowly until 
is almost 4, when
.
5. Conclusions
The dynamics of a fully developed plume of buoyant fluid, in the form of a channel of finite width, 
, rising in a less buoyant fluid contained between two parallel vertical walls, a distance 
apart, and two fluids rotate uniformly about the vertical have been investigated.
In the absence of boundaries ([18] ), it was found that the stability problem depended on the parameters: the Grashoff number, 
, the Taylor number, 
, and the thickness of the plume,
. The magnitude of the growth rate was of the order 
for 
and the instability took one of the two types: sinuous mode or varicose mode.
The presence of the vertical boundaries here introduces two dimensionless parameters: the distance between the plume and the nearest wall, 
, and the distance between the two vertical sidewalls,
. The introduction of the boundaries modifies the two modes to be the modified sinuous mode (MS) and the modified varicose mode (MV). It is shown here that the boundaries introduce two main effects to the rotating Cartesian plume studied by Eltayeb and Hamza [18] . First, the sidewalls tends to stabilise the plume but succeed only reducing the growth rate and the plume remains unstable for all values of the Taylor number and the distance from the nearest sidewall. Second, the presence of the sidewalls suppresses the modified varicose mode and allows the modified sinuous only to be unstable. The preferred mode can be 3-dimensional or 2-dimensional depending on the values of the parameters of the system. When the preferred mode is 2-dimensional, propagation can be upwards or downwards.
Appendix A: The Derivation of the Dispersion Relation
In this appendix, we apply the boundary conditions (26)-(28) when the subscript is 0 for the variables (41), (44) and (45) to evaluate the constants of the solution. Application of the boundary conditions at the interface 
gives
(A:1)
(A:2)
(A:3)
(A:4)
where
, 
, 
and 
are defined by
(A:5)
(A:6)
(A:7)
(A:8)
We use the Equations (A:5)-(A:8) to obtain
(A:9)
(A:10)
(A:11)
(A:12)
Similarly, at the interface
, we find
(A:13)
(A:14)
(A:15)
(A:16)
in which
, 
, 
and 
are given by
(A:17)
(A:18)
(A:19)
(A:20)
Equations (A:17)-(A:20) give
(A:21)
(A:22)
(A:23)
(A:24)
The boundary conditions at the boundaries 
and 
give
(A:25)
(A:26)
(A:27)
(A:28)
such that
, 
, 
and 
are defined by
(A:29)
(A:30)
(A:31)
(A:32)
The equations and can be solved to find
(A:33)
since the roots 
are distinct. Hence Equations (A:3)-(A:5) give
(A:34)
(A:35)
(A:36)
Using the expression for 
in , and the properties of the 3 roots of the cubic Equation (42) to simplify the expressions (A:35), (A:36). The simplifications lead to
(A:37)
Hence the expressions (A:9)-(A:12) reduce to
(A:38)
(A:39)
(A:40)
The same method can be applied to the conditions at
. The Equations (A:13)-(A:16) leads to
(A:41)
thus the expressions (A:21)-(A:24) give
(A:42)
(A:43)
(A:44)
The expressions
, 
, 
and
, given in (A:29)-(A:32), can be simplified by using the relations (A:38), (A:39) and (A:42), (A:43) to find
(A:45)
(A:46)
(A:47)
(A:48)
where we have defined the notation
(A:49)
(A:50)
(A:51)
(A:52)
The Equations (A:49)-(A:52) give
(A:53)
(A:54)
(A:55)
(A:56)
The boundary conditions at 
and
, given in (A:25) and (A:28), can be written by using (A:45)-(A:48) and (A:55) and (A:56) to obtain
(A:57)
(A:58)
in which
, 
, 
, 
, 
and 
are defined by
(A:59)
(A:60)
and we have defined 
and
, 
as
(A:61)
(A:62)
(A:63)
(A:64)
The algebraic system (A:57) and (A:58) can be solved by elimination to find
(A:65)
in which
, 
, 
and 
are given by
(A:66)
(A:67)
(A:68)
(A:69)
(A:70)
(A:71)
and we have defined the following notation
(A:72)
(A:73)
(A:74)
(A:75)
(A:76)
(A:77)
(A:78)
(A:79)
The constants 
and 
for 
can easily evaluated using (A:38), (A:39), (A:42), (A:43), (A:53), (A:54). The next step is to find the expressions of the constants 
and
. The Equations (A:26), (A:27), (A:40) and (A:44) give
(A:80)
(A:81)
where 
and 
are defined by
(A:82)
The application of the boundary condition (29) gives
(A:83)
such that 
and 
are given by
(A:84)
where 
for 
and 
are defined by
(A:85)
(A:86)
and we have used the notation
, 
, 
, 
, 
, 
and 
(A:87)
(A:88)
(A:89)
(A:90)
(A:91)
(A:92)
(A:93)
and 
has the form
(A:94)
Now the growth rate 
and the displacement 
can be evaluated by solving the Equation (A:83) to find
(A:95)
(A:96)
where 
and 
are defined by
(A:97)
(A:98)