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The influence of boundaries on the dynamics of a compositional plume is studied using a simple model in which a column of buoyant fluid rises in a less buoyant fluid bounded by two vertical walls with a finite distance apart. The problem is governed by four dimensionless parameters: The Grashoff number, R, which is a measure of the difference in concentration of light material of the plume to its surrounding fluid, the Prandtl number, σ, which is the ratio of viscosity, ν, to thermal diffusivity, κ, the thickness of the plume, 2x0, and the distance, d, between the two vertical walls relative to the salt-finger length scale. The influence of the boundary on the fluxes of material, heat, and buoyancy is examined to find that the buoyancy flux possesses a local maximum for moderate to small thicknesses of the plume when they lie close to the wall. This has the effect of introducing a region of instability for thin plumes near the wall with an asymptotically larger growth rate. In addition, the presence of the boundary suppresses the three-dimensional instabilities present in the unbounded domain and allows only two-dimensional instabilities for moderate to small distances between the bounding walls.

Studies on the dynamics of fluid alloys are relevant to industrial (e.g., Rees and Worster [

In industrial applications, one of the problems the iron casting industry faces is the appearance of freckles in iron bars causing their weakness. When iron ore is poured into molds or designs, air trapped at the bottom of the design rises in the form of thin filaments into the liquid iron. When the iron solidifies, these filaments form trapped air pockets that appear as very thin black strips along the outer surface of the iron bar and lead to a weakness in the iron bar. The experimental work of Copley et al. [

Theoretical studies of a compositional plume rising in a fluid of infinite extent have shown that the plume is unstable (see, e.g., Eltayeb and Loper [

The main purpose of this study is to examine the influence of boundaries on the dynamics of compositional plumes. For this purpose, we introduce boundaries to the model discussed by Eltayeb and Loper [

In section 2, we formulate the problem, which involves four dimensionless parameters: the Grashoff number,

where

In section 3, we use a top-hat profile of the concentration of light material to obtain a solution representing a plume of thickness

In section 4, we examine the stability of the plume. This poses an eigenvalue problem for the growth rate

We consider a two-component incompressible fluid in which the concentration of the solvent component (light material) is

where

Motivated by the experimental work on plumes rising from mushy layers, we take a temperature profile

where

We now cast the equations - into dimensionless form. It is found that in order to maintain the effects of temperature variations and compositional variations, we use the salt-finger length scale defined by

and a velocity unit with the definition

so that the ensuing motions are driven by the plume flow transporting the light material,

Here the dimensionless parameters

We define a Cartesian coordinate system

a distance

We can now take the flow variables to have the form

such that the variables with subscript

The variables with an “overbar” are basic state variables dependent only on the horizontal coordinate

Substituting the expressions (15)-(18) into the system (11) - (14), the terms independent of

These equations are discussed in section 3 below.

The order

(23)

The perturbation equations are solved in section 4 below.

Equation (14) is automatically satisfied for the basic state and we are free to choose a concentration function

Consider the basic state equations (21) and (22). Define

Then

The equation (29) is subject to the boundary conditions

The solution is

where

A sample of the profiles of the solutions

(i.e., downwards flow) within the plume when it is wide, and this has an effect on the net transport of material by the plume. The wide plume is also associated with a temperature profile that is almost uniform in the main body of the plume. If the position of the plume moves towards a sidewall, symmetry is broken. Here the downward flow outside the plume is partially suppressed in the narrow region between the plume and the nearest wall and strengthened on the far side. Such behavior will lead to the modification of the modes of instability in the absence of the sidewalls.

The basic state solution is associated with fluxes of heat,

(cf. [

where

The fluxes are presented in the

In this section, we solve the eigenvalue problem posed by the perturbation equations (23)-(26) and the relevant boundary conditions to obtain expressions for the growth rate. Our interest lies in the instability produced by the buoyant fluid in the plume. We assume that the interface at the plane

where

The disturbance (39) will propagate into the system, and affect the second interface and the variables of the system to produce the perturbations. The disturbance at the interface

where

The perturbation variables produced by the disturbance (39) can be expressed in the form

where the factors

Substituting the variables (42) into (23)-(26), we obtain the following ordinary differential equations in

Here we have used

The boundary conditions across the interfaces are (see, Eltayeb and Loper [

In addition, the sidewalls are maintained at the hydrostatic temperature so that

Equation (48) gives

It was found that it is useful to derive the following three equations. First, differentiate (45) once and subtract (44) to get

Where

Thirdly, apply the operator

The previous studies on a compositional plume showed that the plume flow is unstable for small value of Grashoff number [

where

Substituting the expressions (59) into the system (43)-(47), (56)-(58) and the associated boundary conditions (50)-(54), and equating the coefficients of

The coefficients of

noting that (56) and the appropriate conditions imply that

We operate on equation (61) with

The solution of the system (60)-(64) subject to the boundary conditions (65)-(67) is given by

where the superscript “i” in the solution refers to the region of the problem defined by

(see figure 1) and

with

The constants

with

The application of the boundary conditions (68) gives an expression for the growth rate

in which

Thus

The properties of the roots of the cubic equation (73) render

It is informative to establish the relationship between the modes of the bounded plume defined by (88) and those of the unbounded one particularly that we expect the modes of the bounded plume to reduce to sinuous and varicose when the plume is positioned half-way between the two sidewalls. We take the limit

and

where

Substituting these expressions into the equation (88) for

The growth rate (94) is the same as the growth rate of the Cartesian plume obtained in Eltayeb and Loper [

The coefficients of

in which

The associated boundary conditions are

The equations and boundary conditions (95)-(105) are solved in the Appendix A. They lead to the growth rate

where

and

in which

It is noteworthy that because of the properties of the cubic equation (73) for

The growth rates given by the expressions (88) and (106) were computed in the parameter space

The maximum value,

First we consider equation (88). This growth rate at this level of approximation is independent of

Computations of the growth rate (106) showed that the plume is always unstable at a growth rate of

In figure 7 we illustrate the dependence of the preferred mode of instability on the Prandtl number,

In contrast with the unbounded plume where instability is

This instability with growth rate

The preferred mode is associated with plume interfaces that are determined by (39) and (41). The amplitude at

determined by the parameters of the preferred mode for any prescribed values of

The dynamics of a plume of buoyant fluid, in the form of a channel of finite width, rising in a less buoyant fluid contained between two parallel sidewalls, a distance

1) The plume is associated with a vertical flow that is balanced by a down flow on either side of the plume, and the flow inside the plume can develop a reverse (downward) flow around the center of the plume if the plume is wide enough.

2) The flow and concomitant temperature transport material upwards and heat downwards in such a way that the net upward buoyancy flux is positive, and possesses two local maxima and a minimum.

3) The instability of the interfaces has the following main properties:

a) The instability can take one of two modes, which are modifications of the sinuous and varicose modes of the plume in the absence of sidewalls but here modified by the lack of symmetry due to the different positions of the plume relative to the sidewallsb) If the plume is close to a sidewall, the instability has a growth rate

4) The relatively large growth rates of the instability when the plume is close to a sidewall may be due to heat flux emitted by the boundary.

The role of diffusion has been neglected in the present study because it is generally very small. However, it can be expected that it maybe potent when the plume is close to a sidewall. This has been analysed (but not included here) and found to provide small correction. Diffusion may also be potent in thin boundary layers at the interfaces of the plume at

An attempt was made to compare the present results with experimental observations but we have not been able to identify a detailed experimental study on the influence of the boundaries on the plumes rising from mushy layers. However, the results obtained here agree with the general observations of Hell a well et al. [

Here we derive the solvability condition for the first order system (i.e., problem 1) in order to obtain an expression for the growth rate

in which

Next, we consider a function

and satisfies the following conditions

It then follows that

where

and

where

Now we define a function

and introduce the function

and note that

and

where

We multiply equation (A.1) by

and have the proprieties

to obtain the relation

where

We now eliminate the integral involving

which is the expression for the growth rate in terms of the zeroth order variables and the basic state only.