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The effect of two fixed vertical boundaries, a finite distance apart, on the dynamics of a column of buoyant fluid rising in a less buoyant fluid is investigated in the presence of vertical rotation. It is shown that the presence of the boundaries introduces two main effects on a rotating plume. They tend to stabilise the plume but succeed only reducing the value of the growth rate and the plume remains unstable for all finite values of the distance between the boundaries and the plume. In the absence of the sidewalls, two modes of the instability were found known as the sinuous mode and the varicose mode. The influence of the boundaries is such that it reduces the growth rate of the varicose mode more than that of the sinuous mode and consequently the modified sinuous mode is always preferred in the presence of the boundaries.

The study of the dynamics of compositional plumes is important for many real life applications in industry ([

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 [

In Section 2, we formulate the model mathematically and state the boundary conditions of the system. The presence of rotation introduces an additional parameter,

where

The geometry of the problem showing the profile of the basic state concentration of light material,

bounded Cartesian plume. The problem of the rotating plume was studied by Eltayeb and Hamza [

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

Here R,

where

We consider a basic concentration profile

which defines a plume of thickness,

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:

where

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

where

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

where

where

The perturbation variables take the form

in which the factors

Substituting the variables (18) into the Equations (14)-(17) and after some arrangements we get the following equations

subject to the boundary conditions

where the variable

and we have introduced the notation

(cf. Al Mashrafi and Eltayeb [

We use the same method adopted in Al Mashrafi and Eltayeb [

where

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

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,

We note that

We operate on Equation (33) with

The general solution of the differential Equation (40) can be written in the form

where the superscript

with

We use the Equations (34)-(36) and the solution (41) to obtain

The Equation (37) can be solved in the form a complementary function and particular solution to find

where

Now we apply the boundary conditions at

The growth rate is given by the quadratic equation

where

Solving Equation (47) yields

and the displacement

in which

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).

The growth rate (48) is evaluated numerically in the

In

Contours of the growth rate ,

Contours of the growth rate,

a minimum with negative growth rate only when

Isolines of the growth rate ,

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

The solution of (50) gives the values

The preferred mode parameters

The preferred mode parameters

The preferred mode parameters

The preferred mode parameters

In

The preferred mode parameters

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

The dynamics of a fully developed plume of buoyant fluid, in the form of a channel of finite width,

In the absence of boundaries ([

The presence of the vertical boundaries here introduces two dimensionless parameters: the distance between the plume and the nearest wall,