JOuRnal Of extensive ReseaRch e-Print ISSN: 2394-0301
IntroductionThe hydrodynamic instability that manifest under appropriate conditions in a static horizontal initially homogeneous viscous and Boussinesq liquid layer of infinite horizontal extension and finite vertical depth which is kept under the simultaneous action of a uniform vertical temperature gradient and a gravitationally opposite uniform vertical concentration gradient in the force field of gravity is known as thermohaline instability or more general-ly double diffusive convection (Banerjee et al. 1992) which have been first of all, studied by Stern (1960) and Veronis (1965). Dou-ble diffusive convection is now well known and has been exten-sively studied. For a broad view of the subject one may be referred to (Nield, 1967; Baines and Gill, 1969; Turner, 1974; Brandt and Fernando, 1996).
All these researchers have considered the case of two compo-
nent systems. However, it has been recognized later on (Griffiths, 1979; Turner, 1985) that there are many situations wherein more than two components are present. Examples of such multiple diffusive convection fluid systems include the solidification of molten alloys, Earth core, geothermally heated lakes, and magmas and their laboratory models and sea water etc. (Griffiths, 1979; Pearlstein et al. 1989; Lopez et al. 1990) have theoretically studied the onset of convection in a triply diffusive fluid layer (where den-sity depends on three independently diffusing agencies with dif-ferent diffusivities). The summary of the investigations of these researchers is that small concentrations of a third component with a smaller diffusivity can have a significant effect upon the nature of diffusive instabilities and diffusive convection (oscillatory con-vection) and salt finger (Stationary convection) modes are simul-taneously unstable under a wide range of conditions when the density gradients due to components with the greatest and small-est diffusivity are of same signs and further that (Turner, 1985) differential or coupled diffusion can produce convective motions that are associated with a decrease in the gravitational potential energy of the system. It has also been well established theoret-ically and observed experimentally that either the finger or the
AbstractThe present paper mathematically establishes that triply diffusive convection with one of the components as heat with dif-fusivity κ cannot manifest as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the two concentration Rayleigh numbers R1 and R2 , the Lewis numbers τ1 and τ2 for the two concentrations with diffusivities κ1 and
κ2 respectively (with no loss of generality κ > κ1 > κ2) and the Prandtl number σ satisfy the inequality .
It is further established that this result is uniformly valid for the quite general nature of the bounding surfaces.
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diffusive convection modes are possible in multiple component systems even if the overall density stratification is hydrostatically stable (Terrones, 1993). Thus multiple component configurations can, further be classified into two classes, namely the first class in which multiple component convection manifests itself when the total density field is initially bottom heavy and the second class in which the multiple component convection manifests itself when the total density field is initially top heavy. Thus oscillatory mo-tions of growing amplitude can occur in a multiple component fluid layer wherein the total density field is either gravitationally stable or unstable. Banerjee et al. (1993) derived a characterization theorem for thermohaline convection (double diffusive convec-tion) which disallow, in certain parameter regime, the existence of oscillatory motions of growing amplitude in an initially bottom configuration. As a further step, in the present communication, an analogous characterization theorem, in the parameter space of the system alone, is derived for physically more important configura-tion of triply diffusive convection which disallow the existence of oscillatory motions of growing amplitude in an initially bottom heavy triply diffusive convection. The following result is obtained in this direction:
Triply diffusive convection with one of the components as heat with diffusivity κ cannot manifest as oscillatory motions of grow-ing amplitude in an initially bottom heavy configuration if the two concentration Rayleigh numbers R1 and R2 , the Lewis numbers τ1 and τ2 for the two concentrations with diffusivities κ1 and κ2 respectively (with no loss of generality κ > κ1 > κ2) and the
Prandtl number σ satisfy the inequality .
It is further established that this result is uniformly valid for the quite general nature of the bounding surfaces.
The essence of the theorem proved herein lies in that it pro-vides a classification of the neutral or unstable triply diffusive convection configuration into two classes namely, the bottom heavy class and the top heavy class, and then strikes a distinction between them my means of characterization theorem which disal-low the existence of oscillatory instability in the former class. The present work will bring a fresh outlook to the subject matter of triply diffusive convection and pave the way for further theoretical and experimental investigations in this field of enquiry.
MethodsProblem formulationA viscous finitely heat conducting Boussinesq fluid of infinite horizontal extension is statistically confined between two horizon-tal boundaries Z=0 and Z=d which are respectively maintained at uniform temperatures T0 and T1 (< T0) and uniform concentra-tions S10, S20 and S11 (< S10) S21 (< S20), as shown in Fig. 1.
The relevant governing equations and boundary conditions of triply diffusive convection are given by (Pearlstein et al. 1989; Prakash et al. 2014).
Figure 1. Physical Configuration
with boundary conditionsand
(Both the boundaries are rigid)or
(Both the boundaries are dynamically free)
and(upper boundary is dynamically free)or (lower boundary is dynamically free)and
(upper boundary is rigid)Where z is the real independent variable such that 0≤ z≤1, D=d/dz is differentiation with respect to z, a2 is a constant, σ > 0 is a constant, τ1 > 0 is a constant, τ2 > 0 is a constant, R, R1 and R2 are constants whose signs depend on the configuration, p=pr+ipi is a complex constant such that pr and pi are real constants and as a consequence the dependent variables
, ,
,
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
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are complex valued functions of the real variable z. The meaning of the symbols involved in equations (1)-(4) together with the boundary conditions (5) or (6) or (7) or (8) from the physical point of view are as follows: z is the vertical coordinate, D is the differentiation w.r.t. z, a2 is square of the wave number, σ is the Prandtl number, τ1 (=κ1/κ) and τ2 (=κ2/κ) are the Lewis numbers for the two concentrations S1 and S2 respectively where k is ther-mal diffusivity, k1 and k2 are respectively the mass diffusivities of S1 and S2, R is the thermal Rayleigh number, R1 and R2 concentra-tion Rayleigh numbers for the two concentration components, p is the complex growth rate, w is the vertical velocity, θ is the tem-perature and ϕ1 and ϕ2 are the respective concentrations of the two components. It may further be noted that in (1)-(4) together with the boundary conditions (5) or (6) or (7) or (8) describe an eigenvalue problem for p and govern triply diffusive convection for any combination of dynamically free and rigid boundaries.
then a necessary condition for the existence of nontrivial solution(w, θ, ϕ1, ϕ2, p) of (1) - (4) together with boundary conditions (5) or (6) or (7) or (8) is that Rs=R1+R2<R.
Proof: Multiplying (1) by w*(* denotes the complex conjugation) throughout, integrating the resulting equation, by parts, over the vertical range of z, we get
Using (2), (3) and (4), we can write
Equating the real and imaginary parts of both sides of (15) and canceling (pi≠0) throughout from the imaginary part, we get
and
we can rearrange (16) in the following form
We derive the validity of the theorem from the inequality obtained by replacing each one of the terms of (18) by its appropriate es-timate.
Since w, θ, ϕ1 and ϕ2 satisfy w(0)=0=w(1), θ(0)=0=θ(1), ϕ1 (0)=0=ϕ1 (1) and ϕ2 (0)=0=ϕ2(1) we obtain by Rayleigh-Ritz in-equality (Schultz, 1973)
Combining (10) – (13), we get
Integrating the various terms of (14) by parts for a suitable num-ber of times and utilizing either of the boundary conditions (5) – (8), we obtain
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
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Further, since w(0)=0=w(1), it follows that
(using Schwartz inequality)
(using (19))Thus we have that
Combining (19) and (23), we get
and thus by using (19) and (24), we have
Secondly, since pr≥0, we obtain
Further, using (21) and (22), we get
(using (17))
Now multiplying (3) and (4) by their respective complex conju-gates, integrating the resulting equations by parts for an appropri-ate number of times and using the boundary conditions on ϕ1 and ϕ2, we obtain
Since ϕ1 (0)=0=ϕ1 (1) and ϕ2 (0)=0=ϕ2 (1), like (24), we have results for ϕ1 and ϕ2 as
,
.
Further, multiplying (2) by θ* and integrating the various terms on the left hand side of the resulting equation by parts for a suitable number of times by making use of the boundary conditions on θ, namely θ(0)=0=θ(1), we obtain from the real part of the final equation
(using Schwartz inequality)
Combining the above inequality with (20) and the fact that pr≥0, we have
which gives that
and thus
and
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
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Thus utilizing (21) and (31) for ϕ1 and (22) and (32) for ϕ2 in (29) and (30) respectively, and since pr≥0, we obtain
and
Now using (33) and (34), we have
Using (19) and (35) in (28), we have
Let thus
Therefore
Thus (36) becomes
(since κ1>κ2, thus τ1>τ2) and thus
Also from (17) and the fact that pr≥0, we obtain
Now, if permissible, let Rs=R1+R2≥R. Then, in that case, we de-rive from (18) and from (25), (26), (27), (37) and (38) that
which clearly implies that
so that we necessarily have
since the minimum value of
(for a2 = ) is
Hence if
then we must have
This establishes the theorem.
The essential content of the theorem from the physical point of view are that triply diffusive convection with one of the com-ponents as heat with diffusivity κ cannot manifest as oscillatory motions of growing amplitude in an initially bottom heavy config-uration if the two concentration Rayleigh numbers R1 and R2, the Lewis numbers τ1 and τ2 for the two concentrations with diffusiv-ities κ1 and κ2 respectively (with no loss of generality κ> κ1> κ2) and the Prandtl number σ satisfy the inequality
.
It is further proved that this result is uniformly valid for the quite general nature of the bounding surfaces.
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
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ConclusionsLinear stability analysis of triply diffusive convection with one component as heat has been performed. A classification of the neutral or unstable triply diffusive convection configuration into two classes namely, the bottom heavy class and top heavy class has been made then a distinction between these classes is made by means of a characterization theorem which disallow the existence of oscillatory motions in the former class. The work done in the present manuscript will certainly pave the way for further theoreti-cal and experimental investigation in this field of enquiry.
Conflict of interestsThe authors declare that they have no conflict of interests.
AcknowledgementsAuthors thank the learned reviewer for his valuable comments. One of the authors (JP) also acknowledges the financial assistance by UGC, New Delhi in the form of major research project.
References1. Baines, P.G., Gill, A.E., 1969. On thermohaline convection
with linear gradient. J. Fluid Mech. 37, 289 – 306.2. Banerjee, M.B., Gupta, J.R., Prakash, J., 1992. Upper limits to
the complex growth rate in thermohaline instability. J. Math. Anal. Appl. 167 (1), 66-73.
3. Banerjee, M.B., Gupta, J.R., Prakash, J., 1993. On thermo-haline convection of Veronis type. J. Math. Anal. Appl. 179, 327-334.
5. Griffiths, R.W., 1979. The influence of a third diffusing com-ponent upon the onset of convection. J. Fluid Mech. 92, 659 – 670.
6. Lopez, A.R., Romero, L.A., Pearlstein, A.J., 1990. Effect of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer. Phys. of Fluids A. 2 (6), 897 – 902.
7. Nield, D.A., 1967. The thermohaline Rayleigh- Jeffreys prob-lem. J. Fluid Mech. 29, 545-558.
8. Pearlstein, A.J., Harris, R.M., Terrones, G., 1989. The onset of convective instability in a triply diffusive fluid layer. J. Fluid Mech. 202, 443 – 465.
9. Prakash, J., Vaid, K., Bala, R., 2014. Upper limits to the com-plex growth rate in triply diffusive convection. Proc. Ind. Nat. Sci. Acad. 80 (1), 115-122.
15. Veronis, G., 1965. On finite amplitude instability in thermo-haline convection. J. Mar. Res. 23, 1-17.
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Article Information:Received: 01 March 2015 Accepted: 27 March 2015Online published: 28 March 2015
Cite this article as:J Prakash et al. 2015. On characterization of triply diffusive convection. International Journal of Extensive Research. Vol. 3: 87-92.
http://www.journalijer.comPrakash et al. 2015. International J Ext Res. 3:87-92
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