The term "nanofluid" was coined by Choi in his seminal paper presented in 1995 at the ASME Winter Annual Meeting [1]. It refers to a liquid containing a dispersion of submicronic solid particles (nanoparticles) with typical length on the order of 1-50 nm [2]. The unique properties of nanofluids include the impressive enhancement of thermal conductivity as well as overall heat transfer [3–7]. Various mechanisms leading to heat transfer enhancement in nanofluids are discussed in numerous publications; see, for example [8–12].

Recent publications show significant interest in applications of nanofluids in various types of microsystems. These include microchannels [20], microheat pipes [21], microchannel heat sinks [22], and microreactors [23]. There is also significant potential in using nanomaterials in different bio-microsystems, such as enzyme biosensors [24]. In [25], the performance of a bioseparation system for capturing nanoparticles was simulated. There is also strong interest in developing chip-size microdevices for evaluating nanoparticle toxicity; Huh et al. [26] suggested a biomimetic microsystem that reconstitutes the critical functional alveolar-capillary interface of the human lung to evaluate toxic and inflammatory responses of the lung to silica nanoparticles.

The aim of this article is to propose a novel type of a nanofluid that contains both nanoparticles and oxytactic microorganisms, such as a soil bacterium *Bacillus subtilis*. These particular microorganisms are oxygen consumers that swim up the oxygen concentration gradient. There are important similarities and differences between nanoparticles and motile microorganisms. In their impressive review of nanofluids research, Wang and Fan [27] pointed out that nanofluids involve four scales: the molecular scale, the microscale, the macroscale, and the megascale. There is interaction between these scales. For example, by manipulating the structure and distribution of nanoparticles the researcher can impact macroscopic properties of the nanofluid, such as its thermal conductivity. Similar to nanofluids, in suspensions of motile microorganisms that exhibit spontaneous formation of flow patterns (this phenomenon is called bioconvection) physical laws that govern smaller scales lead to a phenomenon visible on a larger scale. While superfluidity and superconductivity are quantum phenomena visible at the macroscale, bioconvection is a mesoscale phenomenon, in which the motion of motile microorganisms induces a macroscopic motion (convection) in the fluid. This happens because motile microorganisms are heavier than water and they generally swim in the upward direction, causing an unstable top-heavy density stratification which under certain conditions leads to the development of hydrodynamic instability. Unlike motile microorganisms, nanoparticles are not self-propelled; they just move due to such phenomena as Brownian motion and thermophoresis and are carried by the flow of the base fluid. On the contrary, motile microorganisms can actively swim in the fluid in response to such stimuli as gravity, light, or chemical attraction. Combining nanoparticles and motile microorganisms in a suspension makes it possible to use benefits of both of these microsystems.

One possible application of bioconvection in bio-microsystems is for mass transport enhancement and mixing, which are important issues in many microsystems [28, 29]. Also, the results presented in [30] suggest using bioconvection in a toxic compound sensor due to the ability of some toxic compounds to inhibit the flagella movement and thus suppress bioconvection. Also, preventing nanoparticles from agglomerating and aggregating remains a significant challenge. One of the reasons why this is challenging is because although inducing mixing at the macroscale is easy and can be achieved by stirring, inducing and controlling mixing at the microscale is difficult. Bioconvection can provide both types of mixing. Macroscale mixing is provided by inducing the unstable density stratification due to microorganisms' upswimming. Mixing at the microscale is provided by flagella (or flagella bundle) motion of individual microorganisms. Due to flagella rotation, microorganisms push fluid along their axis of symmetry, and suck it from the sides [31]. While the estimates given in [32] show that the stresslet stress produced by individual microorganisms have negligible effect on macroscopic motion of the fluid (which is rather driven by the buoyancy force induced by the top-heavy density stratification due to microorganisms' upswimming), the effect produced by flagella rotation is not negligible on the microscopic scale (on the scale of a microorganism and a nanoparticle).

In order to use suspensions containing both nanoparticles and motile microorganisms in microsystems, the behavior of such suspensions must be understood at the fundamental level. Bio-thermal convection caused by the combined effect of upswimming of oxytacic microorganisms and temperature variation was investigated in [33–36]. Bioconvection in nanofluids is expected to occur if the concentration of nanoparticles is small, so that nanoparticles do not cause any significant increase of the viscosity of the base fluid. The problem of bioconvection in suspensions containing small solid particles (nanoparticles) was first studied in [37–41] and then recently in [42]. Non-oscillatory bioconvection in suspensions of oxytactic microorganisms was considered in Kuznetsov AV: **Nanofluid bioconvection: Interaction of microorganisms oxytactic upswimming, nanoparticle distribution and heating/cooling from below**. *Theor Comput Fluid Dyn* 2010, submitted. This article extends the theory to the case of oscillatory convection in suspensions containing both nanoparticles and oxytactic microorganisms.

### Governing equations

The governing equations are formulated for a water-based nanofluid containing nanoparticles and oxytactic microorganisms. The nanofluid occupies a horizontal layer of depth *H*. It is assumed that the nanoparticle suspension is stable. According to Choi [2], there are methods (including suspending nanoparticles using either surfactant or surface charge technology) that lead to stable nanofluids. It is further assumed that the presence of nanoparticles has no effect on the direction of microorganisms' swimming and on their swimming velocity. This is a reasonable assumption if the nanoparticle suspension is dilute; the concentration of nanoparticles has to be small anyway for the bioconvection-induced flow to occur (otherwise, a large concentration of nanoparticles would result in a large suspension viscosity which would suppress bioconvection).

In formulating the governing equations, the terms pertaining to nanoparticles are written using the theory developed in Buongiorno [43], while the terms pertaining to oxytactic microorganisms are written using the approach developed by Hillesdon and Pedley [44, 45].

The continuity equation for the nanoparticle-microorganism suspension considered in this research is

where U = (*u,v,w*) is the dimensionless nanofluid velocity, defined as U**H*/*α*_{f}; U* is the dimensional nanofluid velocity; *α*_{f} is the thermal diffusivity of a nanofluid, *k*/(*ρc*)_{f}; *k* is the thermal conductivity of the nanofluid; and (*ρc*)_{f} is the volumetric heat capacity of the nanofluid. The dimensionless coordinates are defined as (*x,y,z*) = (*x*, y*, z**)/*H*, where *z* is the vertically downward coordinate.

The buoyancy force can be considered to be made up of three separate components that result from: the temperature variation of the fluid, the nanoparticle distribution (nanoparticles are heavier than water), and the microorganism distribution (microorganisms are also heavier than water). Utilizing the Boussinesq approximation (which is valid because the inertial effects of the density stratification are negligible, the dominant term multiplying the inertia terms is the density of the base fluid that exceeds by far the density stratification), the momentum equation can be written as:

$\frac{1}{Pr}\left(\frac{\partial U}{\partial t}+U\cdot \nabla U\right)=-\nabla p+{\nabla}^{2}U+Rm\phantom{\rule{0.5em}{0ex}}\widehat{k}-RaT\widehat{k}+Rn\varphi \widehat{k}+\frac{Rb}{Lb}n\widehat{k}$

(2)

where $\widehat{k}$ is the vertically downward unit vector.

The dimensionless variables in Equation

2 are defined as:

$t={t}^{*}{\alpha}_{\text{f}}/{H}^{2},\phantom{\rule{0.5em}{0ex}}p={p}^{*}{H}^{2}/\mu {\alpha}_{\text{f}},\phantom{\rule{0.5em}{0ex}}\varphi =\frac{{\varphi}^{*}-{\varphi}_{0}^{*}}{{\varphi}_{1}^{*}-{\varphi}_{0}^{*}},\phantom{\rule{0.5em}{0ex}}T=\frac{{T}^{*}-{T}_{c}^{*}}{{T}_{h}^{*}-{T}_{c}^{*}},\phantom{\rule{0.5em}{0ex}}n={n}^{*}/{n}_{0}^{\ast}$

(3)

where *t* is the dimensionless time, *p* is the dimensionless pressure, *ϕ* is the relative nanoparticle volume fraction, *T* is the dimensionless temperature, *n* is the dimensionless concentration of microorganisms, *t** is the time, *p*^{
*
}is the pressure, *μ* is the viscosity of the suspension (containing the base fluid, nanoparticles and microorganisms), *ϕ*^{
*
}is the nanoparticle volume fraction, ${\varphi}_{0}^{\ast}$ is the nanoparticle volume fraction at the lower wall, ${\varphi}_{1}^{\ast}$ is the nanoparticle volume fraction at the upper wall, *T** is the nanofluid temperature, ${T}_{c}^{\ast}$ is the temperature at the upper wall (also used as a reference temperature), ${T}_{h}^{\ast}$ is the temperature at the lower wall, *n** is the concentration of microorganisms, and ${n}_{0}^{\ast}$ is the average concentration of microorganisms (concentration of microorganisms in a well-stirred suspension).

The dimensionless parameters in Equation

2, namely, the Prandtl number,

*Pr*; the basic-density Rayleigh number,

*Rm*; the traditional thermal Rayleigh number,

*Ra*; the nanoparticle concentration Rayleigh number,

*Rn*; the bioconvection Rayleigh number,

*Rb*; and the bioconvection Lewis number,

*Lb*, are defined as follows:

$Pr=\frac{\mu}{{\rho}_{\text{f0}}{\alpha}_{\text{f}}},\phantom{\rule{0.5em}{0ex}}Rm=\frac{\left[{\rho}_{\text{p}}{\varphi}_{0}^{*}+{\rho}_{\text{f0}}(1-{\varphi}_{0}^{*})\right]g{H}^{3}}{\mu {\alpha}_{\text{f}}},\phantom{\rule{0.5em}{0ex}}Ra=\frac{{\rho}_{\text{f0}}g\beta {H}^{3}\left({T}_{h}^{*}-{T}_{c}^{*}\right)}{\mu {\alpha}_{\text{f}}}$

(4)

$Rn=\frac{({\rho}_{\text{p}}-{\rho}_{\text{f0}})({\varphi}_{1}^{*}-{\varphi}_{0}^{*})g{H}^{3}}{\mu {\alpha}_{\text{f}}},\phantom{\rule{0.5em}{0ex}}Rb=\frac{\Delta \rho g\theta {n}_{0}^{\ast}{H}^{3}}{\mu {D}_{\text{mo}}},\phantom{\rule{0.5em}{0ex}}Lb=\frac{{\alpha}_{\text{f}}}{{D}_{\text{mo}}}$

(5)

where *ρ*_{
f0
}is the base-fluid density at the reference temperature; *ρ*_{
p
}is the nanoparticle mass density; *g* is the gravity; *β* is the volumetric thermal expansion coefficient of the base fluid; Δ*ρ* is the density difference between microorganisms and a base fluid, *ρ*_{mo}*- ρ*_{f0}; *ρ*_{mo} is the microorganism mass density; *θ* is the average volume of a microorganism; and *D*_{mo} is the diffusivity of microorganisms (in this model, following [44, 45], all random motions of microorganisms are simulated by a diffusion process).

The conservation equation for nanoparticles contains two diffusion terms on the right-hand side, which represent the Brownian diffusion of nanoparticles and their transport by thermophoresis (a detailed derivation is available in [

43,

46]):

$\frac{\partial \varphi}{\partial t}+U\cdot \nabla \varphi =\frac{1}{Ln}{\nabla}^{2}\varphi +\frac{{N}_{A}}{Ln}{\nabla}^{2}T$

(6)

In Equation

6, the nanoparticle Lewis number,

*Ln*, and a modified diffusivity ratio,

*N*_{
A
}(this parameter is somewhat similar to the Soret parameter that arises in cross-diffusion phenomena in solutions), are defined as:

$Ln=\frac{{\alpha}_{\text{f}}}{{D}_{\text{B}}},\phantom{\rule{0.5em}{0ex}}{N}_{A}=\frac{{D}_{\text{T}}\left({T}_{h}^{*}-{T}_{c}^{*}\right)}{{D}_{\text{B}}{T}_{c}^{*}({\varphi}_{1}^{*}-{\varphi}_{0}^{*})}$

(7)

where *D*_{B} is the Brownian diffusion coefficient of nanoparticles and *D*_{T} is the thermophoretic diffusion coefficient.

The right-hand side of the thermal energy equation for a nanofluid accounts for thermal energy transport by conduction in a nanofluid as well as for the energy transport because of the mass flux of nanoparticles (again, a detailed derivation is available in [

43,

46]):

$\frac{\partial T}{\partial t}+U\cdot \nabla T={\nabla}^{2}T+\frac{{N}_{B}}{Ln}\nabla \varphi \cdot \nabla T+\frac{{N}_{A}{N}_{B}}{Ln}\nabla T\cdot \nabla T$

(8)

In Equation

8,

*N*_{
B
}is a modified particle-density increment, defined as:

${N}_{B}=\frac{{(\rho c)}_{\text{p}}}{{(\rho c)}_{\text{f}}}\left({\varphi}_{1}^{*}-{\varphi}_{0}^{*}\right)$

(9)

where (*ρc*)_{p} is the volumetric heat capacity of the nanoparticles.

The right-hand side of the equation expressing the conservation of microorganisms describes three modes of microorganisms transport: due to macroscopic motion (convection) of the fluid, due to self-propelled directional swimming of microorganisms relative to the fluid, and due diffusion, which approximates all stochastic motions of microorganisms:

$\frac{\partial n}{\partial t}=-\nabla \cdot \left(nU+nV-\frac{1}{Lb}\nabla n\right)$

(10)

where

**V** is the dimensionless swimming velocity of a microorganism,

**V***

*H*/

*α*_{f}, which is calculated as [

44,

45]:

$V=\frac{Pe}{Lb}\widehat{H}\left(C\right)\nabla C$

(11)

In Equation

11 $\widehat{H}$ is the Heaviside step function and

*C* is the dimensionless oxygen concentration, defined as:

$C=\frac{{C}^{\ast}-{C}_{\mathrm{min}}^{\ast}}{{C}_{0}^{\ast}-{C}_{\mathrm{min}}^{\ast}}$

(12)

where *C** is the dimensional oxygen concentration, ${C}_{0}^{\ast}$ is the upper-surface oxygen concentration (the upper surface is assumed to be open to atmosphere), and ${C}_{\mathrm{min}}^{\ast}$ is the minimum oxygen concentration that microorganisms need to be active. Equation 11 thus assumes that microorganisms swim up the oxygen concentration gradient and that their swimming velocity is proportional to that gradient; however, in order for microorganisms to be active the oxygen concentration need to be above ${C}_{\mathrm{min}}^{\ast}$. Since this article deals with a shallow layer situation, it is assumed that ${C}^{\ast}>{C}_{\mathrm{min}}^{\ast}$ throughout the layer thickness, and the Heaviside step function, $\widehat{H}\left(C\right)$, in Equation 11 is equal to unity.

Also, the bioconvection Péclet number,

*Pe*, in Equation

11 is defined as:

$Pe=\frac{b{W}_{\text{mo}}}{{D}_{\text{mo}}}$

(13)

where *b* is the chemotaxis constant (which has the dimension of length) and *W*_{mo} is the maximum swimming speed of a microorganism (the product *bW*_{mo} is assumed to be constant).

Finally, the oxygen conservation equation is:

$\frac{\partial C}{\partial t}+U\cdot \nabla C=\frac{\text{1}}{Le}{\nabla}^{2}C-\widehat{\beta}n$

(14)

The first term on the right-hand side of Equation 14 represents oxygen diffusion, while the second term represents oxygen consumption by microorganisms.

The new dimensionless parameters in Equation

14 are

$Le=\frac{{\alpha}_{\text{f}}}{{D}_{S}},\phantom{\rule{0.5em}{0ex}}\widehat{\beta}=\frac{\gamma {H}^{2}{n}_{0}^{\ast}}{\left({C}_{0}^{\ast}-{C}_{\mathrm{min}}^{\ast}\right)\phantom{\rule{0.5em}{0ex}}{\alpha}_{\text{f}}}$

(15)

where *Le* is the traditional Lewis number, $\widehat{\beta}$ is the dimensionless parameter describing oxygen consumption by the microorganisms, *D*_{
S
}is the diffusivity of oxygen, and *γ* is a dimensional constant describing consumption of oxygen by the microorganisms.

According to Hillesdon and Pedley [

45], the layer can be treated as shallow as long as the following condition is satisfied:

$H\le {\left(\frac{2{\left(\mathrm{exp}\left(Pe\right)-1\right)}^{1/2}}{Pe\phantom{\rule{0.5em}{0ex}}Le}\frac{\left({C}_{0}^{\ast}-{C}_{\mathrm{min}}^{\ast}\right)\phantom{\rule{0.5em}{0ex}}{\alpha}_{f}}{\gamma {n}_{0}^{\ast}}{\mathrm{tan}}^{-1}\left[{\left(\mathrm{exp}\left(Pe\right)-1\right)}^{1/2}\right]\right)}^{1/2}$

(16)

Equation 16 gives the maximum layer depth for which the oxygen concentration at the bottom does not drop below ${C}_{\mathrm{min}}^{\ast}$.

The boundary conditions for Equations

1,

2,

6,

8,

10, and 14 are imposed as follows. It is assumed that the temperature and the volumetric fraction of the nanoparticles are constant on the boundaries and the flux of microorganisms through the boundaries is equal to zero. The lower boundary is always assumed rigid and the upper boundary can be either rigid or stress-free. The boundary conditions for case of a rigid upper wall are

$w=0,\phantom{\rule{0.5em}{0ex}}\frac{\partial w}{\partial z}=\text{0,}\phantom{\rule{0.5em}{0ex}}T=1,\phantom{\rule{0.5em}{0ex}}\varphi =0,\phantom{\rule{0.5em}{0ex}}\frac{\text{d}n}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}\frac{\partial C}{\partial z}=\text{0}\phantom{\rule{0.5em}{0ex}}\text{at}z=1\phantom{\rule{0.5em}{0ex}}\left(\text{thelowerwall}\right)$

(17)

$w=0,\phantom{\rule{0.5em}{0ex}}\frac{\partial w}{\partial z}=\text{0,}\phantom{\rule{0.5em}{0ex}}T=0,\phantom{\rule{0.5em}{0ex}}\varphi =1,\phantom{\rule{0.5em}{0ex}}Pe\phantom{\rule{0.5em}{0ex}}n\frac{\text{d}C}{\text{d}z}-\frac{\text{d}n}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}C=\text{1}\phantom{\rule{0.5em}{0ex}}\text{at}z=0\phantom{\rule{0.5em}{0ex}}\left(\text{theupperwall}\right)$

(18)

The fifth equation in (18) is equivalent to the statement that the total flux of microorganisms at the upper surface is equal to zero: the microorganisms swim vertically upward at the top surface but (because their concentration gradient at the top surface is directed vertically upward) they are simultaneously pushed downward by diffusion; the two fluxes are equal but opposite in direction).

If the upper surface is stress-free, the second equation in (18) is replaced with the following equation:

$\frac{{\partial}^{2}w}{\partial {z}^{2}}=\text{0}$

(19)

### Basic state

The solution for the basic state corresponds to a time-independent quiescent situation. The solution is of the following form:

${U}_{\text{b}}=0,\phantom{\rule{0.5em}{0ex}}p={p}_{\text{b}}\text{(}z\text{),}\phantom{\rule{0.5em}{0ex}}T={T}_{\text{b}}(z),\phantom{\rule{0.5em}{0ex}}\varphi ={\varphi}_{\text{b}}\text{(}z\text{),}\phantom{\rule{0.5em}{0ex}}n={n}_{\text{b}}\text{(}z\text{),}\phantom{\rule{0.5em}{0ex}}C={C}_{\text{b}}\left(z\right)$

(20)

In this case, the solution of Equations

6,

8,

10, and 14 subjects to boundary conditions (17) and (18) is (the particular form of hydrodynamic boundary conditions at the upper surface is not important because the solution in the basic state is quiescent):

${\varphi}_{\text{b}}\left(z\right)=-{N}_{A}\frac{\mathrm{exp}\left[\frac{\left(1-{N}_{A}\right){N}_{B}}{Ln}z\right]-1}{\mathrm{exp}\left[\frac{\left(1-{N}_{A}\right){N}_{B}}{Ln}\right]-1}-(1-{N}_{A})z+1$

(21)

${T}_{\text{b}}\left(z\right)=\frac{\mathrm{exp}\left[\frac{\left(1-{N}_{A}\right){N}_{B}}{Ln}z\right]-1}{\mathrm{exp}\left[\frac{\left(1-{N}_{A}\right){N}_{B}}{Ln}\right]-1}$

(22)

${n}_{\text{b}}\left(z\right)=\frac{{A}_{1}^{2}}{\text{2}Pe\phantom{\rule{0.5em}{0ex}}\widehat{\beta}Le}{\mathrm{sec}}^{2}\left(\frac{{A}_{1}\left(1-z\right)}{2}\right)$

(23)

${C}_{\text{b}}\left(z\right)=1-\frac{2}{Pe}\mathrm{ln}\left(\frac{\mathrm{cos}\left\{{A}_{1}\left(1-z\right)/2\right\}}{\mathrm{cos}\left\{{A}_{1}/2\right\}}\right)$

(24)

where

*A*_{1} is the smallest positive root of the transcendental equation

$\mathrm{tan}\left(\frac{{A}_{1}}{2}\right)=Pe\frac{\widehat{\beta}Le}{{A}_{1}}$

(25)

The solutions given by Equations 23 and 24 were first reported in [44].

The pressure distribution in the basic state,

*p*_{b} (

*z*), can then be obtained by integrating the following form of the momentum equation (which follows from Equation

2):

$-\frac{\text{d}{p}_{\text{b}}}{\text{d}z}+Rm-Ra\phantom{\rule{0.5em}{0ex}}{T}_{\text{b}}+Rn\phantom{\rule{0.5em}{0ex}}{\varphi}_{\text{b}}+\frac{Rb}{Lb}{n}_{\text{b}}=0$

(26)

Equations 21 and 22 can be simplified if characteristic parameter values for a typical nanofluid are considered. Based on the data presented in Buongiorno [

43] for an alumina/water nanofluid, the following dimensional parameter values are utilized:

${\varphi}_{0}^{*}=0.01$,

*α*_{f} = 2 × 10

^{-7}m

^{2}/s,

*D*_{B} = 4 × 10

^{-11}m

^{2}/s,

*μ* = 10

^{-3} Pas, and

*ρ*_{f0} = 10

^{3} kg/m

^{3}. The thermophoretic diffusion coefficient,

*D*_{T}, is estimated as

$\tau \frac{\mu}{\rho}{\varphi}_{0}^{\ast}$, where, according to Buongiorno [

43],

*τ* is estimated as 0.006. This results in

*D*_{T} = 6 × 10

^{-11}m

^{2}/s. The nanoparticle Lewis number is then estimated as

*Ln* = 5.0 × 10

^{3}. The modified diffusivity ratio,

*N*_{
A
}, and the modified particle-density increment,

*N*_{
B
}, depend on the temperature difference between the lower and the upper plates and on the nanoparticle fraction decrement. Assuming that

${T}_{h}^{*}-{T}_{c}^{*}=1\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\text{K}$,

${\varphi}_{1}^{*}-{\varphi}_{0}^{*}=0.001$, and

${T}_{c}^{*}=300\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\text{K}$, gives the following estimates:

*N*_{
A
}= 5 and

*N*_{
B
}= 7.5 × 10

^{-4}. This suggests that the exponents in Equations

21 and

22 are small and that these equations can be simplified as:

${\varphi}_{\text{b}}\left(z\right)=1-z$

(27)

${T}_{\text{b}}\left(z\right)=z$

(28)

### Linear instability analysis

Perturbations are superimposed on the basic solution, as follows:

$\begin{array}{l}\left[U,T,\varphi ,n,C,p\right]=\left[0,{T}_{\text{b}}\left(z\right),{\varphi}_{\text{b}}\left(z\right),{n}_{\text{b}}\left(z\right),{C}_{\text{b}}\left(z\right),{p}_{\text{b}}\left(z\right)\right]\\ \phantom{\rule{0.5em}{0ex}}+\epsilon \left[{U}^{\prime}\left(t,x,y,z\right),{T}^{\prime}\left(t,x,y,z\right),{\varphi}^{\prime}\left(t,x,y,z\right),{n}^{\prime}\left(t,x,y,z\right),{C}^{\prime}\left(t,x,y,z\right),{p}^{\prime}\left(t,x,y,z\right)\right]\end{array}$

(29)

Equation 29 is then substituted into Equations

1,

2,

6,

8,

10, and 14, the resulting equations are linearized and the use is made of Equations

27 and

28. This procedure results in the following equations for the perturbation quantities:

$\nabla \cdot {U}^{\prime}=0$

(30)

$\frac{1}{Pr}\frac{\partial {U}^{\prime}}{\partial t}=-\nabla {p}^{\prime}+{\nabla}^{2}{U}^{\prime}-Ra{T}^{\prime}\widehat{k}+Rn{\varphi}^{\prime}\widehat{k}+\frac{Rb}{Lb}{n}^{\prime}\widehat{k}$

(31)

$\frac{\partial {T}^{\prime}}{\partial t}+{w}^{\prime}={\nabla}^{2}{T}^{\prime}+\frac{{N}_{B}}{Ln}\left(\frac{\partial {\varphi}^{\prime}}{\partial z}-\frac{\partial {T}^{\prime}}{\partial z}\right)+\frac{2{N}_{A}{N}_{B}}{Ln}\frac{\partial {T}^{\prime}}{\partial z}$

(32)

$\frac{\partial {\varphi}^{\prime}}{\partial t}-{w}^{\prime}=\frac{1}{Ln}{\nabla}^{2}{\varphi}^{\prime}+\frac{{N}_{A}}{Ln}{\nabla}^{2}{T}^{\prime}$

(33)

$\frac{\partial {n}^{\prime}}{\partial t}+{w}^{\prime}\frac{d{n}_{\text{b}}}{dz}+\frac{Pe}{Lb}\left(\frac{\partial {C}^{\prime}}{\partial z}\frac{d{n}_{\text{b}}}{dz}+\frac{d{C}_{\text{b}}}{dz}\frac{\partial {n}^{\prime}}{\partial z}+{n}^{\prime}\frac{{d}^{2}{C}_{\text{b}}}{d{z}^{2}}+{n}_{\text{b}}{\nabla}^{2}{C}^{\prime}\right)=\frac{1}{Lb}{\nabla}^{2}{n}^{\prime}$

(34)

$\frac{\partial {C}^{\prime}}{\partial t}+{w}^{\prime}\frac{\text{d}{C}_{\text{b}}}{\text{d}z}=\frac{\text{1}}{Le}{\nabla}^{2}{C}^{\prime}-\widehat{\beta}{n}^{\prime}$

(35)

Equations 30 to 35 are independent of

*Rm* since this parameter is just a measure of the basic static pressure gradient. In order to eliminate the pressure and horizontal components of velocity from Equations

30 and

31, Equation

31 (see [

46]) is operated with

$\widehat{k}\cdot \text{curlcurl}$ and the use is made of the identity curl curl ≡ grad div - ∇

^{2} together with Equation

30. This results in the reduction of Equations

30 and

31 to the following scalar equation which involves only one component of the perturbation velocity,

*w*':

$\frac{1}{Pr}\frac{\partial}{\partial t}{\nabla}^{2}{w}^{\prime}-{\nabla}^{4}{w}^{\prime}=-Ra{\nabla}_{\text{H}}^{2}{T}^{\prime}+Rn{\nabla}_{\text{H}}^{2}{\varphi}^{\prime}+\frac{Rb}{Lb}{\nabla}_{\text{H}}^{2}{n}^{\prime}$

(36)

where ${\nabla}_{\text{H}}^{2}$ is the two-dimensional Laplacian operator in the horizontal plane and ∇^{4}*w'* is the Laplacian of the Laplacian of *w'*.

Equations 17 and 18 then lead to the following boundary conditions for the perturbation quantities for the case when both the lower and upper walls are rigid:

${w}^{\prime}=0,\phantom{\rule{0.5em}{0ex}}\frac{\partial {w}^{\prime}}{\partial z}=\text{0,}\phantom{\rule{0.5em}{0ex}}{T}^{\prime}=0,\phantom{\rule{0.5em}{0ex}}{\varphi}^{\prime}=0,\phantom{\rule{0.5em}{0ex}}\frac{\text{d}{n}^{\prime}}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}\frac{\text{d}{C}^{\prime}}{\text{d}z}=\text{0}\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}z=1\phantom{\rule{0.5em}{0ex}}\left(\text{thelowerwall}\right)$

(37)

${w}^{\prime}=0,\phantom{\rule{0.5em}{0ex}}\frac{\partial {w}^{\prime}}{\partial z}=\text{0,}\phantom{\rule{0.5em}{0ex}}{T}^{\prime}=0,\phantom{\rule{0.5em}{0ex}}{\varphi}^{\prime}=0,\phantom{\rule{0.5em}{0ex}}Pe\left[{n}_{\text{b}}\frac{\text{d}{C}^{\prime}}{\text{d}z}+\frac{\text{d}{C}_{\text{b}}}{\text{d}z}{n}^{\prime}\right]-\frac{\text{d}{n}^{\prime}}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}{C}^{\prime}=\text{0}\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}z=0\phantom{\rule{0.5em}{0ex}}\left(\text{theupperwall}\right)$

(38)

If the upper boundary is stress-free, the second equation in Equation

38 is replaced by

$\frac{{\partial}^{2}{w}^{\prime}}{\partial {z}^{2}}=\text{0}\phantom{\rule{0.5em}{0ex}}\text{at}z=0$

(39)

The method of normal modes is used to solve a linear boundary-value problem composed of differential Equations

32 to

36 and boundary conditions (37), (38) (or (39)). A normal mode expansion is introduced as:

$\left[{w}^{\prime},{T}^{\prime},{\varphi}^{\prime},{n}^{\prime},{C}^{\prime}\right]=\left[W(z),\Theta (z),\Phi (z),N\left(z\right),\Xi \left(z\right)\right]f\left(x,y\right)\mathrm{exp}(st),$

(40)

where the function

*f*(

*x,y*) satisfies the following equation:

$\frac{{\partial}^{2}f}{\partial {x}^{2}}+\frac{{\partial}^{2}f}{\partial {y}^{2}}=-{m}^{2}f$

(41)

and *m* is the dimensionless horizontal wavenumber.

Substituting Equation

40 into Equations

36 and

32 to

35, utilizing Equation

41, and letting

$\Xi \to \widehat{\beta}\phantom{\rule{0.5em}{0ex}}\overline{\Xi}$ (so that the resulting equation for amplitudes would depend on the product

$\varpi =Pe\widehat{\beta}$ rather than on

*Pe* and

$\widehat{\beta}$ individually), the following equations for the amplitudes,

*W*, Θ, Φ,

*N*, and

$\overline{\Xi}$, are obtained:

$\frac{{\text{d}}^{4}W}{\text{d}{z}^{4}}-2{m}^{2}\frac{{\text{d}}^{2}W}{\text{d}{z}^{2}}+{m}^{4}W-\frac{s}{Pr}\frac{{\text{d}}^{2}W}{\text{d}{z}^{2}}+{m}^{2}\frac{s}{\mathrm{Pr}}W+Ra\phantom{\rule{0.5em}{0ex}}{m}^{2}\Theta -Rn\phantom{\rule{0.5em}{0ex}}{m}^{2}\Phi -\frac{Rb}{Lb}{m}^{2}N=0$

(42)

$-W+\frac{{\text{d}}^{2}\Theta}{\text{d}{z}^{2}}-\frac{{N}_{B}}{Ln}\frac{\text{d}\Theta}{\text{d}z}+\frac{2{N}_{A}{N}_{B}}{Ln}\frac{\text{d}\Theta}{\text{d}z}-\left({m}^{2}+s\right)\Theta +\frac{{N}_{B}}{Ln}\frac{\text{d}\Phi}{\text{d}z}=0$

(43)

$-W+\frac{{N}_{A}}{Ln}{m}^{2}\Theta +\frac{1}{Ln}{m}^{2}\Phi +s\Phi -\frac{{N}_{A}}{Ln}\frac{{\text{d}}^{2}\Theta}{\text{d}{z}^{2}}-\frac{1}{Ln}\frac{{\text{d}}^{2}\Phi}{\text{d}{z}^{2}}=0$

(44)

$\begin{array}{l}-2{A}_{1}\phantom{\rule{0.5em}{0ex}}Le\phantom{\rule{0.5em}{0ex}}\varpi \mathrm{tan}\left[\frac{1}{2}{A}_{1}\left(1-z\right)\right]\frac{\text{d}N}{\text{d}z}-{A}_{1}^{3}\phantom{\rule{0.5em}{0ex}}{\mathrm{sec}}^{2}\left[\frac{1}{2}{A}_{1}\left(1-z\right)\right]\mathrm{tan}\left[\frac{1}{2}{A}_{1}\left(1-z\right)\right]\left(Lb\phantom{\rule{0.5em}{0ex}}W+\varpi \frac{\text{d}\overline{\Xi}}{\text{d}z}\right)\\ +2\phantom{\rule{0.5em}{0ex}}Le\phantom{\rule{0.5em}{0ex}}\varpi \left({m}^{2}N-\frac{{\text{d}}^{2}N}{\text{d}{z}^{2}}\right)+{A}_{1}^{2}\phantom{\rule{0.5em}{0ex}}\varpi {\mathrm{sec}}^{2}\left[\frac{1}{2}{A}_{1}\left(1-z\right)\right]\left(Le\phantom{\rule{0.5em}{0ex}}N-{m}^{2}\overline{\Xi}+\frac{{\text{d}}^{2}\overline{\Xi}}{\text{d}{z}^{2}}\right)+2Lb\phantom{\rule{0.5em}{0ex}}Le\phantom{\rule{0.5em}{0ex}}\varpi \phantom{\rule{0.5em}{0ex}}s\phantom{\rule{0.5em}{0ex}}N=0\end{array}$

(45)

$\varpi N-{A}_{1}\mathrm{tan}\left(\frac{1}{2}{A}_{1}\left(1-z\right)\right)W+\frac{\varpi {m}^{2}\overline{\Xi}}{\text{Le}}+\varpi s\phantom{\rule{0.5em}{0ex}}\overline{\Xi}-\frac{\varpi}{Le}\frac{{\text{d}}^{2}\overline{\Xi}}{\text{d}{z}^{2}}=0$

(46)

where Equation

25 for

*A*_{1} is reduced to

$\mathrm{tan}\left(\frac{{A}_{1}}{2}\right)=\frac{\varpi Le}{{A}_{1}}$

(47)

In Equations 42 to 46 *s* is a dimensionless growth factor; for neutral stability the real part of *s* is zero, so it is written *s* = *iω*, where *ω* is a dimensionless frequency (it is a real number).

For the case of rigid-rigid walls, the boundary conditions for the amplitudes are

$W=0,\phantom{\rule{0.5em}{0ex}}\frac{\text{d}W}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}\Theta =0,\phantom{\rule{0.5em}{0ex}}\Phi =0,\phantom{\rule{0.5em}{0ex}}\frac{\text{d}N}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}\frac{\text{d}\overline{\Xi}}{\text{d}z}=\text{0}\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}z=1\phantom{\rule{0.5em}{0ex}}\left(\text{thelowerwall}\right)$

(48)

$W=0,\phantom{\rule{0.5em}{0ex}}\frac{\text{d}W}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}\Theta =0,\phantom{\rule{0.5em}{0ex}}\Phi =0,\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{\varpi {n}_{\text{b}}|}_{z=0}\frac{\text{d}\overline{\Xi}}{\text{d}z}+Pe{\frac{\text{d}{C}_{\text{b}}}{\text{d}z}|}_{z=0}\phantom{\rule{0.5em}{0ex}}N-\frac{\text{d}N}{\text{d}z}=\text{0,}\phantom{\rule{0.5em}{0ex}}\overline{\Xi}=\text{0}\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}z=0\phantom{\rule{0.5em}{0ex}}\left(\text{theupperwall}\right)$

(49)

If the upper surface is stress-free, the second equation in (49) is replaced by

$\frac{{\text{d}}^{2}W}{\text{d}{z}^{2}}=\text{0}\phantom{\rule{0.5em}{0ex}}\text{at}\phantom{\rule{0.5em}{0ex}}z=0$

(50)

Equations 42 to 46 are solved by a single-term Galerkin method. For the case of the rigid-rigid boundaries, the trial functions, which satisfy the boundary conditions given by Equations

48 and

49, are

${W}_{1}={z}^{2}{(1-z)}^{2},\phantom{\rule{0.5em}{0ex}}{\Theta}_{1}=z(1-z),\phantom{\rule{0.5em}{0ex}}{\Phi}_{1}=z(1-z),\phantom{\rule{0.5em}{0ex}}{N}_{1}=1+\alpha \left(z-\frac{1}{2}{z}^{2}\right),\phantom{\rule{0.5em}{0ex}}{\overline{\Xi}}_{1}=z-\frac{1}{2}{z}^{2}$

(51)

where

$\alpha =\frac{{A}_{1}\left({A}_{1}-Le\phantom{\rule{0.5em}{0ex}}\mathrm{sin}{A}_{1}\right)}{Le\left(1+\mathrm{cos}{A}_{1}\right)}$

(52)

and *A*_{1} is given by Equation 47.

If the upper boundary is stress-free,

*W*_{1} is replaced by

${W}_{1}=z-3{z}^{3}+2{z}^{4}$

(53)

and the rest of the trial functions are still given by Equation 51. *W*_{1} given by Equation 53 satisfies the boundary condition given by Equation 50.