**3. Conservation equations for a nanofluid layer**

We start this section with the description of Boussinesq approximation which is used to write the conservation equations of nanofluids in simplified form.

As is the case of regular fluid [32], equations of nanofluids are difficult to solve because of their non-linear character. Therefore some mathematical approximations are to be used to simplify the basic equations without violating the physical laws. The contribution of Boussinesq [40] in the solution of thermal instability problems is in the form of approximations which is after his name. This approximation has been used by a many researchers for solving different problems of fluids. Boussinesq suggested that inertial effects of density variations can be neglected as compared to its gravitational effects as such situations exist in the domain of meteorology and oceanography. So, density is assumed to be constant everywhere in the equations of motion except in the term with external force. Therefore, we change *ρ*0½ � 1 þ *α*ð Þ *T*<sup>0</sup> � *T* by *ρ*<sup>0</sup> everywhere in the equations of motion except the term representing the external body force.

Anoop et al. [41] explained various experimental techniques using which nanoparticles can be suspended in the base fluid and that suspension remain stable for several weeks. Buongiorno [6] adopted the formalism of Bird et al. [42] and Chandrasekhar [32] to write conservation equations for nanofluids by considering nanoscale effects; Brownian diffusion and thermophoresis. A model for convective transport in regular fluids was reformulated for nanofluids to accommodate these nanoscale effects as follows.

The random motion of nanoparticles is called Brownian motion and results into the continuous collisions with the base fluid molecules. The Brownian diffusion coefficient due to Brownian motion is given by

$$D\_B = \frac{k\_B T}{3\pi\mu d\_p},\tag{1}$$

temperature across the upper and lower boundaries of the fluid becomes the main reason for the occurrence of instability. **Figure 2** shows the schematic representation of Rayleigh-Bénard convection. Rayleigh [28] was the first person who gave an analytical treatment of the problem related to identifying the conditions responsible for breakdown of basic state. As a subsequent work carried out by Rayleigh and Bénard, thermal instability of fluids is known as Rayleigh Bénard convection. The condition for convective motions (depends on temperature gradient) can be represented in dimensionless form by the critical Rayleigh number. He figured out the condition for the instability of free surfaces by showing that the instability would occur on a large temperature gradient *β* ¼ �*dT=dz* in such a way that the

tion due to gravity is represented by g, coefficient of thermal expansion by *α*, the depth of the layer by d, thermal diffusivity by *κ* and kinematic viscosity is given by *ν*. For the stabilizing viscous force, *RA* parameter gives the force of destabilizing buoyancy. Chandra [29] found discrepancy between the theoretical and experimental work for the convective motions in fluids when heated underside. He explained it by conducting an experiment on the layer in air and observed that instability of the fluid layer was dependent on its depth. A simplification in the partial differential equations describing the flow of compressible fluid is done by Spiegel and Veronis [30] by assuming very small depth of the layer as compared to the height. The basic equations of a fluid layer in porous medium (when heated underside) were formulated and derived by Joseph [31] by using Boussinesq approximation. The problem of thermal convection of a fluid layer has been put forward by Chandrasekhar [32] by considering the implications of various aspects of hydrodynamics and hydromagnetics. He depicted the result that addition of rotation and magnetic field increases the stability of the system. Kim et al. [33] considered the same problem of thermal convection for nanofluids. They showed that convective motion directly depends on the two physical properties (heat capacity and density) of nanoparticles and adversely depends on the conductivity of nanoparticles. Buongiorno [6] was the first scientist who formulated the conservation equations of nanofluids by assimilating the effects of diffusion due to Brownian motion and thermophoresis of nanoparticles. During his analysis, he concluded that Brownian and thermophoretic diffusion play a significant role in the absence of turbulent effects as compared to other seven mechanisms. Hwang et al. [34] treated this problem analytically and put forth the result of thermal instability of water based nanofluid with alumina nanoparticles in a rectangular container which is heated from below. They found that stability of the base fluid is enhanced by adding alumina nanoparticles and further it is enhanced by increasing the volume fraction of nanoparticles, the average temperature of the nanofluids and by decreasing the size of nanoparticles. They observed the decrease in heat transfer coefficient of nanofluids with the increase of the size of nanoparticles and decrease in the

*=κν*, exceeds a certain critical value; where accelera-

Rayleigh number; *RA* <sup>¼</sup> *αβgd*<sup>4</sup>

*A view of Rayleigh-Bénard convection.*

*Applications of Nanobiotechnology*

**Figure 2.**

**116**

where *dp* is the nanoparticle's diameter, *kB* is the Boltzmann's constant and *μ* is the viscosity of the fluid. The nanoparticles mass flux due to Brownian diffusion, *j <sup>p</sup>*,*<sup>B</sup>* is given as

$$\dot{\mathbf{j}}\_{\mathbf{p},\mathbf{B}} = -\rho\_p D\_\mathbf{B} \nabla \phi,\tag{2}$$

**3.1 Equation of state**

*DOI: http://dx.doi.org/10.5772/intechopen.88887*

state can be written as

where *T0* is the temperature at which *ρ = ρ0*.

*Convection Currents in Nanofluids under Small Temperature Gradient*

**3.2 Equation of continuity-conservation of mass**

The equation of continuity for nanofluids is

so that the Eq. (7) reduces to

can be written as

**119**

*∂ρ ∂t* þ *uj ∂ρ ∂xj*

> *∂ρ ∂t* þ *uj ∂ρ ∂xj*

> > *∂uj ∂xj*

and in vector form continuity equation for nanofluid is expressed as

The conservation equation for nanoparticles in absence of chemical reactions is

*ρp* ∇*:j*

*<sup>p</sup>*,*<sup>T</sup>* ¼ �*ρpDB*∇*ϕ* � *ρpDT*

<sup>þ</sup> *<sup>v</sup>:*∇*<sup>ϕ</sup>* ¼ � *<sup>1</sup>*

diffusion terms (Brownian diffusion and thermophoresis) using Eqs. (1) and (4)

Combining Eqs. (11) and (12), nanoparticles conservation equation becomes

where *uj* is the jth component of nanofluid's velocity. For an incompressible flow (using equation of state)

**3.3 Equation of nanoparticles-conservation of mass**

*j <sup>p</sup>* ¼ *j* *∂ϕ ∂t*

where *t* is the time, *ϕ* is the nanoparticles volume fraction and *j*

mass flux for nanoparticles and as external forces are negligible *j*

*<sup>p</sup>*,*<sup>B</sup>* þ *j*

Variables of state depend only upon the state of a system. The physical quantities: *p*; the pressure,*T*; the temperature and *ρ*; the density are the variables of state. We have three thermodynamic variables and a relation between them is given as.

For substances with which we shall be principally concerned, the equation of

¼ �*ρ*

*∂uj ∂xj*

*F p*ð Þ¼ , *ρ*, *T 0*, (5)

*ρ* ¼ *ρ*0½ � 1 þ *α*ð Þ *T*<sup>0</sup> � *T* , (6)

, (7)

¼ 0, (8)

¼ 0, (9)

∇*:v* ¼ 0*:* (10)

∇*T*

*<sup>p</sup>*, (11)

*<sup>p</sup>* is the diffusion

*<sup>p</sup>*, the sum of two

*<sup>T</sup>* , (12)

where *ϕ* is the nanoparticle volume fraction and *ρ<sup>p</sup>* is the nanoparticle mass density.

Thermophoresis is the phenomenon in which particles diffuse due to temperature gradient and the effect is similar to one of well-known effects of solute; Soret effect. The thermophoretic velocity is defined as.

$$\mathbf{V}\_T = -\tilde{\boldsymbol{\beta}} \frac{\mu}{\rho} \frac{\nabla T}{T} \text{ where } \tilde{\boldsymbol{\beta}} = \mathbf{0}.26 \frac{k}{2k + k\_p}. \tag{3}$$

Here, *ρ* is the overall density of the nanofluid, *k* and *kp* are the thermal conductivities of the fluid and the particle material, respectively. The negative sign in thermophoretic velocity represents movement of particles down the temperature gradient (from hot to cold). The nanoparticle mass flux due to thermophoresis, *j <sup>p</sup>*,*<sup>T</sup>* is given as.

$$\dot{\mathbf{y}}\_{\mathbf{p},T} = -\rho\_p \boldsymbol{\phi} \mathbf{V}\_T = -\rho\_p D\_T \frac{\nabla T}{T} \text{ with } D\_T = \tilde{\boldsymbol{\beta}} \frac{\mu}{\rho} \boldsymbol{\phi}, \tag{4}$$

where *DT* represents the thermophoretic diffusion coefficient.

The nanoparticles mass flux due to Brownian diffusion (Eq. (1)) and thermophoresis (Eq. (4)) are used to develop a two-component model for convective transport in nanofluids with the following assumptions:


The seven equations based on basic conservation laws with the above mentioned assumptions are given as follows.


*Convection Currents in Nanofluids under Small Temperature Gradient DOI: http://dx.doi.org/10.5772/intechopen.88887*

### **3.1 Equation of state**

where *dp* is the nanoparticle's diameter, *kB* is the Boltzmann's constant and *μ* is the viscosity of the fluid. The nanoparticles mass flux due to Brownian diffusion,

where *ϕ* is the nanoparticle volume fraction and *ρ<sup>p</sup>* is the nanoparticle mass density. Thermophoresis is the phenomenon in which particles diffuse due to temperature gradient and the effect is similar to one of well-known effects of solute; Soret

*<sup>T</sup>* where *<sup>β</sup>*<sup>~</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>26</sup> *<sup>k</sup>*

∇*T*

Here, *ρ* is the overall density of the nanofluid, *k* and *kp* are the thermal conductivities of the fluid and the particle material, respectively. The negative sign in thermophoretic velocity represents movement of particles down the temperature gradient (from hot to

effect. The thermophoretic velocity is defined as.

*<sup>V</sup><sup>T</sup>* ¼ �*β*<sup>~</sup> *<sup>μ</sup>*

cold). The nanoparticle mass flux due to thermophoresis, *j*

*<sup>p</sup>*,*<sup>T</sup>* ¼ �*ρpϕV<sup>T</sup>* ¼ �*ρpDT*

tive transport in nanofluids with the following assumptions:

• There are no chemical reactions in the fluid layer.

• The viscous dissipation is negligible in the fluid.

• The radiative heat transfer is negligible.

assumptions are given as follows.

• Equation of state (one).

• Equation of continuity (one).

• Equations of motion (three).

• Equation of energy (one).

**118**

• Equation of nanoparticles (one).

where *DT* represents the thermophoretic diffusion coefficient. The nanoparticles mass flux due to Brownian diffusion (Eq. (1)) and thermophoresis (Eq. (4)) are used to develop a two-component model for convec-

• The mixture is dilute with nanoparticle volume fraction less than 1%.

• The nanoparticles and base fluid are locally in thermal equilibrium.

The seven equations based on basic conservation laws with the above mentioned

*j*

• The nanofluid flow is incompressible.

• The external forces are negligible.

*ρ* ∇*T*

*jp*,*<sup>B</sup>* ¼ �*ρpDB*∇*ϕ*, (2)

2*k* þ *kp*

*<sup>T</sup>* with *DT* <sup>¼</sup> *<sup>β</sup>*~*<sup>μ</sup>*

*<sup>p</sup>*,*<sup>T</sup>* is given as.

*ρ*

*:* (3)

*ϕ*, (4)

*j*

*<sup>p</sup>*,*<sup>B</sup>* is given as

*Applications of Nanobiotechnology*

Variables of state depend only upon the state of a system. The physical quantities: *p*; the pressure,*T*; the temperature and *ρ*; the density are the variables of state. We have three thermodynamic variables and a relation between them is given as.

$$F(p,\rho,T) = \mathbf{0},\tag{5}$$

For substances with which we shall be principally concerned, the equation of state can be written as

$$
\rho = \rho\_0 [1 + a(T\_0 - T)],\tag{6}
$$

where *T0* is the temperature at which *ρ = ρ0*.

#### **3.2 Equation of continuity-conservation of mass**

The equation of continuity for nanofluids is

$$\frac{\partial \rho}{\partial t} + u\_j \frac{\partial \rho}{\partial \mathbf{x}\_j} = -\rho \frac{\partial u\_j}{\partial \mathbf{x}\_j},\tag{7}$$

where *uj* is the jth component of nanofluid's velocity. For an incompressible flow (using equation of state)

$$\frac{\partial \rho}{\partial t} + u\_j \frac{\partial \rho}{\partial x\_j} = \mathbf{0},\tag{8}$$

so that the Eq. (7) reduces to

$$\frac{\partial \mu\_j}{\partial \mathbf{x}\_j} = \mathbf{0},\tag{9}$$

and in vector form continuity equation for nanofluid is expressed as

$$
\nabla.\mathfrak{v} = \mathbf{0}.\tag{10}
$$

#### **3.3 Equation of nanoparticles-conservation of mass**

The conservation equation for nanoparticles in absence of chemical reactions is

$$\frac{\partial \phi}{\partial t} + \mathfrak{v} . \nabla \phi = -\frac{1}{\rho\_p} \nabla . \mathfrak{j}\_p,\tag{11}$$

where *t* is the time, *ϕ* is the nanoparticles volume fraction and *j <sup>p</sup>* is the diffusion mass flux for nanoparticles and as external forces are negligible *j <sup>p</sup>*, the sum of two diffusion terms (Brownian diffusion and thermophoresis) using Eqs. (1) and (4) can be written as

$$\dot{j}\_{\mathbf{p}} = \dot{j}\_{\mathbf{p},\mathbf{B}} + \dot{j}\_{\mathbf{p},T} = -\rho\_p D\_\mathbf{B} \nabla \phi - \rho\_p D\_T \frac{\nabla T}{T} \,, \tag{12}$$

Combining Eqs. (11) and (12), nanoparticles conservation equation becomes

$$\frac{\partial \phi}{\partial t} + \boldsymbol{\nu} . \nabla \phi = \nabla . \left[ D\_B \nabla \phi + D\_T \frac{\nabla T}{T} \right]. \tag{13}$$

ð Þ *ρc*

Note that if *j*

parameters.

*x*0 , *y*<sup>0</sup>

*ρα<sup>f</sup> μ*

> *∂T ∂t*

**121**

Rayleigh number as.

where *<sup>α</sup><sup>f</sup>* <sup>¼</sup> *<sup>k</sup>*

*∂v ∂t*

þ *v:*∇*v* 

<sup>þ</sup> *<sup>v</sup>:*∇*<sup>T</sup>* <sup>¼</sup> <sup>∇</sup><sup>2</sup>

, *<sup>z</sup>*<sup>0</sup> ð Þ¼ ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*

*<sup>d</sup>* , *<sup>t</sup>*

*<sup>ρ</sup> <sup>C</sup> :*.

<sup>0</sup> <sup>¼</sup> *<sup>t</sup>α<sup>f</sup>*

¼ �∇*<sup>p</sup>* <sup>þ</sup> <sup>∇</sup><sup>2</sup>

*<sup>T</sup>* <sup>þ</sup> ð Þ *<sup>ρ</sup><sup>C</sup>* <sup>P</sup> *<sup>ρ</sup><sup>C</sup> <sup>ϕ</sup><sup>b</sup>*

where thermal Rayleigh number *RA* <sup>¼</sup> *<sup>ρ</sup> <sup>g</sup> <sup>β</sup>Td*<sup>3</sup>

<sup>þ</sup> *<sup>v</sup>:*∇*<sup>ϕ</sup>* <sup>¼</sup> *DB*

Let us add perturbations to initial solution and write

*∂ϕ ∂t*

**4. Initial and perturbed flow**

*∂T ∂t*

þ *v:*∇*T* 

*DOI: http://dx.doi.org/10.5772/intechopen.88887*

<sup>¼</sup> *<sup>k</sup>*∇<sup>2</sup>

*Convection Currents in Nanofluids under Small Temperature Gradient*

*T* þ ð Þ *ρc <sup>p</sup> DB*∇*ϕ:*∇*T* þ *DT*

equation for regular fluid and therefore last two terms on right-hand side truly account for contributions of nanoparticle motion relative to fluid. Eq. (20) establishes that the transport of heat in nanofluids is possible by convection (second term on left-hand side), by conduction (first term on right-hand side), and also by virtue

Thus, Eqs. (10), (13), (16), (20) constitute the convective transport model for nanofluids which further can be solved for different parameters once the initial and boundary conditions are known. It is interesting to note that all the equations are strongly coupled meaning thereby that the one parameter depends on various other

Let us introduce non-dimensional variables to get the expression for thermal

, *<sup>p</sup>*<sup>0</sup> <sup>¼</sup> *p d*<sup>2</sup> *μα<sup>f</sup>*

, *<sup>ϕ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>ϕ</sup> ϕb*

*k* �

*DT*ð Þ *T*<sup>1</sup> � *T*<sup>0</sup> *DB T*<sup>0</sup> *ϕ<sup>b</sup>*

*DT*ð Þ *T*<sup>1</sup> � *T*<sup>0</sup> *DB T*<sup>0</sup> *ϕ<sup>b</sup>*

> ð Þ *T*1�*T*<sup>0</sup> *μα<sup>f</sup>*

*:*

*vi* ¼ 0, *ϕ<sup>i</sup>* ¼ 1, *Ti* ¼ 1 � *z* (26)

ð Þ¼ *<sup>v</sup>*, *<sup>p</sup>*, *<sup>T</sup>*, *<sup>ϕ</sup> <sup>v</sup><sup>i</sup>* <sup>þ</sup> *<sup>v</sup>*~, *pi* <sup>þ</sup> *<sup>p</sup>*~, *Ti* <sup>þ</sup> *<sup>T</sup>*~, *<sup>ϕ</sup><sup>i</sup>* <sup>þ</sup> *<sup>ϕ</sup>*<sup>~</sup> *:* (27)

∇*:v* ¼ 0, (22)

*ρ<sup>p</sup>* � *ρ <sup>ϕ</sup><sup>b</sup> <sup>g</sup> <sup>d</sup>*<sup>3</sup> *μα<sup>f</sup>*

> *DB αf* ∇2

ð Þ *ρC* <sup>P</sup> *<sup>ρ</sup><sup>C</sup> <sup>ϕ</sup><sup>b</sup>*

, *<sup>T</sup>*<sup>0</sup> <sup>¼</sup> *<sup>T</sup>* � *<sup>T</sup>*<sup>0</sup> *T*<sup>1</sup> � *T*<sup>0</sup>

, (21)

*ϕ* ^

*T*, (25)

*DB αf*

*k*, (23)

∇*T:*∇*T*,

(24)

of nanoparticle diffusion (second and third terms on right-hand side).

*<sup>d</sup>*<sup>2</sup> , *<sup>v</sup>*<sup>0</sup> <sup>¼</sup> *<sup>v</sup> <sup>d</sup> αf*

Using Eqs. (21), (10), (13), (16), (20) after dropping the dashes are

*<sup>v</sup>* � *<sup>ρ</sup>gd*<sup>3</sup> *μα<sup>f</sup>* ^ *<sup>k</sup>* <sup>þ</sup> *RAT*^

∇*ϕ:*∇*T* þ

At the initial state, it is assumed that nanoparticle volume fraction is constant and fluid layer is still while temperature and pressure vary in horizontal direction. We get initial solution of Eqs. (22)–(25) using the fact that thermal diffusivity is very large as compared to Brownian diffusion coefficient (refer Buongiorno [1]) as

*DB αf*

*αf* ∇2 *ϕ* þ

*<sup>p</sup>* is zero, Eq.(19) and hence Eq. (20) reduces to the familiar energy

∇*T:*∇*T T :* (20)

Eq. (13) reveals that the nanoparticles move consistently with fluid (second term of left-hand side) and possess velocity relative to fluid (right-hand side) due to Brownian diffusion and thermophoresis.

#### **3.4 Equations of motion-conservation of momentum**

The equation of motion is derived from Newton's second law of motion which states that

Rate of change of linear momentum ¼ Total force*:*

The momentum equation for nanofluid with negligible external forces is

$$
\rho\_0 \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} . \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g}, \tag{14}
$$

where *ρ*<sup>0</sup> is the nanofluid density at the reference temperature *T*<sup>0</sup> and the overall density of nanofluid; written as

$$
\rho = \phi \rho\_p + (\mathbf{1} - \phi)\rho\_f \cong \phi \rho\_p + (\mathbf{1} - \phi)\{\rho\_0(\mathbf{1} - a(T - T\_0))\}\tag{15}
$$

Thus Eq. (14) becomes

$$\rho\_0 \left( \frac{\partial \mathbf{v}}{\partial t} + \boldsymbol{\nu} . \nabla \boldsymbol{\nu} \right) = -\nabla p + \mu \nabla^2 \boldsymbol{\nu} + \left( \phi \rho\_p + (1 - \phi) \{ \rho\_0 (1 - a(T - T\_0)) \} \right) \mathbf{g}. \tag{16}$$

Note that in the absence of nanoparticles, Eq. (16) reduces to momentum equation for regular fluid.

#### **3.5 Equation of energy-conservation of energy**

The thermal energy equation for nanofluid with the assumptions (i)–(v) is

$$(\rho c) \left[ \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T \right] = -\nabla \cdot \mathbf{q} + h\_p \nabla \cdot \mathbf{j}\_p,\tag{17}$$

where *c* and *hp* are the specific heat of fluid (at constant pressure) and the specific enthalpy of nanoparticles, respectively and *q* is the energy flux, neglecting radiative heat transfer, the sum of heat fluxes due to conduction and nanoparticle diffusion, written as

$$q = -k\nabla T + h\_p j\_p,\tag{18}$$

Substituting Eq. (18) in Eq. (17), we get

$$\mathbb{E}\left(\rho\mathbf{c}\right)\left[\frac{\partial T}{\partial t} + \boldsymbol{\nu}.\nabla T\right] = \nabla.\left(k\nabla T\right) - c\_p \mathbf{j}\_p.\nabla T.\tag{19}$$

with assumption of negligible external forces ∇*hp* ¼ *cp*∇*T*. Substituting Eq. (12) in Eq. (19); gives final form of thermal energy equation as

*Convection Currents in Nanofluids under Small Temperature Gradient DOI: http://dx.doi.org/10.5772/intechopen.88887*

$$(\rho c) \left[ \frac{\partial T}{\partial t} + \mathfrak{v} .\nabla T \right] = k \nabla^2 T + (\rho c)\_p \left[ D\_B \nabla \phi .\nabla T + D\_T \frac{\nabla T .\nabla T}{T} \right]. \tag{20}$$

Note that if *j <sup>p</sup>* is zero, Eq.(19) and hence Eq. (20) reduces to the familiar energy equation for regular fluid and therefore last two terms on right-hand side truly account for contributions of nanoparticle motion relative to fluid. Eq. (20) establishes that the transport of heat in nanofluids is possible by convection (second term on left-hand side), by conduction (first term on right-hand side), and also by virtue of nanoparticle diffusion (second and third terms on right-hand side).

Thus, Eqs. (10), (13), (16), (20) constitute the convective transport model for nanofluids which further can be solved for different parameters once the initial and boundary conditions are known. It is interesting to note that all the equations are strongly coupled meaning thereby that the one parameter depends on various other parameters.

Let us introduce non-dimensional variables to get the expression for thermal Rayleigh number as.

$$(\mathbf{x'}, \mathbf{y'}, \mathbf{z'}) = \frac{(\mathbf{x}, \mathbf{y}, \mathbf{z})}{d}, \mathbf{t'} = \frac{\mathbf{t}a\_f}{d^2}, \mathbf{z'} = \frac{\mathbf{v}\,d}{a\_f}, \mathbf{p'} = \frac{\mathbf{p}\,d^2}{\mu a\_f}, \boldsymbol{\phi'} = \frac{\boldsymbol{\phi}}{\phi\_b}, \mathbf{T'} = \frac{T - T\_0}{T\_1 - T\_0}, \quad \text{(21)}$$

where *<sup>α</sup><sup>f</sup>* <sup>¼</sup> *<sup>k</sup> <sup>ρ</sup> <sup>C</sup> :*.

*∂ϕ ∂t*

**3.4 Equations of motion-conservation of momentum**

*ρ*0 *∂v ∂t*

¼ �∇*<sup>p</sup>* <sup>þ</sup> *<sup>μ</sup>*∇<sup>2</sup>

**3.5 Equation of energy-conservation of energy**

ð Þ *ρc*

Substituting Eq. (18) in Eq. (17), we get

ð Þ *ρc*

*∂T ∂t*

in Eq. (19); gives final form of thermal energy equation as

þ *v:*∇*T* 

*∂T ∂t*

þ *v:*∇*T* 

density of nanofluid; written as

Thus Eq. (14) becomes

þ *v:*∇*v* 

equation for regular fluid.

diffusion, written as

**120**

Brownian diffusion and thermophoresis.

*Applications of Nanobiotechnology*

states that

*ρ*0 *∂v ∂t*

þ *v:*∇*ϕ* ¼ ∇*: DB*∇*ϕ* þ *DT*

Eq. (13) reveals that the nanoparticles move consistently with fluid (second term of left-hand side) and possess velocity relative to fluid (right-hand side) due to

The equation of motion is derived from Newton's second law of motion which

Rate of change of linear momentum ¼ Total force*:*

¼ �∇*<sup>p</sup>* <sup>þ</sup> *<sup>μ</sup>*∇<sup>2</sup>

*ρ* ¼ *ϕρ<sup>p</sup>* þ ð Þ 1 � *ϕ ρ<sup>f</sup>* ffi *ϕρ<sup>p</sup>* þ ð Þ 1 � *ϕ ρ*<sup>0</sup> f g ð Þ 1 � *α*ð Þ *T* � *T*<sup>0</sup> (15)

*v* þ *ϕρ<sup>p</sup>* þ ð Þ 1 � *ϕ ρ*<sup>0</sup> f g ð Þ 1 � *α*ð Þ *T* � *T*<sup>0</sup> 

¼ �∇*:q* þ *hp*∇*:jp*, (17)

*<sup>p</sup>:*∇*T:* (19)

*q* ¼ �*k*∇*T* þ *hpjp*, (18)

¼ ∇*:*ð Þ� *k*∇*T cpj*

with assumption of negligible external forces ∇*hp* ¼ *cp*∇*T*. Substituting Eq. (12)

where *ρ*<sup>0</sup> is the nanofluid density at the reference temperature *T*<sup>0</sup> and the overall

Note that in the absence of nanoparticles, Eq. (16) reduces to momentum

The thermal energy equation for nanofluid with the assumptions (i)–(v) is

where *c* and *hp* are the specific heat of fluid (at constant pressure) and the specific enthalpy of nanoparticles, respectively and *q* is the energy flux, neglecting radiative heat transfer, the sum of heat fluxes due to conduction and nanoparticle

The momentum equation for nanofluid with negligible external forces is

þ *v:*∇*v*  ∇*T T*

*:* (13)

*v* þ *ρg*, (14)

*g:* (16)

Using Eqs. (21), (10), (13), (16), (20) after dropping the dashes are

$$
\nabla \cdot \mathbf{z} = \mathbf{0},
\tag{22}
$$

$$\frac{\rho a\_f}{\mu} \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}.\nabla \mathbf{v} \right) = -\nabla p + \nabla^2 \mathbf{v} - \frac{\rho \mathbf{g} d^3}{\mu a\_f} \hat{k} + R\_A T \hat{k} - \frac{\left(\rho\_p - \rho\right) \phi\_b \mathbf{g} \, d^3}{\mu a\_f} \phi \, \hat{k}, \tag{23}$$

$$\frac{\partial T}{\partial t} + \mathfrak{v}.\nabla T = \nabla^2 T + \frac{(\rho \mathbf{C})\_\mathbf{p}}{\rho \mathbf{C}} \phi\_b \frac{D\_\mathbf{B}}{a\_f} \nabla \phi.\nabla T + \frac{D\_T (T\_1 - T\_0)}{D\_\mathbf{B} T\_0 \phi\_b} \frac{(\rho \mathbf{C})\_\mathbf{p}}{\rho \mathbf{C}} \phi\_b \frac{D\_\mathbf{B}}{a\_f} \nabla T.\nabla T,\tag{24}$$

$$\frac{\partial \phi}{\partial t} + \mathfrak{v} \cdot \nabla \phi = \frac{D\_B}{a\_{\!\!f}} \nabla^2 \phi + \frac{D\_T (T\_1 - T\_0)}{D\_B T\_0 \, \phi\_b} \frac{D\_B}{a\_{\!\!f}} \nabla^2 T,\tag{25}$$

where thermal Rayleigh number *RA* <sup>¼</sup> *<sup>ρ</sup> <sup>g</sup> <sup>β</sup>Td*<sup>3</sup> ð Þ *T*1�*T*<sup>0</sup> *μα<sup>f</sup> :*

### **4. Initial and perturbed flow**

At the initial state, it is assumed that nanoparticle volume fraction is constant and fluid layer is still while temperature and pressure vary in horizontal direction. We get initial solution of Eqs. (22)–(25) using the fact that thermal diffusivity is very large as compared to Brownian diffusion coefficient (refer Buongiorno [1]) as

$$\{v\_i = 0, \phi\_i = 1, T\_i = 1 - z\}\tag{26}$$

Let us add perturbations to initial solution and write

$$\rho(\boldsymbol{\mathfrak{v}}, \boldsymbol{p}, T, \boldsymbol{\phi}) = \left(\boldsymbol{\mathfrak{v}}\_{i} + \boldsymbol{\tilde{\sigma}}, \boldsymbol{p}\_{i} + \boldsymbol{\tilde{p}}, T\_{i} + \boldsymbol{\tilde{T}}, \ \boldsymbol{\phi}\_{i} + \boldsymbol{\tilde{\phi}}\right). \tag{27}$$

The Eq. (27) in Eqs. (22)–(25) give

$$
\nabla \tilde{\boldsymbol{\sigma}} = \mathbf{0},
\tag{28}
$$

using of the orthogonality to the functions; gives eigenvalue equation as

� *<sup>α</sup>*<sup>2</sup> *J DT <sup>ρ</sup><sup>p</sup>* � *<sup>ρ</sup>*

0 @

@

þ *RA*

� �ð Þ *<sup>T</sup>*<sup>1</sup> � *<sup>T</sup>*<sup>0</sup> *<sup>g</sup> <sup>d</sup>*<sup>3</sup>

� �ð Þ *<sup>T</sup>*<sup>1</sup> � *<sup>T</sup>*<sup>0</sup> *<sup>g</sup> <sup>d</sup>*<sup>3</sup>

0 !

þ *RA*

þ *s*

1 A

*:* (40)

<sup>2</sup>*k*þ*kp* as given by Nield and

*A*; (41)

1 A ¼ 0

*J DB αf* þ *s* 1 A

(39)

1 A

*DBμ α*<sup>2</sup> *<sup>f</sup> T*<sup>0</sup>

> *DBμ α*<sup>2</sup> *<sup>f</sup> T*<sup>0</sup>

*J DT ρ<sup>p</sup>* � *ρ*

For non-oscillatory motions *s* ¼ 0, this gives the expression for *RA* from Eq. (39) as

*<sup>ρ</sup> <sup>ϕ</sup>* with *<sup>β</sup>*<sup>~</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>26</sup> *<sup>k</sup>*

*ρ<sup>p</sup>* � *ρ* � �

2*k* þ *kp*

**7. Discussions on analytical results using various metallic/non-metallic**

**Table 2** shows the ratios of density to conductivity of various metallic/nonmetallic nanoparticles and density 997.1 and conductivity 0.613 of water is used. It is observed that ratio of density to conductivity is accountable for hastening the onset of convection in the system. The ratio is more for non-metals than metals establishes the lesser stability of non-metallic nanoparticles than metals. Alumina is

**Physical properties Al Cu Ag Fe Al2O3 SiO2 CuO TiO2** *<sup>ρ</sup>* Kg*=*m<sup>3</sup> ð Þ 2700 9000 10,500 7900 3970 2600 6510 4250 k Wð Þ *=*m*K* 237 401 429 80 40 10.4 18 8.9 *ρp=*k*<sup>p</sup>* 11.3 22.4 24.47 98.7 99.25 250 361.6 477.5

� �ð Þ *<sup>T</sup>*<sup>1</sup> � *<sup>T</sup>*<sup>0</sup> *<sup>g</sup> <sup>d</sup>*<sup>3</sup> *DBμ αfT*<sup>0</sup>

*DT ρ<sup>p</sup>* � *ρ*

*RA* <sup>¼</sup> *<sup>J</sup>* 3 *α*<sup>2</sup> �

ð Þ *J* þ *s*

0 @

0 @

�*α*<sup>2</sup> *J*

*J DB αf* þ *s* !

*J DT ρ<sup>p</sup>* � *ρ*

where *<sup>J</sup>* <sup>¼</sup> *<sup>π</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>*2*:*

**6. Results and discussion**

**6.1 Stationary convection**

where *DB* <sup>¼</sup> *kBT*

**nanoparticles**

**Table 2.**

**123**

Kuznetsov [35]. Also

*J* 2 þ

*DOI: http://dx.doi.org/10.5772/intechopen.88887*

� �ð Þ *<sup>T</sup>*<sup>1</sup> � *<sup>T</sup>*<sup>0</sup> *<sup>g</sup> <sup>d</sup>*<sup>3</sup>

*DBμ α*<sup>2</sup> *<sup>f</sup> T*<sup>0</sup>

*ραfJs μ* � �

*Convection Currents in Nanofluids under Small Temperature Gradient*

*RA* <sup>¼</sup> *<sup>J</sup>* 3 *α*<sup>2</sup> �

<sup>3</sup>*πμdp* and *DT* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *<sup>μ</sup>*

where *A* depends on the base fluid properties.

*Ratios of density to conductivity of metallic and non-metallic nanoparticles.*

$$\frac{\rho a\_f}{\mu} \frac{\partial \bar{\boldsymbol{\sigma}}}{\partial t} = -\nabla \bar{\boldsymbol{p}} + \nabla^2 \bar{\boldsymbol{\sigma}} + R\_A \bar{\boldsymbol{T}} \hat{\boldsymbol{k}} - \frac{\left(\rho\_p - \rho\right) \phi\_b \mathbf{g} \; d^3}{\mu a\_f} \bar{\phi} \hat{\boldsymbol{k}},\tag{29}$$

$$\frac{\partial \tilde{T}}{\partial t} - \tilde{u}\_3 = \nabla^2 \tilde{T} - \frac{(\rho C)\_\text{p}}{\rho \text{C}} \phi\_b \frac{D\_\text{B}}{a\_f} \frac{\partial \tilde{\phi}}{\partial \mathbf{z}} - 2 \frac{D\_T (T\_1 - T\_0)}{D\_\text{B} T\_0 \phi\_b} \frac{(\rho C)\_\text{p}}{\rho \text{C}} \phi\_b \frac{D\_\text{B}}{a\_f} \frac{\partial \tilde{T}}{\partial \mathbf{z}},\tag{30}$$

$$\frac{\partial \tilde{\phi}}{\partial t} = \frac{D\_B}{a\_f} \nabla^2 \tilde{\phi} + \frac{D\_T (T\_1 - T\_0)}{D\_B T\_0 \phi\_b} \frac{D\_B}{a\_f} \nabla^2 \tilde{T}. \tag{31}$$

Making use of the identity *curlcurl* � *graddiv* � <sup>∇</sup><sup>2</sup> curlcurl <sup>¼</sup> graddiv � <sup>∇</sup><sup>2</sup> on Eq. (29) together with Eq. (28), we get

$$
\rho \frac{\rho a\_f}{\mu} \frac{\partial}{\partial t} \left( \nabla^2 \tilde{u}\_3 \right) - \nabla^4 \tilde{u}\_3 = R\_A \nabla\_H^2 \tilde{T} - \frac{\left( \rho\_p - \rho \right) \phi\_b \mathbf{g} \, d^3}{\mu a\_f} \nabla\_H^2 \tilde{\phi}, \tag{32}
$$

where ∇<sup>2</sup> <sup>H</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> *:*.

### **5. Method of normal modes**

To change PDE's to ODE's, Eqs. (30)–(32) are solved using normal mode analysis and perturbed variables are written as

$$\left(\ddot{\mu}\_{\text{3}}, \ddot{T}, \ddot{\phi}\right) = \left(\mathcal{W}(z), \mathcal{T}(z), \Phi(z)\right) \exp\left(ik\_{\text{x}}\varkappa + ik\_{\text{y}}\wp + \text{st}\right), \tag{33}$$

Thus above mentioned equations reduce to

$$\left(\left(D^2 - a^2\right)^2 - \frac{\rho \alpha\_f}{\mu} \left(D^2 - a^2\right)\right) W - R\_A a^2 \mathcal{T} + \frac{\left(\rho\_p - \rho\right) \oint\_b \phi\_b \mathbf{g} \, d^3 \mathbf{r}}{\mu a\_f} a^2 \Phi = 0,\tag{34}$$

$$W + \left(\left(D^2 - a^2\right) - s - 2 \frac{D\_T (T\_1 - T\_0) \left(\rho \mathcal{C}\right)\_\mathbf{p}}{D\_B T\_0 \phi\_b} \phi\_b \frac{D\_B}{\rho \mathcal{C}} D\right) \mathcal{T} - \frac{\left(\rho \mathcal{C}\right)\_\mathbf{p}}{\rho \mathcal{C}} \phi\_b \frac{D\_B}{a\_f} D \Phi = 0,\tag{35}$$

$$
\left(\frac{D\_B}{a\_f}\left(D^2 - a^2\right) - \varsigma\right)\Phi + \frac{D\_T(T\_1 - T\_0)}{D\_B T\_0 \,\phi\_b} \frac{D\_B}{a\_f} \left(D^2 - a^2\right) \mathbf{T} = \mathbf{0},\tag{36}
$$

where *<sup>D</sup>* � *<sup>d</sup> dz*, *α* ¼ *kx* <sup>2</sup> <sup>þ</sup> *ky* <sup>2</sup> � �<sup>1</sup><sup>⁄</sup> <sup>2</sup> *:* Using one term Galerkin weighted residual method and free-free boundaries conditions

$$
\mathcal{W} = \; \mathcal{D}^2 \mathcal{W} = \; \mathcal{T} \; \; = \mathbf{0} \; \text{at} \; \; z = \mathbf{0} \; \text{ and } z = \mathbf{1}. \tag{37}
$$

We write

$$W = A \sin \pi z, \text{ and } T = B \sin \pi z,\tag{38}$$

*Convection Currents in Nanofluids under Small Temperature Gradient DOI: http://dx.doi.org/10.5772/intechopen.88887*

using of the orthogonality to the functions; gives eigenvalue equation as

$$\begin{split} & \left( (J+s) \left( \frac{ID\_B}{a\_f} + s \right) \left( J^2 + \frac{\rho a\_f l \varepsilon}{\mu} \right) - a^2 \left( \frac{ID\_T \left( \rho\_p - \rho \right) (T\_1 - T\_0) \mathbf{g} \cdot d^3}{D\_B \mu \, a\_f^2 T\_0} + R\_A \left( \frac{ID\_B}{a\_f} + s \right) \right) \right) \\ & - a^2 \Im \left( \frac{ID\_T \left( \rho\_p - \rho \right) (T\_1 - T\_0) \mathbf{g} \, d^3}{D\_B \mu \, a\_f^2 T\_0} + R\_A \left( \frac{ID\_T \left( \rho\_p - \rho \right) (T\_1 - T\_0) \mathbf{g} \, d^3}{D\_B \mu \, a\_f^2 T\_0} + s \right) \right) = 0 \end{split} \tag{39}$$

where *<sup>J</sup>* <sup>¼</sup> *<sup>π</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>*2*:*

The Eq. (27) in Eqs. (22)–(25) give

*<sup>∂</sup><sup>t</sup>* ¼ �∇*p*~<sup>þ</sup> <sup>∇</sup><sup>2</sup>

*DB αf*

� � � <sup>∇</sup><sup>4</sup>*u*~<sup>3</sup> <sup>¼</sup> *RA*∇<sup>2</sup>

*∂ϕ*~ *<sup>∂</sup><sup>z</sup>* � <sup>2</sup>

*<sup>T</sup>*<sup>~</sup> � ð Þ *<sup>ρ</sup><sup>C</sup>* <sup>P</sup> *<sup>ρ</sup><sup>C</sup> <sup>ϕ</sup><sup>b</sup>*

Eq. (29) together with Eq. (28), we get

*<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> *:*.

and perturbed variables are written as

*μ <sup>D</sup>*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup> � � � �

> *<sup>D</sup>*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup> � � � *<sup>s</sup>* !

> > *dz*, *α* ¼ *kx*

method and free-free boundaries conditions

*<sup>W</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>2</sup>

Thus above mentioned equations reduce to

*ρα<sup>f</sup> μ ∂ ∂t* ∇2 *u*~3

<sup>H</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

**5. Method of normal modes**

*<sup>D</sup>*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup> � �<sup>2</sup> � *<sup>s</sup>ρα<sup>f</sup>*

*<sup>W</sup>* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup> � � � *<sup>s</sup>* � <sup>2</sup>

*DB αf*

where *<sup>D</sup>* � *<sup>d</sup>*

We write

**122**

*∂ϕ*~ *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *DB αf* ∇2 *<sup>ϕ</sup>*<sup>~</sup> <sup>þ</sup>

*<sup>v</sup>*~<sup>þ</sup> *RAT*<sup>~</sup> ^

*k* �

*DT*ð Þ *T*<sup>1</sup> � *T*<sup>0</sup> *DB T*<sup>0</sup> *ϕ<sup>b</sup>*

Making use of the identity *curlcurl* � *graddiv* � <sup>∇</sup><sup>2</sup> curlcurl <sup>¼</sup> graddiv � <sup>∇</sup><sup>2</sup> on

*HT*<sup>~</sup> �

To change PDE's to ODE's, Eqs. (30)–(32) are solved using normal mode analysis

*<sup>W</sup>* � *RA <sup>α</sup>*<sup>2</sup> <sup>Τ</sup> <sup>þ</sup>

*DT*ð Þ *T*<sup>1</sup> � *T*<sup>0</sup> *DB T*<sup>0</sup> *ϕ<sup>b</sup>*

ð Þ *ρC* <sup>P</sup> *<sup>ρ</sup><sup>C</sup> <sup>ϕ</sup><sup>b</sup>*

*DT*ð Þ *T*<sup>1</sup> � *T*<sup>0</sup> *DB T*<sup>0</sup> *ϕ<sup>b</sup>*

!

Φ þ

<sup>2</sup> <sup>þ</sup> *ky* <sup>2</sup> � �<sup>1</sup><sup>⁄</sup> <sup>2</sup>

*<sup>u</sup>*~3, *<sup>T</sup>*~, *<sup>ϕ</sup>*<sup>~</sup> � � <sup>¼</sup> ð Þ *W z*ð Þ, <sup>Τ</sup>ð Þ*<sup>z</sup>* , <sup>Φ</sup>ð Þ*<sup>z</sup>* exp *ikxx* <sup>þ</sup> *ikyy* <sup>þ</sup> *st* � �, (33)

*ρ<sup>p</sup>* � *ρ* � �

*DB αf D*

*DB αf*

*μα<sup>f</sup>*

*:* Using one term Galerkin weighted residual

*W* ¼ Τ ¼ 0 at *z* ¼ 0 and *z* ¼ 1*:* (37)

*W* ¼ *A* sin *πz*, and *T* ¼ *B* sin *πz*, (38)

*ϕ<sup>b</sup> g d*<sup>3</sup>

<sup>Τ</sup> � ð Þ *<sup>ρ</sup><sup>C</sup>* <sup>P</sup> *<sup>ρ</sup><sup>C</sup> <sup>ϕ</sup><sup>b</sup>*

*DT*ð Þ *T*<sup>1</sup> � *T*<sup>0</sup> *DB T*<sup>0</sup> *ϕ<sup>b</sup>*

*ρ<sup>p</sup>* � *ρ* � �

*μα<sup>f</sup>*

*DB αf* ∇2

*ρ<sup>p</sup>* � *ρ* � �

*μα<sup>f</sup>*

ð Þ *ρC* <sup>P</sup> *<sup>ρ</sup><sup>C</sup> <sup>ϕ</sup><sup>b</sup>*

*ρα<sup>f</sup> μ ∂v*~

*Applications of Nanobiotechnology*

*∂T*~

where ∇<sup>2</sup>

*<sup>∂</sup><sup>t</sup>* � *<sup>u</sup>*~<sup>3</sup> <sup>¼</sup> <sup>∇</sup><sup>2</sup>

∇*:v*~ ¼ 0, (28)

*ϕ<sup>b</sup> g d*<sup>3</sup>

*ϕ<sup>b</sup> g d*<sup>3</sup>

∇2

*ϕ*~^

*DB αf*

*∂T*~ *∂z*

*T*~*:* (31)

*k*, (29)

*<sup>H</sup>ϕ*~, (32)

*<sup>α</sup>*<sup>2</sup> <sup>Φ</sup> <sup>¼</sup> 0, (34)

*D*Φ ¼ 0,

(35)

*DB αf*

*<sup>D</sup>*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup> � �<sup>Τ</sup> <sup>¼</sup> 0, (36)

, (30)

## **6. Results and discussion**

#### **6.1 Stationary convection**

For non-oscillatory motions *s* ¼ 0, this gives the expression for *RA* from Eq. (39) as

$$R\_A = \frac{f^3}{a^2} - \frac{D\_T \left(\rho\_p - \rho\right) (T\_1 - T\_0) \mathbf{g} \ \mathbf{d}^3}{D\_B \mu \, a\_f T\_0}. \tag{40}$$

where *DB* <sup>¼</sup> *kBT* <sup>3</sup>*πμdp* and *DT* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *<sup>μ</sup> <sup>ρ</sup> <sup>ϕ</sup>* with *<sup>β</sup>*<sup>~</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>26</sup> *<sup>k</sup>* <sup>2</sup>*k*þ*kp* as given by Nield and Kuznetsov [35].

Also

$$R\_A = \frac{J^3}{a^2} - \frac{\left(\rho\_p - \rho\right)}{2k + k\_p} A;\tag{41}$$

where *A* depends on the base fluid properties.
