**1. Introduction**

In 1959, the celebrated physicist and Nobel laureate Richard Feynman presented an idea of nanotechnology in his talk "There is a plenty of room at the bottom-An invitation to enter a new field of physics" by emphasizing on the fact that the laws of physics allow us to arrange the atoms the way we want. Almost a century ago, Maxwell [1] initiated working on this issue theoretically and unveiled that the particles of size of micrometer and millimeter, if used in traditional fluids can resolve the motive in a more efficient manner. Yet they had few drawbacks like clogging, erosion in micro channel and settling down which were curbed with the evolution of better substitute; nanosized particles (called as nanoparticles). The suspension of nanoparticles in the regular fluids comprised the nanofluids [2]. Nanofluids have also shown many interesting properties, and the distinctive features (refer **Table 1**) resulting in unprecedented potential for many applications particularly in biological, medical and biomedical applications.

The catalytic role of nanoparticles in intensifying the thermal conductivity of nanofluids is analyzed by many researchers: Masuda et al. [3], Eastman et al. [4], Das et al. [5] and others. In 2006, Buongiorno [6] pioneered the formulation of conservation equations of nanofluids by incorporating the impacts of diffusion due to Brownian motion and thermophoresis of nanoparticles. He made an observation that the velocity of nanoparticles can be perceived as a sum of base fluid and


#### **Table 1.**

*Comparison of particles.*

relative (slip) velocities. To prosecute his research, he considered seven slip mechanisms; inactivity, magnus effect, Brownian motion, diffusiophoresis, thermophoresis, gravitational settling and fluid drainage. Throughout his investigation, he agreed that from all these seven techniques, Brownian diffusion and thermophoresis have a significant part in the absence of turbulent effects. Choi et al. [7] found that carbon nanotubes provide highest thermal conductivity enhancement of nanofluids. There are ample number of evaluations on thermal conductivity of nanofluids [8–11] in which they discussed and analyzed the theoretical as well as experimental results. Heat transfer in nanofluids because of convection has been examined and contemplated by Das and Choi [12], Ding et al. [13] and Das et al. [14].

The ballistic character of heat transfer within nanoparticles has been studied by Chen [15]. Abnormal increase in viscosity is generally observed in relation to the base fluid. The presence of nanoparticles has found to enhance thermal conductivity [4, 7, 16–19]. At very low nanoparticle volume fractions (<0.1%), a heat transfer enhancement up to 40% has been reported [8] and this percentage is found to enhance with temperature [5] and concentration of nanoparticles [16]. The results of Choi et al. [7] established the unexpected non-linear character of measured thermal conductivity with nanotube loadings at low concentration while all theoretical studies concluded a linear relationship. Also, it was discovered that thermal conductivity strongly depends on temperature [5] and particle size [20]. Pak and Cho [21] in their study also reported the heat transfer data for turbulent flow of nanofluids having nanoparticles as aluminum and titanium in circular tubes. They found that Nusselt number is up to 30% more than that of base fluid. Nowadays nanofluids are also used in drug delivery systems [22] and advanced nuclear systems [23] due to enriched thermal properties. The nanofluid technology is still in its early stage and various researchers are using nanofluids as a tool to solve technological riddles of the modern society. **Figure 1** establishes big impact of small particles in view of the diverse applications of nanofluids in fields of industrial, residential, biomedical and transportation.

number in the system. Then small disturbances are added to the initial flow and new set of equations are obtained. Further PDE's are converted into ordinary differential equations using normal modes and expression for Rayleigh number is obtained. It is found that density and conductivity of nanoparticles are important

**2. Instability of fluids under small temperature gradient: Rayleigh**

The convective motions occur in a fluid layer heated underside in which a small temperature gradient is maintained across its boundaries. The maintained temperature across the boundaries must surpass a certain value before the

instability can manifest itself. This Phenomenon was discovered by Bénard [26] in 1900. In most of his experiments, he found that if a fluid layer is heated underside, the layer at the bottom expands due to higher temperature. This makes the fluid density lighter at the bottom than that on the top making the system top heavy. Here viscosity and thermal diffusivity tend to oppose the convective motions but with the application of higher temperature gradient across the fluid layer, the thermal convection process gets initiated showing the pattern of cellular motions (called Bénard convection). Bénard [27] performed an experiment with metallic plate with a thin non-volatile liquid layer of 1 mm depth maintained under

Keeping the upper layer of fluid exposed to free air, he observed that the fluid layer was decomposed into number of cells (showing cellular motion) called Bénard cells. Thus in the standard Bénard problem, density difference due to variation in

parameters in deciding the stability of the system.

*Convection Currents in Nanofluids under Small Temperature Gradient*

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

**Bênard convection**

**Figure 1.**

*Applications of nanofluids.*

constant temperature.

**115**

These days, nanoparticles are used in almost every biomedical application. Recent usage of nanotechnology in medicine and cancer therapy has attracted a lot of interest in thermal properties of nanofluid such as blood with nanoparticles suspension. Researchers have made the efforts to construct a mathematical model that shows the physical system or phenomenon nearly exact behaviour [24, 25]. Motivated by their work, we also intended to form an analytical model for the analysis of the convection currents in a horizontal nanofluid layer which is in accordance with the physical laws. Consequently, the onset of convection currents in the nanofluid layer is investigated mathematically with the help of partial differential equations. To begin with, equations are non-dimensionalized to get Rayleigh

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

**Figure 1.** *Applications of nanofluids.*

relative (slip) velocities. To prosecute his research, he considered seven slip mech-

Surface/Volume ratio One One thousand times more than that of microparticles

thermophoresis, gravitational settling and fluid drainage. Throughout his investigation, he agreed that from all these seven techniques, Brownian diffusion and thermophoresis have a significant part in the absence of turbulent effects. Choi et al. [7] found that carbon nanotubes provide highest thermal conductivity enhancement of nanofluids. There are ample number of evaluations on thermal conductivity of nanofluids [8–11] in which they discussed and analyzed the theoretical as well as experimental results. Heat transfer in nanofluids because of convection has been examined and contemplated by Das and Choi [12], Ding et al. [13] and Das et al. [14]. The ballistic character of heat transfer within nanoparticles has been studied by Chen [15]. Abnormal increase in viscosity is generally observed in relation to the base fluid. The presence of nanoparticles has found to enhance thermal conductivity [4, 7, 16–19]. At very low nanoparticle volume fractions (<0.1%), a heat transfer enhancement up to 40% has been reported [8] and this percentage is found to enhance with temperature [5] and concentration of nanoparticles [16]. The results of Choi et al. [7] established the unexpected non-linear character of measured thermal conductivity with nanotube loadings at low concentration while all theoretical studies concluded a linear relationship. Also, it was discovered that thermal conductivity strongly depends on temperature [5] and particle size [20]. Pak and Cho [21] in their study also reported the heat transfer data for turbulent flow of nanofluids having nanoparticles as aluminum and titanium in circular tubes. They found that Nusselt number is up to 30% more than that of base fluid. Nowadays nanofluids are also used in drug delivery systems [22] and advanced nuclear systems [23] due to enriched thermal properties. The nanofluid technology is still in its early stage and various researchers are using nanofluids as a tool to solve technological riddles of the modern society. **Figure 1** establishes big impact of small particles in view of the diverse applications of nanofluids in fields of industrial,

These days, nanoparticles are used in almost every biomedical application. Recent usage of nanotechnology in medicine and cancer therapy has attracted a lot of interest in thermal properties of nanofluid such as blood with nanoparticles suspension. Researchers have made the efforts to construct a mathematical model that shows the physical system or phenomenon nearly exact behaviour [24, 25]. Motivated by their work, we also intended to form an analytical model for the analysis of the convection currents in a horizontal nanofluid layer which is in accordance with the physical laws. Consequently, the onset of convection currents in the nanofluid layer is investigated mathematically with the help of partial differential equations. To begin with, equations are non-dimensionalized to get Rayleigh

anisms; inactivity, magnus effect, Brownian motion, diffusiophoresis,

**Properties Microparticles Nanoparticles**

Stability Not stable Stable

*Applications of Nanobiotechnology*

Conductivity Less High Clogging More Negligible Erosion Yes No Nanoscale phenomenon No Yes

**Table 1.**

**114**

*Comparison of particles.*

residential, biomedical and transportation.

number in the system. Then small disturbances are added to the initial flow and new set of equations are obtained. Further PDE's are converted into ordinary differential equations using normal modes and expression for Rayleigh number is obtained. It is found that density and conductivity of nanoparticles are important parameters in deciding the stability of the system.
