Space-Dependent Lorenz Forces Influence on Nanofluid Behavior

Space-Dependent Lorenz Forces Influence on Nanofluid Behavior

DOI: 10.4018/978-1-5225-7595-5.ch003


Magnetic nanofluid (Ferrofluid) is a magnetic colloidal suspension consisting of base liquid and magnetic nanoparticles with a size range of 5–15 nm in diameter coated with a surfactant layer. The effect of magnetic field on fluids is worth investigating due to its numerous applications in a wide range of fields. The study of interaction of the magnetic field or the electromagnetic field with fluids have been documented (e.g., among nuclear fusion, chemical engineering, medicine, and transformer cooling). The goal of nanofluid is to achieve the highest possible thermal properties at the smallest possible concentrations by uniform dispersion and stable suspension of nanoparticles in host fluids. In this chapter, the influence of external magnetic field on ferrofluid flow and heat transfer is investigated. Both effects of Ferrohydrodynamic (FHD) and Magnetohydrodynamic (MHD) have been taken in to account. So, the effects of Lorentz and Kelvin forces on hydrothermal behavior are examined.
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1. Effect Of Space Dependent Magnetic Field On Free Convection Of Fe3o4-Water Nanofluid

1.1. Problem Definition

The physical model along with the important geometrical parameters and the mesh of the enclosure used in the present CVFEM program are shown in Figure 1 (Sheikholeslami, 2014). The inner circular wall is maintained at constant hot temperature 978-1-5225-7595-5.ch003.m01, the two horizontal walls are thermally isolated and other walls are maintained at constant cold temperature 978-1-5225-7595-5.ch003.m02 (978-1-5225-7595-5.ch003.m03>978-1-5225-7595-5.ch003.m04). To reach the shape of inner circular and outer rectangular boundary which consists of the right and top walls, a supper elliptic function can be used as follows;


When 978-1-5225-7595-5.ch003.m06 and 978-1-5225-7595-5.ch003.m07 the geometry becomes a circle. As 978-1-5225-7595-5.ch003.m08 increases from 1 the geometry would approach a rectangle for 978-1-5225-7595-5.ch003.m09 and square for 978-1-5225-7595-5.ch003.m10. In this study we assumed that 978-1-5225-7595-5.ch003.m11.

For the expression of the magnetic field strength it can be considered that two magnetic sources at points 978-1-5225-7595-5.ch003.m12 and 978-1-5225-7595-5.ch003.m13. The components of the magnetic field intensity (978-1-5225-7595-5.ch003.m14) and the magnetic field strength (978-1-5225-7595-5.ch003.m15) can be considered as (Sheikholeslami and Rashidi, 2015):

(4) where 978-1-5225-7595-5.ch003.m19 the magnetic field strength at the source (of the wire). The contours of the magnetic field strength are shown in Figure 2. In this study magnetic source is located at

Figure 1.

(a) Geometry and the boundary conditions; (b) the mesh of enclosure considered in this work.

Figure 2.

Contours of the (a) magnetic field strength 978-1-5225-7595-5.ch003.m22; (b) magnetic field intensity component in x direction 978-1-5225-7595-5.ch003.m23; (c) magnetic field intensity component in y direction 978-1-5225-7595-5.ch003.m24.


The flow is two-dimensional, laminar and incompressible. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. Using the Boussinesq approximation, the governing equations of heat transfer and fluid flow for nanofluid can be obtained as follows:


The terms 978-1-5225-7595-5.ch003.m29 and 978-1-5225-7595-5.ch003.m30 in (6) and (7), respectively, represent the components of magnetic force, per unit volume, and depend on the existence of the magnetic gradient on the corresponding x and y directions. These two terms are well known from FHD which is the so-called the Kelvin force. The term s978-1-5225-7595-5.ch003.m31 and 978-1-5225-7595-5.ch003.m32 appearing in (6) and (7), respectively, represent the Lorentz force per unit volume towards the x and y directions and arise due to the electrical conductivity of the fluid. These two terms are known in MHD. The principles of MHD and FHD are combined in the mathematical model presented in Sheikholeslami and Ganji (2014) and the above mentioned terms arise together in the governing equations (6) and (7). The term 978-1-5225-7595-5.ch003.m33 in Equation (8) represents the thermal power per unit volume due to the magneto caloric effect. Also the term 978-1-5225-7595-5.ch003.m34 in (8) represents the Joule heating. For the variation of the magnetization 978-1-5225-7595-5.ch003.m35, with the magnetic field intensity 978-1-5225-7595-5.ch003.m36 and temperature 978-1-5225-7595-5.ch003.m37, the following relation is used Loukopoulos and Tzirtzilakis (2004):

(9) where 978-1-5225-7595-5.ch003.m39 is a constant and 978-1-5225-7595-5.ch003.m40 is the Curie temperature.

In the above equations, 978-1-5225-7595-5.ch003.m41 is the magnetic permeability of vacuum 978-1-5225-7595-5.ch003.m42, 978-1-5225-7595-5.ch003.m43is the magnetic field strength, 978-1-5225-7595-5.ch003.m44 is the magnetic induction 978-1-5225-7595-5.ch003.m45 and the bar above the quantities denotes that they are dimensional. The effective density (978-1-5225-7595-5.ch003.m46) and heat capacitance978-1-5225-7595-5.ch003.m47 of the nanofluid are defined as:

(11) where 978-1-5225-7595-5.ch003.m50 is the solid volume fraction of nanoparticles. Thermal diffusivity of the nanofluid is
(12) and the thermal expansion coefficient of the nanofluid can be determined as


The dynamic viscosity of the nanofluids is given is


The effective thermal conductivity of the nanofluid can be approximated by the Maxwell–Garnetts (MG) model as (Sheikholeslami, 2014):

(15) and the effective electrical conductivity of nanofluid was presented as below:


By introducing the following non-dimensional variables:

(17) where in Equation (17) 978-1-5225-7595-5.ch003.m57 and 978-1-5225-7595-5.ch003.m58. Using the dimensionless parameters, the equations now become:
(21) where
are the Rayleigh number, Prandtl number, Hartmann number arising from MHD, temperature number, curie temperature number, Eckert number and Magnetic number arising from FHD the for the base fluid, respectively. The thermo physical properties of the nanofluid are given in Table 1 (Sheikholeslami and Ganji, 2014).

Table 1.
Thermo physical properties of water and nanoparticles
Pure water997.141790.613210.05

(Sheikholeslami and Ganji, 2014)

The stream function and vorticity are defined as:


The stream function satisfies the continuity Equation (18). The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e. by taking y-derivative of Equation (20) and subtracting from it the x-derivative of Equation (19).

The boundary conditions as shown in Figure 1 are:

978-1-5225-7595-5.ch003.m73 on the inner circular boundary 978-1-5225-7595-5.ch003.m74 on the outer wall boundary 978-1-5225-7595-5.ch003.m75 on all solid boundaries(23)

The values of vorticity on the boundary of the enclosure can be obtained using the stream function formulation and the known velocity conditions during the iterative solution procedure.

The local Nusselt number of the nanofluid along the hot wall can be expressed as:

(24) where r is the radial direction. The average Nusselt number on the hot circular wall is evaluated as:


To estimate the enhancement of heat transfer between the case of 978-1-5225-7595-5.ch003.m78 and the pure fluid (base fluid) case, the heat transfer enhancement is defined as:


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