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Fig. 13. Variation of (a) u-velocity along vertical centerline (b) velocity along horizontal centerline (c) temperature along horizontal centerline and (d)
Nusselt number along the hot wall of the cavity for varying shape factor; Gr = 10°, Pr = 6.2, ¢ = 0.20. — Figure 13 Variation of (a) u-velocity along vertical centerline (b) velocity along horizontal centerline (c) temperature along horizontal centerline and (d) Nusselt number along the hot wall of the cavity for varying shape factor; Gr = 10°, Pr = 6.2, ¢ = 0.20.

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Non-dimensional form of continuity equation

Fig. 1. Schematic diagram of buoyancy driven cavity with nanofluid inside the cavity.
Flow of incompressible copper—-water nanofluid in a thermally driven square cavity (shown in Fig. 1) is considered in the
present work. The physical properties of fluid and solid phase used in the present study are shown in Table 1. The boundaries
have no-slip tangential and zero normal velocity boundary condition. The left wall is hot and kept at constant temperature
The right wall is cold and kept at constant temperature. The bottom and top walls are insulated.
Table 1
Physical properties of liquid and solid phase used in present study. — Fig. 1. Schematic diagram of buoyancy driven cavity with nanofluid inside the cavity. Flow of incompressible copper—-water nanofluid in a thermally driven square cavity (shown in Fig. 1) is considered in the present work. The physical properties of fluid and solid phase used in the present study are shown in Table 1. The boundaries have no-slip tangential and zero normal velocity boundary condition. The left wall is hot and kept at constant temperature The right wall is cold and kept at constant temperature. The bottom and top walls are insulated. Table 1 Physical properties of liquid and solid phase used in present study.

Fig. 2. Comparison of u-velocity along the vertical centreline and temperature along the horizontal center line of the cavity obtained by the preser
simulations (left) with results available in literature (right) [4].
The flow and thermal field in the cavity depends on the effective thermo-physical properties of the nanofluid. The ther-
mo-physical properties of the nanofluid in the present single-phase model are calculated involving the contribution of the
nanoparticles in the base fluid available in literature [4,8,14]. These expressions in terms of volume fraction of nanoparticles
are already discussed in Section 2. Four basic models of nanofluid based on its thermo-physical properties are proposed here.
Model-I is a pure fluid model where the thermo-physical properties of the nanofluid is same as that of the base fluid other — Fig. 2. Comparison of u-velocity along the vertical centreline and temperature along the horizontal center line of the cavity obtained by the preser simulations (left) with results available in literature (right) [4]. The flow and thermal field in the cavity depends on the effective thermo-physical properties of the nanofluid. The ther- mo-physical properties of the nanofluid in the present single-phase model are calculated involving the contribution of the nanoparticles in the base fluid available in literature [4,8,14]. These expressions in terms of volume fraction of nanoparticles are already discussed in Section 2. Four basic models of nanofluid based on its thermo-physical properties are proposed here. Model-I is a pure fluid model where the thermo-physical properties of the nanofluid is same as that of the base fluid other

Fig. 4. Variation of u-velocity along the vertical center line for Gr = 10°, Pr = 6.2, @ = 0.05 and n = 3 obtained with three different grids Fig. 3. Comparison of the temperature and velocity profile in a buoyancy driven cavity with the experimental results of Krane and Jessee [24]; (a) u-velocit variation along vertical centerline; (b) v-velocity variation along horizontal centerline and (c) temperature variation along the horizontal centerline.

than the thermal conductivity which is taken as a function of volume fraction from Eq. (3). Thus nanoparticles do not have
any contribution to the overall viscosity, density and thermal expansion coefficient of the nanofluid in this model. Model-II is
for nanofluids where nanoparticles only contribute to conductivity and viscosity of the nanofluids as per (Eqs. (2a) and (2b))
Brinkman’s model [14]. Model-III considers the contribution of nanoparticles to the conductivity, viscosity as well as density
of the nanofluids. Model-IV is a general model for nanofluid where all the thermo-physical properties of the nanofluid are
influenced by the presence of nanoparticles. The thermo-physical properties and non-dimensional buoyancy and viscous — than the thermal conductivity which is taken as a function of volume fraction from Eq. (3). Thus nanoparticles do not have any contribution to the overall viscosity, density and thermal expansion coefficient of the nanofluid in this model. Model-II is for nanofluids where nanoparticles only contribute to conductivity and viscosity of the nanofluids as per (Eqs. (2a) and (2b)) Brinkman’s model [14]. Model-III considers the contribution of nanoparticles to the conductivity, viscosity as well as density of the nanofluids. Model-IV is a general model for nanofluid where all the thermo-physical properties of the nanofluid are influenced by the presence of nanoparticles. The thermo-physical properties and non-dimensional buoyancy and viscous

Different models of nanofluids based on thermo-physical properties.
Where (p - B)np= b+ ps: Bs +(1 — 6) - pp Bp
Table 2 — Different models of nanofluids based on thermo-physical properties. Where (p - B)np= b+ ps: Bs +(1 — 6) - pp Bp Table 2

Fig. 5. Streamline profile (left) and isotherms (right) obtained for Gr = 10°, Pr =6.2, @ = 0.20, n=3 for different models of nanofluids based on thermo- physical properties; (a, b) Model-I, (c, d) Model-lIl, (e and f) Model-IlI and (g, h) Model-IV.

Average Nusselt number from hot wall for various models of nanofluid based on thermo-physical properties (Gr = 10°, Pr = 6.2, @ = 0.20, n= 3).
Table 3 — Average Nusselt number from hot wall for various models of nanofluid based on thermo-physical properties (Gr = 10°, Pr = 6.2, @ = 0.20, n= 3). Table 3

Fig. 6. Variation of (a) Nusselt number along the hot wall for ¢ = 0.20 (b) averager Nusselt number (Nu) from hot wall for different volume fractions for
Gr = 10°, Pr=6.2, n=3. — Fig. 6. Variation of (a) Nusselt number along the hot wall for ¢ = 0.20 (b) averager Nusselt number (Nu) from hot wall for different volume fractions for Gr = 10°, Pr=6.2, n=3.

Variation of average Nusselt number from the hot wall with and without thermal dispersion for Gr = 10°, Pr = 6.2, n=3
Table 4 — Variation of average Nusselt number from the hot wall with and without thermal dispersion for Gr = 10°, Pr = 6.2, n=3 Table 4

Fig. 7. (a) Effect of thermal dispersion on flow (streamline pattern on left) and (b) thermal field (isotherm on right). Solid lines are for simulation involving
thermal dispersion and dashed line are those neglecting thermal dispersion effect (c) contours for dispersed thermal conductivity (kg) (10-7) in the cavity
for d, = 10°° (d) contours for dispersed thermal conductivity (kg) (10-7) in the cavity for ds =2x10°.
Simulations are extended to observe the influence of Grashof number on the flow of nanofluid and thermal field in the
cavity. The Grashof number (Gr) for these investigations is varied between 10? < Gr < 10°. The volume fraction and shape
factor are kept constant as 0.20 and 3, respectively. The physical properties of Model-IV discussed in the Section 3.2 are used — Fig. 7. (a) Effect of thermal dispersion on flow (streamline pattern on left) and (b) thermal field (isotherm on right). Solid lines are for simulation involving thermal dispersion and dashed line are those neglecting thermal dispersion effect (c) contours for dispersed thermal conductivity (kg) (10-7) in the cavity for d, = 10°° (d) contours for dispersed thermal conductivity (kg) (10-7) in the cavity for ds =2x10°. Simulations are extended to observe the influence of Grashof number on the flow of nanofluid and thermal field in the cavity. The Grashof number (Gr) for these investigations is varied between 10? < Gr < 10°. The volume fraction and shape factor are kept constant as 0.20 and 3, respectively. The physical properties of Model-IV discussed in the Section 3.2 are used

Fig. 8. Effect of Grashof number on flow (streamline pattern on left) and thermal field (isotherm on right) in the cavity for ¢ = 0.20, Pr = 6.2, n = 3; (a, b Gr = 1000, (c, d) Gr = 10%, (e, f) Gr= 10°. with (d,/L) of order 10~°. The thermal dispersion effect is considered in all these simulations. Grashof number accounts for the effect of temperature difference, gravity, and kinematic viscosity of nanofluid, dimensions of cavity and thermal expan- sion coefficient of nanofluid. Increase in Grashof number increases the buoyancy driven term in y momentum equation and results in larger velocity as shown in stream line profile from Fig. 8a, c and e. The central vortex developed for Gr = 1000 is circular in nature and tends to become elliptic as the Grashof number is increased to 10*. For further increase in Grashot number to 10°, the central vortex is observed to break in two parts. The centre of vortex shifted towards the vertical walls of the cavity. The temperature profile in the cavity flattens with increase in Gr. The isotherms are observed to be densely packed near the hot wall indicating higher rate of heat transfer as the Grashof number increases.

Effect of Grashof number on average Nusselt number on hot wall for volume fraction 0.20, Pr = 6.2, n=3.
Table 5 — Effect of Grashof number on average Nusselt number on hot wall for volume fraction 0.20, Pr = 6.2, n=3. Table 5

$Fig. 9. Effect of Grashof number on (a) u-velocity, (b) v-velocity profile (c) centerline temperature distribution (d) Nu from hot wall for ¢ = 0.20, Pr = 6.2 n= 3. Simulations are also carried out to observe the influence of volume fraction of the nanoparticles on flow, thermal field and heat transfer augmentation in the cavity. These results are discussed in this section. The range of the volume fraction ¢ used in this study is varied from 0 to 0.20. For all these simulations the Grashof number and the shape factor of the nanoparticle are kept at 10° and 3, respectively. The thermo-physical properties of nanofluids are considered as that of Model-IV along with the dispersion effect via the dispersion coefficient.$

Fig. 9. Effect of Grashof number on (a) u-velocity, (b) v-velocity profile (c) centerline temperature distribution (d) Nu from hot wall for ¢ = 0.20, Pr = 6.2 n= 3. Simulations are also carried out to observe the influence of volume fraction of the nanoparticles on flow, thermal field and heat transfer augmentation in the cavity. These results are discussed in this section. The range of the volume fraction ¢ used in this study is varied from 0 to 0.20. For all these simulations the Grashof number and the shape factor of the nanoparticle are kept at 10° and 3, respectively. The thermo-physical properties of nanofluids are considered as that of Model-IV along with the dispersion effect via the dispersion coefficient.

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