428
S. Banerjee and R. Ganguly
The nondimensional boundary conditions are correspondingly:
Bottom Wall:
Left Wall:
U = 0, V = 0(0 ≤ X ≤ 1, Y = 0) U = 0, V = 0(X = 0, 0 ≤ Y ≤ 1)
Top Wall:
Right Wall :
U = 1, V = 0(0 ≤ X ≤ 1, Y = 1) U = 0, V = 0(X = 1, 0 ≤ Y ≤ 1)
In Eqs. (10) and (11), the dimensionless group Mn K =
μ 0 χ m m
2
/ρU
2
0 W
4
denotes the magnetic number arising from polarization forces (due to external field
gradients). Its magnitude varies inversely as the fourth power of the pertinent length
scale (= W ) of the system and can be quite significant for miniaturized enclosure
geometries.
Likewise, the nondimensional group Sn L =
μ 0 W
2
σ
μ
2 m
W 2
2 /Re W
is the
Stuart number, which is the ratio of the square of the Hartman number to the Reynolds
number (Tzirtzilakis 2005). It may be observed that when both of these magnetic
numbers are zero, the problem is reduced to the problem of pure hydrodynamic flow
in lid-driven cavity. For a given value of the Reynolds number, increasing the value
of either of these magnetic numbers is equivalent to increasing the magnetic field
strength.
A staggered grid arrangement has been used in the present study, which offers
advantages over the collocated arrangement in convection-dominated flows (Patankar
1980). In addition, sinusoidal grid spacing has been applied, which has enabled (a)
finer computational grid close to the walls and in vicinity of the magnetic source and
(b) coarser grid close to the center of the cavity. The power-law scheme has been used
to discretize the convection terms. The Semi-Implicit Method for Pressure Linked
Equations Revised (SIMPLER) algorithm has been used to solve the system of the
momentum and the pressure correction equations (Patankar 1980).
3 Results and Discussion
The stream function plot of lid-driven flow field within the cavity for purely sheardriven flow, in absence of magnetic field, has been presented in Fig. 2. The simulations
are presented for values of W = 0.05 m, U 0 = 5 cm/s, ν ≈ 3.1 × 10
−6 m
2 /s (Kalion
et al. 2004). The stream Reynolds number is thereby evaluated as Re W ≈ 8.1 × 10
2 .
The plots indicate crowding of contours near the (sliding) top wall, where velocity
gradients are high. The flow rates flag off as the circulation descends down toward
the bottom wall, under the effects of viscous forces. The wall shear effects create
weak secondary vortices, fully conforming to physics of flow.
The stream function plots (in Fig. 3) show that applying the magnetic field
below the bottom plate of the cavity results in bifurcation of the characteristic
primary vortex obtained for pure hydrodynamic flow. This plot corresponds to
S. Banerjee and R. Ganguly
The nondimensional boundary conditions are correspondingly:
Bottom Wall:
Left Wall:
U = 0, V = 0(0 ≤ X ≤ 1, Y = 0) U = 0, V = 0(X = 0, 0 ≤ Y ≤ 1)
Top Wall:
Right Wall :
U = 1, V = 0(0 ≤ X ≤ 1, Y = 1) U = 0, V = 0(X = 1, 0 ≤ Y ≤ 1)
In Eqs. (10) and (11), the dimensionless group Mn K =
μ 0 χ m m
2
/ρU
2
0 W
4
denotes the magnetic number arising from polarization forces (due to external field
gradients). Its magnitude varies inversely as the fourth power of the pertinent length
scale (= W ) of the system and can be quite significant for miniaturized enclosure
geometries.
Likewise, the nondimensional group Sn L =
μ 0 W
2
σ
μ
2 m
W 2
2 /Re W
is the
Stuart number, which is the ratio of the square of the Hartman number to the Reynolds
number (Tzirtzilakis 2005). It may be observed that when both of these magnetic
numbers are zero, the problem is reduced to the problem of pure hydrodynamic flow
in lid-driven cavity. For a given value of the Reynolds number, increasing the value
of either of these magnetic numbers is equivalent to increasing the magnetic field
strength.
A staggered grid arrangement has been used in the present study, which offers
advantages over the collocated arrangement in convection-dominated flows (Patankar
1980). In addition, sinusoidal grid spacing has been applied, which has enabled (a)
finer computational grid close to the walls and in vicinity of the magnetic source and
(b) coarser grid close to the center of the cavity. The power-law scheme has been used
to discretize the convection terms. The Semi-Implicit Method for Pressure Linked
Equations Revised (SIMPLER) algorithm has been used to solve the system of the
momentum and the pressure correction equations (Patankar 1980).
3 Results and Discussion
The stream function plot of lid-driven flow field within the cavity for purely sheardriven flow, in absence of magnetic field, has been presented in Fig. 2. The simulations
are presented for values of W = 0.05 m, U 0 = 5 cm/s, ν ≈ 3.1 × 10
−6 m
2 /s (Kalion
et al. 2004). The stream Reynolds number is thereby evaluated as Re W ≈ 8.1 × 10
2 .
The plots indicate crowding of contours near the (sliding) top wall, where velocity
gradients are high. The flow rates flag off as the circulation descends down toward
the bottom wall, under the effects of viscous forces. The wall shear effects create
weak secondary vortices, fully conforming to physics of flow.
The stream function plots (in Fig. 3) show that applying the magnetic field
below the bottom plate of the cavity results in bifurcation of the characteristic
primary vortex obtained for pure hydrodynamic flow. This plot corresponds to