Three orthogonal components of the velocity (Vx, Vy and Vz) are acquired such that the fluid velocity vector is determined at a pore-scale resolution of 156 μm.
where V is the fluid velocity vector, H is the induced magnetic field vector, P = ( p + μ | H | 2 / 8 π ) is the magneto-hydrodynamic pressure, p is the fluid pressure, μ, ν, σ, ρ and ς = ( 4 π μ σ ) − 1 denote the magnetic permeability, kinematic viscosity, electric conductivity, fluid density and magnetic diffusivity, respectively.
where u ( x, t ) denotes the fluid velocity vector field, b is the magnetic field, P = P ( x, t ) is the scalar pressure; while u 0 ( x ) and b 0 ( x ) are the given initial velocity and initial magnetic fields, respectively, in the sense of distributions, with ∇ ⋅ u 0 = ∇ b 0 = 0 ; α, β > 0 are the parameters.
where u = ( u 1 ( x, t ), u 2 ( x, t ), u 3 ( x, t ) ) denotes the fluid velocity vector field, p = p ( x, t ) is the scalar pressure, e 3 = ( 0, 0, 1 ) T, while u 0 and θ 0 are given initial velocity and initial temperature, respectively, with ∇ ⋅ u 0 = 0.
Substituting the known value of fluid velocity vector v n into (14), pOC n +1 can be obtained and then v n +1 is obtainable from (12).
Then, applying this obtained velocity vector ∂ u/∂ t to the SV, ST, and subtectorial space models as a fluid velocity vector v over a fluid-structure interface, the pressure pOC in the SV, ST, and subtectorial space caused by the movement of the OC is obtained.
The flow behaviours in turbine mode are characterized by examining the velocity magnitude and flow angle at the volute outlet, angle of attack, fluid velocity vectors in the volute, swirling patterns of flow in the draft tube and deviation angle at impeller outlet.
At each age, a section view overlaid with fluid velocity vectors as well as a view of the whole valve seen from the aortic orifice are shown.
If the rotation amplitude A is small, it can be shown that, at leading order, the fluid velocity vectors are everywhere orthogonal to the axis of rotation [ 10, 11].
The Brinkman equation is as follows [ 16]: (1) μ ∇ 2 u s − μ k u s = ∇ p, ∇ u s = 0, where k the permeability of the porous medium, u s denotes the fluid superficial velocity vector, p the fluid pressure, and μ is the effective viscosity in the porous medium.
where ρ, u ¯, and P are the fluid density, velocity vector, and kinetic pressure, respectively, and E ¯ i and V ¯ i are the electric field intensity and self-gravitating potential of the fluid while E ¯ e and V ¯ e are these of tenuous medium surrounding the fluid cylinder, and G is the gravitational constant.
