 # Micro-Bubble Fluid Lab Report

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3 Analytical Solution
We try to simplify the above model and solve it analytically. Since the micro-bubble fluid consist of liquid and bubbles, so it can be assumed as semi-incompressible fluid. Assume that the liquid phase (water) is incompressible in wide ranges of the pressure. i.e. ρ_w=constant . Expanding equation of 19 and rewrite it, we have:

{█(ϕ ∂(S_w )/∂t+∇.((u_w ) ⃗ )=0 @ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ )+1/ρ_m (Dρ_m)/Dt=0 @n(ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ ))+ϕS_m ∂n/∂t =ϕS_m [K_g (n_∞-n)-K_d n] )┤ 20

where the (Dρ_m)/Dt (material derivative of the density) is defined as below:

(Dρ_m)/Dt=ϕS_m (∂ρ_m)/∂t+(u_m
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Equation 23 indicated that the semi-incompressible micro-bubble fluid arise when its pressure remain constant or pressure variation be small.

{█(ϕ ∂(S_i )/∂t+∇.((u_i ) ⃗ )=0 @(u_i ) ⃗= -λ_i ∇P_i @ϕS_m ∂n/∂t+(u_m ) ⃗.∇n=ϕS_m [K_g (n_∞-n)-K_d n] )┤
24

Now, the fractional flow analysis is introduced. The water fractional flow function fw, defined as ratio of water velocity to total velocity. So the equation 17 become:

{█(ϕ (∂S_w)/∂t+(u ⃗ ).∇f_w+f_w ∇.u ⃗=0 @ @n(ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ ))+Dn/Dt ==ϕS_m [K_g (n_∞-n)-K_d n] )┤
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The micro-bubble generation and destruction coefficients i.e. K_g and K_d were assume constant; this assumption is not so far from the reality . Mechanism of the micro-bubble generation depends on the liquid and gas interstitial velocities; micro-bubble fluid apparent viscosity and the bubble density, this is a known concept from the literature, but here, we can see that, bubble density controls the micro-bubble yield stress.
A typical saturation and bubble density profiles are shown in the Figure 1. It shows that the micro-bubble fluid front, flows with constant speed. This is agree with literature experiments, except when the capillary effects occurs. The Figure 2 shows the bubble density versus bubble generation parameter. It indicates that, any increasing in bubble generation, increase the bubble density at specified