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In the laminar regime, this equation basically reduces to the Darcy law, where the pressure gradient is proportional to the flow rate. …

In your first example the term $V_1$ with is not zero. However, it is approximated very small compared to the other terms. You can see the reasoning below in your figure: $V_1^2 \ll V_j^2$. In other w …

The scaling can be exactly derived for laminar flow, but for turbulent flow (which you probably have), it relation is similar, with a different proportionality. … $$\frac{L_A}{L_B}=\frac{Q_B}{Q_A}$$ Example: If pipe A is three times longer than pipe B, than three times more liquid will flow through pipe B. …

Yes, a steady flow is always laminar (but not conversely as you already understood). Turbulent flows are by definition time-dependent (and thus unsteady) flows and therefore not laminar. …

The flow does not become turbulent. It is the interplay between gravity and surface tension that causes the break-up into droplets. Due to gravity, the fluid will accelerate. … Surface tension in the vertical direction is much weaker, and hence the flow breaks up in droplets. …

Therefore, the flow is laminar. For larger Reynolds number the inertia of the flow is a dominant factor, and viscous dissipation will not happen on the big scales. Therefore, the flow is turbulent. … There is no hard limit on when your flow is turbulent, and when it is laminar. That is why we would typically talk about regimes: laminar, transitional and turbulent. …

However, this is a very specific case: laminar pipe flow. … In general, the stress will be a tensiorial quantity, defined as $$ \tau_{ij}= \eta \frac{\partial u_i}{\partial x_j}$$ which is true for turbulent flow, in arbitrary geometries. …

This is covered in the standard convection-diffusion type of equation: $$ \frac{\partial C}{\partial t} + \vec{u} \cdot \nabla C= D \nabla^2 C$$ Where $C$ is the smoke concentration, and $D$ is the …

If you consider the sloped case, you have gravity as a driving force to accelerate the flow downwards. … We know from experience that this kind of laminar flow will give a Poisseuille profile, so we assume that $u(y)=ay^2+by+c$ and boundary conditions. …

It would just mean that water flows out at $O$, but does not flow upward at $I$ anymore. Here you would have to take into account that $\rho_{air}\ll\rho_{water}$. …

You are right, if you assume the velocity of the fluid is more or less constant across the pipe, then conservation of mass dictates that $AV$ is constant. Now, if you have a pipe, with no constraints …

When you exhale, you can consider the flow of air as a turbulent jet. …

The simple answer is pressure. As you state, from the continuity equation you can see that the velocity $v_2>v_1$. The next step is a momentum balance (like any balance in fluid dynamics: $\frac{d}{dt …