| Assumption | What it means | What it lets you do | When to use it | | -------------------------------------- | -------------------------------------------------------- | -------------------------------------------------------------------- | ------------------------------------------------------------- | | **Steady flow** | Flow properties at a fixed point do not change with time | Often removes $\partial/\partial t$ accumulation terms in a fixed CV | Problem says “steady,” or conditions are constant in time | | **Incompressible** | $\rho=\text{constant}$ | $\dot m=\rho Q$; continuity often becomes $A_1V_1=A_2V_2$ | Liquids; low-speed gas flow when stated/appropriate | | **Inviscid / negligible friction** | Viscous effects and losses are neglected | No wall shear/loss terms; commonly enables Bernoulli directly | Problem says frictionless, inviscid, negligible losses | | **Uniform velocity profile** | Velocity is essentially constant across a section | $\dot m=\rho VA$, momentum flux $=\dot mV$ | Explicitly stated, nozzle/jet sections often modeled this way | | **Static fluid** | Fluid is not accelerating | $dp/dz=-\rho g$, $p=p_0+\rho gh$ | Gates, tanks, manometers, buoyancy | | **Atmospheric pressure on both sides** | Same $p_{atm}$ acts on both relevant surfaces | Use gage pressure and cancel $p_{atm}$ | Open tanks, exposed jets | | **Along a streamline** | Bernoulli is applied following one streamline | Allows standard streamline Bernoulli | Inviscid flow when flow is not necessarily irrotational | | **No heat transfer** | $\dot Q=0$ | Removes heat term from energy equation | Adiabatic or when heat transfer is negligible | | **No shaft/work input** | $\dot W=0$ | Removes pump/turbine/work term | Simple pipe/tank flows with no machinery | | **Negligible elevation change** | $z_1\approx z_2$ | Cancel $gz$ terms | Horizontal flow or very small height difference | | **Negligible kinetic-energy change** | $V_1\approx V_2$ or both small | Cancel $V^2/2$ terms | Large reservoirs or same-area slow flow | | **Fully developed** | Velocity profile no longer changes in flow direction | $\partial u/\partial x=0$ for the profile shape | Long straight pipe/channel flow | | **No-slip** | Fluid touching a solid has same velocity as solid | At fixed wall: $u=0$; at moving wall: $u=U_{wall}$ | Viscous flow near solid boundaries | | **Linear velocity profile** | $u(y)$ varies linearly across gap | $du/dy=\Delta u/\Delta y$ | Thin oil films / Couette-type approximations | | **Rigid control volume** | CV boundary does not deform | Standard fixed-CV equations | Stationary pipes/tanks | >[!info] Bernoulli Eqn. Assumptions >- Inviscid >- Incompressible >- Along a streamline >- Steady