MANE 2720 - Fluid Mechanics Midterm and Homework Review

Vocabulary practice flashcards covering core concepts in fluid statics, kinematics, control-volume conservation laws, the Bernoulli equation, and viscous internal/external flows from MANE 2720 course materials.

Continuity Equation (Steady Flow)

The differential conservation of mass relation $\nabla \cdot (\rho \mathbf{V}) = 0$, which is valid at every point in a steady flow field, including inside viscous boundary layers and in compressible or incompressible conditions.

Incompressible Flow Field

A flow field that satisfies $\nabla \cdot \mathbf{V} = 0$, indicating that the divergence of velocity is zero and fluid element volume is conserved along flow paths.

Fully Developed Flow

A flow regime where the velocity profile does not change in the streamwise direction ($\frac{\partial \mathbf{V}}{\partial x} = 0$), which does not require the streamwise pressure gradient to be zero ($\frac{\partial P}{\partial x} \neq 0$).

Vorticity

A vector quantity defined as the curl of the velocity field, $\boldsymbol{\zeta} = \nabla \times \mathbf{V}$, representing twice the local fluid angular velocity ($\boldsymbol{\zeta} = 2\boldsymbol{\omega}$) and describing rigid-body rotation rather than shear deformation.

Energy Grade Line (EGL)

The graphical line representing total mechanical energy head ($\frac{P}{\gamma} + \frac{V^2}{2g} + z$); it stays horizontal in frictionless flow without work, decreases downstream due to head loss, drops across turbines, and increases only across pumps.

Hydraulic Grade Line (HGL)

The graphical line representing piezometric head ($\frac{P}{\gamma} + z$), situated a vertical distance equal to the dynamic velocity head ($\frac{V^2}{2g}$) below the energy grade line.

Stagnation Pressure (Total Pressure)

The pressure attained when a fluid is decelerated to zero velocity isentropically, given by $P_{\text{stag}} = P + \frac{1}{2}\rho V^2$, which equals the sum of static pressure and dynamic pressure.

Pitot-Static Probe

A measurement device that combines a forward-facing stagnation tap with side static taps to measure dynamic pressure ($P_0 - P = \frac{1}{2}\rho V^2$) and calculate flow velocity ($V = \sqrt{\frac{2(P_{\text{stag}} - P)}{\rho}}$).

Couette Flow with Pressure Gradient

Laminar flow of a viscous fluid between two parallel plates moving at relative velocities subjected to an applied streamwise pressure gradient ($\frac{\partial P}{\partial x}$), governed by $\mu \frac{d^2u}{dy^2} = \frac{\partial P}{\partial x}$.

Absolute Pressure

Pressure measured relative to a perfect vacuum ($P_{\text{abs}} = P_{\text{gage}} + P_{\text{atm}}$), which cannot take negative values ($P_{\text{abs}} \ge 0$).

Hydrostatic Equation

The differential equation $\frac{dp}{dz} = -\rho g$, defining vertical pressure variation in a static fluid under gravity, applicable in any gravity field by utilizing the local acceleration $g$.

Pascal's Law

The principle stating that pressure applied to an enclosed static fluid is transmitted undiminished to every portion of the fluid and vessel walls, where hydraulic force multiplication is accompanied by a reciprocal change in displacement ($A_1 \Delta x_1 = A_2 \Delta x_2$).

Archimedes's Principle

The law stating that a body immersed in a static fluid experiences a buoyant force equal to the weight of the fluid it displaces ($F_B = \rho_f g V_{\text{disp}}$), originating directly from the vertical hydrostatic pressure gradient.

Rigid-Body Rotation

A fluid motion where all particles rotate with a uniform angular velocity $\omega$ with zero relative velocity and zero viscous shear stresses, causing isobaric surfaces and the free surface to form upward-opening paraboloids ($z = z_0 + \frac{\omega^2 r^2}{2g}$).

Center of Pressure

The point of application of the resultant hydrostatic force on a submerged surface, located deeper than the area centroid by an offset distance $y_{\text{CP}} - y_c = \frac{I_G}{y_c A}$.

Center of Buoyancy

The centroid of the displaced liquid volume where the upward buoyant force acts; for floating bodies, stability requires that the metacenter remain above the center of gravity.

Irrotational Flow

A flow field in which vorticity is identically zero everywhere ($\nabla \times \mathbf{V} = \mathbf{0}$), ensuring that the Bernoulli constant ($\frac{P}{\rho} + \frac{1}{2}V^2 + gz$) is uniform across all streamlines throughout the flow.

Pelton Wheel Turbine

An impulse hydraulic turbine where a high-velocity jet with speed $V_j$ impinges on peripheral buckets that turn the flow through an angle $\beta$, producing shaft power $\dot{W}_{\text{shaft}} = \rho \omega r \dot{V}(V_j - \omega r)(1 - \cos(\beta))$.

Atomizer

A fluid device that uses a high-speed air jet over an open vertical tube to create a localized low pressure according to Bernoulli's principle, drawing liquid upward from a reservoir to be dispersed as fine droplets.

Irreversible Head Loss ($h_L$)

The loss of usable mechanical energy per unit weight of fluid due to viscous friction and turbulence in piping systems, calculated as $h_L = \frac{\dot{E}_{\text{mech,loss}}}{\dot{m}g}$.

Aerodynamic Lift (Two-Surface Model)

The net upward force produced on an airfoil ($L = \Delta P A = \frac{1}{2}\rho (V_{\text{top}}^2 - V_{\text{bottom}}^2)A$) caused by higher flow velocities over the upper surface inducing a lower static pressure relative to the lower surface.

Moment-of-Momentum Equation

The integral conservation relationship equating the net external torque applied on a fluid system to the net rate of flux of angular momentum leaving the control surface plus accumulation within the control volume: $\sum \mathbf{M} = \frac{d}{dt}\int_{\text{CV}} (\mathbf{r} \times \mathbf{V})\rho\,dV + \int_{\text{CS}} (\mathbf{r} \times \mathbf{V})\rho(\mathbf{V} \cdot \mathbf{n})\,dA$.