Fluid Mechanics

Density

\rho = {m \over V}

Specific weight

\gamma = {m g \over V} = \rho V

Viscosity

Resistance to flow
Ability to resist shear force

Example

Consider a fluid between two plates a distance, b, apart. One is moving with velocity, \vec{u}, and the other is fixed in place. The moving plate applies a force, \vec{p} to the fluid. The fluid has continuously changing velocity between the two plates.

The velocity gradient between the two plates is given by:

{du \over dy}

The shearing stress, T, is given by:

\vec{T} = {\vec{p} \over A} = \mu {du \over dy}

Where \mu is the absolute viscosity.

Pressure in a Fluid

p_2 = p_1 + \rho g z

Pascal's Law

{F_1 \over A_1} = {F_2 \over A_2}

Archemedes Principle

The upward force of bouyancy, F_B, is given by the weight of the displaced fluid.

Continuity Equation

A_1 \vec{v}_1 \Delta t = A_2 \vec{v}_2 \Delta t

Conservation of mass principle applied to a steady flow. The mass flow rate \dot{m} is given by:

\dot{m} = \rho_1 A_1 \vec{v}_1 = \rho_2 A_2 \vec{v}_2

Flowrate:

\vec{Q} = A \vec{v}

Bernoulli's Equation

p_1 + {1\over2}\rho v_1^2 + \rho g h_1 = p_2 + {1\over2}\rho v_2^2 + \rho g h_2

An increase in speed of a fluid occurs with a decrease of static pressure or decrease in potential energy.

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