- the dynamics of fluids are the foundation of the understanding of water movement in streams and in the subsurface
- we need to understand this in order to figure out how to measure river discharge, for example
- the basic principles also apply to the flow of air, lava, glaciers, and the Earth's mantle

- we usually classify matter as either solid, liquid, or gas, based on macroscopic properties
- a
*gas*takes on the shape and volume of a container, - a
*liquid*takes the shape of the portion of the container that it fills but retains a fixed volume - a
*solid*has its own defined shape as well as volume - liquids and gases are called
*fluids* - shear stress is a tangential force per unit area acting on a surface
- the property of a fluid that describes the resistance to motion
under
an
applied shear stress is termed
*viscosity* - experiment to determine the viscosity of water (Fig3.1)
- F/A = μ U
_{plate}/d (dimension/unit of viscosity?)

- viscosity as a function of temperature (FigA2.1)
- unit of viscosity: Pa s, dimension?
- for some fluids, density changes

- the basic forces that make fluids move are gravity, pressure differences, and surface stresses
- pressure is a normal stress

- hydrostatic equation (Fig3.3)
- p = F/A = m*g/A = V*ρ*g/A
= ρ*g*d (d: depth)

- let us look at the movement of a fluid in a pipe (Fig3.5)
- the statement of the conservation of energy for a frictionless fluid along a flowline is the following:
- p*V + m*g*z + 1/2*mu
^{2}= constant, means: - work due to pressure + potential energy + kinetic energy
- by dividing this equation by m = v*ρ*g we get the Bernoulli equation:
- u
^{2}/(2*g) + z +p/(ρ*g) = constant, means: - velocity head + elevation head + pressure head = total head
- this head can be measured by looking at the level to which the water rises in a vertical tube stuck into the pipe
- assumptions we are making:
- no friction (viscosity = 0)
- incompressible fluid
- homogeneous fluid
- flow steady with time
- example: tank with steady flow of water (Fig3.6)
=> u
_{2}= (2*g*d)^{1/2} - perform the actual experiment
- example: expansion joint in a hose (Fig3.7)
- apply continuity equation Q = u*A = const, then Bernoulli equation
- decrease of mean velocity results in increase of pressure
- experiment: blow on sheet of paper

- in reality, if we do an experiment, we do see a loss in head along a pipe due to friction (Fig3.8)
- instead of having a uniform value in the pipe, there is a velocity profile (Fig3.9)
- u
^{2}/(2*g) + z +p/(ρ*g) = constant needs to be modified to: u^{2}/(2*g) + z +p/(ρ*g) +h_{L}= constant - the head loss can be described as: h
_{L}= f*L*u^{2}/(D*2*g), f being the friction factor - the friction factor has been measured under a range of circumstances (Fig3.10 for smooth pipes)
- The key parameter to describe a flow
situation is
the
*Reynolds number*: R = ρ*U*D/μ - at R>2000 something happens, the flow
becomes
*turbulent* - Reynolds experiment, laminar ("layered") flow, turbulent ("restless") flow (Fig3.11)
- is the flow through the NYC water tunnels laminar or turbulent?
- is flow in a straw laminar or turbulent?
- let us calculate R for an example:
- U=2 m/s
- D = 0.03 m
- ρ= 10
^{3}kg/m^{3} - μ =1.139
10
^{-3}Pa s - how can we make the flow laminar?
- reducing the velocity;
- reducing the diameter;
- reducing the density of the fluid; or
- increasing the viscosity. reduce
- where in nature is turbulent flow important?