Physics 5B Lecture 10 - Fluids in Motion

9.5 Fluids in Motion

Assumptions:
- Fluids are incompressible.
- The flow is steady. Fluid velocity is constant; it does not fluctuate or change with time locally.
- Laminar flow

The Equation of Continuity

If a fluid of volume $V$ enters a tube during some time interval $Δt$ , then an equal volume of fluid must leave the tube.
If the fluid is incompressible, the volumes $ΔV1$ and $ΔV2$ must be equal, leading to the equation of continuity.

Volume Flow Rate

Flow is faster in narrower parts of the tube, slower in wider parts.
The rate at which fluid flows through the tube is called volume flow rate $Q$ .
Volume flow rate is constant at all points in a tube.

Ideal Fluid Dynamics: Bernoulli’s Equation

The mass of the fluid section is considered.
This fluid section is accelerating, so it experiences a net force due to a pressure difference across it.
Newton’s second law is applied to this section of fluid for an incompressible ideal fluid that flows without viscous forces.

Bernoulli’s Equation

Using a kinematic equation relating velocity and acceleration, we can derive Bernoulli's Equation.
The pressure is higher at a point on a streamline where the fluid is moving slower, and lower where the fluid is moving faster.
This is Bernoulli’s equation.

Example: Pressure loss at a constriction

Water flows through a 12-mm-diameter pipe at 4.0 m/s and a gauge pressure of 150 kPa.
An imperfection in a 50-cm-long section of the pipe has caused scale to build up on the pipe wall, which constricts the diameter to 8.0 mm.
We need to find the gauge pressure in this section of the pipe, assuming the water can be modeled as a nonviscous fluid when flowing through a relatively large diameter pipe.

9.7 Ideal vs. Viscous Fluid

(a) Ideal fluid: The speed is the same at all points in the tube.
(b) Viscous fluid: The speed is maximum at the center of the tube and decreases away from the center. The speed is zero on the walls of the tube.
Bernoulli's equation only applies to ideal fluids, which have no viscosity.

Viscous Flow

Analogy: Similar to motion with friction.
Ideal fluid: The fluid will just glide along with no external force.
Viscous fluid: Needs a force applied to counter the viscous drag.
A pressure difference is needed to keep a viscous fluid flowing.
The volume flow rate for a viscous fluid depends on that pressure difference.

Poiseuille’s Equation (No Derivation)

The equation for the volume flow rate for a viscous fluid:

$Q = \frac{\pi R^4 \Delta p}{8 \eta L}$

*   Where:
    *    $Q$  is the volume flow rate,
    *    $R$  is the radius of the tube,
    *    $\Delta p$  is the pressure difference,
    *    $\eta$  is the viscosity,
    *    $L$  is the length of the tube.

The quantity $\frac{\Delta p}{L}$ is called the pressure gradient.
We can define an average flow speed $v_{avg} = \frac{Q}{A}$ .
Viscosity is a fluid’s resistance to flow.

Example: Blood flow through a clogged artery

Suppose the diameter of an artery is reduced by 25% due to atherosclerosis.
By what percent is the blood flow reduced if the pressure gradient along the artery stays constant?
By what factor would the pressure difference established by the heart have to increase to keep the blood flow rate?
Assumptions:
- The blood flow is a viscous flow.
- Blood vessels have constant diameters.

Solution: Blood flow through a clogged artery

Suppose the flow rate is $Q1$ through an unclogged artery of radius $R1$ and $Q2$ through a clogged artery with a smaller radius $R2 = 0.75R_1$ .
From Poiseuille’s equation, we find that a 25% reduction in diameter leads to a huge 68% reduction in flow if the pressure gradient is unchanged.
For a constant flow rate $Q$ , the pressure difference is inversely proportional to $R^4$ .
Blood pressure would have to increase by a factor of 3.2 to maintain the flow rate.

Turbulence

Many flowing fluids exhibit turbulence, which is characterized by chaotic and erratic changes in pressure and velocity.
The Reynolds number—the ratio of inertial forces to viscous forces—determines whether a fluid has laminar flow or turbulent flow.

Reynolds Number: How to Tell if the Flow is Laminar

$Re = \frac{\rho v L}{\eta}$
- Where:
  - $Re$ = Reynolds Number
  - $\rho$ : density
  - $L$ : Obstacle length (sometimes referred to as characteristic length)
  - $v$ : flow speed
  - $\eta$ : viscosity of fluid
The lower the Reynolds Number, the more likely the flow is laminar.
The exact point at which the flow becomes turbulent depends on the geometry of the flow itself.

Reynolds Number and Turbulence

Flow through a tube:
- Laminar flow for Re < 2000
Flow around a sphere:
- Laminar flow for Re < 1
- Fully turbulent wake for Re > 1000
- Fully turbulent flow for Re > 4000