Fluid Mechanics: Bernoulli's Equation

Questions and Answers

What does the flow rate Q represent in a pipe or artery?

The speed of the fluid flowing through the pipe

The temperature of the fluid flowing through the pipe

The pressure exerted by the fluid within the pipe

The volume of fluid entering or leaving the pipe per unit time (correct)

What does the equation Q1 = Q2 represent?

The conservation of energy in a fluid system

The conservation of mass in fluid dynamics (correct)

The relationship between fluid pressure and flow rate

The relationship between fluid velocity and cross-sectional area

If the cross-sectional area of a pipe decreases, what happens to the fluid velocity?

The fluid velocity decreases proportionally

The fluid velocity increases (correct)

The fluid velocity becomes zero

The fluid velocity remains constant

How is the volume of fluid leaving the tube over a time interval Δt expressed?

ΔV = A v Δt Signup and view all the answers

What does Bernoulli's equation indicate when fluid velocity increases in a segment of the pipe?

The pressure in that segment must decrease Signup and view all the answers

What does the first term in Bernoulli’s equation represent?

Potential energy per unit volume due to pressure Signup and view all the answers

In the context of Bernoulli’s equation, what happens to the sum of the three energy terms in the absence of friction?

It remains constant Signup and view all the answers

According to the principles of flow in a pipe, if fluid enters one end of a pipe, what must occur at the other end?

The same volume of fluid must exit Signup and view all the answers

What is the unit of pressure mentioned in Bernoulli’s equation?

dyn/cm2 Signup and view all the answers

If blood is flowing at a velocity of 4 m/s and passes through a Venturi tube, at what point is its velocity likely to increase?

In the constriction of the tube Signup and view all the answers

In which scenario is Bernoulli's equation applicable?

For incompressible fluid flow with negligible frictional losses Signup and view all the answers

What does the equation derived from the law of energy conservation in fluid mechanics relate?

Velocity, pressure, and elevation Signup and view all the answers

When a dam leaks at a height of 20 m, what kind of velocity will the fluid have as it emerges?

Velocity influenced by pressure and gravitational potential energy Signup and view all the answers

What is the acceleration due to pressure at rest when the pressure $P_H=13.2$ kPa and height $h=0.5$ m?

25 m/s² Signup and view all the answers

If the blood flow rate is 5 liters/min, what is the flow rate in cm³/sec?

83.4 cm³/sec Signup and view all the answers

What condition allows Bernoulli's equation to be applied with minimal error?

Large area of flow Signup and view all the answers

In a healthy individual, which values are considered high and low pressures respectively during blood flow?

120 torr and 80 torr Signup and view all the answers

How is the total energy of blood per unit volume calculated at a systolic pressure of 120 torr?

Sum of kinetic energy and energy due to fluid pressure Signup and view all the answers

What is the Power $P_H$ produced by the heart if the flow rate $Q$ is 83.4 cm³/sec and total energy per unit volume is $1.63×10^5$ erg/cm³?

13.6 watts Signup and view all the answers

What is the relationship between the flow rate through a needle and its diameter?

Flow rate increases with increasing diameter Signup and view all the answers

What is the acceleration relative to gravity ($g$) when the calculated acceleration $a = 25 m/s²$?

2.5 g Signup and view all the answers

At what point does laminar flow transition to turbulent flow?

When the fluid velocity exceeds a critical point Signup and view all the answers

What factor affects the time taken for blood to be transfused through a needle?

Length of the hypodermic needle Signup and view all the answers

What is the gauge pressure of the blood within the vein during transfusion?

2400 N/m2 Signup and view all the answers

What is a characteristic feature of turbulent flow compared to laminar flow?

Presence of eddies and whirls Signup and view all the answers

Which of the following measurements defines the pressure head for the transfusion scenario?

10388 N/m2 Signup and view all the answers

What is the critical flow velocity above which the flow in a cylindrical pipe becomes turbulent?

Defined by the diameter and viscosity of the fluid Signup and view all the answers

What does the Reynold's number (ℜ) indicate in fluid mechanics?

The relationship between inertial forces and viscous forces Signup and view all the answers

Which pressure gauge reading correctly reflects the relationship among heart pressure (PH), foot pressure (PF), and brain pressure (PB)?

PH < PF < PB Signup and view all the answers

How does standing affect blood pressure readings in different parts of the body?

Blood pressure in the upper body is higher than in the lower body Signup and view all the answers

What is the effect of fluid flow turning turbulent compared to laminar flow?

Frictional forces increase, making fluid movement more difficult Signup and view all the answers

What happens to the pressures in the circulatory system when the heart pressure (PH) is measured standing compared to reclining?

Pressure in lower extremities decreases when standing Signup and view all the answers

Using the given parameters, what is the computed pressure difference between foot pressure (PF) and heart pressure (PH)?

26.8 kPa Signup and view all the answers

In discussions of the circulatory system, how is pressure conventionally measured?

In kilopascals primarily Signup and view all the answers

Study Notes

Bernoulli's Equation

Governs flow of incompressible fluids, linking velocity, pressure, and elevation.
Formula includes terms for pressure (P), height (h), density (ρ), and velocity (v).
First term: Potential energy per unit volume due to pressure.
Second term: Gravitational potential energy per unit volume.
Third term: Kinetic energy per unit volume.
Based on energy conservation, the total energy remains constant without friction.

Continuity Equation

Fluid volume flow rate remains constant in a pipe: Q1 = Q2.
Flow rate (Q) calculates as Q = A × v, where A is cross-sectional area and v is velocity.
Change in cross-section affects velocity: larger area results in lower velocity and vice versa.

Applications of Poiseuille's Law

Blood flow characterized by high (120 mmHg) and low pressure (80 mmHg) during systolic and diastolic phases.
For transfusions, pressures impact flow rate; higher pressure increases speed through smaller diameter tubing.

Turbulent vs. Laminar Flow

Laminar flow occurs under critical velocity; exceeding this causes transition to turbulent flow.
Turbulent flow leads to increased frictional forces and greater difficulty in fluid movement.
Reynolds number (ℜ) helps identify flow type: values between 2000 and 3000 indicate turbulence.

Blood Flow and Pressure in the Body

Heart pressures at different body parts (head, foot) measured in kilopascals (kPa) or torr.
Height differences (h) in body parts affect pressure readings: pF - pH = ρghH.
Normal pressures for healthy adults: high pressure (13.3 kPa), foot pressure (26.8 kPa), brain pressure (9.3 kPa).

Power Produced by the Heart

Heart's energy generation maintains blood circulation.
Power (PH) calculated as product of flow rate (Q) and energy per unit volume (E).
Example calculations include kinetic energy from blood flow and energy from pressure contribute to total energy.

Example Challenges

Blood velocity in a Venturi tube calculated using Bernoulli’s equation.
Pressure head impacts transfusion rate via needle diameter and length.
Governing equations for flow rates and pressures applied to realistic medical scenarios.

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Description

Explore the principles of Bernoulli's equation, which defines the relationship between pressure, velocity, and elevation of incompressible fluids. This quiz will test your understanding of this fundamental concept in fluid mechanics and its applications in real-world scenarios.