Premed · Premed · Physics 1
Lecture 16: Fluid Dynamics — Bernoulli's Equation
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the characteristics of ideal fluid flow (steady, incompressible, irrotational, non-viscous)
- Apply the equation of continuity to relate flow speed and cross-sectional area
- Derive and apply Bernoulli's equation to a variety of fluid flow problems
- Explain and calculate the Venturi effect, lift on an airfoil, and flow from a tank (Torricelli's theorem)
- Discuss real-world deviations from ideal flow: viscosity and turbulence
Lecture Content
I. Ideal Fluid Flow
An ideal fluid is one that is incompressible (constant density, a good approximation for liquids), non-viscous (no internal friction), exhibits steady (laminar) flow (velocity at each point does not change with time), and is irrotational (fluid elements do not spin). In such a flow, streamlines are drawn in the direction of the velocity at each point. In steady flow, streamlines do not cross, the velocity is tangent to the streamline everywhere, and closely spaced streamlines indicate higher speed. The flow rate (Q) is the volume of fluid passing a point per unit time: Q = Av, where A is the cross-sectional area and v is the flow speed, measured in m^3/s.
II. Equation of Continuity
For an incompressible fluid in steady flow, the mass flow rate is constant along a streamline. Since the density is constant, this simplifies to A_1 v_1 = A_2 v_2, known as the equation of continuity. The physical implication is straightforward: where the pipe narrows, the fluid speeds up, and where it widens, the fluid slows down, with v_2 = v_1 (A_1 / A_2).
This principle applies directly to blood flow. When arteries narrow due to atherosclerosis, the blood speed increases locally. However, the total cross-sectional area of the capillary bed is vastly larger than that of the arteries, so blood flows slowly through capillaries, allowing time for gas exchange.
III. Bernoulli's Equation
Bernoulli's equation is a statement of energy conservation for fluid flow: P + (1/2) rho v^2 + rho g y = constant along a streamline. Equivalently, between two points: P_1 + (1/2) rho v_1^2 + rho g y_1 = P_2 + (1/2) rho v_2^2 + rho g y_2. Each term represents energy per unit volume: P is the pressure energy (work done by pressure forces), (1/2) rho v^2 is the kinetic energy per unit volume, and rho g y is the gravitational potential energy per unit volume.
Two important special cases arise frequently. When the fluid is at rest (v = 0), Bernoulli's equation reduces to the hydrostatic pressure equation P_2 = P_1 + rho g (y_1 - y_2). For horizontal flow (y_1 = y_2), the equation becomes P_1 + (1/2)rho v_1^2 = P_2 + (1/2)rho v_2^2, revealing that where speed is higher, pressure is lower, and vice versa.
<image>A pipe that narrows from cross-section A_1 to A_2 and also changes elevation from y_1 to y_2. Fluid flows from left to right. At the wide section: pressure P_1, speed v_1, height y_1. At the narrow section: pressure P_2, speed v_2, height y_2. Streamlines are drawn converging as the pipe narrows. Bernoulli's equation is written connecting the two points. Annotations highlight that speed increases and pressure decreases in the narrow section.</image>
IV. Applications of Bernoulli's Equation
The Venturi effect occurs when a constriction in a pipe causes the fluid to speed up and the pressure to drop. The pressure difference can be measured to determine the flow speed using v_1 = A_2 sqrt[2(P_1 - P_2) / (rho(A_1^2 - A_2^2))]. This effect is exploited in carburetors, aspirators, and medical devices.
Torricelli's theorem describes flow from an open tank through a hole. Since both the surface and the hole are open to the atmosphere (P_1 = P_2 = P_atm) and the surface drops slowly in a large tank (v_1 approximately 0), Bernoulli's equation gives v_2 = sqrt(2g h), where h is the depth of the hole below the surface. This is the same speed an object would reach if it fell freely from height h.
Lift on an airfoil arises because air flows faster over the curved top of a wing than under the flatter bottom. By Bernoulli's equation, P_top < P_bottom, producing a net upward force (lift) equal to (P_bottom - P_top) times the wing area. This is a simplified explanation; real lift also depends on the angle of attack and circulation effects.
V. Pitot Tubes and Flow Measurement
A Pitot tube measures fluid speed by comparing the stagnation pressure to the static pressure. At the stagnation point, where the flow is brought to rest, P_stag = P + (1/2) rho v^2. Solving for speed gives v = sqrt[2(P_stag - P) / rho]. Pitot tubes are used in aircraft to measure airspeed and in engineering flow measurements, often combined with the continuity equation for comprehensive analysis.
VI. Viscosity and Real Fluid Flow (Qualitative)
Real fluids have viscosity (internal friction), meaning energy is lost as the fluid flows. Viscosity (eta) is measured in Pa s, with high-viscosity fluids like honey and blood flowing sluggishly and low-viscosity fluids like water and air flowing freely. Laminar flow is smooth and orderly, occurring at low speeds, while turbulent flow is chaotic and irregular, occurring at high speeds.
The Reynolds number (Re) predicts the flow regime: Re = rho v D / eta, where D is the pipe diameter. For Re < 2000, flow is laminar; for Re > 4000, flow is turbulent; and values between 2000 and 4000 represent a transitional regime.
Poiseuille's law governs laminar flow through a cylindrical pipe: Q = pi r^4 Delta P / (8 eta L). The flow rate depends on the fourth power of the radius, meaning a small reduction in radius dramatically reduces flow. This is critically important for understanding blood flow: a 50% reduction in arterial radius reduces flow to just 1/16 of its original value.
<image>Panel A: A cross-section of a cylindrical pipe with laminar flow, showing velocity profile as a parabola — fastest at the center, zero at the walls (no-slip condition). Arrows of varying lengths illustrate the velocity profile. Panel B: The same pipe with turbulent flow, showing chaotic, swirling streamlines and a flatter velocity profile. Panel C: A graph of flow rate Q vs. radius r following Poiseuille's law (Q proportional to r^4), showing the dramatic effect of radius changes on flow rate. A medical context annotation shows a healthy artery vs. one with plaque buildup.</image>
