In ideal flow, a liquid of density 850 kg / m^3 moves from a horizontal tube of radius 1.00 cm i

step 1 Question number 42, the density is given as row equal to 850 kg per meter cube. R1 is 1 centimet

In ideal flow, a liquid of density 850 kg / m^3 moves from a...

In ideal flow, a liquid of density 850 $\mathrm{kg} / \mathrm{m}^{3}$ moves from a horizontal tube of radius 1.00 $\mathrm{cm}$ into a second horizontal tube of radius $0.500 \mathrm{cm} .$ A pressure difference $\Delta P$ exists between the tubes. (a) Find the volume flow rate as a function of $\Delta P$ . Evaluate the volume flow rate (b) for $\Delta P=6.00 \mathrm{kPa}$ and $(\mathrm{c})$ for $\Delta P=12.0 \mathrm{kPa}$ . (d) State how the volume flow rate depends on $\Delta P .$

Key Concepts

Ideal Fluid Flow

Ideal fluid flow assumes that the fluid is incompressible and non?viscous, meaning that effects such as friction and viscous dissipation are neglected. This assumption simplifies the analysis and allows one to use conservation laws such as Bernoulli’s equation without needing to account for energy losses.

Bernoulli’s Principle

Bernoulli’s principle is a statement of energy conservation for flowing fluids. It relates the pressure, velocity, and elevation between two points along a streamline, implying that a decrease in pressure results in an increase in kinetic energy (or velocity) when other forms of energy remain constant. In horizontal flow, changes in pressure are directly related to changes in the flow’s speed.

Continuity Equation

The continuity equation expresses the conservation of mass in fluid flow and states that the volume flow rate must remain constant along a streamline in an incompressible fluid. This means that the product of cross-sectional area and flow velocity is constant, so when the area decreases, the velocity must increase, and vice versa.

Pressure-Driven Flow and Flow Rate Dependence

Pressure-driven flow is the mechanism by which a pressure difference between two points induces motion in the fluid. In the context of ideal flow, the change in pressure is converted entirely into kinetic energy change. As a result, the volume flow rate is related to the square root of the pressure difference, illustrating that larger pressure differences cause significantly increased flow speeds and flow rates through the constriction.

Cross-sectional Area and Geometric Effects

The cross-sectional area of a tube plays a key role in determining the flow characteristics; for circular tubes it is calculated from the radius. When the fluid transitions between tubes with different radii, the resulting changes in area require adjustments in velocity (as prescribed by the continuity equation) and consequently affect the pressure and kinetic energy distribution, as captured by Bernoulli’s principle.

Transcript

Please give Ace some feedback