An ideal fluid flows within a fixed tube with decreasing cross-sectional area. Assuming constant height and volumetric flow rate, which of the following graphs best illustrates the relationship between fluid pressure and the increase in fluid velocity?

Question

An ideal fluid flows within a fixed tube with decreasing cross-sectional area.  Assuming constant height and volumetric flow rate, which of the following graphs best illustrates the relationship between fluid pressure and the increase in fluid velocity?

Quizplus · Accepted Answer

11ecba2d_1315_7294_a3bf_611380c6ad4a_MD0008_00 One application of Bernoulli's equation relates to the travel of fluids between two points (A and B) within a fixed conduit (a pipe or channel through which a fluid passes). Each side of Bernoulli's equation contains three terms, the first of which is pressure (P), a measure of energy per unit volume. The second and third terms reflect the potential and kinetic energy of a fluid, which both differ from the potential and kinetic energy of solids by being related to fluid density (ρ) rather than to mass: 11ecba2d_1315_7295_a3bf_6b4f8bbbebbc_MD0008_00 PA+ ρghA+12ρvA2=PB+ρghB+12ρvB2 Bernoulli's equation is useful in modeling the flow of ideal fluids because ideal fluids are non-compressible (ie, unchanging density). Fluid incompressibility requires that changes in conduit geometry (ie, cross-sectional area) cause changes in fluid velocity because the volumetric flow rate through interconnected segments of a conduit must be equal. Consequently, changes in conduit geometry are accompanied by changes in fluid pressure or potential energy, maintaining the conservation of energy. In this question, an ideal fluid flows through a fixed tube with a decreasing cross-sectional area. Because the height (h) remains constant, the potential energy terms can be eliminated from Bernoulli's equation: 11ecba2d_1315_7296_a3bf_6fbee9c18408_MD0008_00 11ecba2d_1315_7297_a3bf_55338d0ec176_MD0008_00 Further rearrangement of the remaining terms yields an expression that relates the difference in pressure at both points to the difference in kinetic energy at both points: 11ecba2d_1315_7298_a3bf_e7e9791af00d_MD0008_00 Because the cross-sectional area of the conduit decreases while the volumetric flow rate remains constant, fluid velocity must increase: 11ecba2d_1315_99a9_a3bf_434c5070e0d3_MD0008_00 11ecba2d_1315_99aa_a3bf_c1d0e7ebf06e_MD0008_00 Substituting this inequality into the previous arrangement of Bernoulli's equation yields an expression for pressure: 11ecba2d_1315_99ab_a3bf_0b8e485a431c_MD0008_00 11ecba2d_1315_99ac_a3bf_4150997535fb_MD0008_00 11ecba2d_1315_99ad_a3bf_839fd6702d92_MD0008_00 Therefore, the pressure of ideal fluid in motion will decrease as fluid velocity increases (Choice A), a phenomenon known as the Venturi effect. The Venturi effect has relevance to practical scenarios (eg, aviation, blood flow) and can be observed using a device known as a Venturi tube. Educational objective: The Venturi effect describes the decrease in pressure observed concurrent to an increase in fluid velocity. The Venturi effect is modeled by Bernoulli's equation, which describes the conservation of energy in fluids.

An Ideal Fluid Flows Within a Fixed Tube with Decreasing