The Euler’s equation for the motion of fluids is based on the assumption that (i). The fluid is non-viscous, homogenous, and incompressible. (ii). The velocity of flow is uniform over the section. (iii). The flow is continuous, steady and along the stream line.
- AOnly (i)
- B(i) and (ii)
- C(ii) and (iii)
- D(i), (ii) and (iii)Correct
Solution
Concept
Euler's equation of motion for fluids is derived by considering a small fluid element and applying Newton's second law of motion along a stream tube, assuming specific ideal conditions.
Formula
dp / ρ + v dv + g dz = 0
Step-by-step solution
1. Understand what Euler's equation of motion represents. It relates pressure, velocity, and elevation in a moving fluid. 2. Look at the basic assumptions made during the derivation of Euler's equation. 3. Assumption (i): The fluid is non-viscous (viscosity is zero, so friction is ignored), homogenous (same density throughout), and incompressible (density remains constant). 4. Assumption (ii): The velocity of flow is uniform (same) across any given cross-section of the stream. 5. Assumption (iii): The flow is continuous, steady (does not change with time), and takes place along a stream line. 6. Since all three given statements (i), (ii), and (iii) are correct assumptions for deriving Euler's equation of motion, all of them are valid. Answer: (D) (i), (ii) and (iii)
Explanation
Euler's equation is derived for an ideal fluid flow. This means the fluid must be non-viscous, incompressible, and homogenous. The flow is also assumed to be steady, continuous, along a streamline, and uniform across the section.
Why the other options are wrong
A: Statement (i) is correct, but (ii) and (iii) are also necessary assumptions. B: Statements (i) and (ii) are correct, but (iii) is also a required assumption. C: Statements (ii) and (iii) are correct, but (i) is also a core assumption.
Shortcut
Remember the ideal fluid conditions: Inviscid (non-viscous), incompressible, steady, along a streamline, and uniform.
Common mistake
Students often think real fluids have viscosity, so they reject the non-viscous assumption, forgetting that Euler's equation is for ideal fluids.
Correct answer: D) (i), (ii) and (iii)