Question

For an ideal fluid flow the Reynolds number is

• A. one
• B. zero
• C. infinity
• D. 4000

Hint:

Keep these definitions handy for fluid types: - Ideal Fluid: Viscosity (\(\mu\)) = 0 \(\implies\) Reynolds number (\(Re\)) = \(\infty\). - Real Fluid: Viscosity (\(\mu\)) > 0 \(\implies\) Reynolds number (\(Re\)) is finite.

Answer 1

The Correct Option is C

Concept: The Reynolds number (\(Re\)) is a dimensionless quantity in fluid mechanics that helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulent. The mathematical formulation for the Reynolds number is expressed as the ratio of inertial forces to viscous forces within the fluid system: \[ Re = \frac{\text{Inertial Forces}}{\text{Viscous Forces}} = \frac{\rho \cdot v \cdot D}{\mu} \] Where the parameters are defined as:

• \(\rho\) is the density of the fluid.

• \(v\) is the characteristic velocity of the flow.

• \(D\) is the characteristic linear dimension (such as diameter of a pipe).

• \(\mu\) is the dynamic viscosity of the fluid.
An "ideal fluid" is a theoretical concept used in fluid dynamics to simplify mathematical analysis. By definition, an ideal fluid is assumed to be completely frictionless, meaning it is non-viscous (viscosity is zero) and incompressible.

Step 1: Analyzing the core definition of an ideal fluid.
By fundamental physical definition, an ideal fluid possesses zero viscosity (\(\mu = 0\)). This implies that there are absolutely no internal shear stresses or frictional resistances acting between adjacent layers of the fluid as they slide past one another.

Step 2: Substituting the ideal fluid property into the Reynolds number formula.
Let us write the expression for the Reynolds number and evaluate its behavior as the dynamic viscosity approach or reaches zero: \[ Re = \frac{\rho \cdot v \cdot D}{\mu} \] Substituting \(\mu = 0\) directly into the denominator of this ratio gives: \[ Re = \frac{\rho \cdot v \cdot D}{0} \]

Step 3: Evaluating the mathematical limit.
In mathematical terms, dividing any finite, non-zero positive quantity (representing the inertial forces generated by fluid density, velocity, and dimension) by zero yields an infinitely large value: \[ \lim_{\mu \to 0} \frac{\rho \cdot v \cdot D}{\mu} = \infty \] Therefore, because there are absolutely no viscous forces present to oppose the motion or dampen the momentum of the fluid, the ratio of inertial forces to viscous forces becomes infinitely large. This means that for any ideal fluid flow, the Reynolds number is always infinity.