This question requires matching different types of fluids with the corresponding relationship between shear stress and velocity gradient.

Understanding Fluid Types and Relationships

Let's analyze each fluid type and its characteristics regarding shear stress ($\tau$) and velocity gradient ($\frac{du}{dy}$):

• Newtonian Fluid: These fluids follow Newton's law of viscosity, which states that the shear stress is directly proportional to the rate of shear strain (velocity gradient). The relationship is linear. Mathematically, it's represented as:

  $\tau = \mu \frac{du}{dy}$

  where $\mu$ is the dynamic viscosity, a constant for a given temperature and pressure. This linear relationship corresponds to option (III).

• Non-Newtonian Fluid: These fluids do not follow Newton's law of viscosity. The relationship between shear stress and velocity gradient is non-linear. Examples include shear-thinning fluids (like paint) and shear-thickening fluids (like cornstarch suspension). This corresponds to option (IV).
• Ideal Fluid: An ideal fluid is a theoretical concept. It is defined as a fluid that is both incompressible and inviscid (having zero viscosity). Since viscosity is zero ($\mu = 0$), the shear stress is always zero ($\tau = 0$) regardless of the velocity gradient. Among the given options, "Zero velocity gradient" (I) is the closest description, possibly implying minimal resistance to motion or the state under zero stress.
• Ideal Plastic: An ideal plastic fluid, like a Bingham plastic, behaves like a rigid solid below a certain minimum stress, known as the yield stress ($\tau_y$). Once the applied shear stress exceeds the yield stress, it starts to flow, and the relationship between the shear stress and velocity gradient becomes linear. The constitutive equation is:

  $\tau = \tau_y + \mu_p \frac{du}{dy}$ for $\tau > \tau_y$

  where $\mu_p$ is the plastic viscosity. This matches option (II) "With definite yield stress and linear relationship".

Matching Lists

Based on the analysis, we can establish the following matches:

• (A) Newtonian fluid corresponds to (III) Linear.
• (B) Non-Newtonian fluid corresponds to (IV) Non-linear.
• (C) Ideal Fluid corresponds to (I) Zero velocity gradient.
• (D) Ideal Plastic corresponds to (II) With definite yield stress and linear relationship.

Let's summarize these matches in a table:

Selecting the Correct Option

Comparing our derived matches with the given options, the correct option is the one that lists the pairings as (A)-(III), (B)-(IV), (C)-(I), and (D)-(II).

This corresponds to option 4.