JNTU B.Tech II FLUID MECHANICS Previous Question Paper (September - 2021 - R18)

Capillarity is the phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of the liquid. It is caused by the combined effect of cohesion (attraction between liquid molecules) and adhesion (attraction between liquid and tube wall molecules), mediated by the surface tension of the liquid.

For a non-wetting liquid like mercury, the angle of contact θ is greater than 90°, resulting in a capillary depression. To derive the expression, consider a glass tube of diameter d inserted into a reservoir of mercury. The forces acting on the column of mercury are:

• The downward vertical component of the surface tension force acting along the meniscus: F_s = σ π d cos(θ), where σ is the surface tension.
• The upward force due to the pressure difference (hydrostatic head) created by the depression h: F_p = ρ g h × (π / 4) d², where ρ is the density of mercury and g is the acceleration due to gravity.

At equilibrium, these forces are balanced:

Solving for the capillary depression h:

Since θ > 90° for mercury, cos(θ) is negative, confirming that h represents a depression below the free surface level.

For an elastic solid, the stress-strain relationship is governed by Hooke's Law, which states that stress is directly proportional to strain within the elastic limit. The stress-strain curve is a straight line passing through the origin, where the slope represents the modulus of elasticity (Young's modulus). The relationship is expressed as:

where τ is the shear stress, G is the shear modulus, and γ is the shear strain.

The fundamental differences between a Newtonian fluid and a Hooke's solid are as follows:

• Dependence on variables: A Hooke's solid exhibits a stress that is proportional to the magnitude of deformation (strain), whereas a Newtonian fluid exhibits a stress that is proportional to the rate of deformation (strain rate).
• Time dependency: In a Hooke's solid, the deformation is independent of time; once the stress is removed, the solid returns to its original shape. In a Newtonian fluid, the deformation is time-dependent; the fluid continues to deform as long as a shear stress is applied.
• Energy characteristics: A Hooke's solid stores energy elastically, which is recoverable upon unloading. A Newtonian fluid dissipates energy through internal friction (viscosity), which is non-recoverable.
• Governing equation: For a Newtonian fluid, the relationship is defined by Newton's law of viscosity: τ = μ(du/dy), where μ is the dynamic viscosity and (du/dy) is the velocity gradient or strain rate. In contrast, the Hooke's solid relies on the static displacement of particles from their equilibrium positions.