JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV Year TRANSPORT PHENOMENA Previous Question Paper (October/November - 2020 - R16)

The viscosity of a fluid is a measure of its resistance to flow, and its dependence on temperature and pressure differs significantly between dilute gases and liquids due to differences in molecular structure and intermolecular forces.

1. Dilute Gases:

According to the kinetic theory of gases, the viscosity of a dilute gas is primarily a function of temperature and is largely independent of pressure, provided the pressure is not extremely high or low.

As temperature increases, the average molecular velocity increases, leading to a higher rate of momentum transfer between layers of gas, which results in an increase in viscosity. Mathematically, this is often expressed by the Sutherland formula or power-law models where viscosity is proportional to T^(1/2) or higher powers depending on the intermolecular potential model.

2. Liquids:

In liquids, molecules are closely packed, and viscosity is dominated by strong intermolecular cohesive forces rather than momentum transfer by molecular collisions. As temperature increases, the kinetic energy of the molecules increases, which helps them overcome these cohesive forces, leading to a decrease in viscosity. This is typically described by the Andrade equation:

where A and B are constants specific to the liquid. Regarding pressure, the viscosity of liquids is generally independent of pressure at low to moderate values. However, at very high pressures, the viscosity of liquids increases significantly because the increased pressure reduces the free volume available for molecular movement, thereby hindering the flow process.

To derive the velocity distribution for steady, laminar, fully developed flow of an incompressible Newtonian fluid through a horizontal circular tube of radius R and length L, we perform a shell momentum balance on a cylindrical shell of fluid at radius r with thickness Δr and length L.

The momentum balance states that the rate of momentum in minus the rate of momentum out plus the sum of forces acting on the system equals zero. For steady flow, the momentum flux balance simplifies to the equality of pressure forces and viscous forces:

Dividing by 2πrL and rearranging for the shear stress τᵣᵣ:

Using Newton's law of viscosity, τᵣᵣ = -μ(dvᶻ/dr), where μ is the dynamic viscosity and vᶻ is the velocity in the axial direction:

Separating variables and integrating with respect to r:

Applying the no-slip boundary condition at the wall, where vᶻ = 0 at r = R, we find the constant of integration C:

Substituting C back into the velocity equation yields the final parabolic velocity distribution:

This result shows that the velocity profile is parabolic, with the maximum velocity occurring at the centerline (r = 0).