November 2020Module: 5

ME 631 A

Viscous Flow Theory

Sachin Y. Shinde

Department of Mechanical Engineering,Indian Institute of Technology,

Kanpur, India.

Module: 5Part 02

Flow Instabilities

➢ Fluid Mechanics

- Kundu, Cohen, Dowling

Orr-Sommerfeld Equation

➢ In case of thermal instability and centrifugal instability, we saw that viscosity kills the perturbations and

helps to stabilize the flow

➢ Is that always true?

➢ What about these flows?

Stable or Unstable?

➢ In case of parallel viscous flows, viscosity plays a role in destabilizing flow.

➢ The equation that governs the stability of parallel viscous flows is the Orr-Sommerfeld Equation, which

we will derive in detail.

Philosophy

Philosophy

Squire’s Theorem

➢ Squire’s Theorem - Statement:

➢ “To each unstable three-dimensional disturbance, there corresponds a

more unstable two-dimensional disturbance”.

➢ Inference:

➢ As we increase Reynolds number, the Critical Reynolds Number at which

instability starts is reached first by two-dimensional disturbance.

➢ Advantage:

➢We need to consider only a two-dimensional disturbance to determine

the minimum Reynolds number for the onset of instability (Critical

Reynolds Number).

➢ Ref: Fluid Mechanics - Kundu, Cohen, Dowling

Derivation of Orr-Sommerfeld Equation

➢We will conduct the detailed derivation if the Orr-Sommerfeld equation

using Hand-written Notes.

Orr-Sommerfeld Equation & Boundary Conditions➢ We derived the Orr-Sommerfeld Equation as

➢ Four Boundary Conditions are:

Case 1. Bounded Flow:

Case 2. Unbounded Flow with non-zero shear at y = 0:

Inviscid Stability of Parallel Flows

Inviscid Disturbances

➢ The Orr-Sommerfeld equation that governs the stability of parallel viscous flow is given as

➢ Consider a situation where the disturbances obey the inviscid dynamics.

➢ The viscous term on RHS would be zero

➢ The resulting equation then would be:

➢ This equation is called as the Rayleigh equation.