CH6050: Viscous Fluid Flows

Instructor's details:

Dr. Kirti Chandra Sahu

Department of Chemical Engineering

Room No: 04

Email: ksahu@iith.ac.in

Tel: 040 2301 6053

Attendance is compulsory in this course.

Student with below 80 % attendance will not be allowed to sit in the exam.

Syllabus: Viscous Fluid Flow

Properties of Fluids, Fundamental equations of fluid flow: Derivation of

Navier-Stokes, continuity and energy equations, Boundary conditions for

viscous flow, Some discussion on potential flows: stream function, potential

function, Flow separation, Dimensionless parameters, Laminar boundary

layers, similarity solutions: Blasius velocity profile for flow over a flat plate,

Transition to turbulence: linear stability analysis, Introduction to Turbulence:

RANS equations, modeling, etc.

Books:

1.Viscous fluid flow by Frank M. White.

2.Boundary-layer theory by H. Schlichting and K. Gersten

3.Hydrodynamics by H. Lamb

Proposed Course Outline

Topics No of classes

Introduction 1 Properties of Fluids 1NS/Conti/Energy Equations 6Potential Flows 2Flow separation 2Boundary layer theory 6Linear Stability analysis 6Introduction to turbulence 2

Grading

Mid Term: 30 MarksEnd Term: 50 MarksAssignments: 10 MarksOther (Seminar/attendance): 10 Marks

1. What is a fluid?

Any material deforms when a shear stress is applied.

Solid: Deforms a fixed amount or breaks completely when a stress is applied on it.

Fluid: Deforms continuously as long as any shear stress is applied. This definition

includes gases and liquids.

2. What is Mechanics?

Mechanics: The study of motion and the forces which cause (or prevent) motion.

(a) Kinematics (Kinetics): The description of motion: displacement, velocity,

acceleration.

(b) Statics: The study of force acting on particles or bodies at rest.

(c) Dynamics: The study of force acting on particles or

bodies at motion.

3. Continuum Mechanics?

Three types of mechanics:

(a) Particle mechanics

(b) Rigid body mechanics

(c) Continuum mechanics

The continuum hypothesis

• Matters are aggressions of molecules; the molecules of fluid are more closely

packed then that of gas. Attractive forces between the molecules in liquid are also

much larger than those of gas.

• The molecules in a lattice are not fixed but move freely relative to each other.

• In most engineering applications, there is a very large number of molecules in the

region under consideration.

• The materials behaves in the same as if the mass was distributed continuously.

• In fluid mechanics, we may treat the fluid as a continuum rather than a collection of

discrete molecules.

•It may be mentioned here that most gases have the molecules density of

•Molecules/m3.

2 52 .7 1 0×

4. Body Forces and Surface forces:

Forces exerted on an element of fluid may come from long-range or short-range

interactions.

Body forces: Long range-no direct contact, e.g. gravity, electrostatic. Proportional

to volume of the fluid element.

Surface forces: Short range-direct contact, e.g. pressure,friction. Proportional to

area of the fluid element. e.g., Stress

5. How many components does stress have?

Stress is force (a vector) per unit surface area (a scalar).

Stress depends on the direction of the surface normal.

e.g. Stress in y-direction on a surface perpendicular to x-axis.

First subscript corresponds to direction normal to surface.

Second subscript corresponds to direction of force component.

:xyσ

The quantity stress is a tensor. The sign convection for stress components

on a Cartesian element is shown below.

y

z

x

xxσ

xyσ

xzσ

zxσ

zyσ

zzσ

yxσ

yyσ

yzσ

The stress tensor can be written as

xx xy xz

ij yx yy yz

zx zy zz

σ σ σ

σ σ σ σ

σ σ σ

=

The stress tensor: symmetric.

This symmetric is required to

satisfy equilibrium of moments

about the three axes of the

element.

ij jiσ σ=

6. Type of stresses:

Normal Stresses:

There are three normal stress components:

The pressure is (minus) the average of these:

Shear Stresses: There are three distinct shear stress components since it can

be shown that the stress tensor is symmetric:

, ,xx yy zzσ σ σ

( ) / 3xx yy zzp σ σ σ= − + +

, ,xy yx yz zy zx xzσ σ σ σ σ σ= = =

7. Definition of viscosityIt is a measure of resistance of a fluid which is being deform by the application of

shear stress.

In everyday terms viscosity is “thickness”. Thus, water is “thin” having a lower

viscosity, while honey is “think” having a higher viscosity.

Although it is a fluid property, the effect of this property is understood when the

fluid is in motion.

Newton's law of viscosity:

where the constant of proportionality is known as the coefficient of viscosity

or simply the dynamic viscosity.

Common fluids, e.g., water, air, mercury obey Newton's law of viscosity and

are known as Newtonian fluid. Other classes of fluids, e.g., paints, polymer

solution, blood do not obey the typical linear relationship of and du/dy . They

are known as non-Newtonian fluids.

The study which describe the properties of Newtonian and non-Newtonian

fluids is known as Rheology.

,du

dyτ µ=

τ

Viscosity of common fluids:

Air:

Density= 1.3 kg/m3

Dynamic Viscosity=1.8 X 10-5 Pas

Kinematic viscosity = 1.4 X 10-5 m2/s

Water:

Density= 103 kg/m3

Dynamic Viscosity= 10-3 Pas

Kinematic viscosity = 10-6 m2/s