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Mathematical Modelling Notes - HSMahato
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Undergraduate text onFluid Mechanics
Lecture Notes for
3rd Year Maths & Computing & 2nd Year MSc Students
Course code: MA40011/MA51003
, Chapter 1
Basic Concepts
Fluids are two types:
. Incompressible Fluid
. Compressible Fluid
. Liquids are usually incompressible. Their volume do not change when the pressure changes.
. Gasses are compressible, their volume changes when the pressure changes.
. Continuum Hypothesis: We assume that the fluid is uniformly/macroscopically dis-
tributed in a region.
. Isotropy: A fluid is said to be isotropic with respect to some properties (say pressure,
velocity, density etc.) if that property remains unchanged in all direction.
If that property remains changed, then the fluid is anisotropic.
. Density: The density of fluid is mass per unit volume. Mathematically,
δm
ρ = lim
δV →0 δV
. Specific weight: The specific weight γ of a fluid is defined as the weight per unit
volume. Thus,
γ = ρg
where g is the acceleration due to gravity.
. Pressure: Pressure at a point p is defined as force per unit area.
δF
P = lim
δS→0 δS
. Temperature:
. Thermal conductivity: It is given by Fourier’s Law.
∂T ∂T
qn ∝ = −k
∂n ∂n
where qn is the conductive heat flow per unit area, k is thermal conductivity.
. Viscous an Inviscous Fluid: An infinitesimal fluid element is acted upon by two
types of force: (1) body force (contact force) and (2) Surface force.
2
,Body force is proportional to mass of the body and surface force is proportional to surface
area of the body.
. Viscosity: Viscosity of a fluid is that property which exhibits a certain resistance to
alternation of form.
The upper plate moves with the velocity u in the x-direction, whereas the lower plate is
stationary.
The fluid at y=0 is at rest and at y=h it is at motion and it is moving with the plate,
then the shear stress τ is given by
du
τ =µ
dy
where µ is a constant of proportionality and it is called as the viscosity of the fluid.
. Ideal fluid: Shear stress must be 0, which means that µ = 0,
=⇒ does not exist.
Two types of forces exist on the fluid element-
1. Body force: It is distributed over the entire mass or volume of the element. It is expressed
as the per unit mass of the element.
2. Surface force: Two types-
(i) Normal force: Along the normal to the surface area.
(ii) Shear force: Along the plane of the surface area.
du
τ =µ
dy
. Laminar/ Stream line flow: A flow in which each fluid particle traces out a defnite
curve and the curves traced by any two different particle does not intersect.
.Turbulent flow: Opposite to laminar flow related to Reynolds number. Higher Reynolds
no.
3
, . Steady and unsteady flow: A fluid in which properties as pressure, volume etc are
independent of time t. i.e,
dP
=0
dt
Then such flows are called as steady flow.
If dependent of time is called unsteady flow.
.Rotational and Irrotational flow: A flow in which the fluid is rotating about own axis.
? Lagrangion Approach: Initially at t = 0, let the fluid particle is at P0 .
The current position is given in terms of initial position.
x = f (x0 , y0 , z0 , t)
y = g(x0 , y0 , z0 , t)
z = h(x0 , y0 , z0 , t)
This is the Lagrangian approach. =⇒ ~r = x~i + y~j + z~k
dr X dx~
= i
dt dt
. This approach is very complicated for physical use.
* Eulerian Approach: We fix a point in space, say p and we look how a fluid particle is
behaving at that point.
~r = x~i + y~j + z~k
dr X dx~
= i
dt dt
and
d2 r X d2 x~
= i
dt2 dt2
4
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