MTH- 486: Fluid Mechanics Instructor: Dr. Fahad Munir Abbasi Assistant Professor Department of Mathematics Comsats Institute of Information Technology

MTH- 486: Fluid Mechanics

Instructor:

Dr. Fahad Munir AbbasiAssistant Professor

Department of Mathematics

Comsats Institute of Information Technology

Islamabad, Pakistan

• Highlights of the previous lecture

• “Mass” introduction and physical interpretation

• Law of conservation of mass

• Mathematical form of the law of conservation of mass:

the Continuity equation

• Different forms of the continuity equation

• Rotational and irrotational flows

• Summary

Layout of lecture # 12

• Perfect fluid

• Equations of streamlines

• Equations of streamtubes

• Equations of streamlines or streamtubes in cylindrical

coordinates

• Equations of surface perpendicular to streamlines

• Examples

Previously we studied about the…

Mass:

A collection of incoherent particles, parts, or objects regarded as forming

one body.

Or

Mass (m) is a dimensionless quantity representing the amount of matter in

a particle or object.

The standard unit of mass in the International System ( SI ) is the kilogram

( kg ).

MTH-486, Fluid Mechanics Lec. # 12

Mass is measured by determining the extent to which a particle or object

resists a change in its direction or speed when a force is applied. Isaac

Newton stated: A stationary mass remains stationary, and a mass in

motion at a constant speed and in a constant direction maintains that

state of motion, unless acted on by an outside force. For a given applied

force, large masses are accelerated to a small extent, and small masses

are accelerated to a large extent. The following formula applies:

F = ma

Note: Make it clear that there is a big difference in the mass and weight.


Law of conservation of mass:

The law of conservation of mass states that mass in an isolated system is

neither created nor destroyed.

Or! Conservation of mass implies that matter can be neither created nor

destroyed—i.e., processes that change the physical or chemical properties

of substances within an isolated system (such as conversion of a liquid to a

gas) leave the total mass unchanged.

Or! The law of conservation of mass states that mass in an isolated system

is neither created nor destroyed by chemical reactions or physical

transformations.


The Law of Conservation of Mass states that matter can be changed

from one form into another, mixtures can be separated or made,

and pure substances can be decomposed, but the total amount of

mass remains constant. We can state this important law in another

way. The total mass of the universe is constant within measurable

limits; whenever matter undergoes a change, the total mass of the

products of the change is, within measurable limits, the same as the

total mass of the reactants.


Mathematical form of the law of conservation of mass:

The Continuity equationFor a control volume (CV) the principle of conservation of mass can be stated as:

Rate at which mass enters the system = rate at which mass leaves the system +

rate of accumulation of mass in the system

Or

rate of accumulation of mass in the CV + net rate of mass efflux from the CV = 0

The above equation can be expressed analytically in terms of velocity and density

field of a flow and the resulting expression is known as the continuity equation.


Let us consider a differential control volume with dimensions dx, dy and dz,

which can be fixed in a given flow field with a rectangular coordinate

system. Assume that the general flow directions of the fluid particles is

such that the velocity and the density increase in the increasing

coordinates directions. Assume that the center point of the control volume

has the x-component of velocity “u” and density “ρ”. The outward drawn

unit normal shows the direction of the specific area under consideration.

The densities and velocities at the centers of the faces “abcd” and “oefg”

(perpendicular to the x-axis) are determined (using the Taylor’s theorem)

by considering only the first order derivatives.


The x-component of the velocity at the center of the face “abcd” = .

The density at the center of the face “abcd” = .



Fig. A differential CV in rectangular coordinates fixed in a given flow field

The x-component of the velocity at the center of the face “oefg” = . The

density at the center of the face “oefg” = .

The values of density and velocity at the center of the two faces of

differential CV (for the x-direction flow) are uniformly distributed over the

respective faces. The net mass efflux in the x-direction is


(1)

Similar treatment in the y- and z-directions yields the expressions of net

mass effluxes in these directions, respectively, as:

Adding (1)-(3), we get,


(3)

(2)

(4)

The total net mass efflux is causing reduction in the mass of the differential CV at

a rate of per unit volume. The total mass reduction in the differential control

volume is found by multiplying the rate by the volume , or

The negative sign on the RHS of (5) is necessary as it expresses the reduction of

mass in the differential CV dv. Therefore, total mass efflux out of dv is equal to the

reduction of mass in dv; by equating (4) and (5), we obtain


(5)

(6)

Dividing this equation by throughout and letting (i.e. letting dv shrink dv

to a point volume), we get

Or

Where .


(7)

(7a)

(8)

Or

Or

Equations (7)-(9) are different forms of the equation of continuity in the

differential forms applicable to unsteady compressible flow field.

Where, .


(9)

Different forms of the continuity equation

The continuity equation for incompressible flows:

Recall that, for incompressible flows, , therefore the flow is steady or

unsteady, equation (7) reduces to

In vector notation (10) can be written as


(10)

(11)

In cylindrical coordinates, (10) is written as

For a steady flow, all fluid properties are be definition, independent of

time. Thus, for steady compressible flow the equation of continuity can be

written as


(12)

(*)

Special case:

For two dimensional incompressible flow fields, . Hence (10) reduces to


(13)

Irrotational flow:

If , then the given flow field is irrotational.

Rotational flow:

If , then the given flow field is rotational.

Note: There is a vector identity which states that the curl of a gradient of a

scalar function is always zero. i.e.

If , then from (i)

Thus for irrotational flow, .


(i)

Equation of continuity for an incompressible, irrotational flow:

Being irrotational,

Using (15) in (14),

Or


(16)

(15)

(14)

• “Mass” introduction and physical interpretation

• Law of conservation of mass

• Mathematical form of the law of conservation of mass:

the Continuity equation

• Different forms of the continuity equation

• Rotational and irrotational flows

Summary

Today’s quote:

An effort is never wasted.

Documents

MTH- 486: Fluid Mechanics Instructor: Dr. Fahad Munir Abbasi Assistant Professor Department of Mathematics Comsats Institute of Information Technology