Fluid Mechanics Applications/A28:Fluid Flow

INDRODUCTION

The flow pattern may be described in term of streamlines, streak-line, path lines, stream-tubes, timeline.

Streamline

A stream line is an imaginary curve drawn through a flowing fluid in such a way that the tangent to it at any point gives the direction of the velocity of flow at that point since fluid is composed of fluid particles

P(x,y) is tangential to the velocity vector V at P and u and v are the components of V along x and y directions
Then,
v/u = tan θ = dy/dx
Where dx and dy are the x and y components of the ds(differential displacement) therefore the differential equation of streamlines in x y plane may be written as
(dx/u)=(dy/v) or udy-vdx=0
Same way in general for 3 dimensional flow
(dx/u)=(dy/v)=(dz/w)
Everywhere the streamline is tangent to the velocity vector ,there can be no component of velocity perpendicular to the streamline, hence no flow across any streamline
For a steady flow the streamline pattern remains same for different time
For a unsteady flow the streamline pattern may change from time to time
Stream function(ψ) : is defined as a scalar function of space and time ,such that its partial derivative with respect to any direction gives velocity component at right angles
Example 1)
Let u=(y² –x²)/(x²+y²)² and v=-2xy/(x²+y²)²
(∂ ψ/∂x)=v=-2xy/(x²+y²)²
(∂ ψ/∂y)= -u=-(y² –x²)/(x²+y²)²
ψ = y/(x²+y²) +f(y)
solving (1)&(2) gives f(y) =constant (taken as zero)
therefore ψ = y/(x²+y²)
the streamline is given below

Example 2)
ψ = y cosh(1 + 0.8cos(2∏x))

Stream tube

a stream tube is formed by a group of streamlines passing through a small closed curve, which may be circular.
Since stream tube is bounded by streamlines and the streamline is tangent to the velocity vector ,there can be no component of velocity perpendicular to the streamline, hence no flow across streamline.

PATH-LINE

A path line may be defined as the line traced by a single fluid particle as it flows .
Therefore the pathline will show the velocity of the same fluid particle at successive instant of time

STREAK-LINE

A streak line consist of al particle in a flow that have previously