S.N.P.I.T & R.C,UMRAKH GUJRARAT TECHNICHAL UNIVERSITY

SUBJECT: FLUID MECHANICS (B SUBJECT: FLUID MECHANICS (B.E III SEM-2014) TOPIC NAME: STREAM FUNCTION AND FLOW NET

PATEL FENIL M.:130490106075 Guide by : Bankim R. Joshi Prepared By : PATEL FENIL M.:130490106075 PATEL GHANSHYAM S.:130490106076 PATEL HINAL K.:1304901069078 PATEL KISHAN J.:130490106079 PATEL KISHAN B.:130490106080 Guide by : Bankim R. Joshi Sarika G. Javiya Kartila D. Uchdadiya

Stream line: A function is an imaginary line drawn through the flow field in such a way that the velocity vector of the fluid at each and every point on the streamline is tangent to the streamline at that instant.

Steady flow: When the velocity at each location is constant, the velocity field is invarient with time and the flow is said to be steady. Uniform flow Uniform flow: occurs when the magnitude and direction of velocity do not change from point to point in the fluid. Flow of liquids through long pipelines of constant diameter is uniform whether flow is steady or unsteady. Non-uniform flow occurs when velocity, pressure etc., change from point to point in the fluid.

Steady, unifrom flow: Conditions do not change with position or time. e.g., Flow of liquid through a pipe of uniform bore running completely full at constant velocity. Steady, non-unifrom flow: Conditions change from point to point but do not with time. e.g., Flow of a liquid at constant flow rate through a tapering pipe running completely full.

Unsteady, unifrom Flow: e.g. When a pump starts-up. Unsteady, non-unifrom Flow: e.g. Conditions of liquid during pipetting out of liquid.

Uniform flow: A uniform flow consists of a velocity field where ~V = uˆı + vˆj is a constant. In 2-D, this velocity field is specified either by the freestream velocity components u1, v1, or by the freestream speed V1 and flow angle α. u = u1 = V1 cos α v = v1 = V1 sin α

Velocity potential function: We can define a potential function,! (x, z, t) , as a continuous function that satisfies the basic laws of fluid mechanics: conservation of mass and momentum, assuming incompressible, inviscid and irrotational flow

Mathematically , the velocity potential is defined as i = f(x,y,z) for steady flow such that u=-di / dx v=-di / dy z=-di / dz

The continuity equation for an incompressible steady flow is du / dx + dv / dy + dw /dz. Substituting the values of u , v and w we get “Laplace eqation”.

STREAM FUNCTION: Let A and be the two points lying on the streamlines prescribed by some numerical system is called stream function.

(a) We will show that if conservation of mass (continuity) is: Then for an incompressible or slightly compressible fluid

(b) Iff (c) A function can be defined such that

Check from B (d) Whenever can be defined

From & along a streamline D C

(e) Finally, it can be shown (see Kundu Sec.) that x The volume flowrate between streamlines is numerically equal to the difference in their values.

E.g. flow around a cylinder:

Vortex flows: Vorticity is the curl of the velocity field Vorticity is also the circulation per unit area From Stokes Theorem

- “Component of vorticity through a surface A bounded by C equals the line integral of the velocity around C.” - If we define circulation Then Circulation = Total amount of vorticity ┴ to a given area; or flux of vorticity through a given area.

FLOW NET: A grid obtained by drawing a series of streamlines and equipotential lines is known as a flow net.

From equation:

Let two curves c= constant and s=constant, intersect each other at any point. At the point of intersect the slope are: slope= d y / dx =( -dc / dx ) / (dc / dy)= -u/v for the curve s=constant: Slope= d y /d x=( -ds / dx ) / ( ds /dy)= v/u now , product of the slope of these curves: = -u / v . V / u= - 1

Typical flow nets:

Uses: The velocity at any point can be calculated if the velocity at any reference point is known. It assists in determining efficient boundary shape for which the flow does not separate from boundary shape.

(3) The net analysis developed for ideal fluids can be used for the fluids particularly outside boundary layer as the viscous forces diminish rapidly outside boundary layer.

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