Write condition of existing irrotational motion of gas (liquid), call their members.
a)Irrotational
means that there is no vorticity (when
the vorticity 
has
the magnitude zero everywhere).
b)The flow
velocity 
of
a fluid is a vector field, and the vorticity
of
the flow can be defined by
If
is
irrotational, with
then the flow is
said to be an irrotational
flow. The vorticity of
an irrotational flow is zero.
c) exist velocity potential function Q(x,y,z)
Write condition of existing rotational motion of gas (liquid), call their members.
If
is not equal to zero
Give definitions of one dimensional, two dimensional and three dimensional flows.
Term one, two or three dimensional flow refers to the number of space coordinated required to describe a flow. It appears that any physical flow is generally three-dimensional. But these are difficult to calculate and call for as much simplification as possible. This is achieved by ignoring changes to flow in any of the directions, thus reducing the complexity.
Flow is one dimensional if the flow parameters (such as velocity, pressure, depth etc.) at a given instant in time only vary in the direction of flow and not across the cross-section. The flow may be unsteady, in this case the parameter vary in time but still not across the cross-section. An example of one-dimensional flow is the flow in a pipe.
Flow is two-dimensional if it can be assumed that the flow parameters vary in the direction of flow and in one direction at right angles to this direction. Streamlines in two-dimensional flow are curved lines on a plane and are the same on all parallel planes.
In three-dimensional flow the hydrodynamic parameters are functions of three space coordinates and time. Fluid flow is three-dimensional in nature
Give definitions of the plane flow.
In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of a gradient always being equal to zero.
U(x,y) – velocity potential function
Give the definition of the vortex line, write its differential equation.
Vortex line – line in which point tangent is parallel to angular velocity.
– differential
equation of vortex line
Give the definition of the vortex surface, vortex tube, vortex infinitesimal tube
A vortex tube is the surface in the fluid formed by all vortex-lines passing through a given (reducible) closed curve in the fluid. Vortex surface is the union of all vortex lines seeded densely on a curve. An infinitesimal vortex tube ia a vortex tube whose cross-section is of infinitesimal dimensions.
Give the definition of the vortex filament. Write vortex filament strength calculation formula, call its members.
vortex filament is a vortex tube with infinite small cross section area. In this tube flow has vorticity, . (In the limit as a diameter of the tube is made small, but circulation, Г, is held fixed, this region of orticity is called a vortex filament).
– strength
of vortex filament
Give the definition of velocity circulation . Write the formula for velocity circulation calculation, call its members.
Velocity circulation is the line integral around a closed curve of the velocity field.
The circulation around a closed curve L is the line integral
V is the fluid velocity on a small element of a defined curve, and dl is a vector representing the differential length of that small element, the contribution of that differential length to circulation is dΓ
Give the definition of the vortex. Write the formula for vortex strength calculation, call its members.
A vortex is a region within a fluid where the flow is mostly a spinning motion about an imaginary axis, straight or curved. That motion pattern is called a vortical flow
Formulate Stokes theorem.
a)Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.
b)S- an oriented, piecewise-smooth surface
C - a simple, closed, piecewise-smooth curve that bounds S
F - a vector field whose components have continuous derivatives in an open region of R3 containing S
Stokes'
Theorem:
