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A velocity field is given as \(\vec V = \;Axy\hat i - \;Byzt\hat j\;\) where A and B are constants. x, y, z are in metre and t is in seconds. Which of the following is true of this flow field?

A velocity field is given as \(\vec V = \;Axy\hat i - \;Byzt\hat j\;\) where A and B are constants. x, y, z are in metre and t is in seconds. Which of the following is true of this flow field? Correct Answer Unsteady and 3 – dimensional

Concept:

Steady flow: When the fluid properties do not change with respect to time, then the flow is known as steady flow.

Unsteady flow: When the fluid properties change with respect to time, then the flow is known as unsteady flow.

One-dimensional flow

  • All the flow parameters may be expressed as functions of time and one space coordinate only.
  • The single space coordinate is usually the distance measured along the centre-line (not necessarily straight) in which the fluid is flowing.
  • Example: the flow in a pipe is considered one-dimensional when variations of pressure and velocity occur along the length of the pipe, but any variation over the cross-section is assumed negligible.
  • In reality, flow is never one-dimensional because viscosity causes the velocity to decrease to zero at the solid boundaries.

Two-dimensional flow

  •  All the flow parameters are functions of time and two space coordinates (say x and y).
  •  No variation in the z direction.
  • The same streamline patterns are found in all planes perpendicular to z-direction at any instant.

Three-dimensional flow

The hydrodynamic parameters are functions of three space coordinates and time.

Calculation:

Given:

\(\vec V = \;Axy\hat i - \;Byzt\hat j\;\)

As the velocity changes with time and expression contain x,y and z term. Therefore it is unsteady 3-dimensional flow.

Various other types of flow are:

Uniform flow is defined as the type of flow in which the velocity at any given time does not change with respect to space.

\({\left( {\frac{{\partial V}}{{\partial s}}} \right)_{t = const}} = 0\)

Non-Uniform flow is defined as the type of flow in which the velocity at any given time changes with respect to space.

\({\left( {\frac{{\partial V}}{{\partial s}}} \right)_{t = const}} \ne 0\)

When the velocity and other hydrodynamic parameters changes from one point to another the flow is defined as non-uniform.

Rotational flow: When the fluid particles rotate about their centre of mass, then the flow is known as rotational flow.

Irrotational flow: When the fluid particles do not rotate about their centre of mass, then the flow is known as irrotational flow.

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