Streamline, Pathline and Streakline MCQ Quiz in मराठी - Objective Question with Answer for Streamline, Pathline and Streakline - मोफत PDF डाउनलोड करा

Explanation:

Streamline:

\(\begin{array}{l} u = \frac{{\partial \psi }}{{\partial y}},v = - \frac{{\partial \psi }}{{\partial x}} \end{array}\)

\(d\psi = \frac{{\partial \psi }}{{\partial x}}dx + \frac{{\partial \psi }}{{\partial y}}dy = 0 \)

\(\frac{{dy}}{{dx}} = - \frac{{\frac{{\partial \psi }}{{\partial x}}}}{{\frac{{\partial \psi }}{{\partial y}}}} = \frac{v}{u}\)

Potential line:

\(u = \frac{{\partial \phi }}{{\partial x}},v = \frac{{\partial \phi }}{{\partial y}}\)

\(d\phi = \frac{{\partial \phi }}{{\partial x}}dx + \frac{{\partial \phi }}{{\partial y}}dy = 0\)

\(\frac{{dy}}{{dx}} = - \frac{{\frac{{\partial \phi }}{{\partial x}}}}{{\frac{{\partial \phi }}{{\partial y}}}} = - \frac{u}{v}\)

The slope of the velocity potential \(= {\left( {\frac{{dy}}{{dx}}} \right)_1} = - \frac{u}{v}\)

The slope of the stream-line \({\left( {\frac{{dy}}{{dx}}} \right)_2} = \frac{v}{u}\)

\({\left( {\frac{{dy}}{{dx}}} \right)_1} \times {\left( {\frac{{dy}}{{dx}}} \right)_2} = - \frac{u}{v} \times \frac{v}{u} = - 1\)

Explanation:

Path line:

• The path traced by a single fluid particle throughout the flow is represented by a path line.
• The trajectory of the path line is traced by injecting a dye into the fluid.

• Path lines can intersect infinite times i.e. it can rotate in a circular path and need not travel in a straight line only.
• And there is no limitation like streamlining where the line cannot intersect each other.

Explanation:

Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a massless fluid element will travel at any point in time

Streaklines are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streak line.

Pathlines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time.

Explanation:

When dealing with fluid dynamics, streamlines represent the direction a fluid element will flow in that field at any point in time. If the streamlines are straight and converging, it implies that the velocity of the fluid is increasing, but all in a common, or straight path direction.

The displacement of fluid particles in a flow can be decomposed into normal and tangential components. Acceleration of a fluid particle can also be decomposed into two constituent parts: convective normal acceleration, which occurs as a result of the change in direction of the velocity; and convective tangential acceleration, which results from changes in the magnitude of the velocity.

Convective Tangential Acceleration:

• This refers to the acceleration of a fluid particle in the direction of its instantaneous velocity vector due to the change in its speed along the streamline. For straight converging streamlines, there's a change in magnitude (speed) as the particle moves along, so there's tangential acceleration.

Convective Normal Acceleration:

• This refers to the acceleration of a fluid particle perpendicular to the instantaneous velocity vector due to the change in its direction along the streamline. For straight streamlines, the direction of the particle's velocity does not change; thus, there's no normal acceleration.

Explanation:

Streamlines have the following properties:

1. It is a continuous line such that the tangent at any point on it shows the velocity vector at that point.
2. There is no flow across streamlines.
3.  \(\frac{{{\rm{dx}}}}{{\rm{u}}} = \frac{{{\rm{dy}}}}{{\rm{v}}} = \frac{{{\rm{dz}}}}{{\rm{w}}}\) is the differential equation of a streamline, where u, v and w are velocities in directions x, y and z, respectively.

Explanation:

Pathlines:

• Pathlines are the trajectories that individual fluid particles follow.
• These can be thought of as "recording" the path of a fluid element in the flow over a certain period.
• The direction the path takes will be determined by the streamlines of the fluid at each moment in time.
• For 1-D flow, the shape is a straight line.

Streamlines:

• Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow.
• These show the direction in which a massless fluid element will travel at any point in time.

Streaklines:

• Streaklines are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past.
• Dye steadily injected into the fluid at a fixed point extends along a streak line.

Important Points

• For steady flow, path lines, streamlines, and streaklines coincide

Concept:

Streamlines are the lines drawn through the flow field in such a manner that the velocity vector of the field at each and every point on the streamline is tangent to the streamline at that instant.

So, the curve that is everywhere tangent to the instantaneous local velocity vector is ‘streamline’

The equation of streamline is given by

Explanation:

Streamline:

• It is an imaginary curve drawn in space such that a tangent drawn to it at any point will give the velocity of that fluid particle at a given instant of time. 
• A line along which stream function (ψ) is constant is known as a streamline.

\(\begin{array}{l} u = \frac{{\partial \psi }}{{\partial y}},v = - \frac{{\partial \psi }}{{\partial x}} \end{array}\)

\(d\psi = \frac{{\partial \psi }}{{\partial x}}dx + \frac{{\partial \psi }}{{\partial y}}dy = 0 \)

\(\frac{{dy}}{{dx}} = - \frac{{\frac{{\partial \psi }}{{\partial x}}}}{{\frac{{\partial \psi }}{{\partial y}}}} = \frac{v}{u} \)

Potential line:

\(u = \frac{{\partial \phi }}{{\partial x}},v = \frac{{\partial \phi }}{{\partial y}} \)

\(d\phi = \frac{{\partial \phi }}{{\partial x}}dx + \frac{{\partial \phi }}{{\partial y}}dy = 0 \)

\(\frac{{dy}}{{dx}} = - \frac{{\frac{{\partial \phi }}{{\partial x}}}}{{\frac{{\partial \phi }}{{\partial y}}}} = - \frac{u}{v} \)

The slope of the velocity potential \(= {\left( {\frac{{dy}}{{dx}}} \right)_1} = - \frac{u}{v} \)

The slope of the stream-line \({\left( {\frac{{dy}}{{dx}}} \right)_2} = \frac{v}{u} \)

\({\left( {\frac{{dy}}{{dx}}} \right)_1} \times {\left( {\frac{{dy}}{{dx}}} \right)_2} = - \frac{u}{v} \times \frac{v}{u} = - 1 \)

Concept:

Streamline: It is an imaginary curve drawn in space such that tangent drawn to it at any point will give the velocity of that fluid particle at a given instant of time. A line along which stream function (ψ) is constant is known as streamline.

Equipotential line: A line along which velocity potential function (ϕ) is constant is known as the equipotential line.

\({\left( {\frac{{dy}}{{dx}}} \right)_ϕ } \times {\left( {\frac{{dy}}{{dx}}} \right)_ψ } = - 1\)

Slope of equipotential Line × Slope of stream function = -1

They are orthogonal to each line other.

For a streamline, \(ψ(x,y)=constant\) and the differential of ψ  is zero.

\(dψ=\frac{\partialψ}{\partial x}dx+\frac{\partialψ}{\partial y}dy\)

\(dψ=-vdx+udy\)

\((\frac{{\partial y}}{{\partial x}} )_{ψ=const}= \frac{v}{u}\)

For an equipotential line, \(ϕ(x,y)=constant\) and the differential of ϕ is zero.

\(dϕ=\frac{\partialϕ}{\partial x}dx+\frac{\partialϕ}{\partial y}dy\)

\(dϕ=udx+vdy\)

\((\frac{{\partial y}}{{\partial x}} )_{ϕ=const}= -\frac{u}{v}\)

\((\frac{{\partial y}}{{\partial x}} )_{ψ=const}=- \frac{1}{(\frac{{\partial y}}{{\partial x}} )_{ϕ=const}}\)

Explanation:

Streamline flow

• A flow in which each liquid particle has a definite path and their paths do not cross each other is called streamline flow.
• In streamline flow, the fluid flow can be represented by a streamline pattern defined within an Eulerian description of the flow field.
• These streamlines are drawn such that, at any instant in time, the tangent to the streamline at any one point in space is aligned with the instantaneous velocity vector at that point.

Steady Flow

A flow in which the velocity of the fluid at a particular fixed point does not change with time is called a steady flow.

Uniform flow

The flow of a fluid in which each particle moves along its line of flow with constant speed and in which the cross-section of each stream tube remains unchanged is known as Uniform flow.

Turbulent Flow

• Turbulent flow tends to occur when there is intermixing of fluid between the fluid layers it is characterized by random and rapid fluctuations of swirling regions of fluid, called eddies, throughout the flow.
• These fluctuations provide an additional mechanism for momentum and energy transfer.

Important Points