A two-dimensional flow field has velocities along the x and y directions given by u = x^{2}t and v = - 2xyt respectively, where t is time. The equation of streamlines is:
x^{2}y = constant
xy^{2} = constant
xy = constant
Not possible to determine
Answer (Detailed Solution Below)
Option 1 : x^{2}y = constant
Streamline, Pathline and Streakline MCQ Question 1 Detailed Solution
Fluid particle is rotating about their mass own center.
For rotation, some torque is required and this is created by shear force due to viscosity.
A non-viscous fluid can never be rotational.
Vorticity exists due to the rotational component.
Stream function (ψ ) does not satisfy the Laplace equation for rotational flow.
Irrotational flow:
A fluid particle does notrotate about its mass own center.
Vorticity is zero because there is no rotational component.
Stream function (ψ ) satisfy the Laplace equation for irrotational flow.
Velocity potential function (ϕ) exist only for irrotational flow.
Singularities:
When different types of fluid flow merge at a single point then singularities are formed and at this point, velocity is zero or infinite.
Streamline:
Streamlines are the lines drawn through the flow field in such a manner that the velocity vector of the fluid at every point on the streamline is tangent to the streamline at that instant.
Using the continuity equation it can be shown that the speed of flow is inversely proportional to the spacing between streamlines.
The streamline also gets divided at the upstream of the body and again joins on the downstream.
streamline that follows the flow division is called a dividing streamline.
The point at which the division takes place is called a stagnation point, and the velocity of the fluid at the stagnation point is zero.
Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a mass-less 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.
For steady flow, path lines, streamlines and streaklines coincide.
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.
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
Steady Uniform flow
Flow at a constant rate through a duct of uniform cross-section (The region close to the walls of the duct is disregarded)
Steady non-uniform flow
Flow at a constant rate through a duct of the non-uniform cross-section (tapering pipe)
Unsteady Uniform flow
Flow at varying rates through a long straight pipe of uniform cross-section. (Again the region close to the walls is ignored.)
Unsteady non-uniform flow
Flow at varying rates through a duct of a non-uniform cross-section.
Consider the following statements regarding streamline(s) :
i) It is a continuous line such that the tangent at any point on it shows the velocity vector at that point
ii) There is no flow across streamlines
iii) \(\frac{{{\rm{dx}}}}{{\rm{u}}} = \frac{{{\rm{dy}}}}{{\rm{v}}} = \frac{{{\rm{dz}}}}{{\rm{w}}}\) is the differential equation of a streamline, where u, v and w are velocities in directions x, y and z, respectively
iv) In an unsteady flow, the path of a particle is a streamline
Which one of the following combinations of the statements is true?
(i), (ii), (iv)
(ii), (iii), (iv)
(i), (iii), (iv)
(i), (ii), (iii)
Answer (Detailed Solution Below)
Option 4 : (i), (ii), (iii)
Streamline, Pathline and Streakline MCQ Question 9 Detailed Solution
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 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.
Smoke emitting from a lighted cigarette represents streakline
Additional Information
Streamline:
It is an imaginary line or curve drawn in space such that the tangent drawn at any point to it will give the direction of velocity.
As there is no component of velocity in the perpendicular direction, therefore there is no flow across the streamline i.e. there is always flow occurs along the streamline.
Which of the following characteristics regarding fluid kinematics is/are correct?
1. Streamline represents an imaginary curve in the flow field so that the tangent to the curve at any point represents the direction of instantaneous velocity at that point.
2. Path lines, streamlines and streak lines are identical in steady flow.
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Answer (Detailed Solution Below)
Option 3 : Both 1 and 2
Streamline, Pathline and Streakline MCQ Question 13 Detailed Solution
1. A streamline is a line everywhere tangent to the velocity vector at a given instant.
2. A pathline is the actual path traversed by a given fluid particle.
3. A streakline is the locus of particles that have earlier passed through a
prescribed point.
4. A timeline is a set of fluid particles that form a line at a given instant.
Additional Information
Streamlines, pathlines, and streaklines are identical in a steady flow.
A line along which a water particle moves through a permeable soil medium. (also called streamline).
Flow Channel:
The strip between any two adjacent Flow Lines.
Equipotential Lines:
A line along which the potential head at all points is equal.
Flow net:
A flow net is a grid obtained by drawing a series of equipotential lines and streamlines. Flow net is very much useful in analyzing the two dimensional, irrotational flow problems.
The flow nets can be constructed only in the following situations -
The flow should be steady. This is so because streamline pattern for unsteady flow does not remain constant, it changes from instant to instant.
The flow should be irrotational, which is possible only when flowing fluid is an ideal fluid (having no viscosity) or it has negligible viscosity.
The flow should not be governed by the force of gravity, because under the action of gravity, the shape of the free surface changes constantly and hence no fixed flow net pattern can be obtained.
∴ Thus flow net cannot be drawn when the flow is governed by gravity.
(i) Streak is the type of test which is used todetermine the color of a mineral in powdered form. The colour of a mineral's powder is often a very important property for identifying the mineral.
(ii) The streak test is done by scraping a specimen of the mineral across a piece of unglazed porcelain known as a "streak plate." This can produce a small amount of powdered mineral on the surface of the plate. The powder colour of that mineral is known as its "streak."
Streamline is an imaginary line or series of imaginary lines in a flow field, such that a tangent to this line at any point at any instant represents the direction of the instantaneous velocity vector at that point.
There is no flow across streamlines.
\(\frac{{{\rm{dx}}}}{{\rm{u}}} = \frac{{{\rm{dy}}}}{{\rm{v}}} \) is the differential equation of a streamline for 2D flow, with slope \(\frac{{{\rm{dy}}}}{{\rm{dx}}} = \frac{{{\rm{v}}}}{{\rm{u}}} \)
\(\frac{{{\rm{dx}}}}{{\rm{u}}} = \frac{{{\rm{dy}}}}{{\rm{v}}} = \frac{{{\rm{dz}}}}{{\rm{w}}}\) is the differential equation of a streamline for 3D flow, where u, v and w are velocities in directions x, y, and z, respectively.
Streamline flow is also called laminar flow.
This type of flow is more viscous than turbulent flow.
Streamline never intersects each other because if they intersect then there will be two tangents for two curves that mean there will be two velocity vector but it is not possible as at a given instant or at a given point there will be a unique velocity vector only.
Important Points
Path line is the actual path traversed by a given fluid particle.
Streak line is the locus of particles that have earlier passed through a prescribed point.
For steady flow, streamlines, path lines and streak lines are identical because
For a steady flow, the velocity vector at any point is invariant with time.
The path line of the particles with different identities passing through a point will not differ.
The path line could coincide with one another in a single curve which will indicate the streak line too.