**Fluid Mechanics**

**Part #3**

**Velocity of Fluid**

➢ Velocity is an important basic parameter governing a flow field. Other field variables such as the pressure and temperature are all influenced by the velocity of the fluid flow.

➢ In general, velocity is a function of both the location and time. The velocity vector can be expressed in Cartesian coordinates as :

➢ Where the velocity components (u, v and w) are functions of

**Acceleration of Fluid**

➢ The acceleration is the change in velocity, δV, over the change in time, δt.

➢ But it is not just a simple derivative of just time since the velocity is a function time, and space (x,y,z).The change in velocity must be track in both time and space.

➢ Using the chain rule of calculus, the change in velocity is

➢ Let ax , ay and az are the total acceleration in x , y and z directions respectively, Then :

➢ As acceleration of fluid is the function of both time and space, it can be classified as follows :

**Local Acceleration**

➢ Local acceleration is defined as the rate of increase of velocity with respect to time at a given point in a flow field.

➢ So space derivatives will become zero as there is no change in position, so expression for acceleration will be :

**Convective Acceleration **

➢ Convective acceleration is defined as the rate of change of velocity due to the change of position of fluid particles in a fluid flow at a particular point of time.

➢ So time derivatives will become zero, expression for acceleration will be :

**Rate of Flow or Discharge (Q)**

➢ It is defined as the quantity(volume) of a fluid flowing per second through a section of a pipe or a channel.

➢ Consider a liquid flowing through a pipe in which

➢ A = Cross – sectional area of pipe

➢ V = Average velocity of fluid across the section

➢ Then discharge :

**Continuity Equation**

➢ Continuity Equation is derived from conservation of mass of fluid.

**Generalized form of Continuity Equation in 3D**

➢ Let us consider a fluid flow with velocities u , v and w in x , y and z direction respectively and

density of the fluid = ρ

➢ So continuity equation is written as :

**For Steady Flow**

➢ For steady flow density does not change with time so

➢ So continuity equation is written as :**If fluid is Incompressible**

➢ If fluid is incompressible then density of fluid will be constant through out the flow. ρ = constant

➢ So continuity equation becomes:

➢ Above equation is always valid for a steady and incompressible flow, If a fluid is flowing it must satisfy continuity equation.

**Continuity Equation for 1D Flow**

➢ For a 1D fluid flow continuity equation can be written as : ρAV = constant

➢ If fluid is incompressible : AV = constant

**Flow Lines**

➢ Flow lines are used for analysis and representation of the flow.

**Stream lines**

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

**Path lines**

➢ Path lines 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.

**Streak lines**

➢ Streak lines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past.

➢ In Steady flow all the three lines are become same.

**Types of Displacements of Fluid Element**

➢ A fluid particle while moving may undergo anyone or combination of following four types of displacements.

➢ Linear Translation or Pure Translation

➢ Linear deformation

➢ Angular Deformation

➢ Rotation

**Linear Translation or Pure Translation**

➢ It is defined as the movement of a fluid element in such a way that it moves bodily from one position to another position and the two axes AB and CD represented in new position by *ab* and *cd* are parallel.

**Linear Deformation**

➢ It is defined as the deformation of a fluid element in linear direction when the element moves. The axes of the element in the deformed position and un deformed position are parallel, but their lengths change.

**Angular deformation or Shear Deformation**

➢ It is defined as the average change in the angle contained by two adjacent sides. Let Δθ₁ and Δθ₂ is the change in angle between two adjacent sides of a fluid element as shown in figure then angular

deformation or shear strain rate.

➢ As we know :

**ROTATION**

➢ It is defined as the movement of a fluid element in such a way that both of its axes (horizontal as well as vertical) rotate in the same direction as shown in figure. The rotational components (angular velocities) are:

➢ For given fluid element it is rotating about z axis so :

➢ Similarly if fluid element rotates about x and y axis :

**Vorticity**

➢ It is defined as the value twice of the rotation and hence it is given as 2 ω.

**VELOCITY POTENTIAL FUNCTION**

➢ It is defined as a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. It is defined by φ = f (x, y, z) for steady flow such that

➢ Where u v and w are the components of velocity in x, y and z directions respectively

➢ The velocity components in cylindrical polar co-ordinates in terms of velocity potential function are given by➢ Where ur = velocity component in radial direction (i.e. In r direction)

➢ And uθ = velocity component in tangential direction (i.e., in θ direction)

**The properties of the potential function**

➢ If velocity potential function (φ) exists, the flow should be irrotational.

➢ If velocity potential (φ) satisfies the Laplace equation, it represents the possible steady incompressible irrotational flow.

**Stream Function**

➢ It is defined as the scalar function of space and time, such that its partial derivative with respect to any direction gives the velocity component a right angles to that direction.

➢ It is denoted by Ψ (Psi) and defined only for two dimensional flow. Mathematically, for steady flow it is defined as Ψ = f (x, y) such that and

➢ The velocity components in cylindrical polar coordinates in term of stream function are given as➢ Where ur = velocity component in radial direction (i.e. In r direction)

➢ And uθ = velocity component in tangential direction (i.e., in θ direction)

**The properties of the stream function**

➢ If stream function (Ψ) exists, it is a possible case of fluid flow which may be rotational or irrotational.

➢ It stream function (Ψ) satisfies the Laplace equation, it is a possible case of an irrotational flow

**Relation between Velocity Potential Function and Stream Function**

**Equipotential line**

➢ A line along which the velocity potential φ is constant, is called equipotential line.

**Constant steam function line**

➢ A line along which the stream function is constant, is called constant stream function line.

*✓ Equipotential line and Constant Stream function line are always perpendicular to each other*

**SOME QUESTION FROM THESE TOPICS**

1.A flow whose stream line is represented by a curve, is called _____ .

(A) One dimensional flow

(B) Three dimensional flow

(C) Two dimensional flow

(D) Four dimensional flow

2.In a steady flow of a fluid, the total acceleration of any fluid particle ___.

(A) Can be zero

(B) Is never zero

(C) Is always zero

(D) Is independent of coordinates

3.A type of flow in which he fluid particles while moving in the direction of flow rotate about their mass centre, is called _____ .

(A) Steady flow

(B) Uniform flow

(C) Laminar flow

(D) Rotational flow

4.The flow in which the velocity vector is identical in magnitude and direction at every point,for any given instant, is knowns as

(A) One dimensional flow

(B) Uniform flow

(C) Steady flow

(D) Turbulent flow

5.Which of the following represents steady uniform flow?

(A) Flow through an expanding tube at an increasing rate

(B) Flow through an expanding tube at constant

(C) Flow through a long pipe at decreasing rate

(D) Flow through a long pipe at constant rate

6.The continuity equation is connected with

(A) Viscous / in viscous fluids

(B) Compressibility of fluids

(C) Conservation of mass

(D) steady/unsteady flow

7.The components of velocity in a two dimensional frictionless incompressible flow are u = t2 + 3y and v = 3t + 3x. What is the approximate resultant total acceleration at the point (3, 2) and t = 2?

(a) 5 (b) 49 (c) 59 (d) 54

8.A streamline is a line:

(a) Which is along the path of the particle

(b) Which is always parallel to the main direction of flow

(c) Along which there is no flow

(d) On which tangent drawn at any point given the direction of velocity

9.For a flow to be rotational, vorticity should be equal to the ______ .

(A) Angular velocity vector

(B) Half the angular velocity vector

(C) Twice the angular velocity vector

(D) Zero

10.Two flows are specified as

(A) u = y, v = - (3/2) x (B) u = xy2 , v = x2y Which one of the following can be concluded?

(a) Both flows are rotational

(b) Both flows are irrotational

(c) Flow A is rotational while flow B is irrotational

(d) Flow A is irrotational while flow B is rotational

11.Which one of the following statements is correct? Irrotational flow is characterized as the one in which

(a) The fluid flows along a straight line

(b) The fluid does not rotate as it moves along

(c) The net rotation of fluid particles about their mass centres remains zero.

(d) The streamlines of flow are curved and closely spaced

12.The streamlines and the lines of constant velocity potential in an inviscid irrotational flow field form.

(a) Parallel grid lines placed in accordance with their magnitude.

(b) Intersecting grid-net with arbitrary orientation.

(c) An orthogonal grid system

(d) None of the above.

13.A velocity field is given by u = 3xy and v = 3/2 (x

^{2}− y

^{2}) What is the relevant equation of a streamline

(b) Flow is uniform

(c) Flow velocities do not change steadily with time

(d) Flow is neither steady nor uniform

[Fluid Mechanics] WITH IMPORTANT QUESTION.

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