Motion of fluid particles and streams

Name: Dhyey Shukla

1. Subject: Fluid Mechanics (2141906) Chapter: Motion of Fluid Particles and Streams Department Mechanical Engineering Name of Subject Teacher Mr. Shivang Ahir

2. Team Members Name Enrollment Number Dhyey Shukla 170990119016 Safiuddin Siddique 170990119017 Aman Singh 170990119018 Mihir Suman 170990119019 Aakash Tamakuwala 170990119020

3. FLUID MECHANICS :- Motion of fluid particles and stream

4.   Fluid Flow :- The study of fluid in motion is called Fluid Kinematics .   The motion of fluid is usually extremely complex.   The study of fluid at rest was simplified by absence of shear force, but when fluid flows over boundary a velocity gradient created.   the resulting change in velocity from one layer to other parallel to boundary gives rise to shear stress.   So, the study of fluid flow needs study of shear stresses .

5.   The following terms are considered suitable to describe the motion of the fluid:-   Path line :- The path traced by a single fluid particle over a period of time is called its path line .   Streak line : - It is an instantaneous picture of the position of all fluid particles in the flow which have passed through a given fixed point.   Streamline : - a streamline is an imaginary line drawn through flow field such that the velocity vector of the fluid flow at each pint is tangent to streamline.   Stream tube : - it is defined as tubular space formed by the collection of streamlines passing through the perimeter of closed curve.

6. Stream Lines Stream Tube

7.   Classification of fluid flow : -   Steady flow :-it is defined as that type of flow in which fluid characteristics like pressure, density at a point do not change with time.  Unsteady flow :-it is defined as flow in which fluid characteristics at a point change with time .   Uniform flow :-it is defined as flow in which flow parameters like pressure, velocity do not change with respect to space(length of flow).

8.   Non uniform flow :-it is the flow in which flow parameters like pressure , velocity at a given time change with respect to space .   Laminar Flow :-It is the flow in which fluid particles flow in layers or lamina with one layer sliding over the other .Fluid elements move in well defined path and they retain the same relative position at successive cross section of the fluid passage . The laminar flow is also called the streamline or viscous flow. This type of flow occurs when the velocity of flow is low and liqiud having high viscosity.

9.   Turbulent Flow :-It is that type of flow in which fluid particles move in zig-zag way.   All the fluid particles are disturbed and they mix with each other.   Thus there is continuous transfer of momentum to adjacent layers.   Due to movement of fluid particles in a zig-zag way, the eddies formation takes place which are responsible for high energy loss.   At Reynold number less than 2000 the flow is laminar and Re greater than 4000 the flow is turbulent.

10.   Compressible flow :- When the volume and thereby density of fluid changes appreciably during flow the flow is said to be compressible flow   Incompressible flow :-Flow is incompressible if the volume and thereby the density of fluid changes insignificantly in fluid flow. For all practical purposes liquids can be considered as incompressible. This means that pressure and temperature changes have little effect on their volume .

11.   Real and Ideal fluids   When a real fluid flows over a boundary the adjacent layer of fluid in contact with the boundary will have the same velocity as the boundary.   The velocity of successive layer increases as we move away from boundary   So, it indicates that shear stresses are produced between the layers of fluid moving with different velocities as a result of viscosity and the interchange of momentum due to turbulence causing particles fluid move from one layer to another.

12.   An ideal fluid is defined as fluid in which there are no viscosity and shear stresses .   If the viscosity of fluid is less and velocity is high the boundary layer is comparatively thin and the assumption that a real fluid is treated as an ideal fluid.

13.   Motion of fluid particle :-   As fluid is composed of particles each one of them has its own velocity and acceleration.   Further these both quantities may change with respect to time as well as with respect to position of the particle in the flow passages .   Its behaviour can be predicted from newton’s law of motion when a force is applied.   Newton’s law can be written , Force = mass×acceleration

14.   The relationship between acceleration and velocity is as follows :- V’ = V + at S = Vt + ½ at^2 V’^2 = V^2 + 2aS   Consider two points A and B δs apart so cahnge of velocity which occurs when particle moves from A to B in time δt .

15.   Difference in velocity between A and B at the given time is as follows : - difference in velocity = (әv/әs)(δs) Also , Change of velocity at B in time t, t = (әv/әt)δt . Thus total change of velocity , dv = (әv/әs)(δs) + (әv/әt)δt

16.   Lagrangian frame of reference :-   This approach refers to description of the behaviour of individual fluid particles during their motion through space .   The observer travels with the particle beeing studied The fluid velocity and acceleration are then determi- ned as function of position and time .

17.   Let the initial coordinates of fluid particles are a,b,c and final coordinates after time interval are x,y,z   The kinetic flow pattern is described by following equations of motion , x = x(a,b,c,t) y = y(a,b,c,t) z = z(a,b,c,t)   These equations can be stated as “final position x of a fluid particle is function of initial space coordinates and time”   The initial space coordinates a,b,c and time t are known as the Lagrangian variables .

18.   The acceleration components are : a = du/dt   Here we get acceleration in all three direction or in all three planes   The motion of one individual fluid particle is not sufficient to describe the entire flow field , motion of all the fluid particles has to be considered simultaneously .

19.   Eulerian frame of reference : -   In this method our co-ordinates are fixed in space, and we observe the fluid as it passes by as if we had described a set of coordinate lines on a glass window.   The observer remains stationary and observe what happens at some particular point.

20.   Let x,y,z be the space coordinates at time t.   Then the component of velocity vector are functions of these space coordinates and time . u = u(x,y,z,t) v = v(x,y,z,t) w = w(x,y,z,t)   Each component is represented as the rate of change of displacement , u = dx/dt , v = dy/dt , w = dz/dt

21.   The acceleration in Eulerian frame of reference is given by : - 1. a(x) = (әu/әt) + u(әu/әx) + v(әu/әy) + w(әu/әz) 2. a(y) = (әv/әt) + u(әv/әx) + v(әv/әy) + w(әv/әz) 3. a(z) = (әw/әt) + u(әw/әx) + v(әw/әy) + w(әw/әz)

22.   Discharge and mean velocity : -   The discharge is defined as the total quantity of fluid flowing per unit time at any particular cross section of a stream .   It is also called flow rate . It can be measured in terms of mass, in which case it is referred as the mass flow rate (kg/s) or in terms of volume, when it is called as the volume flow rate .

23.   In an ideal fluid in which there is no fluid friction the velocity of the fluid would be the same at every point of cross section .   The fluid stream would pass thee given cross section per unit time   Let the cross sectional area normal to direction of flow is A So, discharge Q = AV

24.   In real fluid the velocity of adjacent layer of fluid to a solid boundary will be zero or equal to velocity velocity of solid boundary in the flow direction, a condition called ‘ no slip ’   Let us say V is velocity at any radius r , the flow δQ through an annular element of radius r and thickness dr will be δQ = 2πr dr × V   Total discharge Q = 2πᶘ Vr dr

25.   Continuity of Flow :-  According to the principle of mass conservation , matter can be neither created nor destroyed except nuclear processes. This principle can be applied to a flowing fluid.   Considering any fixed region in the constituting a control volume.

26. The mass of fluid in the control volume remains constant for steady flow and the relation is reduced to Mass of fluid entering = Mass of fluid leaving per per unit time unit time

27. Let = Average velocity of fluid at section - 1 = Area of stream tube at section-1 = Density of fluid at section-1

28. And , , are corresponding values at section-2 Then, mass flow rate at section-1= mass flow rate at section-2= According to law of conservation of mass, mass flow rate at section-1= mass flow rate at section-2 =

29. • If fluid is incompressible , and continuity equation reduces to Volume flow rate, • The continuity equation can also be applied to determine the relation between the flows into and out of a junction. fig.(2)

30. From fig.(2), for steady conditions, Total inflow to junction=Total outflow from junction For an incompressible fluid,

31.  Continuity Equation for 3-D flow :- Mass flux out of differential volume Rate of change of mass in differential volume = Mass flux into differential volume

32. Out In Rate of mass decrease One dimensional equation Mass flux out of differential volume Higher Order term

33. 3-D Continuity Equation,

34. Questions and Suggestions are Welcome Thank You