Fluid Kinematics Highlights

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Fluid Kinematics Highlights

Fluid Kinematics Highlights

Fluid Kinematics is the branch of fluid mechanics studying fluids in motion, where the governing forces are not considered.

Fluid kinematics in points :

1- If the fluid characteristics like velocity, pressure, density etc do not change at a point with respect to time, the fluid flow is called steady flow. If they change with respect to time, the fluid flow is called unsteady flow.

For steady flow :

\[\frac{{\partial v}}{{\partial t}} = 0\]

For unsteady flow :

\[\frac{{\partial v}}{{\partial t}} \ne 0\]

2- If the velocity in a fluid flow does not change with respect to space (length of direction of flow), the flow is said to be uniform otherwise non-uniform.

For uniform flow :

\[\frac{{\partial v}}{{\partial s}} = 0\]

For non-uniform flow :

\[\frac{{\partial v}}{{\partial s}} \ne 0\]

3- If the Reynolds number in a pipe is less than 2000, the flow is said to be laminar and of Reynolds number is more than 4000, the flow is said to be turbulent.

4- For compressible flow :

\[\rho \ne const\]

For incompressible flow :

\[\rho = const\]

5- Rate of discharge for incompressible fluid (liquid) is given by :

\[Q = A \times v\]

6- Continuity equation is written as :

\[{A_1} \times {v_1} = {A_2} \times {v_2} = {A_3} \times {v_3}\]

7- Continuity equation in differential form :

For three dimensional flow :

\[\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\]

For two dimensional flow :

\[\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\]

8- The components of acceleration in x, y and z directions are :

\[{a_x} = u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}} + \frac{{\partial u}}{{\partial t}}\]

\[{a_y} = u\frac{{\partial v}}{{\partial x}} + v\frac{{\partial v}}{{\partial y}} + w\frac{{\partial v}}{{\partial z}} + \frac{{\partial v}}{{\partial t}}\]

\[{a_z} = u\frac{{\partial w}}{{\partial x}} + v\frac{{\partial w}}{{\partial y}} + w\frac{{\partial w}}{{\partial z}} + \frac{{\partial w}}{{\partial t}}\]

9- The components of velocity in x, y and z directions in terms of velocity potential Φ are :

\[u = - \frac{{\partial \phi }}{{\partial x}}\]

\[v = - \frac{{\partial \phi }}{{\partial y}}\]

\[w = - \frac{{\partial \phi }}{{\partial z}}\]

10- The stream function Ψ is defined only for two dimensional flow. The velocity components in x and y directions in terms of stream function are :

\[u = - \frac{{\partial \psi }}{{\partial x}}\]

\[v = \frac{{\partial \psi }}{{\partial y}}\]

11- Angular deformation or shear strain rate is given as :

\[\frac{1}{2}\left[ {\frac{{\partial v}}{{\partial x}} + \frac{{\partial u}}{{\partial y}}} \right]\]

12- Rotational components of a fluid particle are :

\[{\omega _z} = \frac{1}{2}\left[ {\frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}}} \right]\]

\[{\omega _x} = \frac{1}{2}\left[ {\frac{{\partial w}}{{\partial y}} - \frac{{\partial v}}{{\partial z}}} \right]\]

\[{\omega _y} = \frac{1}{2}\left[ {\frac{{\partial u}}{{\partial z}} - \frac{{\partial w}}{{\partial x}}} \right]\]

13- Vorticity is two times the value of rotation.

14- Flow of a fluid along a curved path is known as vortex flow. If the particles are moving round in curved path with the help of some external torque, the flow is called forced vortex flow. And if no external torque is required to rotate the fluid particles, the flow is called free vortex flow.

15- The relation between tangential velocity and radius :

For forced vortex :

\[v = \omega \times r\]

For free vortex :

\[v \times r = const\]

16- The pressure variation along the radial direction for vortex flow a long horizontal plane :

\[\frac{{\partial p}}{{\partial r}} = \rho \frac{{{v^2}}}{r}\]

The pressure variation in the vertical plane :

\[\frac{{\partial p}}{{\partial z}} = - \rho g\]

17- For the forced vortex flow :

\[Z = \frac{{{v^2}}}{{2g}} = \frac{{{w^2}{r^2}}}{{2g}} = \frac{{{w^2}{R^2}}}{{2g}}\]

where :

Z : height of paraboloid formed

ω : angular velocity

18- For a forced vortex flow in an open tank, fall of liquid level at the center is equal to the rise of liquid level at the ends.

19- In case of closed cylinder, the volume of air before rotation is equal to the volume of air after rotation.

20- If a closed cylindrical vessel completely filled with water is rotated about its vertical axis, the total pressure forces acting on the top and bottom are :

\[{F_T} = \frac{\rho }{4}{\omega ^2}\pi {R^4}\]

\[{F_B} = {F_T} + Weight\]

where :

FT : pressure force on the top of cylinder

FB : pressure force on the bottom of cylinder

ω : angular velocity

R : radius of the vessel

ρ : density of fluid

21- For a free vortex, the equation is :

\[\frac{{{p_1}}}{{\rho g}} + \frac{{{v_1}^2}}{{2g}} + {z_1} = \frac{{{p_2}}}{{\rho g}} + \frac{{{v_2}^2}}{{2g}} + {z_2}\]

You can also check other Fluid Mechanics Highlights on the following links :

Fluid Mechanics - Fluid Properties Highlights

Fluid Mechanics - Fluid Statics Highlights

Fluid Mechanics - Hydrostatic Forces on Surfaces Highlights

Fluid Mechanics - Buoyancy & Floatation Highlights

Fluid Mechanics - Dynamics of Fluid Flow Highlights

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