Fluid Kinematics

Fluid kinematics involves velocity, acceleration of the fluid, description and visualisation of fluid motion without considering forces that produce motion. There are two general approaches in analysis fluid mechanics:

1. Eulerian Approach: Fluid motion is given by prescribing the necessary properties (p, V, T etc) in terms of space and time. 
2. Lagrangian Approach: Information of the fluid in terms of what happens at fixed points in space. 

• Steady Flow: A flow is said to be steady if fluid properties do not vary with respect to time.
• Unsteady Flow: Fluid properties vary with time.
• Uniform Flow: if fluid properties do not change point to point at any instant of time.
• Non-Uniform flow: When fluid properties changes from point to point at any instant of time, flow is defined as non-uniform flow.
• Laminar flow: Fluid particles moves in layers or lamina. No mixing in normal direction.
• Turbulent Flow: Fluid particles have random movement, intermixing in layers.
• Incompressible Flow: Density variation is negligible.

Stream Line

An imaginary line or curve such that it is tangent to the velocity field at a given instant, given 

Equation of streamline is given by:

\[\frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w}\]

for 2D flow : vdx - udy = 0

Path Line

It is the path traced by a single fluid particle at different instant of time. 

Streak Line 

It is defined as locus of various fluid particles that have passed through a fixed point.

NOTE: In steady flow, streakline, pathline and streamline are identical.

Continuity Equation (Conservation of mass)

\[ρAV = Constant\]

where, ρ = Density

A = Area

V = Velocity

For incompressible fluid it reduces to $A_1V_1 = A_2V_2$.

Generalised Continuity Equation

\[\frac{\partial ρ}{\partial t} + \frac{\partial (ρ u)}{\partial x} + \frac{\partial (ρ v)}{\partial y} +\frac{\partial (ρw)}{\partial z} = 0\]

where u,v,w are component of velocity in x, y, z direction.

• If flow is steady $\frac{\partial ρ}{\partial t} = 0$.
• For steady, imcompressible flow continuity equation $\boxed{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} = 0}$.

Total acceleration of fluid

\[\frac{D\vec{V}}{Dt} = a_x i + a_y j + a_z k = \frac{Du}{Dt}i + \frac{Dv}{Dt}j + \frac{Dw}{Dt}k\]

$\frac{D}{Dt}$ is called Material Derivative or Substantial Derivative or Total Derivative with respect to time.

• In steady flow, local acceleration will be zero.
• In uniform flow, convective acceleration will be zero.

Rotational Compenent ($\omega$)

\[ \omega =  \frac{1}{2} \nabla \times \vec{V} = \frac{1}{2}
\begin{vmatrix}
 i & j & k \\
 \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}
\\ u & v & w
\end{vmatrix} \]

\[\omega_z = \frac{1}{2}(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})\]

• Vorticity ($\Omega$) is as the vector that is twice the rotational vector.

\[\Omega = 2 \omega = \nabla \times \vec{V}\]

• Flow said to be irrotational if vorticity or rotational vector is zero at all points in region.
• Circulation ($\Gamma$) is line integral of tangential component of velocity around a closed curve.

$\Gamma$ = Vorticity X Area 

Velocity Potential Function ($\phi$)

Velocity potential function exist only for irrotational flow i.e. the existence of velocity potential function implies the flow is irrorational.

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

Stream Line Function ($\psi$)

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

• Volume flow rate between $\psi_1$ and $\psi_2$ is equal to $\psi_2 - \psi_1$.
• Velocity potential function can be defined for 3D flow but stream line function is defined only for 2D.

Along the stream line $d\psi = 0$,

\[d\psi = \frac{\partial \psi}{\partial x}dx + \frac{\partial \psi}{\partial y}dy = 0\]

\[-vdx + udy = 0 \]

\[\Bigg( \frac{dy}{dx}\Bigg)_{\psi = const.} = \frac{v}{u}\]

Along Equipotential line $\phi(x,y)$ is constant,

\[d\phi = \frac{\partial \phi}{\partial x}dx + \frac{\partial \phi}{\partial y}dy = 0\]

\[udx + vdy = 0 \]

\[\Bigg( \frac{dy}{dx}\Bigg)_{\phi = const.} = - \frac{u}{v}\]

\[\Bigg( \frac{dy}{dx}\Bigg)_{\psi = const.} *\Bigg( \frac{dy}{dx}\Bigg)_{\phi = const.} = -1\]

Hence, the streamlines and equipotential lines are orthogonal to each other except stagnation point.