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:
- Eulerian Approach: Fluid motion is given by prescribing the necessary properties (p, V, T etc) in terms of space and time.
- 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.
