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:
dxu=dyv=dzw
ρAV=Constant
∂ρ∂t+∂(ρu)∂x+∂(ρv)∂y+∂(ρw)∂z=0
DV⃗ Dt=axi+ayj+azk=DuDti+DvDtj+DwDtk
DDt is called Material Derivative or Substantial Derivative or Total Derivative with respect to time.
Rotational Compenent (
ω=12∇×V⃗ =12∣∣∣∣∣i∂∂xuj∂∂yvk∂∂zw∣∣∣∣∣ ωz=12(∂v∂x−∂u∂y)
Ω=2ω=∇×V⃗
Γ = Vorticity X Area
Velocity Potential Function (
u=∂ϕ∂x , v=∂ϕ∂y , w=∂ϕ∂z
Stream Line Function (
u=∂ψ∂y , v=−∂ψ∂x
dψ=∂ψ∂xdx+∂ψ∂ydy=0 −vdx+udy=0 (dydx)ψ=const.=vu
dϕ=∂ϕ∂xdx+∂ϕ∂ydy=0 udx+vdy=0 (dydx)ϕ=const.=−uv
(dydx)ψ=const.∗(dydx)ϕ=const.=−1
- 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:
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.
NOTE: In steady flow, streakline, pathline and streamline are identical.
Continuity Equation (Conservation of mass)
where, ρ = Density
A = Area
V = Velocity
For incompressible fluid it reduces to A1V1=A2V2 .
Generalised Continuity Equation
where u,v,w are component of velocity in x, y, z direction.
- If flow is steady
∂ρ∂t=0 . - For steady, imcompressible flow continuity equation
∂u∂x+∂v∂y+∂w∂z=0 .
Total acceleration of fluid
- In steady flow, local acceleration will be zero.
- In uniform flow, convective acceleration will be zero.
Rotational Compenent (ω )
- Vorticity (
Ω ) is as the vector that is twice the rotational vector.
- Flow said to be irrotational if vorticity or rotational vector is zero at all points in region.
- Circulation (
Γ ) is line integral of tangential component of velocity around a closed curve.
Velocity Potential Function (ϕ )
Velocity potential function exist only for irrotational flow i.e. the existence of velocity potential function implies the flow is irrorational.
Stream Line Function (ψ )
- Volume flow rate between
ψ1 andψ2 is equal toψ2−ψ1 . - Velocity potential function can be defined for 3D flow but stream line function is defined only for 2D.
Along the stream line dψ=0 ,
Along Equipotential line ϕ(x,y) is constant,
Hence, the streamlines and equipotential lines are orthogonal to each other except stagnation point.
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