FLUID KINEMATICS. BY GP CAPT NC CHATTOPADHYAY. Fluid Kinematics. Velocity Field Continuity Equation. Fluid Kinematics. What is fluid kinematics? Fluid kinematics is the study on fluid motion in space and time without considering the force which causes the fluid motion.
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GP CAPT NC CHATTOPADHYAY
What is fluid kinematics?
3-dimensional and unsteady : u (x,y,z,t)
2-dimensional unsteady and steady flow
1-dimensional unsteady and steady flow
ALSO, u = ∂ ψ/∂ y and v= - ∂ Ψ/∂ x,
u = - ∂ Φ /∂ x and v = - ∂ Φ /∂y, w = - ∂ Φ/∂z
A liquid is flowing from left to right and the pipe is narrowing in the same direction. By the continuity principle, the mass flow rate must be the same at each section - the mass going into the pipe is equal to the mass going out of the pipe. So we can write
Considering a stream-tube of cylindrical cross sections with velocities perpendicular to the cross sections and densities at the respective cross sections and assuming the velocities and densities are constant across the whole cross section , a fluid mass closed between cross section 1 and 2 at an instant t will be moved after a time interval dt by to the cross section 1’ and 2’ respectively.
Because the closed mass between 1 and 2 must be the same between 1’ and 2’, and the mass between 1’ and 2 for a steady flow can not change from t and t+dt, the mass between 1 and 1’ moved in dt, i.e must be the same as the mass between 2 and 2’ moved in the same time dt i.e :
Discharge in a pipe
The rate of mass entering a face is the product of the density, the fluid velocity and the face area. For example, on the side facing the reader, the density (r) is multiplied by the velocity in the x direction (u) and the area of the face Dy Dz. Thus, the mass flux entering the volume through this face is
The application of basic calculus (taking the limit as Δt tends to 0) allows us to write this equation as
The Continuity Equation may be simplified for some common flow situations as follows. If the fluid may be treated as incompressible (as is the case with water or in low velocity air flows), the density will be constant. The Continuity Equation then becomes
Total mass flow into the junction = Total mass flow out of the junction
r1Q1 = r2Q2 + r3Q3
When the flow is incompressible (e.g. if it is water) r1 = r2 = r