Calculating Acceleration Along Streamline XY=4: Problems at Point B?

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SUMMARY

The discussion focuses on calculating acceleration along the streamline defined by the equation xy = 4, specifically at points A(1,4), B(2,2), and C(4,1). The user encountered discrepancies in acceleration calculations at point B, where initial calculations indicated non-zero acceleration, contradicting lecture notes stating it should be zero. Two methods were explored: using the velocity vector components and an alternative approach involving the magnitude of the velocity vector. The latter confirmed zero acceleration at point B, highlighting the importance of method selection in fluid dynamics calculations.

PREREQUISITES
  • Understanding of fluid dynamics principles, specifically acceleration along streamlines.
  • Familiarity with vector calculus, particularly in the context of velocity and acceleration vectors.
  • Knowledge of the equations of motion for fluids, including the continuity equation.
  • Experience with mathematical modeling of fluid flow, including cross-sectional area variations.
NEXT STEPS
  • Study the derivation and application of the continuity equation in fluid dynamics.
  • Learn about the Navier-Stokes equations and their implications for fluid acceleration.
  • Explore the concept of streamline curvature and its effect on fluid acceleration.
  • Investigate the relationship between cross-sectional area changes and velocity in fluid flow scenarios.
USEFUL FOR

Fluid dynamics students, engineers working with fluid systems, and anyone involved in calculating fluid motion and acceleration in various applications.

tommywan410
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Along a streamline xy = 4, given u = 2x, v = -2y,
I was trying to calculate the acceleration at three different points A(1,4), B(2,2), C(4,1),
In my lecture notes, it is said that at A, the fluid is decelerating, at B, the acceleration is zero, and at C it is accelerating. However, i got some problem with my calculation at B.
V=ui+vj therefore the a = 4xi+4yj, when i plug in B, the acceleration is not zero. Why would it happen? Is there any problems with my calculation?

If i use another approach (i am not sure), V = sqrt(u^2+v^2), then
a = 2(u^2-v^2)/sqrt(u^2+v^2), when i plug in B, it is zero now. What is the difference between these two methods?


Another question is that in one of my practice problems, it is said that in a nozzle the cross-sectional area changes linearly from the base to the tip, does it mean that
A = A(in) + kx, where A is the cross-sectional area x m away from the inlet?
 
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What is V, u,v ?
 
ajayguhan said:
What is V, u,v ?
Big V is the velocity vector. V=ui+vj+wk, where i,j, and k are unit vectors in the x y and z directions.
 

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