Determine the acceleration field for a three-dimensional flow with velocity components

  Determine the acceleration field for a three-dimensional flow with velocity components u= -x, v = 4x^2y^2, and w = x-y

Answer

General guidance

Concepts and Reason

Velocity of a fluid particle is a function of location and time. Velocity is a vector which has both magnitude and direction. The acceleration of a fluid particle is the time rate of change of velocity.

Fundamentals

The velocity vector for a fluid particle, V=ui + vj+wk The acceleration vector for a fluid particle in three dimensional flow, DV
Dt
öttöxt oyt az The scalar components of acceleration vector, g%29 心
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Step-by-step

Step 1 of 2

The x-component of acceleration, -(-) 4(4)4xy2) (-2)
=(0) + (-x) (-1)+4xy² (0)+(x-y)(0)
=X The y-component of acceleration, - (ar v)+(-7) (ap ya) ar yn olaf y) = (x ->)>(axy)
ду
=(0)+(-x)(8xy?) + 4x’y? (8x’y)+(x- y)(0)
= -8x’y+32x*y? The z-component of acceleration, -+
Ow
-
@x
+-
w
Oy
= (x+y)+(+)0(4,7+ 4x?p: 0(8+3)+(-»)O(x-»)
= (0)+(-x)(1-0) + 4x’y?(0-1)+(x-y)(0)
=-x-4x²y2

Find the components of acceleration along the three Cartesian coordinate axes using the spatial and time derivatives of velocity components.

Step 2 of 2

Find the acceleration field using the relation,

a =q i+a, j+ak
= xi +(+8x”y? +32x*y);+(-x-4x’y?)k

The acceleration field is xi +(-8x’y? +32x*y);+(-x-4x’y?)k.


Write the acceleration field vector using the components of acceleration of the fluid particle.

Answer

The acceleration field is xi +(-8x’y? +32x*y);+(-x-4x’y?)k.

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