The velocity field for a two dimensional flow is given by : V(x,y,t) = (x/t)i-(y/t)j. The total acceleration is?

Question

Answer

Given: Velocity field for a two dimensional flow is V(x,y,t) = (x/t)i-(y/t)j.

To find: Total acceleration.

Solution:

Step 1: Calculation of partial derivatives:

We have V(x,y,t) = (x/t)i-(y/t)j.

Partial derivative with respect to x: Vx = i/t

Partial derivative with respect to y: Vy = -j/t

Partial derivative with respect to t: Vt = -xi/t² - yj/t²

Step 2: Calculation of acceleration:

Acceleration is given by the formula: a = ∂v/∂t + (v . ∇)v

where,

∇ is the del operator
v is the velocity vector

We have ∂v/∂t = -xi/t² - yj/t²

(v . ∇)v = (Vx i + Vy j) (Vx ∂/∂x + Vy ∂/∂y)

= [(x/t)i - (y/t)j] . [i/t ∂/∂x - j/t ∂/∂y]

= (x²+y²)/t³

Therefore, the total acceleration is:

a = -xi/t² - yj/t² + (x²+y²)/t³

Final Answer: The total acceleration for the given velocity field is -xi/t² - yj/t² + (x²+y²)/t³.

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