Answer to Question #248947 in Mechanical Engineering for Aura

Question

The velocity field of a flow is given 2 2 V x ti y t x t j m s = + −+ 2 4 ( 1) 2 / ⎡ ⎤ ⎣ ⎦ K JK , where x and y are in meters and t is seconds. For fluid particles on the x axis, determine the speed and direction of flow.

Answer 1

velocity field of a flow: In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity

If a fluid particle moves along the curve

$x ( t ) = ( x ( t ) , y ( t ) )$

then its velocity at time t  is the derivative

$\mathbf{v}= \frac{d\mathbf{x}}{dt}$

of its position with respect to t. Thus, for a time-independent velocity vector field

$\mathbf{v}(x, y) = ( v_1(x, y), v_2(x, y) )$

the fluid particles will move in accordance with an autonomous, first order system of ordinary differential equations

$\frac{dx}{dt}= v_1(x, y),\qquad \frac{dy}{dt}= v_2(x, y).$