The motion of a fluid particle is specified by the equations. x = et and y = e -

Question

The motion of a fluid particle is specified by the equations. x = et and y = e -t in the Lagrangian description. Determine the equation for the path of the fluid particle. Determine the velocity of the fluid particle. Determine the acceleration of the fluid particle. Determine the velocity of the flow field. State under what conditions such a flow field can be defined. Determine whether the flow field is steady or not. Determine the equation of the streamline on which the fluid particle is located. Determine the equation of the streakline on which the fluid particle is located. Determine whether the flow field is compressible or not. Determine whether the flow field is rotational or not. Determine the equation of the streamline passing through point Q(2, 2). Is it possible to define a stream function for this flow field? If yes, state why determine the stream function such that (psi = 0 at the origin.

Explanation / Answer

A) eliminating t from both eqns

path of the particle : x*y =1

B)velocity of fluid particle vx = dx/dt = e^t ; vy = dy/dt = -e^t

velocity of fluid particle at time t = e^t i + (-e^-t) j

C)acceleration = dv/dt = e^t i + (e^-t) j

D)xpress velocity of fluid particle in terms of x,y,t

velocity of flow field V(x,y,t) = x i - y j ........(i)

E)the flow field can be only in case of steady laminar flows as it is a 2D flow(involving x,y and not z)

F)as the velocity of flow field (i) is only a function of x,y and not of t steady flow field or dV/dt = 0

G)to get equation of streamline:

dy/dx = dy/dt / dx/dt = Vy/Vx = -y/x (from (i))

integrating = x^2 + y^2 = C

H) let us consider the line joining all particles that go through the point (x0,y0) at t=t0

let x,y lie on such a curve

then integrating dx/dt = e^t and dy/dt= -e^t from x=x0 to x ,y=y0 to y, t=t0 to t

x-x0 = e^t - e^t0

y -y0 = e^-t - e^t0

(as x0=e^t0 ; y0=e^-t0)

x*y = 1 is equation of the streakline same as that of pathline

I)conservation of mass ==> grad(.V) = 0

grad()*V + *grad(V) = 0

grad(V) = 0 (from (i) as   x/x + (-y)/y = 1-1=0)

grad() = 0 ==> incompressible fluid

J)for irrotational flow gradient vector X V = 0 (vector multiplication /cross product with velocity field)

or v/x - u/y = 0-0 =0 ==> irrotational flow

K) equation of streamline through (2,2)

x^2+y^2 = 4

L) as flow is irrotational stream function exists

v = -y = /y ==> = -y^2/2 + C(x)

u = x = /x = C'(x) ==> C(x) = x^2/2

   = 0.5*(x^2 - y^2)

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