The velocity field of a permanent, incompressible and isothermal flow is given by u = a...

The velocity field of a permanent, incompressible and isothermal flow is given by

u = a (x^2 - y^2)   (1)
v = −2axy   (2)
w = 0    (3)

Gravitational acceleration acts on the z axis only.

a) Write the Navier-Stokes equations for this flow, canceling the appropriate terms
and replacing the non-zero derivatives with their detailed expression. Be sure to indicate
clearly your assumptions. (4 points)

b) From the result obtained in (a), demonstrate that the pressure field is continuous and therefore
physically possible (2 points)

c) Get the expression for the flow pressure field (4 points)

Expert Solution

Ally Wells answered 1 year ago

The velocity components of an incompressible, two-dimensional velocity field are given by the equations u=y^2-x(1+x) v=y(2x+1)...

The velocity components of an incompressible, two-dimensional velocity field are given by the equations u=y^2-x(1+x) v=y(2x+1) (a)Show that the flow satisfies continuity. (b) Determine the corresponding stream function for this flow field. (c) Determine if the flow is irrotational.

Consider an incompressible flow in cylindrical coordinates with velocity field v = Cr^(a) θ^ where C...

Consider an incompressible flow in cylindrical coordinates with velocity field v = Cr^(a) θ^ where C is a constant (a) Show that the flow satisfies the Navier-Stokes equations for only two values of a. (b) Given that p(r; θ; z) = p(r) only, and neglecting gravity, find the pressure distribution for each case, assuming that the pressure at r = R is p0

Velocity potential function in the velocity field is given asØ=1axyt:a-Is the flow compressible ?(3p)b-Show the flow...

Velocity potential function in the velocity field is given asØ=1axyt:a-Is the flow compressible ?(3p)b-Show the flow is steady ornot?(3p)c-In this flow field, find the resultant velocity and acceleration and their components at point A(8,8) at time t=1.(5p)d-Find the stream function for this flow.(7p)e-If the particle passing through the point (x0, y0)=(8, 8) is marked at time t=0, determine the location of the particle at time t=4s.(7p)

A two-dimensional transient velocity field is given by u = ax(b + ct) v = ey(f+...

A two-dimensional transient velocity field is given by u = ax(b + ct) v = ey(f+ ht) where u is the x velocity component and v, the y component. Find: The trajectory x(t), y(t) if x = x0, y = y0 at t = 0. The streamline that passes through x0, y0 when t=0 and plot it. The acceleration field. a (-) = 5 b (1/s) = 5 c (m) = -1 e (-) = 5 f (1/s) = 5...

Question: The velocity components for a fluid flow are given as follow: z(x-2)2 u = ty...

Question: The velocity components for a fluid flow are given as follow: z(x-2)2 u = ty w = xy (t is time) t+1 a.) Is the flow incompressible or not? b.) How many dimensions does the flow have? c.) Is the flow irrotational or rotational? d.) Find the local acceleration of the flow along x-direction at the points A(x,y,z,t)=A(0,1,1,4); B(x,y,z,t)=B{2,2,1,0)? e.) Find the convective acceleration of the flow along y-direction at the points A(x.y,z,t)=A(0,1,1,4); B(x,y,z,t)=B(2,2,1,0)?

The stream function for an incompressible flow is given by Y(x,y) = 2x2 – 2y2 –...

The stream function for an incompressible flow is given by Y(x,y) = 2x2 – 2y2 – xy. Determine if this flow satisfies the equation of continuity (the mass balance). On the graph below (or on your own paper), pick a point in each quadrant and determine the magnitude and direction of the velocity at that point. Indicate the magnitude and direction by drawing an arrow on the graph for each point. Use numbers.

1) Distinguish between blood flow rate and blood flow velocity. When an expert in the field...

1) Distinguish between blood flow rate and blood flow velocity. When an expert in the field uses the term blood flow, does that term usually mean rate or velocity? 2) Explain why a heart can keep beating after it has been removed from a living body.

A velocity vector field is given by F (x, y) = - i + xj a)...

A velocity vector field is given by F (x, y) = - i + xj a) Draw this vector field. b) Find parametric equations that describe the current lines in this field. c) Obtain the current line which passes through point (2,0) at time t = 0 and represent it graphically. d) Obtain the representation y = f(x) of the current line in part c).

The stream function ?? in a two-dimensional flow field is given as ?? = 4?? −...

The stream function ?? in a two-dimensional flow field is given as ?? = 4?? − 3?? + 7???? (a) Prove that this flow field is irrotational and that it satisfies the continuity equation. (b) Find the potential flow function Ф(??, ??) for this flow field with boundary condition Ф = 0 at x = 2, y = 1.

An incompressible fluid oscillates harmonically (V = V0 sin(wt), where V is the velocity) with a...

An incompressible fluid oscillates harmonically (V = V0 sin(wt), where V is the velocity) with a certain frequency in a 19 cm diameter pipe. A 1/4 scale model is to be used to determine the pressure difference per unit length, ∆?? at any instant along the pipe. Assume that ∆?? = ?(?, ?0,?,?,?,?) Where D is the pipe diameter, ω the frequency, t the time, μ the fluid viscosity, and ρ the fluid density. a) Determine the similarity requirements for...

Question