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#### Concept explainers

An incompressible frictionless flow field is given by
*A* = 2 s^{−1} and *B* = 2 *s*^{−1}, and the coordinates are measured in meters. Find the magnitude and direction of the acceleration of a fluid particle at point (*x*, *y*) = (2, 2). Find the pressure gradient at the same point, if

The magnitude of the acceleration of a fluid particle at point

The direction of the acceleration of a fluid particle at point

The pressure gradient of fluid at point

### Explanation of Solution

**Given:**

The flow field

The constants

Fluid particle point

Consider the density of water

**Calculations:**

From the flow field

Calculate the acceleration of the particle along *x* direction

Calculate the acceleration of the particle along *y* direction

Calculate the resultant acceleration of a fluid particle at point.

Thus, the magnitude of the acceleration of a fluid particle at point

Calculate the direction of the acceleration of fluid particle

Thus, the direction of the acceleration of a fluid particle at point

Calculate the pressure gradient in *x* direction.

Thus, the pressure gradient of fluid at point *x* direction is

Calculate the pressure gradient in *y* direction.

Thus, the pressure gradient of fluid at point *y* direction is

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# Chapter 6 Solutions

Fox and McDonald's Introduction to Fluid Mechanics

- According to the potential equation of a two-dimensional flow in the horizontal plane defined as; i-) Is this current physically possible? Prove ii-) Determine the current function ψ (x, y) of this current. [ψ (0,0) = 0] iii-) Calculate the resultant velocity and resultant acceleration at point A (e, f) in this flow field. iv-) Calculate the flow rate passing between the streamlines ψ (a, a) and ψ (c, c).
*arrow_forward*A two-dimensional flow field has an x-component of velocity given in Cartesian coordinates by u = 2x − 3y. (a) Find v, the y-component of velocity, if the flow is incompressible and v = 0 when x = 0. (b) If the flow follows the Bernoulli equation, find an expression for the pressure distribution as a function of x and y, given that the pressure is p0 at the stagnation point.*arrow_forward*The velocity components of an incompressible, two-dimensional field are given bythe following equations: u(x,y) =y^2 -x (1+x) v(x,y) = y(2x+1) Show that the flow field is (a) irrotational and (b) satisfies conservation of mass.*arrow_forward* - If an incompressible fluid flows in a corner bounded by walls meeting at the origin at an angle of 60◦, the streamlines of the flow satisfy the equation 2xy dx+ (x2−y2) dy = 0. Find the streamlines.
*arrow_forward*1. The components of velocity in a flow field are given byu=x2+y2+z2v=xy+yz+z2w=-3xz- 0.5z2+4a) Determine the volumetric dilatation rate and interpret the result.b) Determine an expression for the rotation vector. Is this an irrotational flow field?*arrow_forward*Consider the velocity field given by u = y/(x2 + y2) and v = −x/(x2 + y2). For the velocity field given , calculate the circulation around a circular path of radius 5 m. Assume that u and v given are in units of meters per second.*arrow_forward* - Once it has been started by sufficient suction, the siphon in the Fig. will run continuously as long as reservoir fluid available. Using Bernoulli's equation with no losses, show (a) that the exit velocity V2 depends only on gravity and the distance H and (b) that the lowest vacuum pressure occurs at point 3 and depends on the distance L+H
*arrow_forward*A sink of strength 20 m2/s is situated 3 m upstream of a source of 40 m²/s in a uniform stream. It is found that, at a point 2.5 m from both source and sink, the local velocity is normal to the line joining the source and sink. Find the velocity at this point and the velocity of the uniform stream. Locate any stagnation points and sketch the flow field.*arrow_forward*A2 A Newtonian fluid with viscosity (Mu) flows upward at a steady rate between two parallel plates that make an angle y with the horizontal. The fluid thickness h is much smaller than the width of the channel W. The pressures at each end, P(0) and P(L), are known and the pressure variations in the y-direction are small. Assume that Vy and Vz = 0 and Vx is a function of y alone. Use the shell balance to approach : - the Total flowrate, QV*arrow_forward* - A 2D incompressible velocity field is defined by . If both u and v are 0 at point (x = 0, y = 0), determine a) v component of the velocity and b) an expression for pressure field. Assume gravity is in the – y-direction and pressure is 0 at point (x=0, y=0). The flow is defined by: u=2(x2-y2)
*arrow_forward*If a hole is drilled at the bottom of a full bucket of water of height. H 60 meter what is the second velocity and first if we have pressure 1 in the air is 103 kpa and pressure 2 760pa (proof the equation you used to solve it ) Kindly re-draw the diagram.*arrow_forward*Find the stagnation point in the following two-dimensional velocity field: V=(3+x-y)i + (5+x+y)j*arrow_forward*

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