Fluid Mechanics
Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
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Chapter 4, Problem 4.29P
To determine

(a)

Whether the flow satisfy conservation of mass.

Expert Solution
Check Mark

Answer to Problem 4.29P

Yes, the flow satisfy conservation of mass

Explanation of Solution

Given information:

The velocity in x -direction is 2xy, the velocity in y direction is y2x2.

Write the expression for dot product of the gradient and the velocity vector.

V=ux+vy+wz ... (I)

Here, velocity component along x direction is u, velocity component along y direction is v, the velocity component along z direction is w.

Calculation:

Substitute 2xy for u, y2x2 for v, 0 for w in Equation (I).

V=x2xy+yy2x2+z0=x2xy+yy2x2=2y+2y=0

Conclusion:

Since the dot product of the gradient and velocity vector is zero therefore the field satisfies conservation of mass.

To determine

(b)

The pressure field p(x,y), if the pressure at the point x=0,y=0 is equal to pa.

Expert Solution
Check Mark

Answer to Problem 4.29P

The pressure field p(x,y), if the pressure at the point x=0,y=0 is equal to pa is px,y=ρ22x2y2+x4+y4+pa.

Explanation of Solution

Given Information:

The pressure at the point x=0,y=0 is pa.

Write the expression for incompressible Navier stoke Equation using x relation.

ρuux+vuy+wuz=ρgxpx+μ2ux2+2uy2+2uz2 ... (II)

Here the density of the fluid is ρ The acceleration due to gravity in the x direction is gx, Here, pressure change in x direction is px, viscosity of the fluid is μ.

Write the expression for incompressible Navier stoke Equation using y relation.

ρuvx+vvy+wvz=ρgypy+μ2vx2+2vy2+2vz2 ... (III)

Here the density of the fluid is ρ the acceleration due to gravity in the y direction is gy, Here, pressure change in y direction is py.

Calculation:

Substitute 2xy for u, y2x2 for v, 0 for w, 0 for gx in Equation (II).

ρ 2xy 2xy x + y 2 x 2 2xy y + 0 2xy z =ρ×0px+μ 2 2xy x 2 + 2 2xy y 2 + 2 2xy z 2 ρ 2xy+ 2y+ y 2 x 2 2x+0=0px+μ0+0+0ρ4xy2+ 2x y 2 +2 x 3 =pxρ2xy2+2x3=px

2ρxy2+x3=pxpx=2ρxy2+x3 ... (IV)

Substitute 2xy for u, y2x2 for v, 0 for w, 0 for gy in Equation (III)

ρ 2xy y 2 x 2 x + y 2 x 2 y 2 x 2 y +0× y 2 x 2 z =ρ×0py+μ 2 y 2 x 2 x 2 + 2 y 2 x 2 y 2 + 2 y 2 x 2 z 2 ρ 2xy+ 2x+ y 2 x 2 2y+0=0py+μ 2+2+0ρ4xy2+ 2 y 3 2 x 2 y=py

2ρx2y+y3y=pp=2ρx2y+y3y

Integrate the equation on both sides

p=2ρ x 2 y+ y 3 yp=2ρx2 y 2 2+ y 4 4+fx+constant ... (V)

Differentiate the Equation with respect to x.

px=x2ρ x 2 y 2 2 + y 4 4 +fx+constant=2ρ× 2x y 2 2 +0+ f x x+0=2ρ× x y 2 +0+ f x x=2ρxy2+ f x x

Substitute 2ρxy2+fxx for px in Equation (IV).

2ρxy2+ f x x=2ρxy2+x3 f x x=2ρxy2+x3+2ρxy2 f x x=2ρxy22ρx3+2ρxy2 f x x=2ρx3

fx=2ρx3x

Integrating the above equation.

f x =2ρ x 3 xfx=2ρ x 4 4

Substitute 2ρx44 for fx in Equation (V).

p=2ρx2 y 2 2+ y 4 4+2ρ x 4 4+constant=2ρx2 y 2 2+ y 4 4+ x 4 4+constant=2ρ×142x2y2+y4+x4+constant=ρ22x2y2+y4+x4+constant ... (VI).

Substitute 0 for x, 0 for y, pa for p in Equation (VI).

pa=ρ22×0+0+0+constantpa=constant

Substitute pa for constant in Equation (VI).

p=ρ22x2y2+y4+x4+pa

Conclusion:

The pressure field for px,y is ρ22x2y2+x4+y4+pa ..

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Chapter 4 Solutions

Fluid Mechanics

Ch. 4 - Prob. 4.11PCh. 4 - Prob. 4.12PCh. 4 - Prob. 4.13PCh. 4 - Prob. 4.14PCh. 4 - What is the most general form of a purely radial...Ch. 4 - Prob. 4.16PCh. 4 - An excellent approximation for the two-dimensional...Ch. 4 - Prob. 4.18PCh. 4 - A proposed incompressible plane flow in polar...Ch. 4 - Prob. 4.20PCh. 4 - Prob. 4.21PCh. 4 - Prob. 4.22PCh. 4 - Prob. 4.23PCh. 4 - Prob. 4.24PCh. 4 - An incompressible flow in polar coordinates is...Ch. 4 - Prob. 4.26PCh. 4 - Prob. 4.27PCh. 4 - P4.28 For the velocity distribution of Prob. 4.10,...Ch. 4 - Prob. 4.29PCh. 4 - Prob. 4.30PCh. 4 - Prob. 4.31PCh. 4 - Prob. 4.32PCh. 4 - Prob. 4.33PCh. 4 - Prob. 4.34PCh. 4 - P4.35 From the Navier-Stokes equations for...Ch. 4 - A constant-thickness film of viscous liquid flows...Ch. 4 - Prob. 4.37PCh. 4 - Prob. 4.38PCh. 4 - Reconsider the angular momentum balance of Fig....Ch. 4 - Prob. 4.40PCh. 4 - Prob. 4.41PCh. 4 - Prob. 4.42PCh. 4 - Prob. 4.43PCh. 4 - Prob. 4.44PCh. 4 - Prob. 4.45PCh. 4 - Prob. 4.46PCh. 4 - Prob. 4.47PCh. 4 - Consider the following two-dimensional...Ch. 4 - Prob. 4.49PCh. 4 - Prob. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - P4.54 An incompressible stream function is...Ch. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - A two-dimensional incompressible flow field is...Ch. 4 - P4.58 Show that the incompressible velocity...Ch. 4 - Prob. 4.59PCh. 4 - Prob. 4.60PCh. 4 - An incompressible stream function is given by...Ch. 4 - Prob. 4.62PCh. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Prob. 4.65PCh. 4 - Prob. 4.66PCh. 4 - A stream function for a plane, irrotational, polar...Ch. 4 - Prob. 4.68PCh. 4 - A steady, two-dimensional flow has the following...Ch. 4 - A CFD model of steady two-dimensional...Ch. 4 - Consider the following two-dimensional function...Ch. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.74PCh. 4 - Given the following steady axisymmetric stream...Ch. 4 - Prob. 4.76PCh. 4 - Prob. 4.77PCh. 4 - Prob. 4.78PCh. 4 - Prob. 4.79PCh. 4 - Oil, of density and viscosity , drains steadily...Ch. 4 - Prob. 4.81PCh. 4 - Prob. 4.82PCh. 4 - P4.83 The flow pattern in bearing Lubrication can...Ch. 4 - Consider a viscous film of liquid draining...Ch. 4 - Prob. 4.85PCh. 4 - Prob. 4.86PCh. 4 - Prob. 4.87PCh. 4 - The viscous oil in Fig. P4.88 is set into steady...Ch. 4 - Oil flows steadily between two fixed plates that...Ch. 4 - Prob. 4.90PCh. 4 - Prob. 4.91PCh. 4 - Prob. 4.92PCh. 4 - Prob. 4.93PCh. 4 - Prob. 4.94PCh. 4 - Two immiscible liquids of equal thickness h are...Ch. 4 - Prob. 4.96PCh. 4 - Prob. 4.97PCh. 4 - Prob. 4.98PCh. 4 - For the pressure-gradient flow in a circular tube...Ch. 4 - W4.1 The total acceleration of a fluid particle is...Ch. 4 - Is it true that the continuity relation, Eq....Ch. 4 - Prob. 4.3WPCh. 4 - Prob. 4.4WPCh. 4 - W4.5 State the conditions (there are more than...Ch. 4 - Prob. 4.6WPCh. 4 - W4.7 What is the difference between the stream...Ch. 4 - Under what conditions do both the stream function...Ch. 4 - Prob. 4.9WPCh. 4 - Consider an irrotational, incompressible,...Ch. 4 - Prob. 4.1FEEPCh. 4 - Prob. 4.2FEEPCh. 4 - Prob. 4.3FEEPCh. 4 - Given the steady, incompressible velocity...Ch. 4 - Prob. 4.5FEEPCh. 4 - Prob. 4.6FEEPCh. 4 - C4.1 In a certain medical application, water at...Ch. 4 - Prob. 4.2CP
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