To determine The acceleration vector . Explanation Write the expression of velocity vector. V = − 2 x y z ( x 2 + y 2 ) 2 i + ( x 2 − y 2 ) z ( x 2 + y 2 ) 2 j + y ( x 2 + y 2 ) k (I) Here, the displacement in x axis is x , the displacement in y axis is y , the displacement in z axis is z , the unit vector in x direction is i , the acceleration vector in y direction is j and unit vector in z direction is k . Write the expression of velocity vector. V = u i + v j + w k (II) Here, the x component of the velocity is u , the y component of velocity is v and the z component of the velocity is w . Write the expression of acceleration vector. a = ∂ V ∂ t + u ∂ V ∂ x + v ∂ V ∂ y + w ∂ V ∂ z (III) Conclusion: From Equation (I) and (II). u = − 2 x y z ( x 2 + y 2 ) 2 v = ( x 2 − y 2 ) z ( x 2 + y 2 ) 2 w = y ( x 2 + y 2 ) Differentiate Equation (I) with respect to t on both sides. ∂ V ∂ t = ∂ ∂ t { − 2 x y z ( x 2 + y 2 ) 2 i + ( x 2 − y 2 ) z ( x 2 + y 2 ) 2 j + y ( x 2 + y 2 ) k } = 0 + 0 + 0 = 0 Differentiate Equation (I) with respect to x on both sides. ∂ V ∂ x = ∂ ∂ x [ − 2 x y z ( x 2 + y 2 ) 2 i + ( x 2 − y 2 ) z ( x 2 + y 2 ) 2 j + y ( x 2 + y 2 ) k ] = [ − 2 y z ∂ ∂ x ( x ( x 2 + y 2 ) 2 ) i + z ∂ ∂ x ( ( x 2 − y 2 ) ( x 2 + y 2 ) 2 ) j + y ∂ ∂ x ( 1 ( x 2 + y 2 ) ) k ] = − 2 y z ( y 2 − 3 x 2 ) ( x 2 + y 2 ) 3 i + 2 z x ( − x 2 + 3 y 2 ) ( x 2 + y 2 ) 3 j − 2 y x ( x 2 + y 2 ) 2 k Differentiate Equation (I) with respect to y on both sides. ∂ V ∂ y = ∂ ∂ y { − 2 x y z ( x 2 + y 2 ) 2 i + ( x 2 − y 2 ) z ( x 2 + y 2 ) 2 j + y ( x 2 + y 2 ) k } = [ − 2 x z ∂ ∂ y ( y ( x 2 + y 2 ) 2 ) i + z ∂ ∂ y ( ( x 2 − y 2 ) ( x 2 + y 2 ) 2 ) j + ∂ ∂ y ( y ( x 2 + y 2 ) ) k ] = − 2 x z ( x 2 − 3 y 2 ) ( x 2 + y 2 ) 3 i − 2 z y ( − y 2 + 3 x 2 ) ( y 2 + x 2 ) 3 j + x 2 − y 2 ( x 2 + y 2 ) 2 k Differentiate Equation (I) with respect to z on both sides. ∂ V ∂ z = ∂ ∂ z { − 2 x y z ( x 2 + y 2 ) 2 i + ( x 2 − y 2 ) z ( x 2 + y 2 ) 2 j + y ( x 2 + y 2 ) k } = [ − 2 x y ( x 2 + y 2 ) 2 ∂ ∂ z ( z ) i + ( x 2 − y 2 ) ( x 2 + y 2 ) 2 ∂ ∂ z ( z ) ( ( x 2 − y 2 ) ( x 2 + y 2 ) 2 ) j + ∂ ∂ z ( y ( x 2 + y 2 ) ) k ] = [ − 2 x y ( x 2 + y 2 ) 2 i + ( x 2 − y 2 ) ( x 2 + y 2 ) 2 j +

Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

8th Edition

ISBN: 9781119080701

Author: Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Publisher: WILEY

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Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

Ch. 4.1 - Prob. 2P Ch. 4.1 - The velocity field of a flow is given by m/s,...Ch. 4.1 - A two-dimensional velocity field is given by u = 1...Ch. 4.1 - Streamlines are given in Cartesian coordinates by...Ch. 4.1 - Prob. 6P Ch. 4.1 - Prob. 8P Ch. 4.1 - Prob. 9P Ch. 4.1 - Prob. 10P Ch. 4.1 - Prob. 11P Ch. 4.1 - Prob. 12P

Ch. 4.1 - The x and y components of a velocity field are...Ch. 4.1 - Prob. 14P Ch. 4.1 - Prob. 15P Ch. 4.1 - For any steady flow the streamlines and...Ch. 4.1 - Prob. 17P Ch. 4.1 - Prob. 18P Ch. 4.1 - Prob. 19P Ch. 4.1 - Prob. 21P Ch. 4.1 - Classify the following flows as one-, two-, or...Ch. 4.2 - Prob. 23P Ch. 4.2 - Air is delivered through a constant-diameter duct...Ch. 4.2 - Water flows through a constant diameter pipe with...Ch. 4.2 - The velocity of air in the diverging pipe shown in...Ch. 4.2 - A certain flow field has the velocity...Ch. 4.2 - Prob. 28P Ch. 4.2 - Prob. 29P Ch. 4.2 - A shock wave is a very thin layer (thickness = ℓ)...Ch. 4.2 - Estimate the average acceleration of water as it...Ch. 4.2 - Prob. 32P Ch. 4.2 - As a valve is opened, water flows through the...Ch. 4.2 - The fluid velocity along the x axis shown in Fig....Ch. 4.2 - A fluid flows along the x axis with a velocity...Ch. 4.2 - A constant-density fluid flows through a...Ch. 4.2 - Prob. 37P Ch. 4.2 - Prob. 38P Ch. 4.2 - Prob. 39P Ch. 4.2 - An incompressible fluid flows through the...Ch. 4.2 - Prob. 41P Ch. 4.2 - Prob. 42P Ch. 4.2 - Prob. 43P Ch. 4.2 - Prob. 44P Ch. 4.2 - Prob. 45P Ch. 4.2 - Prob. 46P Ch. 4.2 - Assume that the streamlines for the wingtip...Ch. 4.2 - The velocity components for steady flow through...Ch. 4.2 - Water flows through the curved hose shown in Fig....Ch. 4.2 - Water flows though the slit at the bottom of a two...Ch. 4.2 - Prob. 51P Ch. 4.2 - Prob. 52P Ch. 4.2 - Fluid flows through a pipe with a velocity of 2.0...Ch. 4.2 - A gas flows along the x axis with a speed of V =...Ch. 4.2 - Assume the temperature of the exhaust in an...Ch. 4.2 - A bicyclist leases from her home at 9 a.m. and...Ch. 4.2 - The following pressures for the air flow in...Ch. 4.4 - In the region just downstream of a sluice gate,...Ch. 4.4 - At time t = 0 the valve on an initially empty...Ch. 4.4 - From calculus, one obtains the following formula...Ch. 4.4 - Air enters an elbow with a uniform speed of 10 m/s...Ch. 4.4 - A layer of oil flows down a vertical plate as...Ch. 4.4 - Figure P4.64 shows a fixed control volume. It has...Ch. 4.4 - Water enters a 5-ft-wide, 1-ft-deep channel as...Ch. 4.4 - The wind blows across a field with an approximate...Ch. 4.4 - Water flows from a nozzle with a speed of V = 10...

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