Explanation Given information: The Cartesian co-ordinate ( x , y ) is ( 0.4 m,0 .20 m ) and velocity component ( u , v ) is ( 10.3 m / s , − 5.6 m / s ) . Write the expression for the radial distance from the origin. r = x 2 + y 2 ........... (I) Here, x and y are Cartesian co-ordinate. Write the expression for the angle measured from the x-axis. θ = tan − 1 ( y x ) ...... (II) Write the expression for the u r component of the velocity. u r = u cos θ + v sin θ ........... (III) Write the expression for the u θ component of the velocity. u θ = − u sin θ + v sin θ ........... (IV) Calculation: Substitute 0.4 m for x , 0.2 m for y in the Equation (I). r = ( 0.4 m ) 2 + ( 0.2 m ) 2 = ( 0.2 m ) 2 = 0.447 m Substitute 0.4 m for x , 0.2 m for y in the Equation (II). θ = tan − 1 ( 0.2 m 0.4 m ) = tan − 1 ( 0

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Chapter 9, Problem 12P

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The cylindrical coordinates r, θ, ur, and uθ of the fluid at the given point.

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Problem (17): water flowing in an open channel of a rectangular cross-section with width (b) transitions from a mild slope to a steep slope (i.e., from subcritical to supercritical flow) with normal water depths of (y₁) and (y2), respectively. Given the values of y₁ [m], y₂ [m], and b [m], calculate the discharge in the channel (Q) in [Lit/s]. Givens: y1 = 4.112 m y2 = 0.387 m b = 0.942 m Answers: ( 1 ) 1880.186 lit/s ( 2 ) 4042.945 lit/s ( 3 ) 2553.11 lit/s ( 4 ) 3130.448 lit/s

Problem (14): A pump is being used to lift water from an underground tank through a pipe of diameter (d) at discharge (Q). The total head loss until the pump entrance can be calculated as (h₁ = K[V²/2g]), h where (V) is the flow velocity in the pipe. The elevation difference between the pump and tank surface is (h). Given the values of h [cm], d [cm], and K [-], calculate the maximum discharge Q [Lit/s] beyond which cavitation would take place at the pump entrance. Assume Turbulent flow conditions. Givens: h = 120.31 cm d = 14.455 cm K = 8.976 Q Answers: (1) 94.917 lit/s (2) 49.048 lit/s ( 3 ) 80.722 lit/s 68.588 lit/s 4

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The cylindrical coordinates r , θ , u r , and u θ of the fluid at the given point.

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The cylindrical coordinates r, θ, ur, and uθ of the fluid at the given point.

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