Show that the velocity potential (r, z) in axisymmetriccylindrical coordinates (see Fig. 4.2) is defined such thatафдфv,дrazFurther show that for incompressible flow this potentialsatisfies Laplace's equation in (r, z) coordinates. Typicalinfinitesimalelement- Typical point (r,0, z)Basedrlinedzr deCylindrical axis

Write the function in 3d coordinate system:

Answered: Show that the velocity potential (r, z)…

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Consider the two-dimensional flow of an inviscid, incompressible fluid described by the superposition of a parallel flow of velocity V0, a source of strength q, and a sink of strength −q, separated by a distance b in the direction of the parallel flow, the source being upstream of the sink. (a) Find the resultant stream function and velocity potential. (b) Sketch the streamline pattern. (c) Find the location of the upstream stagnation point relative to the source.
6 Consider a fully developed laminar flow of a fluid through 8027 m long and 4 cm diameter horizontal and circular pipe. The dynamic density and the viscosity of the fluid are 1252 kg/m³ and 0.3073 kg/(m.s). The velocity profile at a cross-section is given by: u(r)= 6[1 − ( 7 )²] - m/s Where r is the axial distance from the centre and R is the radius of the pipe. Determine the following: (i) (ii) the maximum velocity at a cross-section of the pipe, Umax the average velocity at a cross-section of the pipe, Vave the volume flow rate, Q (iv) Reynolds number, Re (v) friction factor, f (vi) head loss, hi (vii) pressure loss, AP (viii) pumping power required, (ix) for the same pumping power, the percentage decrease of the flow rate if the pipe is inclined 10° upward (assume the head loss, hò, calculated in part (vi) does not change) Useful formulae: 64 f Re h₂ = f ( =) ( 2² ) 2g =
e1
m Problem 6: The top wall moves with velocity Vtop = 1 S and the bottom wall is stationary. Inside there is a layer of water and a layer of oil on top. Both layers have the same thickness and the total thickness is h=2m. The flow is incompressible, laminar and steady and both fluids are Newtonian. The viscosity of water is uw =|10-3)Pa s and the density is Pw 103 Kg. The viscosity of oil is He 10-1 Pas and the density is Po 3° kg tit 700- {There is no pressure gradient./The width is w= 5m. m3 Veap h=2m water
Water is flowing in a rectangular channel with a base of 4.0 ft. The velocity gradient from the bottom of the channel to the water surface is given by the equation: v=3.5y^0.6 Where y is the distance from the bottom of the channel. The total depth of the flow in the channel is 3.0ft. a) Graph the velocity profile from the bottom of the channel to the water surface. B) Calculate the flow rate by integration. The first step is to draw a cross-section of channel and on that cross-section, Draw on your figure an elemental area DA that you will then ise in the equation: Q=integral V.dA

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