In Chap. 9, we define the stream function
where u and v ate the velocity components in the x- and y-directions, respectively. (a) What are the primary dimensions of
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Fluid Mechanics: Fundamentals and Applications
- Assume an inviscid, incompressible flow. Also, standard sea level density and pressure are 1.23 kg/m3 (0.002377 slug/ft3) and 1.01 × 105 N/m2 (2116 lb/ft2), respectively. Derive the velocity potential for a doublet.arrow_forwardLiquid flows out of a hole in the bottom of a tank as in FinConsider the case in which the hole is very small compared to the tank (d ≪ D). Experiments reveal that average jet velocity V is nearly independent of d, D, ? , or ? . In fact, for a wide range of these parameters, it turns out that V depends only on liquid surface height h and gravitational acceleration g. If the liquid surface height is doubled, all else being equal, by what factor will the average jet velocity increase?arrow_forwardVolumetric strain rate is zero for a steady incompressible flow. In Cartesian coordinates we express this as ∂u/∂x + ∂?/∂y + ∂w/∂z = 0Suppose the characteristic speed and characteristic length for a given flow field are V and L, respectively. Define the following dimensionless variables, Nondimensionalize the equation, and identify any established (named) dimensionless parameters that may appear. Discuss.arrow_forward
- Consider liquid flow of density ρ , viscosity μ , and velocityU over a very small model spillway of length scale L ,such that the liquid surface tension coefficient Y is important.The quantity ρ U 2 L / Y, in this case, is important and iscalled the( a ) capillary rise, ( b ) Froude number, ( c ) Prandtl number,( d ) Weber number, ( e ) Bond numberarrow_forwardAssume an inviscid, incompressible flow. Also, standard sea level density and pressure are 1.23 kg/m3 (0.002377 slug/ft3) and 1.01 × 105 N/m2(2116 lb/ft2), respectively. Prove that the flow field specified is not incompressible;i.e., it is a compressible flow as stated without proof .arrow_forwardFor the flow field given in Cartesian coordinates by u = 2xt + y, v = −2yt, w = 0: (a) Is the flow one-, two- or three-dimensional? (b) Is the flow steady? (c) Is the flow compressible? (d) What is the vector acceleration following a fluid particle?arrow_forward
- Consider the steady, two-dimensional velocity field of : V-›= (u, ? ) = (0.5 + 0.8 x) i-›+ (1.5 − 0.8 y) j-› where lengths are in units of m, time in s, and velocities in m/s. There is a stagnation point at (−0.625, 1.875) as shown in Fig. Streamlines of the flow are also plotted in Fig. Calculate the various kinematic properties, namely, the rate of translation, rate of rotation, linear strain rate, shear strain rate, and volumetric strain rate. Verify that this flow is incompressible.arrow_forwardIn CFD lingo, the stream function is often called a non-primitive variable, while velocity and pressure are called primitive variables. Why do you suppose this is the case?arrow_forwardConsider steady, incompressible, laminar, fully developed, planar Poiseuille flow between two parallel, horizontal plates (velocity and pressure profiles are shown in Fig. At some horizontal location x = x1, the pressure varies linearly with vertical distance z, as sketched. Choose an appropriate datum plane (z = 0), sketch the profile of modi fied pressure all along the vertical slice, and shade in the region representing the hydrostatic pressure component. Discuss.arrow_forward
- Consider laminar flow through a long section of pipe, as in Fig. For laminar flow it turns out that wall roughness is not a relevant parameter unless ? is very large. The volume flow rate V· through the pipe is a function of pipe diameter D, fluid viscosity ? , and axial pressure gradient dP/dx. If pipe diameter is doubled, all else being equal, by what factor will volume flow rate increase? Use dimensional analysis.arrow_forwardFor the flow field given in Cartesian coordinates by u = 0, v = x2z, w = z3: (a) Is the flow compressible? (b) What is the y-component of the acceleration following a fluid particle?arrow_forwardThe open tank in Fig. contains water at 20 ° C and isbeing filled through section 1. Assume incompressibleflow. First derive an analytic expression for the water-levelchange dh / dt in terms of arbitrary volume flows ( Q 1 , Q 2 ,Q 3 ) and tank diameter d . Then, if the water level h is constant,determine the exit velocity V 2 for the given data V 1 =3 m/s and Q 3 = 0.01 m 3 /s.arrow_forward
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