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
- One of the conditions in using the Bernoulli equation is the requirement of inviscid flow. However there is no fluid with zero viscosity in the world except some peculiar fluid at very low temperature. Bernoulli equation or inviscid flow theory is still a very important branch of fluid dynamics for the following reasons: (i) (ii) There is wide region of flow where the velocity gradient is zero and so the viscous effect does not manifest itself, such as in external flow past an un- stalled aerofoil. The conservation of useful energy allows the conversion of kinetic and potential energy to pressure and hence pressure force acting normal to the control volume or system boundary even though the tangential friction stress is absent. It allows the estimation of losses in internal pipe flow. (A) (i) and (ii) (B) (i) and (iii) (ii) and (iii) All of the above (C) (D)arrow_forwardConsider steady, incompressible, two-dimensional flow due to a line source at the origin. Fluid is created at the origin and spreads out radially in all directions in the xy-plane. The net volume flow rate of created fluid per unit width is V·/L (into the page of Fig), where L is the width of the line source into the page in Fig Since mass must be conserved everywhere except at the origin (a singular point), the volume flow rate per unit width through a circle of any radius r must also be V·/L. If we (arbitrarily) specify stream function ? to be zero along the positive x-axis (? = 0), what is the value of ? along the positive y-axis (? = 90°)? What is the value of ? along the negative x-axis (? = 180°)?arrow_forwardA football, meant to be thrown at 60 mi/h in sea-level air( ρ = 1.22 kg/m 3 , μ = 1.78 E-5 N ? s/m 2 ), is to be testedusing a one-quarter scale model in a water tunnel ( ρ =998 kg/m 3 , μ =0.0010 N . s/m 2 ). For dynamic similarity,what is the proper model water velocity?( a ) 7.5 mi/h, ( b ) 15.0 mi/h, ( c ) 15.6 mi/h,( d ) 16.5 mi/h, ( e ) 30 mi/harrow_forward
- A liquid of density ? and viscosity ? flows by gravity through a hole of diameter d in the bottom of a tank of diameter DFig. . At the start of the experiment, the liquid surface is at height h above the bottom of the tank, as sketched. The liquid exits the tank as a jet with average velocity V straight down as also sketched. Using dimensional analysis, generate a dimensionless relationship for V as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result. (Hint: There are three length scales in this problem. For consistency, choose h as your length scale.) except for a different dependent parameter, namely, the time required to empty the tank tempty. Generate a dimensionless relationship for tempty as a function of the following independent parameters: hole diameter d, tank diameter D, density ? , viscosity ? , initial liquid surface height h, and gravitational acceleration g.arrow_forwardQ1: Consider laminar flow over a flat plate. The boundary layer thickness o grows with distance x down the plate and is also a function of free-stream velocity U, fluid viscosity u, and fluid density p. Find the dimensionless parameters for this problem, being sure to rearrange if neessary to agree with the standard dimensionless groups in fluid mechanics. Answer: Q2: The power input P to a centrifugal pump is assumed to be a function of the volume flow Q, impeller diameter D, rotational rate 2, and the density p and viscosity u of the fluid. Rewrite these variables as a dimensionless relationship. Hint: Take 2, p, and D as repeating variables. P e paD? = f( Answer:arrow_forwardA Fluid Mechanics, Third Edition - Free PDF Reader E3 Thumbnails 138 FLUID KINEMATICS Fluid Mechanies Fundamenteis and Applicationu acceleration); this term can be nonzero even for steady flows. It accounts for the effect of the fluid particle moving (advecting or convecting) to a new location in the flow, where the velocity field is different. For example, nunan A Çengel | John M. Cinbala consider steady flow of water through a garden hose nozzle (Fig. 4-8). We define steady in the Eulerian frame of reference to be when properties at any point in the flow field do not change with respect to time. Since the velocity at the exit of the nozzle is larger than that at the nozzle entrance, fluid particles clearly accelerate, even though the flow is steady. The accel- eration is nonzero because of the advective acceleration terms in Eq. 4-9. FLUID MECHANICS FIGURE 4-8 Flow of water through the nozzle of a garden hose illustrates that fluid par- Note that while the flow is steady from the…arrow_forward
- The time t d to drain a liquid from a hole in the bottom of atank is a function of the hole diameter d , the initial fluidvolume y 0 , the initial liquid depth h 0 , and the density ρ andviscosity μ of the fluid. Rewrite this relation as a dimensionlessfunction, using Ipsen’s method.arrow_forwardQ.5 A plate 1 mm distance from a fixed plate, is moving at 500 mm/s by a force induces a 2 shear stress of 0.3 kg(f)/m. The kinematic viscosity of the fluid (mass density 1000 kg/ 3. m) flowing between two plates (in Stokes) isarrow_forwardUse the first principle (dimensional analysis) to generate a dimensionless relationship for the x-component of fluid velocity u as a function of fluid viscosity μ, top plate speed v, distance h, fluid density ρ, and distance y.arrow_forward
- [1] Consider steady flow of air through the diffuser portion of a wind tunnel. Along the centerline of the diffuser, the air speed decreases from uentrance to ut as sketched. Measurements reveal that the centerline air speed decreases parabolically through the diffuser. Write an equation for centerline speed u(x), based on the parameters given here, Dee x=0 to x=L.arrow_forwardThe Stokes number, St, used in particle dynamics studies,is a dimensionless combination of five variables: accelerationof gravity g , viscosity μ , density ρ , particle velocity U ,and particle diameter D . ( a ) If St is proportional to μand inversely proportional to g , find its form . ( b ) Showthat St is actually the quotient of two more traditionaldimensionless groups.arrow_forward1ODiem # The side thrust F, for a smooth spinning ball in a fluid is a function of the ball diameter D, the free-stream velocity V, the densityp, the viscosityu, and the angular velocity of spino. F= f( D, ρ, μ, V, ω) Using the Buckingham Pi theorem to express this relation in dimensionless form. Farrow_forward
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