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A parabolic approximate velocity profile was used in Problem 5.11 to model flow in a laminar incompressible boundary layer on a flat plate. For this profile, find the x component of acceleration, ax, of a fluid particle within the boundary layer. Plot ax at location x = 0.8 m, where δ = 1.2 mm, for a flow with U = 6 m/s. Find the maximum value of ax at this x location.
5.11 A useful approximation for the x component of velocity in an incompressible laminar boundary layer is a parabolic variation from u = 0 at the surface (y = 0) to the freestream velocity, U, at the edge of the boundary layer (y = δ). The equation for the profile is u/U = 2 (y/δ) − (y/δ)2, where δ = cx1/2 and c is a constant. Show that the simplest expression for the y component of velocity is
Plot υ/ U versus y/δ to find the location of the maximum value of the ratio u/U. Evaluate the ratio where δ = 5 mm and x = 0.5 m.
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Chapter 5 Solutions
Fox And Mcdonald's Introduction To Fluid Mechanics
- An incompressible fluid flows between two flat plates, driven by a constant pressure gradient, dP/dx, and the constant motion of one of the plates, with speed, U. The spacing between the plates is b and the viscosity of the fluid is µu. Define a coordinate system where -b/2 < y< b/2 (y = 0 midway between the plates). a. What is the Navier-Stokes equation for x-direction momentum for parallel flow. b. What are the boundary conditions for this case? c. Solve for the x-direction velocity profile between the plates, u(y). d. Integrate to show that the relationship between the volume flow rate (per unit width of the plate), the speed of the plate and the pressure gradient is given by: dP b3 + U 12µ 2 barrow_forwardFrom the laminar boundary layer the velocity distributions given below, find the momentum thickness θ, boundary layer thickness δ, wall shear stress τw, skin friction coefficient Cf , and displacement thickness δ*1. A linear profile, u(x, y) = a + by 2. von K ́arm ́an’s second-order, parabolic profile,u(x, y) = a + by + cy2 3. A third-order, cubic function,u(x, y) = a + by + cy2+ dy3 4. Pohlhausen’s fourth-order, quartic profile,u(x, y) = a + by + cy2+ dy3+ ey4 5. A sinusoidal profile,u = U sin (π/2*y/δ)arrow_forwardIn your own words,a. Describe what is viscosity and explain why viscosity of fluid is related to Navier-Stokes equation.b. Describe what is vorticity and explain its relationship with circulation.c. Explain why the stream function ψ is restricted to 2-D flows and you have to use the velocity potential ζ to define 3-D flows.arrow_forward
- for a steady incomprssible two dimensional flow, represented in cartesian coordinates (x,y), a student correctly writes the equation of pathline of any arbitrary particle as dx/dt =ax and dy/dt= by where a and b are constants having unit of second‐¹. if value of a is 5 determine the value if b.arrow_forwardIn chapter 12, we found the velocity profile for flow around a sphere using the creeping flow approximation. For the flow, derive the velocity profile for V, and Ve. Also, find the pressure distribution P. Finally, find the drag force acting on the sphere. (Hint: use the following integration ranges (1) 0<0<â and (2) 0<ô<2à). You can use all the assumptions that we made for this flow in the class.arrow_forwardfor a steady incompresible two dimensional flow, represented in cartesian coordinates (x,y), a student correctly writes the equation of pathline of any arbitrary particle as dx/dt =ax and dy/dt= by where a and b are constants having unit of second‐¹. if value of a is 5 determine the value if b.arrow_forward
- Please indicate the given, assumption and illustration. A source with strength 0.25 m2/s and a vortex with strength 1 m2/s (counter-clockwise) are located at the origin. After working out the equations for the stream function and velocity potential components, determine the following velocity components at a point P(1, 0.5): A) The Radial Velocity component in meters/second. B) The Tangential Velocity Component in meters/second.arrow_forwardConsider steady flow of water through an axisymmetric garden hose nozzle. Along the centerline of the nozzle, the water speed increases from uentrance to uexit as sketched. Measurements reveal that the centerline water speed increases parabolically through the nozzle. Write an equation for centerline speed u(x), based on the parameters given here, from x = 0 to x = Larrow_forwardQ.2 A flow is described by the stream function v = 25xv, The coordinates of the point at which velocity vector has a magnitude of 4 units and makes an angle 150 ° with the X-axis is A x=1.0, y=0.5774 B X=0.5774, Y=1.0 WRONG C X=1, Y=-0.5774 D X=-1, Y=0.5774arrow_forward
- How would I calculate the fluid acceleration along the nozzle centerline. Here, there is steady flow of water through an axisymmetric garden hose nozzle and alongthe centerline the water speed increases from uentrance to uexit . The centerline water speed increases parabolically through the nozzle. What would be an equation for centerline speed u(x), based on the parameters given in the drawing from x = 0 to x = L ?arrow_forwardA flow is non-uniform when the magnitude of the parameters varies from point to point along the flow path. TRUE FALSEarrow_forwardConsider 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 uexit 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, from x = 0 to x = L.arrow_forward
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