Algebraic equations such as Bernoulli's relation, Bq. (1) of Example 1.3, are dimensionally consistent, but what about differential equations? Consider, for example, the boundary-layer x-momentum equation, first derived by Ludwig Prandtl in 1904:
where t is the boundary-layer shear stress and gx is the component of gravity in the x direction. Is this equation dimen-sionally consistent? Can you draw a general conclusion?
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Fluid Mechanics
- As measured by NASA's Viking landers, the atmosphere of Mars, where g = 3.71 m/s2, is almost entirely carbon dioxide, and the surface pressure averages 700 Pa. The temperature is cold and drops off exponentially: T≈ TO e-Cz, where C 1.3 \times 10-5 m-1 and TO≈ 250 K. For example, at 20, 000 m altitude, T≈ 193 K. (a) Find an analytic formula for the variation of pressure with altitude. (b) Find the altitude where pressure on Mars has dropped to 1 Pascal.arrow_forwardSteam and water flow in a 25 mm tube at 300 C, 3500 kg/m2 s, and X = 0.4. Use the drift flux model and find α and S. [Ans. 0.68 and 4.7]arrow_forwardA seA A, soA A solid cylinder of diameter d, length and density p, falls due to gravity inside a pipe of diameter D. The clearance between the solid cylinder and the pipe is filled with a Newtonian fluid of density p and u. For this clearance fluid, the terminal velocity of the cylinder is determined to be V, assuming a linear velocity profile. However, if the clearance fluid was changed to a Newtonian fluid of density 2p and viscosity 2u, then for an assumed linear velocity profile, the terminal velocity of the cylinder was determined to be V,. From the results of these experiments, one may write that (A) V = V (C) 2 V= V (B) V=2 V, (D) V= 4 Varrow_forward
- If a vertical wall at temperature T, is surrounded by a fluid at temperature T, a natural convection boundary layer flow will form. For laminar flow, the momentum equation is au ди. p(u-+ v) = PB(T – T)g + µ- ди ay to be solved, along with continuity and energy, for (u, v, T) with appropriate boundary conditions. The quantity B is the thermal expansion coefficient of the fluid. Use p, g, L, and (Tw- To) to nondimensionalize this equation. Note that there is no “stream" velocity in this type of flow.arrow_forwardFor the flow of a viscous fluid, with the velocity V = f(x)g(y)h(z)i (where f, g, h are arbitrary functions), the following conditions are given: . The flow is adiabatic. • The quantities v = 2 and 3 = $ are constants. • The velocity circulation is conserved for the flow, irrespective of the values of vand 3. What is the general solution for the functions f, g, h?arrow_forward(b) For each flow description: (i) Steady, compressible flow of air. (ii) Arbitrary flow (Lagrangian perspective). (iii) Unsteady, incompressible flow of viscous oil. (iv) Arbitrary flow (Eulerian perspective). choose from the list below the form of mass conservation you would use for that situation and explain your choice: 1. V · ū = 0 Dp 2. = -pV · ū Dt др 3. + V· (pū) = 0 4. V · (pū) = 0arrow_forward
- To good approximation, the thermal conductivity k of a gasdepends only on the density ρ ,mean free path l , gas constant R , and absolute temperatureT . For air at 20 8 C and 1 atm, k ≈0.026 W/(m.K) and l ≈6.5 E-8 m. Use this information to determine k for hydrogenat 20 ° C and 1 atm if l ≈ 1.2 E-7 m.arrow_forwardd²u dy² pg where g is the acceleration due to gravity Harrow_forward1.6 An incompressible Newtonian fluid flows in the z-direction in space between two par- allel plates that are separated by a distance 2B as shown in Figure 1.3(a). The length and the width of each plate are L and W, respectively. The velocity distribution under steady conditions is given by JAP|B² Vz = 2µL B a) For the coordinate system shown in Figure 1.3(b), show that the velocity distribution takes the form JAP|B? v, = 2μL Problems 11 - 2B --– €. (a) 2B (b) Figure 1.3. Flow between parallel plates. b) Calculate the volumetric flow rate by using the velocity distributions given above. What is your conclusion? 2|A P|B³W Answer: b) For both cases Q = 3µLarrow_forward
- Fluid Mechanics a thin plate is separated from two fixed plates by very viscous liquids μ1 =0.10 pa.S and μ2 = 0.5 Pa.s respectively. the spacings between the center nplate and fixed plate are also hi = 16 cm and h2 = 11 cm, while the contact area between the center plate and each fluid is A= 340 cm^2. Assuming a linear velocity distribution in each fluid, determine the force F(in newton) required to pull the plate at velocity V=3.5 m/s.arrow_forwardConsider the system presented in the figure x1(t) x2(1) 1 N/m 0000 At) 1 kg 1 N-s/m 1 kg Frictionless The matrix form *+2++1 -(+2) X, (6))(F) X,(s) -(s +2) s +s+1X,(s), F(s) a.arrow_forwardThe wind flutter on the wing of a newly proposed jet fighter is given by the following 1st order differential equation: dy/dx = 2yx With the Boundary Condition: y(0) = 1 (remember this means that y = 1 when x = 0) Determine the vertical motion (y) in terms of the span (x) of the wing. The frequency of fluctuations of the wing at mach 2 is given by the non-homogenous 2nd order differential equation: y'' + 3y' - 10y = 100x With the boundary conditions: y(0) = 1 and y(1) = 0 (i.e., y = 1 when x = 0 and y = 0 when x = 1) By solving the homogenous form of this equation, complete the analysis and determine the amplitude (y) of vibration of the wing tip at mach 2. Critically evaluate wing flutter and fluctuation frequency amplitude determined by solving the two differential equations above.arrow_forward
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