An excellent approximation for the two-dimensional incompressible laminar boundary layer on the flat surface in Fig, P4.17 is
(a) Assuming a no-slip condition at the wall, find an expression for the velocity component v(x, y) for y ??. (b) Then mid the maximum value of v at the station x = 1 m, for the particular case of airflow, when U = 3 m/s and
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Fluid Mechanics
- Algebraic equations such as Bernoulli's relation, are dimensionally consistent, but what about differential equations? Consider, for example, the boundary-layer x-momentum equation, first derived by Ludwig Prandtl in 1904: ди ди ap ат ри — + pu Әх + pg: + дх ày ду where T is the boundary-layer shear stress and g, is the com- ponent of gravity in the x direction. Is this equation dimen- sionally consistent? Can you draw a general conclusion?arrow_forwardAn incompressible velocity field is given by u=a(x°y²-y), v unknown, w=bxyz where a and b are constants. (a)What is the form of the velocity component for that the flow conserves mass? (b) Write Navier- Stokes's equation in 2-dimensional space with x-y coordinate system.arrow_forwardFlat Couette Flow An incompressible fluid of viscosity n is located between two plates infinite, spaced 2d. The top plate has velocity V x and the bottom plate low - V x*. (Gravity is neglected and pressure is constant.) YA +d -d -Va X +Vâ (a) Calculate the velocity of the fluid. (b) Sketch the forces acting on the fluid and calculate them.arrow_forward
- Required information A simple flow model for a two-dimensional converging nozzle is the distribution u = U₁(1+z) v = − U₁7/10 w = 0. Find the pressure distribution p(x, y) when the pressure at the origin equals po. Neglect gravity. Multiple Choice p= Po -² (2 + + 7) + P p=-2/²2 (2-) + Po p PU =_/²2(x + 1 + 1) + P. 3L Karrow_forwardVelocity components u = (Axy³ – x²y), v = xy² . possible flow field involving steady incompressible flow is then value of 2 for (a) 0 (b) 1 (c) 2 (d) 3arrow_forwardA proposed harmonic function F(x, y, z) is given byF = 2x2 + y3 - 4xz +f(y)(a) If possible, fi nd a function f (y) for which the laplacianof F is zero. If you do indeed solve part (a), can your fi nalfunction F serve as (b) a velocity potential or (c) a streamfunction?arrow_forward
- Oil, of density ρ and viscosity μ , drains steadily down theside of a vertical plate, as in Fig. P4.80. After a developmentregion near the top of the plate, the oil fi lm willbecome independent of z and of constant thickness δ .Assume that w = w ( x ) only and that the atmosphere offersno shear resistance to the surface of the fi lm. ( a ) Solve theNavier-Stokes equation for w ( x ), and sketch its approximateshape. ( b ) Suppose that fi lm thickness δ and the slopeof the velocity profi le at the wall [ ∂ w /∂ x ] wall are measuredwith a laser-Doppler anemometer (Chap. 6). Find anexpression for oil viscosity μ as a function of ( ρ , δ , g ,[ ∂ w / ∂ x ] wall ).arrow_forwardQ2 Fluid mechanics With respect to the stream function, develop the superposition for two cases: - Uniform + Dipole - Uniform + Dipole (a->0)arrow_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_forward
- 3.4 Specification of a laminar boundary layer profile as an inflow condition is often used in incompressible flow simulations. Let us consider the Blasius profile, which is a similarity solution for the steady laminar boundary layer on a flat plate. Fig. 3.25 Flat-plate laminar boundary layer Dv 2. Consider the streamfunction(with u = and v=- transform from (x, y) to (x, n), where n = y √ux/U% = Rex u(x, y) and a coordinate Using the above and a streamfunction in the form = √xUf(n), show that the governing equations can be reduced to ff" +2f"" = 0 with boundary conditions of f(0) = f'(0) = 0 and lim, f(n) = 1. This equation is referred to as the Blasius equation and its solution is known as the Blasius profile.arrow_forwardFind stream function and velocity potential.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_forward
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