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
- A constant-thickness film of viscous liquid flows in laminar motion down a plate inclined at angle 0, as in Fig. P4.36. The velocity profile is u =Cy(2h-y) v=w=0 Find the constant C in terms of the specific weight and viscosity and the angle 0. Find the volume flux Q per unit width in terms of these parameters. Also, what are the appropriate boundary conditions (i) at the bottom y = 0 and (ii) at the surface y = h? €arrow_forwardcan i get help with all parts, thanksarrow_forwardProblem 3. Numerical results for the Blasius solution ✓ for laminar boundary layer flow can be obtained using a Matlab program on my ME 347 Canvas page. (a) Derive the equation for the dimensionless y-component of velocity, v/U, as a function of the similarity parameter (or dimensionless location), n = y√√U/(vx), the dimensionless stream function, f(n) = 4/√Uv x, and the Reynolds number, Re = Ux/v. (b) From the Blasius solution results, calculate the Reynolds number that corresponds to a boundary layer thickness of 8 = 5.0 mm at x = 0.50 m. (c) Plot v/U (on the x-axis) versus y/d = n/5.0 by editing the Matlab code above to include the needed calculations (a tutorial that describes the code above and indicates how to edit the code is available here). Add to your plot the polynomial profile from Homework 2, Problem 1 which is an approximate fit to the Blasius solution. NOTE: Be careful to differentiate between lowercase for the y-component of velocity and the Greek letter v = μ/p (called…arrow_forward
- Flat 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_forwardA velocity field is given by u = 5y2, v = 3x, w = 0. (a) Is this flow steady or unsteady? Is it two- or three- dimensional? (b) At (x,y,z) = (3,2,–3), compute the velocity vector. (c) At (x,y,z) = (3,2,–3), compute the local (i.e., unsteady part) of the acceleration vector. (d ) At (x,y,z) = (3,2,–3), compute the convective (or advective) part of the acceleration vector. (e) At (x,y,z) = (3,2,–3), compute the (total) acceleration vector.arrow_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_forward
- can someone explain why P is a function if time only for this questionarrow_forwardRequired 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_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_forward
- 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_forwardOil, 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_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
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