Question

Problem 1. (40 pts.) A film of water flows down a moving vertical wall under the force of gravity as shown in the figure below. The air layer can be assumed to produce negligible shear stress on the water film. Wall velocity Wo The following assumptions can be made for this situation: Water film P. Moving Wall 1) Plates are infinite in y and z 2) Flow is steady 3) Streamlines are parallel to the z-axis 4) Flow is incompressible 5) Flow is laminar 6) Pressure=Patm=constant at free surface 7) 2-D flow, v=0,2 = 0 Air P=Patm Mass continuity and the z-direction Navier Stokes equation apply: (6.27) δεν θα dydy (6.1278) Ⓡ= pūdA = PVA a) In the table below, identify which of the circled terms of the governing equations can be neglected by the given assumption. Write the number of the term in the table. Some assumptions relate to multiple terms, include them all. Corresponding term(s) that can be neglected Assumption Flow is incompressible Flow is steady Flow is laminar Analysis of continuity equation Pressure=Patm=constant at free surface

A. In the table below, identify which of the circled terms of the governing equations can be neglected
by the given assumption. Write the number of the term in the table. Some assumptions relate
to multiple terms, include them all.

B.  Write the mathematical equations describing the appropriate boundary conditions and identify
them in words.

C.  Applying the appropriate boundary conditions, solve the differential equation remaining after
appropriate terms have been neglected to determine the velocity profile in the film: d^2(w)/dx^2 = (rho*g)/u

Answer 1

Corresponding team Both are based on this assumption Assumption Flow is incompressible Flow is steady Flow is laminar Analysis of continuity equation Pressure Patm = Constant Both are entirely based on this assumption 6) Boundary conditions i Moveing wall condition at n=0 w=U (") At free surface at n= h dw) -o Tanah [ Newtons law of viscosity] I ie, Negligible shear at boundary o mel og at de una veya wami. Pola + u duvaw + wow oy d on time derivative o pufoow is steady); sw = 0 ; 2020; D O ( Continuity equation, partalled | I flow A=V=0 I No applied pressure gradient flow is I due to the balance of gravitational and ma viscous forces

Idu St 7 - mgz + Moto geong (downwards) - - rg + How to flend i ĝ moving wall.

Integrate the equation twice fast, t mida +G integrate again =wo [wang x + ( a + (a -11 G and C, are constants of integration Apply Boundary conditions (i) and (ii) (") =) n=0 w-wo Wo = or Ot (2 [C2 = Wo] (ii) > pw - vagy a 4 67 4.& de la PAC 0-2 lucha pogut + c => Carvalho ar on 7 Substitute required C. velocity in protile con is, The Pax cat ghnt we W = vaxt