Derive an expression for the velocity profile v x y

This preview shows page 6 - 10 out of 12 pages.

Derive an expression for the velocity profile vx(y) in the backflow region and find L2c)Determine the coating thickness hd)Determine the shear force per unit width (in the z-direction) that the liquid exerts on the sheet...
μd2vxdy2=ΔPFL1.a)Integrating the governing equation twice yieldsvx(y)=1μΔPFL1y22+C1y+C2.Since the velocity is U at y=0,C2=U,and since the velocity is 0 at y=h,0=1μΔPFL1h22+C1h+UorC1=1μΔPFL1h2Uh.The velocity profile is thereforevx(y)=1μΔPFL1h22yhy2h2+U 1yh.This expression gives the velocity profile in both regions (downstream and upstream of the verticalslot).The only difference is the negative pressure gradient in the two regions.In the forward flowregion,ΔPFL1=P1P0L1,whereas in the backflow regionΔPBL2=P0P1L2.b)In the backflow region, the velocity profile is againvx(y)=1μΔPBL2h22yhy2h2+U 1yh.
To find the length of the backflow region, we use the fact that the volumetric flow across any cross-section there must be zero.At steady state, if there is no flow at the left air/liquid boundary, thenthere must be no flow at any x-position.We therefore haveQ=vx(y)=00hor1μΔPBL2h22yhy2h2+U 1yhdy=00h.Doing the integral yieldsP0P1L2=μU6h2orL2=h2P1P0()6μU.c)The flow rate (per unit width) in the forward flow region isQ=1μΔPFL1h22yhy2h2+U 1yhdy0hwith the negative pressure gradient as specified in (a).ThenQ=1μΔPFL1h312+Uh2=Uh,where the last equality reflects the fact that, far downstream, the velocity of the film will equal Ueverywhere.The final coating thickness is thereforeh=h312μUΔPFL1+h2.
d)In both the forward flow and backflow regions, the force per unit width is given byFx=τyxy=0dxL20+τyxy=0dx0L1.The shear stresses in each region have the same form, but the pressure gradient terms differ.Theytake the formτyxy=0= μdvxdyy=o=ΔPLh2+ μUh.The shear stresses are functions only of y, so they are constant at y=0, and the integrals are trivialand yieldFx=L2P0P1L2h2+ μUhL1P1P0L2h2+ μUhorFx=μUhL2+L1().The pressure gradient terms apparently cancel each other, so the net force is in the –x direction, andis the same as it would be for a linear shear flow with no pressure gradient.

Upload your study docs or become a

Course Hero member to access this document

Upload your study docs or become a

Course Hero member to access this document

End of preview. Want to read all 12 pages?

Upload your study docs or become a

Course Hero member to access this document

Term
Winter
Professor
PieterStroeve

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture