Question

A steady, incompressible, and laminar flow of a fluid of viscosity u flows through an inclined narrow gap of a crack in the wall of length L and a constant width W shown in Figure Q1(b). Assume that the gap has a constant thickness of 7. The fluid flows down the inclined gap at an angle and in the positive x-direction. No pressure gradient is applied throughout the flow but there is gravitational effect. Derive an expression for the velocity field through the gap of the crack in the wall. Fluid 8 g

fluid mechanics

Answer 1

Mass is not given so i assume mass m to analyze gravitational effects

Solution with serial page number. (Area of wall =L×w)
SOLUTION page 1 to Nekition's Law of viscosity Acording T = ll v h ga ß (very h small = Linear velocity profile. We use V = Velocity Беса"34р is u viscosity small. T = shear stress => a Bove. Formula is applicable here We Linear Velocity profile gap is narrow . » Let mass = m ( because not given аррlу натqviqti- nal assume as because Now we effect). m (mass) masing mgc cuso do mg The FLUID & is offer Frictional resistance at a particular time against the obo Motion downward of crack klall. CS Scannedhana haning yelocity CamScanner is X

page2 V= Velocity (Linear profil maxmat Crackwall cap per surfan. F-B.D to shearstress at all mg sino ng caso HD from above FB-D EF=0 latinstany) T. mg sino khen wall y = Velocity of crack Just start wall = It is also known V as terminal Velocity. to a velocity CS Scanned with CamScanner

page 3 so the frictional resistance = T (shear stress) F TA ( frictional force against motion) F = mgsing TA mg sino T 0 → А A V S " mg (sino) s - gap Y Xelocity + Mx Х LXu9 -- mg (sino) s L7 Length w widh. Y = mg.s(sino) M X Ly w Velocity o crack wall, CS Scanned with CamScanner