CP502 Advanced Fluid Mechanics

CP502 Advanced Fluid Mechanics Flow of Viscous Fluids and Boundary Layer Flow Lectures 5 and 6

υ ρ Continuity and Navier-Stokes equationsfor incompressible flow of Newtonian fluid

z direction of flow y θ x Steady, fully developed, laminar, incompressible flow of a Newtonian fluid down an inclined plane under gravity Exercise 1: Show that, for steady, fully developed laminar flow down the slope (shown in the figure), the Navier-Stokes equations reduces to where u is the velocity in the x-direction, ρ is the density, μ is the dynamic viscosity, g is acceleration due to gravity, and θ is the angle of the plane to the horizontal. Solve the above equation to obtain the velocity profile u and obtain the expression for the volumetric flow rate for a flowing film of thickness h. Exercise 2: If there is another solid boundary instead of the free-surface at y = h and the flow occurs with no pressure gradient, what will be the volumetric flow rate?

z direction of flow y } θ x (1) } (2) Step 1: Choose the equation to describe the flow Navier-Stokes equation is already chosen since the system considered is incompressible flow of a Newtonian fluid. Step 2: Choose the coordinate system Cartesian coordinate system is already chosen. Step 3: Decide upon the functional dependence of the velocity components Steady, fully developed flow and therefore no change in time and in the flow direction. Channel is not bounded in the z-direction and therefore nothing happens in the z-direction.

z direction of flow y θ x Step 4: Use the continuity equation in Cartesian coordinates Flow geometry shows that v can not be a constant, and therefore we choose

} xdirection: u =function of (y) (3) ydirection: v =0 zdirection: w =0 The functional dependence of the velocity components therefore reduces to Step 5: Using the N-S equation, we get x - component: y - component: z - component:

z direction of flow y θ x N-S equation therefore reduces to x - component: y - component: z - component: No applied pressure gradient to drive the flow. Flow is driven by gravity alone. Therefore, we get (4) x - component: y - component: What was asked to be derived in Exercise 1 p is not a function of z z - component:

z direction of flow y θ x Steady, fully developed, laminar, incompressible flow of a Newtonian fluid down an inclined plane under gravity Exercise 1: Show that, for steady, fully developed laminar flow down the slope (shown in the figure), the Navier-Stokes equation reduces to √done where u is the velocity in the x-direction, ρ is the density, μ is the dynamic viscosity, g is acceleration due to gravity, and θ is the angle of the plane to the horizontal. Solve the above equation to obtain the velocity profile u and obtain the expression for the volumetric flow rate for a flowing film of thickness h. Exercise 2: If there is another solid boundary instead of the free-surface at y = hand the flow occurs with no pressure gradient, what will be the volumetric flow rate?

z y x (4) Equation (4) is a second order equation in u with respect to y. Therefore, we require two boundary conditions (BC) of u with respect to y. h BC 1: At y = 0, u = 0 (no-slip boundary condition) θ BC 2: At y = h, (free-surface boundary condition) (5) Integrating equation (4), we get (6) Applying BC 2, we get (7) Combining equations (5) and (6), we get

z h y θ x (8) Integrating equation (7), we get (9) Applying BC 1, we get B = 0 (10) Combining equations (8) and (9), we get

z h y θ x (10) Volumetric flow rate through one unit width fluid film along the z-direction is given by (11)

z direction of flow y θ x Steady, fully developed, laminar, incompressible flow of a Newtonian fluid down an inclined plane under gravity Exercise 1: Show that, for steady, fully developed laminar flow down the slope (shown in the figure), the Navier-Stokes equation reduces to √done where u is the velocity in the x-direction, ρ is the density, μ is the dynamic viscosity, g is acceleration due to gravity, and θ is the angle of the plane to the horizontal. Solve the above equation to obtain the velocity profile u and obtain the expression for the volumetric flow rate for a flowing film of thickness h. √done Exercise 2: If there is another solid boundary instead of the free-surface at y = h and the flow occurs with no pressure gradient, what will be the volumetric flow rate?

z y x Equation does not change. BCs change. (4) BC 1: At y = 0, u = 0 (no-slip boundary condition) h BC 2: At y = h, (free-surface boundary condition) θ u = 0 (no-slip boundary condition) (12) Integrating equation (4), we get (13) Integrating equation (12), we get Applying the BCs in (13), we get B = 0 and

z y x Therefore, equation (13) becomes (14) Volumetric flow rate through one unit width fluid film along the z-direction is given by h θ (15)

Summary of Exercises 1 and 2 (10) (11) z z d d y y θ θ x x (14) (15) Gravity flow through two planes Free surface gravity flow Why the volumetric flow rate of the free surface gravity flow is 4 times larger than the gravity flow through two planes?

Any clarification?

y z x Steady, fully developed, laminar, incompressible flow of a Newtonian fluid down a vertical plane under gravity Exercise 3: A viscous film of liquid draining down the side of a wide vertical wall is shown in the figure. At some distance down the wall, the film approaches steady conditions with fully developed flow. The thickness of the film is h. Assuming that the atmosphere offers no shear resistance to the motion of the film, obtain an expression for the velocity distribution across the film and show that h where ν is the kinematic viscosity of the liquid, Q is the volumetric flow rate per unit width of the plate and g is acceleration due to gravity.

Workout Exercise 3 in 5 minutes!

z V U v U h y u x Steady, fully developed, laminar, incompressible flow of a Newtonian fluid over a porous plate sucking the fluid Exercise 4: An incompressible, viscous fluid (of kinematic viscosity ν) flows between two straight walls at a distance h apart. One wall is moving at a constant velocity U in x-direction while the other is at rest as shown in the figure. The flow is caused by the movement of the wall. The walls are porous and a steady uniform flow is imposed across the walls to create a constant velocity V through the walls. Assuming fully developed flow, show that the velocity profile is given by Also, show that (i) u approaches Uy/h for small V,and (ii) u approaches for very large Vh/ν.

z V U v U h y u } x (1) Step 1: Choose the equation to describe the flow done Step 2: Choose the coordinate system done Step 3: Decide upon the functional dependence of the velocity components Steady, fully developed flow and therefore no change in time and in the flow direction. Channel is not bounded in the z-direction and therefore nothing happens in the z-direction. Step 4: Use the continuity equation in Cartesian coordinates

} xdirection: u =function of (y) (2) ydirection: v =V zdirection: w =0 The functional dependence of the velocity components therefore reduces to Step 5: Using the N-S equation, we get x - component: y - component: z - component:

z V U v U h y u x N-S equation therefore reduces to x - component: y - component: z - component: No applied pressure gradient to drive the flow. Flow is caused by the movement of the wall. Therefore, we get (3) x - component:

(3) Equation (3) is a second order equation in u with respect to y. Therefore, we require two boundary conditions (BC) of u with respect to y. BC 1: At y = 0, u = 0 (no-slip boundary condition) BC 2: At y = h, u = U (no-slip boundary condition) (4) Integrating equation (3), we get (5) Integrating equation (4), we get Applying the BCs in equation (5), we get (6) (7)

z V U v U h y u x From equations (6) and (7), we get Substituting the above in equation (5), we get (8)

z V U v U h y u x (8) (i) For small V, expand exp(Vy/ν) and exp(Vh/ν) using Taylor series as follows: For small V, we can ignore the terms with power.We then get Could you recognize the above profile?

z V U v U h y u x (8) For very large Vh/ν, exp(Vh/ν) goes to infinity. Therefore. Divide equation (8) by exp(Vh/ν). We then get For very large Vh/ν, exp(-Vh/ν) goes to zero. Therefore, we get

CP502 Advanced Fluid Mechanics