A non-Newtonian liquid flows down an inclined flat plate of length L and width W. Its...

    

Transcribed Image Text:

A non-Newtonian liquid flows down an inclined flat plate of length L and width W. Its density p is constant and its viscosity n obeys the power law model, being equal to min-1, where m and n are constants and is the shear rate. The thickness & of the liquid film is constant and far smaller than L and W; accordingly, the disturbances at the edges of the system (that is, for y = 0, y = L, z = 0 and z = W) may be assumed to be negligible. Thus, we assume that: V = 0; vy=vy(x); v = 0; p=p(x) In this assumption - which holds if the flow is steady and laminar - the only nonvanishing components of the viscous stress tensor are Try = Tyr. The system is represented in Fig. 1. Air L Liquid Solid wall 9 Edge Vertical direction Edge Figure 1: Sketch of the system. The z axis is normal to the sheet of paper. Answer all the questions below in the reported order. a) Select an appropriate control volume; then, derive the linear momentum balance equation governing the spatial profile of the xy-component of the viscous stress tensor. To answer this question, do not employ the general linear momentum balance equations. b) Integrate the equation derived in part a) and find the function Try(x). State the boundary condition used and explain the physical grounds on which it is based. c) Using the power law constitutive equation that relates the viscous stress Try (x) to the velocity gradient, derive the equation governing vy(x). Then, solve the equation and prove that it is: 8(pgdcos /m)/n (1/n) + 1 1 - (x/8)(1/n)+1 (1.1) vy (x) where denotes the angle of inclination of the plate, that is, the angle between the y coordinate axis and the direction of gravity, and g is the magnitude of the gravitational field. State the boundary condition employed, explaining the physical grounds on which it is based. d) Using the velocity profile, derive the expressions yielding the (total) fluid volume flow rate and the force exerted by the fluid on the flat plate. A non-Newtonian liquid flows down an inclined flat plate of length L and width W. Its density p is constant and its viscosity n obeys the power law model, being equal to min-1, where m and n are constants and is the shear rate. The thickness & of the liquid film is constant and far smaller than L and W; accordingly, the disturbances at the edges of the system (that is, for y = 0, y = L, z = 0 and z = W) may be assumed to be negligible. Thus, we assume that: V = 0; vy=vy(x); v = 0; p=p(x) In this assumption - which holds if the flow is steady and laminar - the only nonvanishing components of the viscous stress tensor are Try = Tyr. The system is represented in Fig. 1. Air L Liquid Solid wall 9 Edge Vertical direction Edge Figure 1: Sketch of the system. The z axis is normal to the sheet of paper. Answer all the questions below in the reported order. a) Select an appropriate control volume; then, derive the linear momentum balance equation governing the spatial profile of the xy-component of the viscous stress tensor. To answer this question, do not employ the general linear momentum balance equations. b) Integrate the equation derived in part a) and find the function Try(x). State the boundary condition used and explain the physical grounds on which it is based. c) Using the power law constitutive equation that relates the viscous stress Try (x) to the velocity gradient, derive the equation governing vy(x). Then, solve the equation and prove that it is: 8(pgdcos /m)/n (1/n) + 1 1 - (x/8)(1/n)+1 (1.1) vy (x) where denotes the angle of inclination of the plate, that is, the angle between the y coordinate axis and the direction of gravity, and g is the magnitude of the gravitational field. State the boundary condition employed, explaining the physical grounds on which it is based. d) Using the velocity profile, derive the expressions yielding the (total) fluid volume flow rate and the force exerted by the fluid on the flat plate.