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We consider a laminar flow induced by an impulsively started infinite flat plate. The y-axis is normal to the plate. The x- and z-axes form a plane parallel to the plate. The plate is defined by y = 0. For time t <0, the plate and the flow are at rest. For t≥0, the velocity of the plate is parallel to the 2-coordinate; its value is constant and equal to uw. At infinity, the flow is at rest. The flow induced by the motion of the plate is independent of z. (a) From the continuity equation, show that v=0 everywhere in the flow and the resulting momentum equation is მu Ət Note that this equation has the form of a diffusion equation (the same form as the heat equation). (b) We introduce the new variables T, Y and U such that T=kt, Y=k/2y, U = u where k is an arbitrary constant. In the new system of variables, the solution is U(Y,T). The solution U(Y,T) is expressed by a function of Y and T and the solution u(y, t) is expressed by a function of y and t. Show that the functions are identical. Deduce that the solution is a function of y/√t only. (Hint: consider that U can be expressed as U=G(Y/T1/2,T) and u can be expressed as u = g(y/t1/2, t). Write that G and g are the same functions for any value of k and deduce that G can not depend on T). (c) We introduce the variable ŋ and the function F by η y น η F 2(t)1/21 Uw From the preceding result, we have F = F(n). Show that the momentum equa- tion becomes d² F dF +27- 0 dn² d (d) Show that the solution of Eq. (P3.14.4) is 2 F We consider that the viscous effects are significant if F < 0.01. Numerically, F = 0.01 when n = 1.82. (e) An observer is placed at distance d from the plate. Show that the observer feels the motion of the plate after a time t of order d²/v. This time is called a characteristic time of viscosity. It is the time necessary to transfer momentum by diffusion through the field.

We consider a laminar flow induced by an impulsively started infinite flat plate. The y-axis is normal to the plate. The x- and z-axes form a plane parallel to the plate. The plate is defined by y = 0. For time t <0, the plate and the flow are at rest. For t≥0, the velocity of the plate is parallel to the 2-coordinate; its value is constant and equal to uw. At infinity, the flow is at rest. The flow induced by the motion of the plate is independent of z. (a) From the continuity equation, show that v=0 everywhere in the flow and the resulting momentum equation is მu Ət Note that this equation has the form of a diffusion equation (the same form as the heat equation). (b) We introduce the new variables T, Y and U such that T=kt, Y=k/2y, U = u where k is an arbitrary constant. In the new system of variables, the solution is U(Y,T). The solution U(Y,T) is expressed by a function of Y and T and the solution u(y, t) is expressed by a function of y and t. Show that the functions are identical. Deduce that the solution is a function of y/√t only. (Hint: consider that U can be expressed as U=G(Y/T1/2,T) and u can be expressed as u = g(y/t1/2, t). Write that G and g are the same functions for any value of k and deduce that G can not depend on T). (c) We introduce the variable ŋ and the function F by η y น η F 2(t)1/21 Uw From the preceding result, we have F = F(n). Show that the momentum equa- tion becomes d² F dF +27- 0 dn² d (d) Show that the solution of Eq. (P3.14.4) is 2 F We consider that the viscous effects are significant if F < 0.01. Numerically, F = 0.01 when n = 1.82. (e) An observer is placed at distance d from the plate. Show that the observer feels the motion of the plate after a time t of order d²/v. This time is called a characteristic time of viscosity. It is the time necessary to transfer momentum by diffusion through the field.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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We consider a laminar flow induced by an impulsively started infinite flat plate. The y-axis is normal to the plate. The x- and z-axes form a plane parallel to the plate. The plate is defined by y = 0. For time t <0, the plate and the flow are at rest. For t≥0, the velocity of the plate is parallel to the 2-coordinate; its value is constant and equal to uw. At infinity, the flow is at rest. The flow induced by the motion of the plate is independent of z. (a) From the continuity equation, show that v=0 everywhere in the flow and the resulting momentum equation is მu Ət Note that this equation has the form of a diffusion equation (the same form as the heat equation). (b) We introduce the new variables T, Y and U such that T=kt, Y=k/2y, U = u where k is an arbitrary constant. In the new system of variables, the solution is U(Y,T). The solution U(Y,T) is expressed by a function of Y and T and the solution u(y, t) is expressed by a function of y and t. Show that the functions are identical. Deduce that the solution is a function of y/√t only. (Hint: consider that U can be expressed as U=G(Y/T1/2,T) and u can be expressed as u = g(y/t1/2, t). Write that G and g are the same functions for any value of k and deduce that G can not depend on T). (c) We introduce the variable ŋ and the function F by η y น η F 2(t)1/21 Uw From the preceding result, we have F = F(n). Show that the momentum equa- tion becomes d² F dF +27- 0 dn² d (d) Show that the solution of Eq. (P3.14.4) is 2 F We consider that the viscous effects are significant if F < 0.01. Numerically, F = 0.01 when n = 1.82. (e) An observer is placed at distance d from the plate. Show that the observer feels the motion of the plate after a time t of order d²/v. This time is called a characteristic time of viscosity. It is the time necessary to transfer momentum by diffusion through the field.
Transcribed Image Text:We consider a laminar flow induced by an impulsively started infinite flat plate. The y-axis is normal to the plate. The x- and z-axes form a plane parallel to the plate. The plate is defined by y = 0. For time t <0, the plate and the flow are at rest. For t≥0, the velocity of the plate is parallel to the 2-coordinate; its value is constant and equal to uw. At infinity, the flow is at rest. The flow induced by the motion of the plate is independent of z. (a) From the continuity equation, show that v=0 everywhere in the flow and the resulting momentum equation is მu Ət Note that this equation has the form of a diffusion equation (the same form as the heat equation). (b) We introduce the new variables T, Y and U such that T=kt, Y=k/2y, U = u where k is an arbitrary constant. In the new system of variables, the solution is U(Y,T). The solution U(Y,T) is expressed by a function of Y and T and the solution u(y, t) is expressed by a function of y and t. Show that the functions are identical. Deduce that the solution is a function of y/√t only. (Hint: consider that U can be expressed as U=G(Y/T1/2,T) and u can be expressed as u = g(y/t1/2, t). Write that G and g are the same functions for any value of k and deduce that G can not depend on T). (c) We introduce the variable ŋ and the function F by η y น η F 2(t)1/21 Uw From the preceding result, we have F = F(n). Show that the momentum equa- tion becomes d² F dF +27- 0 dn² d (d) Show that the solution of Eq. (P3.14.4) is 2 F We consider that the viscous effects are significant if F < 0.01. Numerically, F = 0.01 when n = 1.82. (e) An observer is placed at distance d from the plate. Show that the observer feels the motion of the plate after a time t of order d²/v. This time is called a characteristic time of viscosity. It is the time necessary to transfer momentum by diffusion through the field.
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