Chapter 2 Governing Equations In the present an the subsequent chapters, we shall, either irectly or inirectly, be concerne with the bounary-layer flow of an incompressible viscous flui without any involvement of heat an mass transfer. Therefore, our governing laws will be the conservation of mass an momentum only which are commonly known as the equations of continuity an the Navier Stokes equations, respectively. In usual notation, they are written as (in Cartesian coorinates): @x þ @v @y þ @w @z ¼ ; q @t þ u @x þ v @y þ w ¼ @p @z @x þ l @2 u @x 2 þ @2 u @y 2 þ @2 u @z 2 þ F x ; q @v @t þ u @v @x þ v @v @y þ w @v ¼ @p @z @y þ l @2 v @x 2 þ @2 v @y 2 þ @2 v @z 2 þ F y ; q @w @t þ u @w @x þ v @w @y þ w @w @z ¼ @p @z þ l @2 w @x 2 þ @2 w @y 2 þ @2 w @z 2 þ F z ; ð2:1þ ð2:2þ ð2:3þ ð2:4þ an in cylinrical coorinates: q @t þ v r @r þ v h r ¼ @p @r þ l @2 v r @r 2 þ 1 r @r þ v r r þ 1 @v h r @h þ @v z @z ¼ ; @h v2 h r þ v z @z @r v r r 2 þ 1 @ 2 v r r 2 @h 2 2 @v h r 2 @h þ @2 v r @z 2 þ F r ; ð2:5þ ð2:6þ Springer International Publishing AG 217 A. Mehmoo, Viscous Flows, Mathematical Engineering, DOI 1.17/978-3-319-55432-7_2 13
14 2 Governing Equations q @v h @v h þ v r @t @r þ v h @v h r @h þ v rv h @v h þ v z r @z ¼ 1 @p r @h þ l @2 v h @r 2 þ 1 @v h r @r v h r 2 þ 1 @ 2 v h r 2 @h 2 þ 2 r 2 @h þ @2 v h @z 2 þ F h ; ð2:7þ q @v z @v z þ v r @t @r þ v h @v z r @h þ v @v z z ¼ @p @z @z þ l @2 v z @r 2 þ 1 @v z r @r þ 1 @ 2 v z r 2 @h 2 þ @2 v z @z 2 þ F z ; ð2:8þ where F ¼ F x ; F y ; F z or ¼ ð Fr ; F h ; F z Þ enotes the boy force. However, the flows consiere in the subsequent chapters o not involve any boy force because of which F will be taken ientically equal to zero in all those chapters. 2.1 Bounary-Layer Equations 2.1.1 The Bounary-Layer Assumption In the stuy of viscous flow ue to the motion of a continuous surface, a natural question arises whether the bounary-layer assumptions consiere by Prantl are applicable to this case where the flow is establishe by the motion of the soli surface in the absence of any potential flow? The same question arose in the min of Sakiais [1] in 196 an he first confirme the bounary-layer character of the viscous flow establishe by the motion of a continuous surface. This question, however, fins some crue justification from Sect. 1.1 of Chap. 1, where the Sakiais flow has been explaine, but a theoretical proof or experimental evience is still require. In this connection, Sakiais [1] performe an experiment an confirme the formation of bounary-layer near the surface of the moving continuous surface in accorance with the explanation given in Sect. 2.1. He consiere the rectangular Lucite acrylic resin tank an fille it with a water-base solution of milling yellow. He passe two parallel threas issuing from the little holes in the blocks immerse completely in the tank as shown in Fig. 2.1. For the sake of comparison, he move the left-han threa from top to bottom an kept the right one fixe within the tank. After a while, when the steay state reache, the photograph (see Fig. 2.2) shows the formation of the bounary-layer near the moving threa, starting right from the hole an thus eveloping in the irection of motion of the threa. This experiment confirms that the viscous flow establishe by the motion of a continuous surface oes exhibit the bounary-layer character. Therefore, the Prantl s bounary-layer assumptions are equally vali in this case an the orer of magnitue analysis also works in the same manner as in the case of finite surfaces.
2.1 Bounary-Layer Equations 15 Fig. 2.1 Schematic of experimental setup of Sakiais [1] Fig. 2.2 Photograph of the Sakiais experiment [1]
16 2 Governing Equations 2.1.2 The Pressure Graient Term Accoring to the Prantl s bounary-layer theory, the pressure within the bounarylayer oes not become very much ifferent from the pressure in the invisci potential flow. That is, there are no significant changes in pressure across the bounary-layer because of which the term @p=@y is simply ignore in the two-imensional case; however, the changes in pressure along the lateral irections can be of any significance. Therefore, the pressure variation along the lateral irections within the bounary-layer is assume to be the same as they are in the invisci potential flow outsie the bounary-layer an are etermine with the use of Bernoulli s equation, for example, in the two-imensional steay-state case, as p x ¼ u u 1 1 x Thus, the constancy or absence of the external potential flow makes the pressure graient term equal to zero in the bounary-layer equations governing the flow ue to a moving continuous soli surface. The external pressure in the subsequently stuie flows will be consiere absent, thus making the pressure graient term equal to zero, i.e., @p @s l ¼ ; ð2:9þ where s l enotes any lateral coorinate. 2.1.3 Bounary-Layer Equations in Cartesian Coorinates However, it has now been establishe that the bounary-layer equations, present alreay in the literature, for the two-imensional an three-imensional flows are equally applicable to the flows ue to moving continuous surfaces too. Therefore, the erivation of these equations here again oes not make any sense. We, therefore, prefer to follow Schlichting [2] an write the bounary-layer equations irectly here @x þ @v @y þ @w @z ¼ ; @t þ u @x þ v @y þ w @z ¼ m @2 u @y 2 ; ð2:1þ ð2:11þ
2.1 Bounary-Layer Equations 17 Fig. 2.3 Cartesian coorinate system @w @t þ u @w @x þ v @w @y þ w @w @z ¼ m @2 w @y 2 ; ð2:12þ where the system of coorinates is shown in Fig. 2.3. The velocity components u; v an w have been taken along the x-, y-, an z-axes, respectively. 2.1.4 Bounary-Layer Equations in Cylinrical Coorinates The geometrical objects, of our interest, owing to the axial symmetry are the continuous circular cyliner an the circular isk of infinite raius. The schematic of the flow ue to a continuous circular cyliner an the associate system of coorinates is shown in Fig. 2.4. Since there is no circular rotation in the flow therefore, the bounary-layer equations in this case look like @ @z ðruþþ @ ð @r rv Þ ¼ ; ð2:13þ @t þ u @z þ v @r ¼ m 1 r @ @r r @r ; ð2:14þ whereas in the case of circular isk, the governing system takes the form @ @r ðruþþ @ ð @z rw Þ ¼ ; ð2:15þ @t þ u @r þ w @z ¼ m @2 u @z 2 : ð2:16þ The chosen system of coorinates, corresponing to the isk geometry, is shown in Fig. 2.5.
18 2 Governing Equations Fig. 2.4 Schematic of the continuous cyliner an the associate coorinate system Fig. 2.5 Coorinate system for the axially symmetric isk geometry 2.2 Momentum Integral Equations The integral form of the momentum bounary-layer equations, either in Cartesian coorinates or in cylinrical coorinates, comes irectly from their respective ifferential forms, such as Eqs. (2.1) (2.12), (2.13) (2.14), (2.15) (2.16) corresponing to the planner two- or three-imensional flow, axially symmetric flow near a cyliner an isk, respectively. 2.2.1 In Cartesian Coorinates Consier a steay three-imensional flow cause ue to the bilateral motion of a flexible continuous flat sheet. The appropriate bounary conitions for this flow rea as: u ¼ u w ðx; zþ; w ¼ w w ðx; zþ; v ¼ v w ðx; zþ; at y ¼ ; ð2:17þ u ¼ ; w ¼ ; at y ¼1 where u w ðx; zþ an w w ðx; zþ enote the stretching/shrinking wall velocities in x- an z-irections, respectively. In the case of porous flexible sheet, suction/injection may also be allowe through the sheet surface which is enote by v w ðx; zþ. Outsie the bounary-layer, the flui is suppose to be at rest in the absence of any potential
2.2 Momentum Integral Equations 19 flow. The momentum integral equations in this case are erive by the integration of Eqs. (2.1) (2.12) w.r.t. y between the limits an ðx; zþ; where ðx; zþ enotes the bounary-layer thickness. Integration of Eq. (2.1) w.r.t. y yiels Z y v ¼ v w @x þ @w y: @z ð2:18þ Integration of Eqs. (2.11) an (2.12) with respect to y between the limits y results in the following integral equations: Z Z u @x þ v @y þ w y ¼ 1 @z q s x;; u @w @x þ v @w @y þ w @w y ¼ 1 @z q s z;; ð2:19þ ð2:2þ where s x; ¼l @y an y¼ s z; ¼l @w @y : y¼ ð2:21þ Substituting Eq. (2.18) in Eqs. (2.19) an (2.2) by noting that 2 4u Z y 3 @x þ @w y5 ¼ ; @z ð2:22þ an we finally arrive at 2 4w Z y 3 @x þ @w y5 ¼ ; @z Z @ u 2 y þ @ Z uwy ¼ v w u w þ 1 @x @z q s x;; ð2:23þ ð2:24þ
2 2 Governing Equations Z @ uwy þ @ Z w 2 y ¼ v w w w þ 1 @x @z q s z;: ð2:25þ The system (2.24) (2.25) represents the momentum integral equations for the steay three-imensional viscous flow ue to a moving continuous porous sheet. For the two-imensional case, Eq. (2.25) just vanishes out an Eq. (2.24) in the absence of wall suction/injection simplifies to Z u 2 y ¼ 1 x q s x;; ð2:26þ which is the same as erive by Sakiais [1]. 2.2.2 In Cylinrical Coorinates In Sect. 2.1.4, the axial flow ue to a moving continuous surface has been represente in cylinrical coorinates by splitting it into two particular flow geometries, namely the circular cyliner an the circular flat isk. The integral formulation for the sai two cases follows immeiately by the integration of Eqs. (2.13), (2.14), (2.15), an (2.16), respectively. Following the same proceure (of Sect. 2.1.4), we erive integral momentum equations for these two cases separately. The appropriate bounary conitions for a uniformly stretching/shrinking long continuous cyliner, as shown in Fig. 2.4, with porous surface rea as u ¼ u w ðþ; z v ¼ v w ðþ; z u ¼ ; at r ¼ R ; ð2:27þ at r ¼1 where R enotes the raius of cyliner. Integrating Eq. (2.13) w.r.t. r, we reaily get v ¼ R r v w 1 r Z r @z rr: ð2:28þ Substituting Eq. (2.28) in Eq. (2.14) an integrating between the limits R r ðzþ, we have
2.2 Momentum Integral Equations 21 Z u 2 rr ¼ R u w v w þ s z; ; ð2:29þ z q which is the momentum integral equation for the uniformly stretching/shrinking cyliner. This equation immeiately reuces to that, first, erive by Sakiais [1] for a continuous cyliner of impermeable soli wall by substituting v w ¼. Here, s z; ¼l @r r¼r is the tangential shear stress at the cylinrical surface. The similar proceure applies to the continuous circular isk of infinite raius. In this case, the escribing conitions at the isk surface shoul be of the form u ¼ u w ðþ; r w ¼ w w ðþ; r at z ¼ ; ð2:3þ u ¼ ; at z ¼1 for a permeable stretching/shrinking isk. Notice that, Eq. (2.16) in steay-state form can be rewritten, with the utilization of Eq. (2.15), as: @ @ @r ru2 þ @z ðruwþ ¼ mr @2 u @z 2 ; ð2:31þ which upon integration between the limits z ðrþ, in view of Eq. (2.3), simplifies to 2 1 4r r r Z 3 u 2 z5 ¼ u w v w þ s r; q ; ð2:32þ where s r; ¼l @z z¼ is the raial component of wall tangential shear stress. Equation (2.32) constitutes the momentum integral equation for a raially stretching/ shrinking isk. References 1. B.C. Sakiais, Bounary-layer behavior on continuous soli surface: I bounary-layer equations for two-imensional an axisymmetric flow. AIChE 7(1), 26 28 (1961) 2. H. Schlichting, Bounary-Layer theory, 6th en. (McGraw-Hill Book Company, 1968). Translate by J. Kestin
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