FLUID MECHANICS FUNDAMENTALS+APPS
4th Edition
ISBN: 2810022150991
Author: CENGEL
Publisher: MCG
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Chapter 9, Problem 106P
To determine
Weather the both equation is same or not.
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The Cauchy stress tensor for a (compressible) Newtonian fluid is given by
S(x, t) = -p(x, t)I+ A (V · v(x, t)) I+ 2µ sym Vv(x, t).
Choose from the list below the expression corresponding to the divergence of the Cauchy stress.
V. S(x, t) = - Vp(x,t) + µ Av(x, t)
|
V. S(x, t) = - Vp(x, t) + AV (V · v(x, t))+ µAv(x, t)
None of the above
V (x, t) = - Vp(x, t) + (A + µ)▼ (V · v(x, t)) + µ Av(x, t)
[Q1] Find the resultant for the force system shown below:
700 N
450 N
300 Ν
|30
60°
В
1500 N-m
to
- 3 m -
-2 m -
-4 m
1. Given the directional cosine matrix [A]=[aj]. Prove the transformation of a stress tensor
follows o; = A¡manjOmn
ij
Chapter 9 Solutions
FLUID MECHANICS FUNDAMENTALS+APPS
Ch. 9 - Explain the fundamental differences between a flow...Ch. 9 - What does it mean when we say that two more...Ch. 9 - The divergence theorem is v.cdv=A c . n dACh. 9 - Prob. 4CPCh. 9 - Prob. 5CPCh. 9 - Prob. 6CPCh. 9 - Prob. 7PCh. 9 - Prob. 8PCh. 9 - Let vector G=2xzi12x2jz2kk . Calculate the...Ch. 9 - Prob. 10P
Ch. 9 - Prob. 11PCh. 9 - Prob. 12PCh. 9 - Prob. 13PCh. 9 - Alex is measuring the time-averaged velocity...Ch. 9 - Let vector c be given G=4xziy2i+yzkand let V be...Ch. 9 - The product rule can be applied to the divergence...Ch. 9 - Prob. 18PCh. 9 - Prob. 19PCh. 9 - Prob. 20CPCh. 9 - In this chapter we derive the continuity equation...Ch. 9 - Repeat Example 9-1(gas compressed in a cylinder by...Ch. 9 - Consider the steady, two-dimensional velocity...Ch. 9 - The compressible from of the continuity equation...Ch. 9 - In Example 9-6 we derive the equation for...Ch. 9 - Consider a spiraling line vortex/sink flow in the...Ch. 9 - Verify that the steady; two-dimensional,...Ch. 9 - Consider steady flow of water through an...Ch. 9 - Consider the following steady, three-dimensional...Ch. 9 - Consider the following steady, three-dimensional...Ch. 9 - Two velocity components of a steady,...Ch. 9 - Imagine a steady, two-dimensional, incompressible...Ch. 9 - The u velocity component of a steady,...Ch. 9 - Imagine a steady, two-dimensional, incompressible...Ch. 9 - The u velocity component of a steady,...Ch. 9 - What is significant about curves of constant...Ch. 9 - In CFD lingo, the stream function is often called...Ch. 9 - Prob. 39CPCh. 9 - Prob. 40CPCh. 9 - Prob. 41PCh. 9 - Prob. 42PCh. 9 - Prob. 44PCh. 9 - Prob. 45PCh. 9 - As a follow-up to Prob. 9-45, calculate the volume...Ch. 9 - Consider the Couette flow of Fig.9-45. For the...Ch. 9 - Prob. 48PCh. 9 - AS a follow-up to Prob. 9-48, calculate the volume...Ch. 9 - Consider the channel flow of Fig. 9-45. The fluid...Ch. 9 - In the field of air pollution control, one often...Ch. 9 - Suppose the suction applied to the sampling...Ch. 9 - Prob. 53PCh. 9 - Flow separates at a shap corner along a wall and...Ch. 9 - Prob. 55PCh. 9 - Prob. 56PCh. 9 - Prob. 58PCh. 9 - Prob. 59PCh. 9 - Prob. 60PCh. 9 - Prob. 61PCh. 9 - Prob. 62PCh. 9 - Prob. 63EPCh. 9 - Prob. 64PCh. 9 - Prob. 65EPCh. 9 - Prob. 66PCh. 9 - Prob. 68EPCh. 9 - Prob. 69PCh. 9 - Prob. 71PCh. 9 - Prob. 72PCh. 9 - Prob. 73PCh. 9 - Prob. 74PCh. 9 - Prob. 75PCh. 9 - Wht in the main distionction between Newtormine...Ch. 9 - Prob. 77CPCh. 9 - What are constitutive equations, and to the fluid...Ch. 9 - An airplane flies at constant velocity Vairplane...Ch. 9 - Define or describe each type of fluid: (a)...Ch. 9 - The general cool volume from of linearmomentum...Ch. 9 - Consider the steady, two-dimensional,...Ch. 9 - Consider the following steady, two-dimensional,...Ch. 9 - Consider the following steady, two-dimensional,...Ch. 9 - Consider liquid in a cylindrical tank. Both the...Ch. 9 - Engine oil at T=60C is forced to flow between two...Ch. 9 - Consider steady, two-dimensional, incompressible...Ch. 9 - Consider steady, incompressible, parallel, laminar...Ch. 9 - Prob. 89PCh. 9 - Prob. 90PCh. 9 - Prob. 91PCh. 9 - The first viscous terms in -comonent of the...Ch. 9 - An incompressible Newtonian liquid is confined...Ch. 9 - Prob. 94PCh. 9 - Prob. 95PCh. 9 - Prob. 96PCh. 9 - Prob. 97PCh. 9 - Consider steady, incompressible, laminar flow of a...Ch. 9 - Consider again the pipe annulus sketched in Fig...Ch. 9 - Repeat Prob. 9-99 except swap the stationary and...Ch. 9 - Consider a modified form of Couette flow in which...Ch. 9 - Consider dimensionless velocity distribution in...Ch. 9 - Consider steady, incompressible, laminar flow of a...Ch. 9 - Prob. 104PCh. 9 - Prob. 105PCh. 9 - Prob. 106PCh. 9 - Prob. 107CPCh. 9 - Prob. 108CPCh. 9 - Discuss the relationship between volumetric strain...Ch. 9 - Prob. 110CPCh. 9 - Prob. 111CPCh. 9 - Prob. 112PCh. 9 - Prob. 113PCh. 9 - Look up the definition of Poisson’s equation in...Ch. 9 - Prob. 115PCh. 9 - Prob. 116PCh. 9 - Prob. 117PCh. 9 - For each of the listed equation, write down the...Ch. 9 - Prob. 119PCh. 9 - Prob. 120PCh. 9 - A block slides down along, straight inclined wall...Ch. 9 - Water flows down a long, straight, inclined pipe...Ch. 9 - Prob. 124PCh. 9 - Prob. 125PCh. 9 - Prob. 126PCh. 9 - Prob. 128PCh. 9 - The Navier-Stokes equation is also known as (a)...Ch. 9 - Which choice is not correct regarding the...Ch. 9 - In thud flow analyses, which boundary condition...Ch. 9 - Which choice is the genera1 differential equation...Ch. 9 - Which choice is the differential , incompressible,...Ch. 9 - A steady, two-dimensional, incompressible flow...Ch. 9 - A steady, two-dimensional, incompressible flow...Ch. 9 - A steady velocity field is given by...Ch. 9 - Prob. 137P
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- The stress tensor below is given for x, y, z coordinate system. Find the stress tensor if x, ycoordinates are rotated 45° clockwise.[?] = 4 1 2 1 6 0 2 0 8arrow_forwardIn the conventional vector basis {ei, e, e}, the stress tensor o at a given point is represented by the matrix 0 [g] = -30 25 a. Find the traction vector allel to vector u, with {u} = {1, 1,0}. 1 -30 25 -40 -15 MPa -15 0 (2) (force per unit area) relative to the element of area with normal vector în par- b. Find the stress components 032 = 023 in a new basis, such that the new basis is obtained by a CCW rotation of the old basis by a 0 = 60° about x3 axis, i.e., the rotation is only in ₁ - 2 plane.arrow_forward[7] Given the following stress tensor in Cartesian coordinates: -2.12 3.69 [2.74 [0] = - Sym 4.07 -3.02| · 103 ΚΡα 5.05 (a) Find the stress tensor in a new coordinate system rotated 30° around the z-axis. (b) Find the principal stresses and the direction of the 011.arrow_forward
- Transforming the coordinates of the shear stress from cartesian to spherical coordinate system allows to better visualize the dynamics of the tensor on a spherical geometry. True or Falsearrow_forwardPlease solve step by steparrow_forwardThe tension tensor at a point on the body is 21 1 Oij = |1 022 1 12 1 Set 022 so that there is a stress-free plate at the desired point.arrow_forward
- These are corrects 100%, but please tell me how to solve that.arrow_forwardforce in three dimensions is given by (a) F = FO(yx^ - xy^) (b) F= FO(yx^ + xy ^) where x^ and y^ are the cartesian basic vectors and F0 is a constant. Find if possible the corresponding potential energyarrow_forward4.35 The equilibrium configuration of a deformed body is described by the mapping X(X) = AX1 ê1 - BX302 + CX203, where A, B, and C are constants. If the Cauchy stress tensor for this body is го о о [a] 0 0 0 MPa, 10 0 60. where do is a constant, determine (a) the deformation tensor and its inverse in matrix form, (b) the matrices of the first and second Piola-Kirchhoff stress tensors, and (c) the pseudo stress vectors associated with the first and second Piola-Kirchhoff stress tensors on the ê3-plane in the deformed configuration.arrow_forward
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