The critical velocity v_c of a fluid flowing through a cylindrical pipe of diameter D depe… — JAMB UTME

The critical velocity $v_c$ of a fluid flowing through a cylindrical pipe of diameter $D$ depends on the dynamic viscosity $\eta$ of the fluid, its density $\rho$ , and the pipe diameter $D$ according to the empirical relationship $v_c = R_e \eta^x \rho^y D^z$ , where $R_e$ is the dimensionless Reynolds number. Using dimensional analysis, calculate the numerical value of the sum of the exponents $x + y + z$ .

Cevap: -1

Cevap

The sum of the exponents

x + y + z

-1

Using the principle of dimensional homogeneity, the dimensions of both sides of the formula

v_c = R_e \eta^x \rho^y D^z

must be equal. Equating the powers of Mass, Length, and Time yields

x = 1

y = -1

, and

z = -1

. Summing these three values gives

1 + (-1) + (-1) = -1

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Determine the dimensions of all physical quantities involved

Critical velocity

[v_c] = \text{L T}^{-1}

, dynamic viscosity

[\eta] = \text{M L}^{-1} \text{T}^{-1}

, fluid density

[\rho] = \text{M L}^{-3}

, and diameter

[D] = \text{L}

. The Reynolds number

R_e

is dimensionless.

Dimensional analysis requires replacing physical quantities with their base SI dimensions.

Formulate the dimensional homogeneity equation

\text{M}^0 \text{L}^1 \text{T}^{-1} = (\text{M L}^{-1} \text{T}^{-1})^x (\text{M L}^{-3})^y (\text{L})^z = \text{M}^{x+y} \text{L}^{-x-3y+z} \text{T}^{-x}.

By the principle of dimensional homogeneity, the total exponent of each fundamental dimension must match on both sides of the equation.

Solve the system of simultaneous linear equations for

x

y

, and

z

From

\text{T}

-x = -1 \implies x = 1

.
From

\text{M}

x + y = 0 \implies y = -1

.
From

\text{L}

-x - 3y + z = 1 \implies -1 + 3 + z = 1 \implies z = -1

Equating powers of fundamental quantities yields explicit values for each dimensional power.

Calculate the target sum

x + y + z

x + y + z = 1 + (-1) + (-1) = -1.

Combining the calculated exponents gives the required numerical value.

Anahtar Kavram

Principle of Dimensional Homogeneity and Derivation of Physical Formulas

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