Unsteady Laminar Convection on a Radiating Plate

V. V. Salomatov, E. M. Puzyrev

(Presented by the Department of Thermophysics and Atomic Power Engineering)

The influence of thermal radiation on the laminar boundary layer of a nonabsorbing fluid flowing over a heat-radiating plate under conditions of natural or forced convection is investigated.

Questions of unsteady convection and, in particular, of the interaction of thermal radiation with an unsteady laminar boundary layer of a nonabsorbing fluid are of considerable interest. Problems of this kind arise in the operation of heat-exchange devices under transient regimes, of various elements of electronic instruments during their forced start-up, in the flow over heat-releasing surfaces under conditions of substantial unsteadiness, and so on.

Among studies concerning particular aspects of this problem, the articles [1–3] should be noted. In [1] the problem of unsteady forced laminar convection with a boundary condition of the first kind is solved. It is shown that, for a stepwise jump in velocity and temperature, the transition from the unsteady regime to the steady one occurs through the passage of the front of thermal and dynamic waves moving from the leading edge of the plate. Experimental investigations by Gebhart [3] indicate the same character of the process in the case of natural convection. In article [2] the problem of the interaction of thermal radiation with the laminar boundary layer of a nonabsorbing fluid under conditions of forced and natural steady convection is considered.

It is shown that, for a constant boundary condition of the second kind along the length of the plate, there is a transition from heat exchange with a boundary condition of the second kind to heat exchange with a boundary condition of the first kind, and analytical solutions are obtained, but only for limiting cases: small and large values of the radiation parameter $\zeta$.

In the present work, a more general problem is considered: the interaction of thermal radiation with the laminar boundary layer of a nonabsorbing fluid for unsteady natural and forced convection.

Natural convection. Unsteady laminar natural convection of a nonabsorbing fluid with constant thermophysical properties is considered, arising near a vertical plate when a constant heat flux \(q_w\) is supplied stepwise to its surface. Heat transfer takes place through the interaction of radiant heat exchange (with the ambient medium at temperature \(T_e\)) and convection (to the fluid washing the plate, at temperature \(T_\infty\)).

Mathematically, the problem is described by the equations of conservation of mass, momentum, and energy:

\[ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1} \]

\[ \frac{\partial u}{\partial \tau}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} =\nu\frac{\partial^2 u}{\partial y^2}+g\beta(T-T_\infty), \tag{2} \]

\[ \frac{\partial T}{\partial \tau}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y} =a\frac{\partial^2 T}{\partial y^2} \tag{3} \]

with boundary conditions, under the assumption that the thicknesses of the thermal and hydrodynamic boundary layers are equal:

for \(\tau \le 0\)

\[ u=0,\quad T=T_\infty, \tag{4} \]

for \(\tau>0\)

\[ \text{at } y=0 \quad \frac{\partial T}{\partial y} =-\frac{q_k}{\lambda} =-\frac{q_w-\sigma\varepsilon\left(T_w^4-T_e^4\right)}{\lambda}, \tag{5} \]

\[ u=v=0, \tag{6} \]

at \(y=\delta\)

\[ T=T_\infty,\quad \frac{\partial T}{\partial y}=0 \tag{7} \]

\[ u=0,\quad \frac{\partial u}{\partial y}=0. \tag{8} \]

The boundary conditions (5), (7) are satisfied by the temperature profile

\[ T-T_\infty=\frac{q_k\delta}{2\lambda} \left(\frac{y^2}{\delta^2}-2\frac{y}{\delta}+1\right), \tag{9} \]

which, at \(y=0\), gives the relation

\[ \delta= \frac{2\lambda\left(T_w-T_\infty\right)}{q_k} = \frac{2\lambda\left(\Theta_w-1\right)} {\sigma\varepsilon T_\infty^3\left(\Theta_l^4-\Theta_w^4\right)}. \tag{10} \]

Writing equation (2) in the form

\[ \nu\frac{\partial^2 u}{\partial y^2} +g\beta(T-T_\infty)=C_1 \tag{11} \]

and, integrating it with allowance for (9), after determining the constants from the boundary conditions (6), (8), we obtain the velocity profile:

\[ u=\frac{g\beta q_k\delta^3}{6\lambda\nu} \left( -\frac{y^4}{4\delta^4}+\frac{y^3}{\delta^3} -\frac{5}{4}\frac{y^2}{\delta^2}+\frac{y}{2\delta} \right). \tag{12} \]

Integrating (3) from \(y=0\) to \(y=\delta\) with the use of (1), (6), (8), we obtain the heat-flow equation

\[ \frac{\partial}{\partial \tau}\int_0^\delta (T-T_\infty)\,dy+ \frac{\partial}{\partial x}\int_0^\delta u(T-T_\infty)\,dy =-a\left(\frac{\partial T}{\partial y}\right)_{y=0}, \tag{13} \]

from which it follows that

\[ \frac{\partial}{\partial \tau}q_k\delta^2+ \frac{g\beta}{140\lambda\nu}\frac{\partial}{\partial x}q_k^2\delta^5 =6aq_k . \tag{14} \]

Using (10), we write the characteristic system of the resulting equation [6]:

\[ \frac{2(\Theta_w-1)d\Theta_w}{(\Theta_l^4-\Theta_w^4)^2} +\frac{(\Theta_w-1)^2d\Theta_w^4}{(\Theta_l^4-\Theta_w^4)^3} = 6a\left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2 d\tau, \tag{15} \]

\[ \frac{5(\Theta_w-1)^4d\Theta_w}{(\Theta_l^4-\Theta_w^4)^4} +\frac{3(\Theta_w-1)^5d\Theta_w^4}{(\Theta_l^4-\Theta_w^4)^5} = \frac{420\nu a}{g\beta T_\infty} \left(\frac{\sigma\varepsilon T_w^3}{2\lambda}\right)^4 dx, \tag{16} \]

\[ \frac{5(\Theta_l^4-\Theta_w^4)+12(\Theta_w-1)\Theta_w^3} {2(\Theta_l^4-\Theta_w^4)+4(\Theta_w-1)\Theta_w^3} \cdot \frac{(\Theta_w-1)^3}{(\Theta_l^4-\Theta_w^4)^2}\,d\tau = \frac{70\nu\,dx}{g\beta T_\infty} \left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2 . \tag{17} \]

The integral of differential equation (15) with the boundary condition \(\Theta_w(\tau=0)=1\) gives a transcendental equation for calculating the change in the surface temperature of the plate under unsteady heat exchange

\[ \frac{(\Theta_w-1)^2}{2(\Theta_l^4-\Theta_w^4)^2} +F(\Theta_l,\Theta_w)-F(\Theta_l,1) = 6a\left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2 \tau . \tag{18} \]

and convective heat transfer

\[ \mathrm{Nu}_x= \frac{q_k x_*}{(T_w-T_\infty)\lambda} = \frac{2x_*}{\delta} = \]

\[ = \frac{1}{ \sqrt{ 3F_0-\frac{1}{2} \left(\frac{2\lambda}{\sigma\varepsilon\cdot T_\infty^3\cdot x_*}\right)^2 \left[F(\Theta_l,\Theta_w)-F(\Theta_l,1)\right] } }, \tag{19} \]

where

\[ F(\Theta_l,\Theta_w)= \int \frac{(\Theta_w-1)d\Theta_w}{(\Theta_l^4-\Theta_w^4)^2} = \frac{\Theta_w(\Theta_w-1)}{4\Theta_l^4(\Theta_l^4-\Theta_w^4)} + \]

\[ +\frac{1}{8\Theta_l^6}\ln\left|\frac{\Theta_l^2+\Theta_w^2}{\Theta_l^2-\Theta_w^2}\right| -\frac{3}{16\Theta_l^7}\left[ \ln\left|\frac{\Theta_l+\Theta_w}{\Theta_l-\Theta_w}\right| +2\arctan\frac{\Theta_w}{\Theta_l} \right], \tag{20} \]

\(x_*\) is a certain fictitious linear dimension, serving only for writing the criterion dependence (19).

Integrating the differential equation (16) under the boundary condition at the leading edge of the plate \(\Theta_w(x=0)=1\), we obtain a stationary solution in the form of a transcendental equation for calculating the temperature variation along the length of the plate,

\[ \frac{3}{5}\frac{(\Theta_w-1)^5}{(\Theta_l^4-\Theta_w^4)^4} +G(\Theta_l;\Theta_w)-G(\Theta_l;1) = \frac{336\alpha\nu x}{g\beta T_\infty} \left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^4, \tag{21} \]

and, consequently, also the convective heat transfer

\[ \mathrm{Nu}_x=\frac{q_k x}{(T_w-T_\infty)\lambda}= \]

\[ = \frac{1}{ \sqrt[4]{ \frac{35}{\mathrm{Gr}_x\mathrm{Pr}} - \frac{5}{3(\Theta_w-1)} \left(\frac{\lambda}{\sigma\varepsilon T_\infty^3 x}\right)^4 \left[\,G(\Theta_l;\Theta_w)+G(\Theta_l;1)\,\right] } }, \tag{22} \]

where

\[ G(\Theta_l,\Theta_w) = \int \frac{(\Theta_w-1)^4\,d\Theta_w}{(\Theta_l^4-\Theta_w^4)^4} = \frac{\Theta_w(\Theta_w-1)^4}{12\Theta_l^4(\Theta_l^4-\Theta_w^4)^3} + \]

\[ +\frac{1}{384\Theta_l^{12}(\Theta_l^4-\Theta_w^4)^2} \left( 21\Theta_l^8\Theta_w +7\Theta_l^4\Theta_w^5 -128\Theta_l^8 +486\Theta_l^4\Theta_w^3 -270\Theta_w^7 -400\Theta_l^4\Theta_w^2 \right. \]

\[ \left. +240\Theta_w^6 +121\Theta_l^4\Theta_w -770\Theta_w^5 \right) + \frac{(-7\Theta_l^4+270\Theta_l^2+77)}{512\Theta_l^{15}} \ln\left|\frac{\Theta_l+\Theta_w}{\Theta_l-\Theta_w}\right| \]

\[ -\frac{5}{16\Theta_l^{24}} \ln\left|\frac{\Theta_l^2+\Theta_w^2}{\Theta_l^2-\Theta_w^2}\right| + \frac{(-7\Theta_l^4-270\Theta_l^2+77)}{256\Theta_l^{15}} \arctan\frac{\Theta_w}{\Theta_l}. \tag{23} \]

To analyze the solutions obtained, let us determine, by L’Hospital’s rule, the relative contribution of the terms \([F(\Theta_l;\Theta_w)-F(\Theta_l;1)]\) and \([G(\Theta_l;\Theta_w)-G(\Theta_l;1)]\) in equations (18), (21) for the limiting cases: small ( \(\tau\) or \(x\to0\); \(\Theta_w\to1\) ) and large ( \(\tau\) or \(x\) large: \(\Theta_w\to\Theta_l\), since \(q_k\to0\) ) boundary-layer thickness:

\[ \lim \frac{ \displaystyle \int_{1}^{\Theta_w} \frac{(\Theta_w-1)\,d\Theta_w}{(\Theta_l^4-\Theta_w^4)^2} }{ \displaystyle \frac{(\Theta_w-1)^2}{2(\Theta_l^4-\Theta_w^4)^2} } = \begin{cases} 1, & (\Theta_w\to1),\\ 0, & (\Theta_w\to\Theta_l), \end{cases} \tag{24} \]

\[ \lim \frac{ \displaystyle \int_{1}^{\Theta_w} \frac{(\Theta_w-1)^4\,d\Theta_w}{(\Theta_l^4-\Theta_w^4)^4} }{ \displaystyle \frac{3}{5}\frac{(\Theta_w-1)^5}{(\Theta_l^4-\Theta_w^4)^4} } = \begin{cases} \dfrac{1}{3}, & (\Theta_w\to1),\\ 0, & (\Theta_w\to\Theta_l). \end{cases} \tag{25} \]

Relation (24) makes it possible to represent equation (19) in the form

\[ \mathrm{Nu}_{\tau}=\sqrt{\frac{2}{3F_{0}}},\quad \left(\delta(\tau)=\sqrt{6a\tau}\right)-\Theta\to 1\;(\tau\to 0) \tag{19a} \]

and

\[ \mathrm{Nu}_{\tau}=\frac{1}{\sqrt{3F_{0}}},\quad \left(\delta(\tau)=\sqrt{12a\tau}\right)-\Theta_{w}\to\Theta_{\ell}\;(\tau\text{ large}), \tag{19b} \]

i.e., equation (19), which describes the process of unsteady convection, coincides in the limiting cases, to within 7%, with the equation of unsteady heat conduction in a semi-infinite body with boundary condition (5). Thermal radiation causes a transition from heat transfer with a boundary condition of the second kind

\[ \left(\mathrm{Bi}=\frac{1}{2i\,\mathrm{erfc}(0)}F_{0}^{-1/2}\right) \]

to heat transfer with a boundary condition of the first kind

\[ \left(\mathrm{Bi}=\frac{1}{\sqrt{\pi}}\,F_{0}^{-1/2}\right) \]

[4].

Similarly, taking (25) into account, we represent (23) in the form:

\[ \mathrm{Nu}_{x}=0.442(\mathrm{Gr}_{x}\cdot \mathrm{Pr})^{1/4} \quad \Theta_{w}\to 1\;(x\to 0) \tag{22a} \]

and

\[ \mathrm{Nu}_{x}=0.409(\mathrm{Gr}_{x}\cdot \mathrm{Pr})^{1/4} \quad \Theta_{w}\to\Theta_{\ell}\;(x\text{ large}), \tag{22b} \]

thus, equation (22) describes the transition from heat transfer with a boundary condition of the second kind (22a) to heat transfer with a boundary condition of the first kind (22b), caused by the presence of thermal radiation. The difference between the limiting solutions and the exact values is no more than 4% [2].

The solution of equation (17), which describes the motion of the front of the thermal wave (the boundary of transition from the unsteady solution to the steady one), can be found only for limiting cases, since equations (18) and (21) do not give explicit dependences \(\Theta_{w}(\tau)\) or \(\Theta_{w}(x)\):

\[ x_{\tau}=\frac{0.105\,g\beta q_{k}\sqrt{a}}{\lambda\cdot \mathrm{Pr}}\cdot \tau^{5/2} \quad \Theta_{w}\to 1 \tag{17a} \]

and

\[ x_{\tau}=\frac{0.257\,g\beta(T_{w}-T_{\infty})}{\mathrm{Pr}}\cdot \tau^{2} \quad \Theta_{w}\to\Theta_{\ell}. \tag{17b} \]

The criterion form (17a) agrees with the experimental data [3] (Fig. 2).

Forced convection. We consider the unsteady flow over a plate by a laminar flow of a nonabsorbing fluid with constant thermophysical properties, arising under a step change in velocity. Simultaneously, a step change in the heat flux \(q_{w}\) is prescribed on the plate surface; this flux is transferred through the interaction of radiant heat exchange (to the surroundings of temperature \(T_{e}\)) and convection (to the incoming flow with temperature \(T\)). Compressibility and viscous dissipation are not taken into account.

The mathematical formulation of this problem coincides with the preceding one, except for relations (2), (8), which take the form:

\[ \frac{\partial u}{\partial \tau} +u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y} = \nu\frac{\partial^{2}u}{\partial y^{2}}, \tag{2′} \]

\[ \tau>0,\quad y=\delta_g,\quad u=U_\infty,\quad \frac{\partial u}{\partial y}=0, \tag{8'} \]

and the difference in the thicknesses of the thermal and hydrodynamic boundary layers is also taken into account.

The boundary conditions (6), (8′) are satisfied by the polynomial

\[ u=U_\infty\frac{y}{\delta_g}\left(2-\frac{y}{\delta_g}\right). \tag{26} \]

After its substitution into the Kármán integral relation, this gives the differential equation

\[ \frac{\partial \delta_g}{\partial \tau} + U_\infty \frac{2}{5}\frac{\partial \delta_g}{\partial x} = \frac{6\nu}{\delta_g}. \tag{27} \]

The characteristic system of this equation [6],

\[ d\tau= \frac{dx}{\frac{2}{5}U_\infty} = \frac{\delta_g d\delta_g}{6\nu}, \tag{28} \]

makes it possible to obtain, respectively, the unsteady and steady solutions, as well as the equation of motion of the dynamic wave:

\[ \delta_g(\tau)=\sqrt{12\nu\tau},\quad \delta_g(x)=\sqrt{\frac{30\nu x}{U_\infty}},\quad x_{\mathrm{H}}=0.4U_\infty\tau. \tag{29} \]

Carrying out integration in the heat-flux equation (13) from \(y=0\) to \(y=\delta_m\), taking (9) and (26) into account, we obtain the differential equation:

\[ \frac{\partial}{\partial \tau}q_k\delta_m^2 + U_\infty \frac{\partial}{\partial x}q_k\delta_m^2 H(\Delta) = 6aq_k, \tag{30} \]

where

\[ H(\Delta)=\frac{\Delta}{2}-\frac{\Delta^2}{10} \qquad \text{for } \Delta=\frac{\delta_m}{\delta_g}<1, \]

\[ H(\Delta)=1-\frac{1}{\Delta}+\frac{1}{2\Delta^2}-\frac{1}{10\Delta^3} \qquad \text{for } \Delta>1. \]

Equation (30) has four solutions, depending on the zone under consideration.

Steady zone I occurs after the passage of the thermal and dynamic waves—\(\delta_g(x)\), \(\delta_m(x)\),

\[ U_\infty \frac{\partial}{\partial x}q_k\delta_m^2 H(\Delta)=6aq_k. \tag{30—1} \]

Equation (30—1), like (16), describes the transition along the length of the plate from heat transfer with a boundary condition of the second kind to heat transfer with a boundary condition of the first kind. In the general case, equation (30—1) can be solved only numerically. To construct its approximate solution, we replace the ratio \(\Delta=\dfrac{\delta_m}{\delta_g}\), equal to \(\Pr^{-1/3}\) for \(T_w=\mathrm{const}\) [5], and

$0.78\,\Pr^{-1/3}$ for $q_k=\mathrm{const}$, (solution (30—I) for $q=\mathrm{const}$, by the mean value $-\Delta=0.89\,\Pr^{-1/3}$ and, taking (10) into account, we reduce (30—I) to a form analogous to (16):

\[ \frac{2(\Theta_w-1)\,d\Theta_w}{(\Theta_\ell^4-\Theta_w^4)^2} + \frac{(\Theta_w-1)^2\,d\Theta_w^4}{(\Theta_\ell^4-\Theta_w^4)^3} = \left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2 \frac{6a\,dx}{U_\infty H(0.89\Pr^{-1/3})}. \tag{31} \]

Integrating the equation obtained, with the boundary condition $\Theta_w(x=0)=1$, we obtain relations for calculating the change in temperature and convective heat transfer along the length of the plate:

\[ \frac{1}{2}\, \frac{(\Theta_w-1)^2}{(\Theta_\ell^4-\Theta_w^4)^2} + F(\Theta_\ell;\Theta_w)-F(\Theta_\ell;1) = \left(\frac{\sigma\varepsilon T_w^3}{2\lambda}\right)^2 \frac{6ax}{U_\infty H(0.89\Pr^{-1/3})}, \tag{32} \]

\[ \mathrm{Nu}_x=\frac{q_k x}{\lambda(T_w-T_\infty)} = \frac{1}{ \sqrt{ \frac{3}{H(0.89\Pr^{-1/3})\mathrm{Re}_x\Pr} -\frac{1}{2} \left(\frac{2\lambda}{\sigma\varepsilon T_\infty^3 x}\right)^2 \left[F(\Theta_\ell;\Theta_w)-F(\Theta_\ell;1)\right] } }. \tag{33} \]

The limiting solutions (33) differ by 5% from the exact values [2]:

\[ \mathrm{Nu}_x= \sqrt{ H(0.89\Pr^{-1/3})\frac{2}{3}\mathrm{Re}_x\Pr }, \qquad \mathrm{Nu}_x=0.43\,\mathrm{Re}_x^{1/2}\;(\Pr=0.7), \tag{33a} \]

\[ \mathrm{Nu}_x= \sqrt{ H(0.89\Pr^{-1/3})\frac{1}{3}\mathrm{Re}_x\Pr }, \qquad \mathrm{Nu}_x=0.304\,\mathrm{Re}_x^{1/2}\;(\Pr=0.7). \tag{33b} \]

The unsteady zone II is located ahead of the fronts of the thermal and dynamic waves—$\delta_g(\tau)$, $\delta_m(\tau)$,

\[ \frac{\partial}{\partial \tau}q_k^2\delta_m^2=6a q_k. \tag{30—II} \]

Equation (30—II) coincides with equation (15) and has solutions (18), (19).

Using the solutions obtained, one can determine the propagation of the thermal-wave front when it coincides with the front of the dynamic wave for the limiting cases from the equality $\delta_m(\tau)=\delta_m(x_\tau)$: near the leading edge of the plate, for small $\tau$ (boundary condition of the second kind), $x_\tau=0.33U_\infty\Pr^{-1/3}$; for large $x$ and $\tau$ (boundary condition of the first kind), $x_\tau=0.4U_\infty\Pr^{-1/3}$.

Thus, the propagation velocity of the thermal wave increases as it moves from $0.33\,\Pr^{-1/3}$ to $0.4\,\Pr^{-1/3}$, i.e., for $\Pr>1$, $x_\tau<x_n$; for $\Pr<0.56$, $x_\tau>x_n$; and for $1>\Pr>0.56$, the thermal wave gradually catches up with and overtakes the dynamic wave.

The solution for the transition zone III, located between the dynamic wave and the thermal wave following it ($\Delta<1$), can be found only in

limiting cases: \(q_k=\mathrm{const}\), or \(T_w=\mathrm{const}\). Neglecting, for the purpose of simplification, the term \(\Delta^2/10\) and introducing the new variable \(\xi=U_\infty\tau/x\), taking into account

\[ \delta_g(x)=\sqrt{\frac{30\nu x}{U_\infty}}, \]

we obtain the differential equations

\[ \Delta\cdot \Delta^{1}\xi\left(2-\frac{3}{2}\Delta\xi\right)=\frac{1}{5\Pr}, \qquad q_k=\mathrm{const}, \tag{34 a} \]

\[ \Delta\cdot \Delta^{1}\xi(1-\Delta\cdot \xi)=\frac{1}{5\Pr}, \qquad T_w=\mathrm{const}, \tag{34 б} \]

which have the solutions

\[ \xi=10\Pr\exp(-2.5\Delta^3\Pr)\int_{0}^{\Delta}\exp(2.5\Delta^3\Pr)\Delta\,d\Delta, \qquad q_k=\mathrm{const} \tag{35 a} \]

\[ \xi=5\Pr\exp(-1.67\Delta^3\Pr)\int_{0}^{\Delta}\exp(1.67^3\Pr)\Delta\,d\Delta, \qquad T_w=\mathrm{const}. \tag{35 б} \]

For computations at small \(\Delta\), the expansion (35) in a Maclaurin series may be used.

The solution for zone IV, located between the thermal wave and the following dynamic wave, \(\Delta>1\), will be determined from the characteristic system of equation (30), which has the form (since \(\delta_g=\sqrt{12a\tau}\)):

\[ \frac{d\tau}{ \dfrac{\partial}{\partial\tau} \left[ \dfrac{(\Theta_w-1)^2}{(\Theta_\ell^4-\Theta_w^4)} \right]d\tau} = \frac{dx}{ U_\infty\dfrac{\partial}{\partial x} \left[ \dfrac{(\Theta_w-1)^2}{(\Theta_\ell^4-\Theta_w^4)}H(\Delta) \right]dx} = \]

\[ = \left(\frac{2\lambda}{\sigma\varepsilon T_\infty^3}\right)^2 \frac{1}{6a(\Theta_\ell^4-\Theta_w^4)}. \tag{36} \]

Neglecting, for the purpose of simplification, the term \(1/(10\Delta^3)\) in \(H(\Delta)\), we write the differential equation for zone IV:

\[ \frac{2(\Theta_w-1)d\Theta_w}{(\Theta_\ell^4-\Theta_w^4)^2} + \frac{(\Theta_w-1)d\Theta_w^4}{(\Theta_\ell^4-\Theta_w^4)^3} - \delta_g\frac{\sigma\varepsilon T_\infty^3}{2\lambda} \frac{d\Theta_w}{\Theta_\ell^4-\Theta_w^4} - \]

\[ - \frac{\delta_g^2}{2} \left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2 \frac{d\Theta_w^4}{\Theta_\ell^4-\Theta_w^4} = \frac{6a}{U_\infty} \left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2dx, \tag{37} \]

which, after integration with the boundary condition at the overtaking point \(\Theta_w=(x=x_T=x_n)=\Theta_0\), gives a transcendental equation for calculating the temperature of the plate surface:

\[ \frac{1}{2}\frac{(\Theta_w-1)^2}{(\Theta_\ell^4-\Theta_w^4)^2} + F(\Theta_\ell;\Theta_w)-F(\Theta_\ell;\Theta_0) - \frac{\sigma\varepsilon T_\infty^3}{2\lambda} \frac{\delta_g}{4\Theta_\ell^3} \left[ \ln\left|\frac{\Theta_\ell+\Theta_w}{\Theta_\ell-\Theta_w}\right| -\right. \]

\[ \left. -\ln\left|\frac{\Theta_\ell+\Theta_0}{\Theta_\ell-\Theta_0}\right| +2\arctg\frac{\Theta_w}{\Theta_\ell} -2\arctg\frac{\Theta_0}{\Theta_\ell} \right] + \]

\[ + \frac{\delta_g^2}{2} \left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2 \left[ \ln|\Theta_\ell^4-\Theta_w^4| -\ln|\Theta_\ell^4-\Theta_0^4| \right] = \frac{6ax}{U_\infty} \left(\frac{\sigma\varepsilon T_\infty^3}{2\lambda}\right)^2. \tag{38} \]

In conclusion we note that the solutions for the transition zones (35) and (38) near the front of the first wave pass into the unsteady solutions (18), (19), and near the front of the lagging wave—into the steady solutions (32), (33), [1].

NOTATION

$T_w$, $T_e$, $T_\infty$ — absolute temperatures of the wall, the surroundings, and the washing fluid, respectively; $\Theta_w=\dfrac{T_w}{T_\infty}$ — dimensionless wall temperature; $\Theta_{\ell}=\sqrt[4]{\dfrac{q_w+\sigma\varepsilon T_e^4}{\sigma\varepsilon T_\infty^4}}$ — dimensionless wall temperature in the case of heat exchange only by radiation; $\delta$, $\delta_T$, $\delta_w$ — thicknesses of the boundary layer, hydrodynamic and thermal, respectively; $q_w$, $q_k$ — heat fluxes at the wall and convective flux, respectively; $x_T$, $x_u$ — coordinates of the front of the thermal and dynamic waves; $\beta=\dfrac{1}{T_\infty}$ — coefficient of thermal expansion;

\[ \operatorname{Gr}_x=\frac{g\beta (T_w-T_\infty)x^3}{\nu^2} \]

Grashof number; $\sigma$ — Stefan–Boltzmann constant.

REFERENCES

1. Yu. L. Rozenshtok. In: Heat and Mass Transfer, 1. Convective Heat Transfer in a Homogeneous Medium. “Nauka i tekhnika,” Minsk, 1965, p. 277.

2. R. D. Cess. In: Problems of Heat Transfer. “Atomizdat,” Moscow, 1967, p. 7.

3. G. Gebhart. Heat Transfer 89, Series C, No. 3, 1967.

4. A. V. Lykov. Theory of Heat Conduction. “Vysshaya shkola,” Moscow, 1967.

5. H. Schlichting. Boundary-Layer Theory. Foreign Literature Publishing House, 1956.

6. E. Kamke. Handbook of First-Order Partial Differential Equations. “Nauka,” Moscow, 1966.